Philosophiæ Naturalis Principia Mathematica

Philosophiae Naturalis Principia Mathematica. 3st Ed. Isaac NEWTON.

Translated and Annotated by Ian Bruce. 34

THE MATHEMATICAL PRINCIPLES OF

NATURAL PHILOSOPHY



DEFINITIONS.

DEFINITION I

The quantity of matter is a measure of the same arising jointly from the density and

magnitude [volume].

AIR with the density doubled, in a volume also doubled, shall be quadrupled; in triple the volume, six times as great. You understand the same about snow and powders with the condensing from melting or by compression. And the account of all bodies is the same which are condensed in different ways by whatever causes. Here meanwhile I have no account of a medium, if which there were, freely pervading the interstitia of the parts [of the body]. But I understand this quantity everywhere in what follows under the name of the body or of the mass. That becomes known through the weight of any body: for the proportion to the weight is to be found through experiments with the most accurate of pendulums set up, as will be shown later.

DEFINITION II.

The quantity of motion is a measure of the same arising from the velocity and quantity of matter jointly.

The whole motion is the sum of the motions within the single parts ; and therefore in a body twice as great, with equal velocity, it is doubled, & with the velocity doubled four times as great.

DEFINITION III. The innate force of matter is the resisting force, by which each individual body, however great it is in itself, persists in its state either of rest or of moving uniformly straight forwards. This innate [or vis insita ] force is proportional always to its body, and nor does it differ at all from the inertia of the mass, unless in the required manner of being considered. From the inertia of matter it arises, that each body may be disturbed with difficulty either from its state of rest or from its state of motion. From which the vis insita will be possible also to be called by a most significant name the vis inertiae [force of inertia]. Truly the body exercises this force only in the change made of its state by some other force impressed on itself ; and the exercise is under that difference with respect to



resistance and impetus: Resistance: in as much as the body is resisting a change in its state by the force acting; Impetus: in as much as the same body, with the force of resistance requiring to concede to the obstacle, tries to change the state of this obstacle. One commonly attributes resistance to states of rest and impetus to states of movement : but motion and rest, as they are considered commonly, are distinguished only in turn from each other; nor are [bodies] truly at rest which may be regarded commonly as being in a state of rest. [The use of the word directum , direct or straight forward rather than straight line, as is given in texts on mechanics removes a circular argument from the definition, as the body can only so move in the absence of forces, and cannot be part of the definition as well as a consequence; and there are of course no lines drawn in space, although we could in principle detect deviations of motion along a given direction. Clearly, this was Newton's original meaning, where he uses the word directum, and his thoughts on the predominance of Mechanics over Geometry are set out in the Preface to the first edition. ]

DEFINITION IV.

The impressed force is the action exercised on the body, to changing the state either of rest or of motion uniform in direction.

This force is in position only during the action, nor remains in the body after the action. For the body may persevere in any new state by the force of inertia only. Moreover the impressed force is of diverse origins, as from a blow, from pressure, or from the centripetal force.

DEFINITION V.

It is the centripetal force, by which bodies are drawn, impelled, or tend in some manner from all sides towards some point, as towards a centre.

Gravity is [a force] of this kind, by which bodies tend towards the centre of the earth ; the magnetic force, by which iron seeks a loadstone; and that force, whatever it may be, by which the planets are drawn perpetually from rectilinear motion, and are forced to revolve along curved lines. A stone rotating in a sling is trying to depart from the turning hand; and in its attempt has stretched the sling, and with that the more the faster it revolves, and it flies off as soon as it is released. I call the force contrary to that endeavour the centripetal force, by which the sling continually pulls the stone back to the hand and keeps it in orbit, as it is directed to the hand or the centre of the orbit. And the account is the same of all bodies, which are driven in a circle. All these are trying to recede from the centre of the orbit; and unless some other force shall be present trying the opposite to this, by which they may be confined and retained in the orbits, and each thus I call centripetal, they will depart with a uniform motion in straight lines. A projectile, if it were abandoned by the force of gravity, would not be deflected towards the earth, but would go in a straight line to the heavens; and that with a uniform motion, but only if the



air resistance may be removed. By its gravity it is drawn from the rectilinear course and always is deflected to the earth, and that more or less for its gravity and with the velocity of the motion. So that the smaller were the gravity for a quantity of matter or the greater the velocity with which it was projected, by that the less will it deviate from a rectilinear course and the further it will go on. If a leaden sphere is projected from the peak of some mountain with a given velocity along a horizontal line by the force of gunpowder, it may go on in a curved line for a distance of two miles, before it falls to earth : since here with the velocity doubled it may go on twice as far as it were, and with ten times the velocity ten times as far as it were: but only if the resistance of the air is removed. And by increasing the velocity it may be possible to increase the distance to any desired distance in which it is projected, and the curvature of the line that it may describe be lessened , thus so that it may fall only according to a distance of ten or thirty or ninety degrees ; or also so that it may encircle the whole earth or finally depart into the heavens, and from the departing speed to go on indefinitely. And by the same account, by which the projectile may be turned by the force of gravity in obit and may be able to encircle the whole earth, also the moon is able, either by the force of gravity, but only it shall be of gravity, or some other force , by which it may be acted on, always to be drawn back from a rectilinear course towards the earth, and to be turning in its orbit : and without such a force the moon would not be able to be retained in its orbit. This force, if it should be just a little less, would not be enough to turn the moon from a rectilinear course : if just a little greater, would turn the moon more and it would be led from its orbit towards the earth. Certainly it is required, that it shall be of a just magnitude : and it is required of mathematicians to find the force, by which a body will be able to be retained carefully in some given orbit ; and in turn to find the curved path, in which a body departing from some given place with a given velocity may be deflected by a given force. Moreover the magnitude of this centripetal force is of three kinds, absolute, accelerative, and motive. [ Newton uses some of his later dynamical ideas to refine the centripetal force acting on a body under the influence of a large mass into three parts: the absolute force, which depends primarily on the magnitudes of the large mass and small mass, e.g. if the centripetal force were produced on a body near the sun, or near the earth, all else being equal; the accelerative force is simply the acceleration due to gravity on a small mass at some location: the force of gravity on a unit mass (i.e. g) ; by motive force Newton means

the rate of change ( )d mvdt of the quantity of motion mv or momentum, which in turn he

calls simply motion.]

DEFINITION VI.

The absolute magnitude of a centripetal force is a measure of the same, greater or less, for the effectiveness of the cause of propagating that from the centre into the orbital

regions.



So that the magnetic force for the size of the loadstone either extends more in one loadstone of greater strength, or lesser in another.

DEFINITION VII.

The accelerative magnitude of the centripetal force is the measure of this proportional to velocity, that it generates in a given time.

As the strength of the same loadstone is greater in a smaller distance, smaller in greater : or the force of gravity is greater in valleys, less at the tops of high mountains, and small still (as it will become apparent afterwards) at greater distances from the globe of the earth ; but at equal distances it is the same on all sides, because therefore all falling bodies (heavy or light, large or small) with the air resistance removed, accelerate equally.

DEFINITION VIII.

The motive magnitude of the centripetal force is the measure of this, proportional to the motion, which it generates in a given time.

As the weight is greater in a greater body, less in a smaller ; and in the same body greater near the earth, less in the heavens. This magnitude is the centripetency or the propensity of the whole body to move towards the centre, and (as thus I have said) the weight; and it becomes known always by that force equal and opposite to itself, by which the descent of the body can be impeded. And the magnitudes of these forces for the sake of brevity can be called the motive, accelerative, and absolute forces, and for the sake of being distinct refer to bodies attracted towards the centre, to the locations of these [moving] bodies, and to the centre of the forces: there is no doubt that the motive force for a body, as the attempt of the whole towards the centre [of the attracting body] is composed from the attempts of all the parts ; and the accelerative force at the position of the body, as a certain effectiveness, spread out in the orbit from the centre through the individual locations to the bodies towards moving the bodies which are in these places ; but the absolute force towards the centre, as being provided by some cause, without which the motive forces may not be propagated through the regions in the revolution; or for that cause there shall be some central body (such is the loadstone at the centre of the magnetic force, or the earth from the centre of the force of gravity) or some other [cause] which may not be apparent. The concept here is only mathematical : For I do not consider the causes and physical seats of the forces. Therefore the accelerative force is to the motive force as the velocity is to the motion. For the quantity of the motion arises from the velocity and also from the quantity of matter; the motive force arises from the accelerative force taken jointly with the same quantity of matter. For the sum of the actions of the accelerative force on the individual particles of the body is the whole motive force [i.e. the weight]. From which next to the



surface of the earth, where the accelerative gravity or the gravitating force is the same in all bodies, the motive gravity or weight is as the body : but if it may ascend to regions were the accelerative weight shall be less, the weight equally may be diminished, and it will be always as the body and accelerative gravity jointly. Thus in regions were the accelerative gravity is twice as small, the weight of the body small by two or three times will be four or six times as small. Again I name attraction and impulses, in the same sense, accelerative and motive forces. But for these words attraction, impulse, or of any propensity towards the centre, I use indifferently and interchangeably among themselves; these forces are required only to be considered from the mathematical point of view and not physically. From which the reader may be warned, lest by words of this kind he may think me to define somewhere either a kind or manner of action or a physical account, or to attribute truly real forces to the centres (which are mathematical points); if perhaps I have said to draw from the centre or to be forces of the centres.

Scholium. Up to this stage it has been considered to explain a few notable words, and in the following in what sense they shall be required to be understood. Time, space, position and motion, are on the whole the most notable. Yet it is required to note that ordinary people may not conceive these quantities otherwise than from the relation they bear to perception. And thence certain prejudices may arise, with which removed it will be agreed to distinguish between the absolute and the relative, the true and the apparent, the mathematical and the common usage. l. Absolute time, true and mathematical, flows equably in itself and by its nature without a relation to anything external, and by another name is called duration. Relative, apparent, and common time is some sensible external measure of duration you please (whether with accurate or with unequal intervals) which commonly is used in place of true time; as in the hour, day, month, year. II. Absolute space, by its own nature without relation to anything external, always remains similar and immovable: relative [space] is some mobile measure or dimension of this [absolute] space, which is defined by its position to bodies according to our senses, and by ordinary people is taken for an unmoving space: as in the dimension of a space either underground, in the air, or in the heavens, defined by its situation relative to the earth. Absolute and relative spaces are likewise in kind and magnitude; but they do not always endure in the same position. For if the earth may move, for example, the space of our air, because relative to and with respect to the earth it always remains the same, now there will be one part of absolute space into which the air moves, now another part of this; and thus always it will be moving absolutely. llI. The position [or place] is a part of space which a body occupies, and for that reason it is either an absolute or relative space. A part of space, I say, not the situation of [places within] the body, nor of the surrounding surface. For there are always equal



positions[within] equal solid shapes; but not so surfaces as most are unequal on account of dissimilarities of the figures; [for a surface is liable to change, due to air resistance, etc.] Truly positions may not have a magnitude on speaking properly, nor are they [to be considered] as places rather than as affectations of places [i.e. the position is not a physical property of the body, but rather an indication of where the body is situated at some time in space]. The whole motion is the same as the sum of the motions of the parts, that is, the translation of all from its place is the same as the sum of the translations of the parts from their places ; and thus the place of the whole is the same as the sum of all the parts of the places and therefore both inside and with the whole body. IV. An absolute motion is the translation of a body from one absolute place into another absolute place, a relative [motion] from a relative [place] into a relative [place]. Thus in a ship which is carried along in full sail, the relative position of the body is that region of the ship in which the body moves about, or the part of the whole cavity of the ship [hold] which the body fills up, and which thus is moving together with the ship : and relative quiet is the state of being of the body in that same ship or in the part of the hold. But the persistence of the body is true rest in the same part of space in which the ship is not moving, in which the ship itself together with the hold and all the contents may be moving. From which if the earth truly is at rest, the body which relatively at rest in the ship, truly will be moving and absolutely with that velocity by which the ship is moving on the earth. If the earth also is moving; there is the true and absolute motion of the body, partially from the motion of the ship truly in an unmoving space, partially from the motion of the ship relative to the earth: and if the body is moving relatively in the ship, the true motion of this arises, partially from the true motion of the earth in motionless space, partially from the relative motion both of the ship on the earth as well as of the body in the ship ; and from these relative motions the motion of the body relative to the earth arises. So that if that part of the earth, where the ship is moving, truly is moving to the east with a speed of 10010 parts; and by the wind in the sails the ship is carried to the west with a velocity of ten parts ; moreover a sailor may be walking on the ship towards the east with a velocity of one part : truly the sailor will be moving and absolutely in the immobile space with a velocity of 10001 to the east, and relative to the earth towards the east with a speed of nine parts. Absolute time is distinguished from relative time in astronomy by the common equation of time. For the natural days are unequal, which commonly may be taken as equal for the measure of time. Astronomers correct this inequality, so that they measure the motion of the heavens from the truer time. It is possible, that there shall be no uniform motion, by which the time may be measured accurately. All motions are able to be accelerated and retarded, but the flow of absolute time is unable to change. The duration or the perseverance of the existence of things is the same, either the movement shall be fast or slow or none at all: hence this is distinguished by merit from the sensibilities of their measurement, and from the same [the passage of time] is deduced through an astronomical equation. But a need prevails for phenomena in the determination of this equation, at some stage through an experiment with pendulum clocks, then also by the eclipses of a satellite of Jupiter.



As the order of the parts of time is unchangeable, thus too the parts of space. These could be moved from their own places (as thus I may say), and they will be moved away from each other [i.e. out of sequence]. For the times and the spaces are themselves of this [kind] and as if the places of all things: in time according to an order of successions, and in space according to an order of positions, to be put in place everywhere. Concerning the essence of these things, it is that they shall regarded as places : and it is absurd to move the first places. These therefore are absolute places ; and only the translations from these places are absolute motions. In truth since these parts of space are unable to be seen, and to be distinguished from each other by our senses; we use in turn perceptible measures of these. For we define all places from the positions and distances of things from some body, which we regard as fixed : and then also we may consider all motion with respect to the aforementioned place, as far as we may conceive bodies to be transferred from the same. Thus in exchange of absolute places and motions we make use of relative ones ; not to be an inconvenience in human affairs : but required to be abstracted from the senses in [natural] philosophical matters. And indeed it can happen, that actually no body may be at rest, to which the position and motion may be referred to. But rest and motion, both absolute and relative, are distinguished in turn from each other by their properties, causes and effects. The property of rest is, that bodies truly at rest are at rest among themselves. And thus since it shall be possible, that some body in the regions of fixed [stars], or far beyond, may remain absolutely at rest ; moreover it is impossible to know in turn from the situation of bodies in our regions, whether or not any of these given at a remote position may serve [to determine true rest in the absolute space for local bodies]; true rest cannot be defined from the situation of these bodies between themselves. A property of motion is, that the parts which maintain given positions to the whole, share in the motion of the whole. For all the rotating parts are trying to recede from the axis of the motion, and the impetus of the forwards motion arises from the impetus of the individual parts taken together. Therefore with the motion for circulating bodies [e.g. planets], they do move in circles [i.e. orbits] in which they are relatively at rest. And therefore true and absolute motion cannot be defined by a translation from the vicinity of such bodies, which [otherwise] may be regarded as being in a state of rest. For external bodies [introduced by way of example] must not only seem as being in a state of rest, but also truly to be at rest. Otherwise everything included also participates in the true orbiting motion, besides a translation from the vicinity of the orbiting body ; and with that translation taken away they are not truly at rest, but they may be seen only at rest in this manner. For the orbiting bodies are to the included, as the total exterior part to the interior part, or as a shell to the kernel. But with the shell moving also the kernel is moving, or a part of the whole, without a translation from the vicinity of the shell. The relation to the preceding property is this, because in the place moved a single location is moved : and thus a body, which is moved with the place moved, also shares the motion of its place. Therefore all [relative] motions, which are made from moved places, are only parts of both the whole and absolute motions, and every whole motion is composed from the motion of the body from its first place, and from the motion of this place from its own place in turn, and thus henceforth ; until at last it may arrive at a



stationary place, as in the example of the sailor mentioned above. From which whole and absolute motions can be defined only from unmoved places: and therefore above I have referred to these as immovable places, and relative places to be moveable places. But they are not immovable places, unless all the given positions may serve in turn from infinity to infinity ; and so always they remain immovable, and I call the space which they constitute immovable. The causes, by which true and relative motions can be distinguished from each other in turn, are the forces impressed on bodies according to the motion required to be generated. True motion neither can be generated nor changed, other than by forces impressed on the motion of the body itself: but relative motion can be generated and changed without forces being impressed on this body. For it suffices that they be impressed on other bodies alone to which the motion shall be relative, so that with these yielding, that relation may be changed from which it consisted, of rest or relative motion. On the contrary true motion always is changed from the forces impressed on a moving body ; but the relative motion from these forces is not changed by necessity. For if the same forces thus may be impressed on other bodies also, for which a relation is made, thus so that the relative situation will be conserved on which the relative motion is founded. Therefore all relative motion can be changed where the true may be conserved, and to be conserved where the true may be changed ; and therefore true motion in relations of this kind are considered the least. The effects, by which absolute and relative motions are to be distinguished from each other, are the forces of receding from the axis of circular motion. For none of these forces in circular motion are in mere relative motion, but are in a true [circular] motion greater from true absolute motion for a quantity of motion. If a vessel may hang from a long thread, and always is turned in a circle, while the thread becomes very stiff, then it may be filled with water, and together with the water remains at rest; then by another force it is set in motion suddenly in the opposite sense and with the thread loosening itself, it may persevere a long time in this motion; the surface of the water from the beginning was flat, just before the motion of the vessel: But after the vessel, with the force impressed a little on the water, has the effect that this too begins to rotate sensibly; itself to recede a little from the middle, and to ascend the sides of the vessel, adopting a concave figure, (as I have itself tested) and by moving faster from the motion it will rise always more and more, while the revolutions by being required to be completed in the same times with the vessel, it may come to relative rest with the same vessel. Here the ascent indicates an attempt to recede from the axis of the motion, and by such an attempt it becomes known, and the true and absolute circular motion of the water is measured, and this generally is contrary to the relative motion. In the beginning, when the motion of the water was a maximum relative to the vessel, that motion did not incite any attempt to recede from the axis: the water did not seek circumference by requiring to ascend the sides of the vessel, but remained flat, and therefore the true circular motion had not yet begun. Truly later, when the relative motion had decreased, the ascent of this to the sides of the vessel indicated an attempt of receding from the axis; and this trial showed this true circular motion always increasing, and finally made a maximum when the water remained at rest relative to the vessel. Whereby this trial does not depend on the translation of the water



with respect to orbiting bodies, and therefore true circular motion cannot be defined by such translations. Truly the circular motion of each revolving body is unique, corresponding to a singular and adequate effort to be performed : but relative motions are for innumerable and varied external relations ; and corresponding to a relation, generally they are lacking in true effects, unless in as much as they share in that true and single motion. And by those who wish, within a system of these [rotational motions], our heavens [i.e. local space] to revolve in a circle below the heavens of the fixed stars [i.e. distant interstellar space], and the planets to defer with it ; the individual parts of the heavens, and the planets which truly are moving, which indeed within their nearby heavens [i.e. the local part of space relative to themselves] are relatively at rest, truly are moving. For they change their positions in turn (as otherwise the system truly passes into rest) and together with the deferred heavens they participate in the motion of these, and so that the parts of the revolving total are trying to recede from the axes of these. Relative quantities are not therefore these quantities themselves, the names of which they bear, but those perceptible measures (true or mistaken) of them which are used by ordinary people in place of the measured quantities. [Thus, a length is related to a standard length, etc.] But if the significances of words are required to be defined from the use; these measures perceptible [to the senses] are to be particularly understood by these names : Time, Space, Location and Motion ; and the discourse will be contrary to custom and purely mathematical, if measured quantities here are understood. [Here Newton is expressing the fact that he uses such quantities in our sense as abstract variables, rather than as mere units for measuring the amounts of physical quantities, as one might use in arithmetic.] Hence they carry the strength of holy scriptures, which may be interpreted there by these names regarding measured quantities. Nor do they corrupt mathematics or [natural] philosophy any less, who combine true quantities with the relations of these and with common measures. Indeed it is most difficult to know the true motion of bodies and actually to discriminate from apparent motion ; therefore because the parts of that immobile space, in which bodies truly are moving, do not meet the senses. Yet the cause is not yet quite desperate. For arguments are able to be chosen, partially from apparent motions which are the differences of true motions, partially from forces which are the causes and effects of true motions. So that if two globes, to be connected in turn at a given distance from the intervening thread, may be revolving about the common centre of gravity; the exertion of the globes to recede from the axis of the motion might become known from the tension in the thread, and thence the quantity of the circular motion can be computed. Then if any forces acting equally likewise may be impressed mutually on the faces of the globes to increase or diminish the circular motion, the increase or decrease in the circular motion may become known from the increase or decrease in the tension of the thread ; and thence finally the faces of the globes on which the impressed forces must be impressed, so that the motion may be increased maximally; that is, the faces to the rear, or which are following in the circular motion. But with the faces which are following known, and with the opposite faces which precede, the determination of the motion may be known. In this manner both the quantity and the determination of the motion of this circle may be found in a vacuum however great, where nothing may stand out externally and perceptibly by which the globes may be able to be brought together [in comparison]. If now bodies may



be put in place in that space with a long distance maintained between themselves, such as the fixed stars are in regions of the heavens : indeed it may not be possible to know from the relative translation of the globes among the bodies, whether from these or those a motion may be required to be given. But if attention is turned to the string, and the tension of that itself is taken to be as the required motion of the globes ; it is possible to conclude that the motion is that of the globes, and [the distant] bodies to be at rest ; & then finally from the translation of the globes among the bodies, the determination of this motion can be deduced. But the true motion from these causes, are to be deduced from the effects and from the apparent differences, or on the contrary from the motions or forces, either true or apparent, the causes and effects of these to be found, will be taught in greater detail in the following. For towards this end I have composed the following treatise.



AXIOMS,

OR

THE LAWS OF MOTION.

LAW I.

Every body perseveres either in its state of resting or of moving uniformly in a direction, unless that is compelled to change its state by impressed forces. PRojectiles persevere in their motion, unless in as much as they may be retarded by the resistance of the air, and they are impelled downwards by the force of gravity. A child's spinning top, the parts of which by requiring to stick together always, withdraw themselves from circular motion, does not stop rotating, unless perhaps it may be slowed down by the air. But the greater bodies of the planets and comets preserve both their progressive and circular motions for a long time made in spaces with less resistance .

LAW II.

The change of motion is proportional to the [magnitude of the] impressed motive force, and to be made along the right line by which that force is impressed.

If a force may generate some motion ; twice the force will double it, three times triples, if it were impressed either once at the same time, or successively and gradually. And this motion (because it is determined always in the same direction generated by the same force) if the body were moving before, either is added to the motion of that in the same direction, or in the contrary direction is taken away, or the oblique is added to the oblique, and where from that each successive determination is composed.



LAW III.

To an action there is always an equal and contrary reaction : or the actions of two bodies between themselves are always mutually equal and directed in opposite directions.

Anything pressing or pulling another, by that is pressed or pulled just as much. If anyone presses a stone with a finger, the finger of this person is pressed by the stone. If a horse pulls a stone tied to a rope, and also the horse (as thus I may say) is drawn back equally by the stone: for the rope stretched the same on both sided requiring itself to be loosened will draw upon the horse towards the stone, and the stone towards the horse; and yet it may impede the progress of the one as much as it advances the progress of the other. If some body striking another body will have changed the latter's motion in some manner by the former's force, the same change too will be undergone in turn on its own motion in the contrary direction by the force of the other (on account of the mutual pressing together). By these actions equal changes are made, not of the velocities but of the motions ; obviously in bodies not impeded otherwise. For changes of the velocity, are made likewise in the contrary parts, because the motions are changed equally, they are inversely proportional with the bodies [i.e. with their masses]. This law is obtained with attractions also, as will be approved in a nearby scholium.

COROLLARY I.

A body with forces added together describes the diagonal of the parallelogram in the same time, in which the separate sides are described.

If the body in a given time, by the force alone M impressed at the place A, may be carried with a uniform motion from A to B; and by the single force N impressed at the same place, may be carried from A to C: the parallelogram ABDC may be completed, and by each force that body may be carried in the same time on the diagonal from A to D. For because the force N acts along the line AC parallel to BD itself, this force by law II will not at all change the velocity required to approach that line BD generated by the other force. Therefore the body approaches the line BD in the same time, whether the force N may be impressed or not ; and thus at the end of the time it may be found somewhere on that line BD. By the same argument at the end of the same time the body will be found somewhere on the line CD, and on that account it is necessary to be found at the concurrence D of each of the lines. Moreover it will go in rectilinear motion from A to D by law I.



COROLLARY II. And hence the composition of the force directed along AD is apparent from any oblique forces AB and BD, and in turn the resolution of any force directed along AD into any oblique forces AB and BD. Which composition and resolution indeed is confirmed abundantly from mechanics. So that if from the centre of some wheel O the unequal radii OM, ON emerge , the weights A and P may be sustained by the threads MA, NP, and the forces of the weights are required towards moving the wheel: Through the centre O a right line KOL is drawn, meeting the threads perpendicularly at K and L, and with the centre O, and OL the greater of the intervals OK, OL, a circle is described meeting the thread MA in D: and AC shall be parallel to the right line made OD , and the perpendicular DC. Because it is of no importance, whether the points of the threads K, L, D shall be fastened or not to the plane of the wheel; the weights will prevail the same, and if they may be suspended from the points K and L or D and L. But the total force of the weight A is set out by the line AD, and this is resolved into the forces AC, CD, of which AC by pulling [drawing in the original text] the radius OD directly from the centre provides no force to the required wheel movement ; but the other force DC, by pulling on the radius DO perpendicularly, accomplishes the same, as if it pulls the radius OL the equal of OD itself; and that is, the same weight P, but only if that weight shall be to the weight A as the force DC to the force DA, that is (on account of the similar triangles ADC, DOK,) as OK to OD or OL. Therefore the weights A and P, which are inversely as the radii OK and OL placed in line, will exert the same influence, and thus remain in equilibrium : which is the most noticeable property of scales, levers, and of a wheel and axle. If either weight shall be greater than in this ratio, the force of this will be so much greater requiring the wheel to rotate. But if a weight p, equal to the weight P, is suspended partly by the thread Np, and partly by resting on the oblique plane pG : [the forces] pH and NH are acting, the former perpendicular to the horizontal, the latter perpendicular to the plane pG; and if the force of the weight p acting downwards, is set out by the line pH, this can be resolved into the forces pN, HN. If some plane pQ shall be perpendicular to pN, cutting the other plane pG in a line parallel to the horizontal [here we have to think in 3 dimensions]; and the weight p lies only on the planes pQ, pG ; that it may press on these planes by these forces pN, HN, without doubt the plane pQ perpendicularly by the force pN, and the plane pG by the force HN. And thus if the plane pQ may be removed, so that the weight may stretch the thread ; because now the thread in turn by being required to sustain the weight, performs the function of the plane removed, may be extended by that same force pN, which before



acted on the plane. From which, the tension of this oblique thread will be to the tension of the thread of the other perpendicular PN, as pN to pH. And thus if the weight p shall be to the weight A in a ratio, which is composed from the reciprocal ratio of the minimum distances of their threads pN, AM from the centre of the wheel, and in the direct ratio pH to pN; the same weights likewise will prevail for the wheel being moved, and thus mutually will sustain each other, as any test can prove. But the weight p, pressing on these two oblique planes, can be compared to a wedge making a split between the internal surfaces: and thence the forces of the wedge and of the hammer become known: seeing that since the force by which the weight p presses hard on the plane pQ to the force, by which the same either by its weight or impelled by the blow of the hammer acting along the line pH in the plane, shall be as pN to pH; and to the force, by which it presses hard on the other plane pG, as pN to NH. But also the force of a screw can be deduced by a similar division of the forces; obviously which wedge is pushed against by a lever. Therefore the uses of this corollary appear to be the widest, and by extending widely the truth of this prevails; since from now with what has been said, all mechanical devices may depend on explanations shown in different ways by authors. And from these indeed we may derive easily the forces of machines, which from wheels, revolving cylinders, pulleys, levers, stretched cords and weights directly or obliquely ascending, and with the rest from the powers of mechanics are accustomed to be assembled, and as also the forces of tendons requiring the bones of animals to be moved. [The diagram has been altered a little, as in the original the oblique angle appears to be 900. Note that Newton pays a little attention here to statics and simple machines.]

COROLLARY III.

The quantity of motion which is deduced by taking the sum of the motions of the contributing factors in the same direction and the difference of the contributing factors in

the opposite direction , may not change from the action of bodies among themselves. And indeed the action and the contrary reaction to that are equal by law III, and thus by law II bring about equal changes in the motions towards the contrary directions. Therefore if the motions are made in the same direction; whatever is added to the motion of the departing body, is taken from the motion of the following body thus, so that the sum may remain the same as before. If the bodies may get in each other's way, an equal among will be taken from the motion of each, and thus the difference of the contributing factors of the motions in the opposite directions will remain the same. So that if a spherical body A shall be three times greater than a spherical body B, and it may have two parts of velocity ; and B may follow on the same right line with a velocity of ten parts, and thus the motion of A shall be to the motion of B itself, as 6 to 10 : the motion from these may be put to be of 6 parts and of 10 parts, and the sum will be of 16 parts. Therefore in the meeting of the bodies, if the body A may gain a motion of 3, 4, or 5 parts, the body B will lose just as many parts, and thus the body A will go on after the reflection with 9, 10 or 11 parts, and B with 7, 6, or 5 parts, with the same sum of the parts present as before. If the body A may gain 9, 10, 11, or 12, and thus may move past



the meeting with 15, 16, 17, or 18 parts ; the body B, by losing as many parts as A may gain, may be progressing with one part, with 9 parts lost, or it may be at rest with the 10 parts of its motion missing, or with one part it may be moving backwards with one part more (as thus I may say) missing from its motion, or it may be moving backwards with two parts on account of subtracting 12 parts of the motion forwards. And thus the sum of the motions in the same direction 15 1+ or 16 0+ , and the differences in the opposite directions 17 1 and 18 2− − will be always 16 parts, as before the meeting and the reflection. But with the motions known with which bodies go on after the reflection, the velocities of which may be found, on putting that to be to the velocity before the reflection, as the motion after is to the motion before. So that in the final case, where the motion of the body A were of six parts before reflection and of eight parts afterwards, and the velocity of two parts before the reflection; the velocity of six parts after the reflection may be found, on being required to say, that the motion of 6 parts before the reflection to the 18 parts afterwards, thus of 2 parts of velocity before the reflection to 6 parts of velocity afterwards. But if bodies are incident between themselves mutually obliquely either non spherical or with differing rectilinear motions, and the motions of these may be required after reflection ; the position of the plane in which the meeting bodies are touching at the point of concurrence is required to be known: then the motion of each body (by Corol.II) is required to be separated into two parts, one perpendicular to this plane, the other parallel to the same : but the parallel motions, therefore because the bodies act in turn between themselves along a line perpendicular to this plane, are required to have retained the same motion before and after the reflection, and the changes in the perpendicular motions thus required are to be attributed equally in opposite directions, so that both the sum of the acting together and the difference of the contrary may remain the same as before. From reflections of this kind also circular motions are accustomed to arise about their own centres. But I will not consider these cases in the following, and it would be exceedingly long to consider showing all this here.

COROLLARY IV.

The common centre of gravity of two or more bodies, from the actions of the bodies between themselves, does not change its state either of motion or of rest ; and therefore

the common centre of gravity of all bodies in the mutual actions between themselves (with external actions and impediments excluded) either is at rest or is moving uniformly in

direction. For if two points may be progressing with a uniform motion in right lines, and the distances of these is divided in a give ratio, the dividing point either is at rest or it is progressing uniformly in the right line. This is shown later in a corollary to lemma XXIII of this work, if the motion of the points is made in the same plane ; and by the same account can be demonstrated, if these motions are not made in the same plane. Hence if some bodies are moving uniformly in right lines, the common centre of gravity of any two either is at rest or progresses uniformly in a right line; because the line joining the centres of these bodies therefore is required to be progressing uniformly in right lines, it



is divided by this common centre in a given ratio. And similarly the common centre of these two and of any third either is at rest or progressing uniformly in a right line ; because the distances between the common centre of the two bodies and of the third body from that therefore is divided in a given ratio . In the same manner also the common centre of these three and of some fourth body either is at rest or is moving uniformly in a straight line; because from that the distance between the common centre of the three and the centre of the fourth body therefore is divided in a given ratio, and thus ad infinitum. Therefore in a system of bodies, which in the interactions among themselves in turn and in general are free from all extrinsic forces, and therefore singly are moving uniformly in individual right lines, the common centre of gravity of all either is at rest or is moving in a direction uniformly. Again in a system of two bodies acting on each other in turn, since the distances of the centres of each from the common centre of gravity shall be reciprocally as the bodies [i.e. the masses of the bodies]; the relative motions of the bodies from the same shall be equally between themselves either approaching to that centre or receding from the same. Hence that centre, from the equal changes of the motions made in opposite directions, and thus from the actions of these bodies between themselves, neither will move forwards nor be retarded nor suffers a change in its state as far as motion or rest is concerned. But in a system of several bodies, because the common centre of gravity of any two bodies acting mutually between themselves on account of that action allows no change at all in its state ; and of the remaining , the action from which does not hinder these, the common centre of gravity thence suffers nothing; but the distance of these two centres of gravity is divided by the common centre of all the bodies into parts to which the total [masses] of the bodies are reciprocally proportional ; and thus with the state of their moving or resting maintained from these two centres, the common centre of all also maintains its own state: because it is evident that common centre of all on account of the actions of two bodies between themselves at no time changes its own state as far as motion and rest are concerned. But in such a system all the actions of the bodies between each other, are either between two bodies, or between the actions of two composite bodies ; and therefore at no time do they adopt a change of everything from the common centre in the state of this of motion or rest. Whereby since that centre where the bodies do not act among themselves in turn, either is at rest, or is progressing uniformly in some right motion, will go on the same, without the opposition of bodies, without actions between themselves, either to be at rest or always to be progressing uniformly in a direction ; unless it may be disturbed from that state by some external forces. Therefore it is the same law of a system of many bodies, which of a solitary body, as far as persevering in a state of motion or of rest. Indeed the progressive motion either of a solitary body or of a system of bodies must always be considered from the motion of the centre of gravity.

COROLLARY V.

The motions of bodies between themselves included in a given space are the same, whether that space be at rest, or the same may be moving in a direction without circular

motion.



For the differences of the motions given in the same direction , and the sums to be given in the opposite directions, are the same initially in each case (by hypothesis), and with these sums or differences the collisions and impulses arise with which the bodies strike each other. Therefore by law II the effects of the collisions are equal in each case ; and therefore the motion between themselves in one case will remain equal to the motions between themselves in another case. The same may be proven clearly by an experiment. All motions between bodies occur in the same way on a ship, whither that is at rest or is moving uniformly in a direction.

COROLLARY VI.

If bodies may be moving in some manner among themselves, and from the forces with equal accelerations may be impressed to move along parallel lines ; all will go on to

move in the same manner between themselves, and if by these forces they are not to be disturbed.

For these forces equally (for the quantities of the bodies required to be moved) and by acting along parallel lines, all the bodies will be moved equally (as far as velocity is concerned) by law II, and thus at no time will the positions and motions of these between each other be changed.

Scholium.

Thus far I have treated the fundamentals with the usual mathematics and confirmed many times by experiment. By the first two laws and from the first two corollaries Galileo found the fall of weights to be in the duplicate ration of the time, and the motion of projectiles to be in a parabola; by agreeing with experiment, unless as far as the those motions are retarded a little by the resistance of the air. With a body falling uniform gravity, by acting equally in equal small intervals of time, will impress equal forces on that body, and generates equal velocities: and in the total time the total force impressed and the total velocity it generates is proportional to the time. And the distances described in the proportional times, are as the velocities and times taken together ; that is in the duplicate ratio of the times. And with the body projected up uniform gravity impresses forces and velocities taken proportional to the times ; and the times required to rise the greatest heights are as the velocities required to be taken away, and these heights are as the velocities and the times taken together or, in the duplicate ratio of the velocities. [Duplicate ratio means of course, as the square.] And the motion arising of a body projected along some right line is composed from the motion arising from gravity. So that if the body A by its motion of projection only in a given time can describe the right line A B and from its motion only of requiring to fall, can describe the height AC in the same time: the parallelogram ABDC may be completed, and that body will be found at the end of the time at the place D from the composed motion ; and the curved line AED, which that body



describes, will be a parabola which the right line AB touches at A, and the ordinate BD of which is as AB squared. The demonstrations of the times of oscillations of pendulums will depend on the same laws and corollaries, from the daily experience with clocks. From the very same, and with the third law Sir Christopher Wren, Dr. John Wallis and Christian Huygens, easily the principle outstanding geometers of the age, have discovered separately the rules of hard bodies colliding and rebounding, and almost at the same time they communicated the same with the Royal Society, among themselves (as regards these laws) everything is in agreement: and indeed first Wallis, then Wren and Huygens produced the discovery. But the truth has been established by Wren in person before the Royal Society through an experiment with pendulums : which also the most illustrious Mariotte soon deemed worthy to explain in a whole book. Truly, so that this experiment may agree to the precision with theories, and account is required to be had, both of the resistance of the air, as well as of the elasticity of the colliding bodies. The spherical bodies A, B may hang from parallel threads and with AC, BD equal from the centres C, D. From these centres and intervals the bisected semicircles EAF, GBH are described, with radii CA, DB. The body A may be drawn to some point R of the arc EAF, and thence may be released (with the body B taken aside), and after one oscillation it may return to the point V. RV is the retardation from the air resistance. ST is made the fourth part of RV placed in the middle, thus evidently so that R S and TV are equal, and R S to ST shall be as 3 to 2. And thus ST will show the retardation in falling from S to A approximately. [A rule gained from experience perhaps for light damping.] Body B may be restored to its place B. Body A may fall from the point S, and the velocity of that at the place of reflection A will be without so great a sensible error, and as if it had fallen from the location T in a vacuum. Therefore this velocity is set out by the chord of the arc TA. For the velocity of the pendulum at the lowest point is as the chord of the arc, that it has described on falling, the proposition is well known from geometry. After the reflection the body A may arrive at the location s, and the body B at the location k. The body B may be taken away and the position v may be found ; from which if the body A may be sent off and after one oscillation may return to the location r, let st be the fourth part of that rv placed in the middle, thus so that it may be considered that rs and tv are equal ; and the velocity may be set out by the chord of the arc tA, that the body A had approximately after the reflection at the place A. For t will be that true and correct place, to which the body A, with the resistance of the air removed, ought to be able to rise. The location k to which the body B has risen is required to be corrected by a similar method, and requiring to find the location l, to which that body ought to ascend in a vacuum. With this done it is possible to test everything, in the same way as if we were placed in a vacuum. Yet the body A will be required to adopt (as thus I may say) the chord TA of the arc, which shows the velocity of this, so that the motion may be had approximately at the place A before the reflection ; then the chord tA of the arc, so that the motion of this may be had approximately at the place A after the reflection. And thus the body B will be



required to adopt the chord of the arc Bl, so that the approximate motion of this may be had after the reflection. And by a similar method, where the bodies are sent off at the same time from two different locations, the motions of each are required to be found both before, as well as after the reflection ; and then finally the motions among themselves are to be brought together and the effect of the reflection deduced. In this manner with the matter requiring to be tested with ten feet pendulums, and that with bodies both unequal as well as equal, and by arranging so that bodies may concur from the greatest intervals, such as 8, 12, or 16 feet ; I have found [the text has reperi or you find in the singular command mode, which has been taken as a misprint, rather than repperi which has been adopted for translation] always within an error of 3 inches in the measurement, where the bodies themselves were meeting each other directly, equal changes were obtained in the contrary parts of the motions for the bodies, and thus the actions and reactions always to be equal. So that if the body A were incident on body B at rest with 9 parts of motion, and with 7 parts removed it went on with 2 parts after the reflection ; body B was rebounding with these 7 parts. If the bodies were going against each other, A with 12 parts and B with 6, and A was returning with 2 parts ; B was returning with 8, each with the removal of 14 parts. From the motion of A 12 parts are removed and nothing remains: 2 other parts are taken away , and there is made a motion of 2 parts in the opposite direction: and thus from the motion of the body B of 6 parts by requiring 14 parts to be taken away, 8 parts are made in the opposite direction. But if the bodies were going in the same direction, A faster with 14 parts, and B slower with 5 parts, and after the reflection A was going on with 5 parts ; B was going on with 14 parts, with the translation made of 9 parts from A to B. And thus for the rest. From a running together and collision of bodies at no time does the quantity of motion change, which is deduced from the sum of the motions acting in the same directions or from the differences in contrary directions. For the error of an inch or two I may attribute to the difficulty required in performing the individual measurements accurately enough. It was with difficulty, not only that the pendulums thus be dropped at the same time, so that the bodies could strike each other at the lowest position A B; but also the locations s, k to be noted, to which the bodies were ascending after the collision. But the unequal density of the parts of the pendulous bodies, and the construction from other causes of irregularity, were leading to errors. Again lest anyone may object to the rule requiring approval which this experiment has found, to presuppose the bodies either to be completely hard, or perhaps perfectly elastic, none are to be found of this kind in natural compositions ; because I add now experiments described which succeed equally with soft or hard bodies, without doubt by no means depending on the condition of hardness. For if that rule is required to be extended to bodies which are not perfectly hard, the reflection is to be diminished only in a certain proportion to the magnitude of the elastic force. In the theory of Wren and Huygens absolutely hard bodies return with the speed of the encounter. Most surely that will be proven with perfectly elastic bodies. In imperfectly elastic bodies the return speed is required to be diminished likewise with the elastic force ; therefore because that elastic force, (unless where the parts of bodies are struck by their coming together, or extended a little as if they suffer under a hammer,) and certainly shall be required to be determined (as much as I know) and may be made so that bodies may return with a relative velocity in turn, which shall be in a given ratio to the relative velocity of approach. This I have



tested thus with balls of wool closely piled together and strongly constricted. First by dropping pendulums and by measuring the reflection, I have found the magnitude of the elastic force; then with this force I have determined the reflections in other cases of concurrence, and they have answered the trials. Always the wool have returned with a relative velocity, which is to be to the relative velocity of concurrence as around 5 to 9. Balls of steel return with almost the same velocity ; others from cork with a little less : but with glass the proportion was around 15 to 16. And with this agreed upon, the third law has agreed with theory as far as impacts and reflections are concerned, which clearly agree with experiment. I briefly show the matter for attractions thus. With any two bodies A and B mutually attracting each other, consider some obstacle placed between each, by which the meeting of these may be impeded. If either body A is drawn more towards the other body B, than that other B towards the first A, the obstacle will be urged more by the pressing of body A than by the pressing of body B; and hence will not stay in equilibrium. The pressing will prevail stronger, and it will act so that the system of the two bodies and the obstacle may move in the direction towards B, and in motions in free spaces always by accelerating, may depart to infinity. Which is absurd and contrary to the first law. For by the first law the system must persevere in it state of rest or of uniform motion in a direction, and hence the bodies will press equally on the obstacle, and on that account are drawn equally in turn. I have tested this with a loadstone and iron. If these placed in their own vessels touching separately they may float next to each other in still water ; neither propels the other, but from the equality of the attraction they sustain mutual attempts between

themselves, and finally they remain in an established equilibrium. Thus also is the gravity between the earth and the mutual parts of this. The earth FI is cut by some place EG into two parts EGF and EGI: and the mutual weights of these shall be equal mutually between themselves. For if by another plane HK which shall be parallel to the first part EG, the greater part EGI shall be cut into the two parts EGKH and HKI, of which HKI shall be equal to the first part cut EFG: it is evident that

the middle part EGKH by its own weight will not be inclined to either of the extreme parts, but between each in equilibrium, thus so that I may say, it may be suspended and it is at rest. But the extreme part HKI by its own weight presses on the middle part, and will urge that into the other extreme part EGF; and thus the force by which the sum of the parts HKI & EGKH , EGI tends towards the third part EGF, is equal to the weight of the third part HKI, that is to the weight of the third part EGF. And therefore the weights of the two parts EGI, EGF are mutually in equilibrium, as I had wished to show. And unless these weights shall be equal, the whole earth floating on the free aether may go towards the greater weight, and from that required flight would go off to infinity. Just as bodies in coming together and reflecting may exert the same influence [on each other], the velocities of which are reciprocally as their innate forces: thus the agents exert the same influence in the movements of mechanical devices, and by contrary exertions mutually sustain each other, the velocities of which, following the determination of the forces considered, are reciprocally as the forces. Thus the weights exert the same



influence towards moving the arms of scales, which with the scales oscillating are reciprocally as the velocities of these up and down: that is, the weights, if they ascend up and down rightly [i.e. vertically], exert the same influences, which are reciprocally as the distances of the points from which they are suspended from the axis of the scales; if, by an oblique plane or from other obstacles to the motion, the ascents or descents are oblique, they exert the same influence, which are reciprocally as the ascent and descent, just as made along the perpendicular: and that on account of the determination of the weight acting downwards. Similarly with a pulley or a pulley system, the force of the hand directly drawing on the rope which shall be to the weight , ascending either directly or obliquely, as the perpendicular speed of ascent to the speed of the hand pulling directly on the rope will sustain the weight. In clocks and similar devices, which have been constructed from little wheels joined together, the forces required contrary to promoting and retarding the motions of the wheels if they are reciprocally as the speeds of the wheels on which they are impressed, will mutually sustain each other. The force of a screw required to press upon a body is to the force of hand turning the handle, as the rotational speed of the handle in that part where it is pressed on by the hand, to the speed of progress of the screw towards the body pressed. The forces by which a wedge urges the two parts of wood to be split are to the force of the hammer to the wedge, as the progress of the wedge following a determined force impressed by the hammer on itself, to the speed by which the parts of the wood, following lines perpendicular to the faces of the wedge. And an account of all machines is the same. The effectiveness and use of these consists in this only, that by diminishing the speed we augment the force, and vice versa: From which the general problem is solved in all kinds of suitable mechanical devices : a given weight is to be moved by a given force, or some given resistance is to be overcome by a given force. For if machines may be formed thus, so that the speeds of the driving force and of the resistance shall be reciprocally as the forces ; the driving force will sustain the resistance : and it may overcome the same with a greater difference of the speeds. Certainly if the disparity of the speeds shall be so great, so that all resistance may also overcome, which is accustomed to arise both from the slipping and friction of nearby bodies between each other, as well as from the cohesion and in turn of the separation and continued elevation of bodies ; with all that resistance overcome, the excess force will produce an acceleration motion proportional to itself, partially within the parts of the machine, and partially within the resisting body. It is not the intention of this work to treat everything mechanical. I have wished only to show, both how wide and sure the third law of motion shall be. For if the action of the driving force may be estimated from the speed and this force taken jointly ; and similarly the reaction of the resistance may be estimated conjointly from the velocities of the individual parts of this, and from the friction of these, from the cohesion, and from the weight, and from the acceleration arising ; the action and the reaction will always be equal to each other in turn, in every use of instruments. And as far as the action is propagated by the instrument and finally may be impressed on any resisting body, the final determination of this will always be contrary to the determined reaction.

Isaac NEWTON: Philosophiae Naturalis Principia Mathematica. 3rd Ed.

Book I Section I. Translated and Annotated by Ian Bruce. Page 76

THE MATHEMATICAL PRINCIPLES OF

NATURAL PHILOSOPHY

CONCERNING THE

MOTION OF BODIES

BOOK ONE.

SECTION I.

Concerning the method of first and last ratios, with the aid of which the following are demonstrated.

LEMMA I.

Quantities, and so the ratios of quantities, which tend steadily in some finite time to

equality, and before the end of that time approach more closely than to any given differences, finally become equal.

If you say no; so that at last they may become unequal, and there shall be a final difference D of these. Therefore they are unable to approach closer to equality than to the given difference a D: contrary to the hypothesis.

Q. E. D. LEMMA II.

If in some figure AacE, with the right lines Aa, AE and the curve acE in place, some number of parallelograms are inscribed Ab, Bc, Cd, &c. with equal bases AB, BC, CD, &c. below, and with the sides Bb, Cc, Dd, &c. maintained parallel to the side of the figure Aa; & with the parallelograms aKbl, bLcm, cMdn, &c. filled in. Then the width of these parallelograms may be diminished and the number may be increased to infinity : I say that the final ratios which the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and to the curvilinear figure AabcdE have in turn between each other, are ratios of equality.



For the difference of the inscribed and circumscribed figures is the sum of the parallelograms Kl, Lm, Mn, Do, that is (on account of the equal bases) the rectangle under only one of the bases Kb and the sum of the heights Aa, that is, the rectangle ABla. But this rectangle, because with the width of this AB diminished indefinitely, becomes less than any given [rectangle] you please. Therefore (by lemma I) both the inscribed and circumscribed figures finally become equal, and much more [importantly] to the intermediate curvilinear figure.

Q. E. D.

LEMMA III.

Also the final ratios are the same ratios of equality, when the widths of the parallelograms AB, BC, CD, &c. are unequal, and all are diminished indefinitely.

For let AF be equal to the maximum width, and the parallelogram FAaf may be completed. This will be greater than the difference of the inscribed and of the circumscribed figure ; but with its own width AF diminished indefinitely, it is made less than any given rectangle.

Q. E. D.

Corol. I. Hence the final sum of the vanishing parallelograms coincides in every part with the curvilinear figure. Corol. 2. And much more [to the point] a rectilinear figure, which is taken together with the vanishing chords of the arcs ab, bc, cd, &c., finally coincides with the curvilinear figure. Corol. 3. And in order that the circumscribed rectilinear figure which is taken together with the tangents of the same arcs. Corol. 4. And therefore these final figures (as far as the perimeter acE,) are not rectilinear, but the curvilinear limits of the rectilinear [figures].



LEMMA IV.

If in the two figures AacE, PprT, there are inscribed (as above ) two series of parallelograms, and the number of both shall be the same, and where the widths are diminished indefinitely, the final ratios of the parallelograms in the one figure to the parallelograms in the other, of the single to the single, shall be the same ; I say on which account, the two figures in turn AacE, PprT are in that same ratio. And indeed as the parallelograms are one to one, thus (on being taken together) shall be the sum of all to the sum of all, and thus figure to figure; without doubt with the former figure present (by lemma III) to the first sum, and with the latter figure to the latter sum in the ratio of equality.

Q. E. D. Corol. Hence if two quantities of any kind may be divided into the same number of parts in some manner; and these parts, where the number of these is increased and the magnitude diminished indefinitely, may maintain a given ratio in turn, the first to the first, the second to the second, and with the others in their order for the remaining : the whole will be in turn in that same given ratio. For if, in the figures of this lemma the parallelograms are taken as the parts between themselves, the sums of the parts always will be as the sum of the parallelograms ; and thus, when the number of parts and of parallelograms is increased and the magnitude is diminished indefinitely, to be in the final ratio of parallelograms to parallelograms, that is (by hypothesis) in the final ratio of part to part.

LEMMA V.

All the sides of similar figures, which correspond mutually to each other, are in proportion, both curvilinear as well as rectilinear: and the areas shall be in the squared

ratio of the sides.



LEMMA VI.

If some arc in the given position ACB is subtended by the chord AB, and at some point A, in the middle of the continued curve, it may be touched by the right line AD produced on both sides; then the points A, B in turn may approach and coalesce ; I say that the angle BAD, contained within the chord and tangent, may be diminished indefinitely and vanishes finally. For if that angle does not vanish, the arc ACB together with the tangent AD will contain an angle equal to a rectilinear angle, and therefore the curve will not be continuous at the point A, contrary to the hypothesis.

Q. E. D.

LEMMA VII. With the same in place; I say that the final ratio of the arc, the chord, and of the tangent in turn is one of equality. For while the point B approaches towards the point A, AB and AD are understood always to be produced to distant points b and d, and bd is drawn parallel to the section BD. And the arc Acb always shall be similar to the arc ACB. And with the points A, B coming together, the angle dAb vanishes, by the above lemma ; and thus the finite arcs Ab, Ad and the intermediate arc Acb coincide always, and therefore are equal. And thence from these always the proportion of the right lines AB, AD, and of the intermediate arc ACB vanish always, and they will have the final ratio of equality.

Q. E. D. Corol. I. From which if through B there is drawn BF parallel to the tangent, some line AF is drawn passing through A always cutting at F, this line BF finally will have the ratio of equality to the vanishing arc ACB, because from which on completing the parallelogram AFBD, AD will have always the ratio of equality to AD. Corol. 2. And if through B and A several right lines BE, BD, AF, AG, are drawn cutting the tangent AD and the line parallel to itself BF; the final ratio of the cuts of all AD, AE, BF, BG, and of the chords and of the arc AB in turn will be in the ratio of equality. Corol. 3. And therefore all these lines can be taken among themselves in turn, in the whole argument concerning the final ratios.



LEMMA VIII.

If the given right lines AR, BR, with the arc ACB, the chord AB and the tangent

AD, constitute the three triangles RAB, RACB, RAD, then the points A and B approach together: I say that the final form of the vanishing triangles is one of

similitude, and the final ratio one of equality. For while the point B approaches towards the point A, always AB, AD, AR are understood to be produced a great distance away to the points b, d and r, and with rbd itself to be made parallel to RD, and the arc Acb always shall be similar to the arc ACB. And with the points A and B merging, the angle bAd vanishes, and therefore the three finite triangles coincide rAb, rAcb, rAd, and by the same name they are similar and equal. From which and with these RAB, RACB, RAD always similar and proportional, finally become similar and equal to each other in turn.

Q. E. D. Corol. And hence those triangles, in the whole argument about the final ratios, can be taken for each other in turn.

LEMMA IX.

If the right line AE and the curve ABC, for a given position, mutually cut each other in the given angle A, and to that right line AE at another given angle, BD and CE may be the applied ordinates, crossing the curve at B and C, then the points B and C likewise approach towards the point A: I say that the areas of the triangles ABD and ACE will be in the final ratio in turn, in the square ratio of the sides . And indeed while the points B and C approach towards the point A, it is understood always that the points AD and AE are to be produced to the distant points d and e, so that Ad and Ae shall be proportional to AD and AE themselves, and the ordinates db and ec are erected parallel to the ordinates DB and EC , which occur for AB and AC themselves produced to b and c. It is understood that there be drawn, both the curve Abc similar to ABC itself, as well as the right line Ag, which touches each curve at A, and which cuts the applied ordinates DB, EC, db, ec in F, G, f, g. Then with the length Ae remaining fixed, the points B and C come together at the point A; and with the angle cAg vanishing, the curvilinear areas Abd to Ace coincide with the rectilinear areas Afd to Age; and thus (by lemma V.) they will be in the square ratio of the sides Ad and Ae : But with these areas there shall always be the



proportional areas ABD to ACE, and with these sides the sides AD to A E. And therefore the areas ABD to ACE shall be in the final ratio as the squares of the sides AD to AE.

Q. E. D.

LEMMA X.

The finite distances which some body will describe on being pushed by some force, shall be from the beginning of the motions in the square ratio of the times, that force

either shall be determined and unchanged, or the same may be augmented or diminished continually.

The times are set out by the lines AD to AE, and the velocities generated by the ordinates DB to EC; and the distances described by these velocities will be as the areas ABD to ACE described by these ordinates, that is, from the beginning of the motion itself (by lemma IX) in the square ratio of the times AD to AE.

Q. E. D. Cor I. And hence it is deduced easily, that the errors of bodies describing similar parts of similar figures in proportional times, which are generated by whatever equal forces applied similarly to bodies, and are measured by the distances of bodies of similar figures from these places of these, to which the bodies would arrive in the same times with the same proportionals without these forces, are almost as the squares of the times in which they are generated. [These corollaries examine the effect of small resistive forces on otherwise uniformly accelerated motion; the initial motion being free from such velocity-related resistance.] Corol. 2. Moreover the errors which are generated by proportional forces similarly applied to similar parts of similar figures, are as the forces and the squares of the times jointly. Corol. 3. The same is to be understood from that concerning any distances whatsoever will describe which bodies acted on by diverse forces. These are, from the beginning of the motion, as the forces and the squares of the times jointly. Corol. 4. And thus the forces are described directly as the distances, from the start of the motion, and inversely as the squares of the times. Corol.5. And the squares of the times are directly as the distances described and inversely with the forces.

Scholium.

If indeterminate quantities of different kinds between themselves are brought together, and of these any may be said to be as some other directly or inversely : it is in the sense, that the first is increased or decreased in the same ratio as the second, or with the reciprocal of this. And if some one of these is said to be as another two or more directly



or inversely : it is in the sense, that the first may be increased or diminished in the ratio which is composed from the ratios in which the others, or the reciprocals of the others, are increased or diminished. So that if A may be said to be as B directly and C directly and D inversely: it is in the sense, that A is increased or diminished in the same ratio with

1B C D

× × , that is, that BCA and D

are in turn in a given ratio.

LEMMA XI.

The vanishing subtense of the angle of contact, in all curves having a finite curvature at the point of contact, is finally in the square ratio of the neighbouring subtensed arcs. Case I. Let that arc AB be [called] the subtensed arc [of the chord] AB, the subtense of the angle of contact is the perpendicular BD to the tangent. To this subtense AB and to the tangent AD the perpendiculars AG and BG are erected, concurring in G; then the points D, B, G may approach the points d,b,g, and let J be the intersection of the lines BG, AG finally made when the points D and B approach as far as to A. It is evident that the distance GJ can be less than any assigned distance. But (from the nature of the circles passing through the points ABG, Abg) AB squared is equal to AG BD× , and Ab squared is equal to Ag bd× ; and thus the ratio AB squared to Ab squared is composed from the ratios AG to Ag and BD to bd. But because GJ can be assumed less than any length assigned, it comes about that the ratio AG to Ag may differ less from the ratio of equality than for any assigned difference, and thus so that ratio AB squared to Ab squared may differ less from the ratio BD to bd than for any assigned difference. Therefore, by lemma I, the final ratio AB squared to Ab squared is the same as with the final ratio BD to bd.

Q. E. D.

[Thus, in the limit, 2 2: : BD bd AB Ab= ; the term subtensed is one not used now in geometry, and does not get a mention in the CRC Handbook of Mathematics, etc., and as it is used here, it means simply the chord subtending the smaller arc in a circle. The related versed sine or sagitta that we will meet soon is the maximum distance of the arc beyond the chord, given by 2

21 2cos sin αα− = , where α is the angle subtended by the arc of the unit circle.] Case 2. Now BD to AD may be inclined at some given angle, and always there will be the same final ratio BD as bd as before, and thus the same AB squared to Ab squared.

Q. E. D.



Case 3. And although the angle D may not be given, but the right line BD may converge to a given point, or it may be put in place by some other law; yet the angles D, d are constituted by a common rule and always will incline towards equality, and therefore approach closer in turn than for any assigned difference, and thus finally will be equal, by lemma I, and therefore the lines BD to bd are in the same ratio in turn and as in the prior proposition.

Q. E. D. Corol. 1. From which since the tangents AD to Ad, the arcs AB to Ab, and the sines of these BC to bc become equal finally to the chords AB to Ab ; also the squares of these finally shall be as the subtenses BD to bd. [Thus,

2

2; ; ; and arcAB BCAD AB AB AB AB BDAd Ab arcAb Ab bc Ab bdAb

.→ → → = Care must be taken to note that

some ratios are equal in the limit only, while others are true more generally.] Carol. 2. The squares of the same also are finally as the versed sines [sagittae] of the arcs, which bisect the chords and they converge to a given point. For these versed sines are as the subtenses BD to bd. Corol. 3. And thus the sagitta is in the square ratio of the time in which the body will describe the arc with a given velocity. Carol. 4. The [areas of the] rectilinear triangles ADB to Adb are finally in the cubic ratio of the sides AD to Ad, and in the three on two ratio [of the powers] of the sides DB to db; as in the combined ratio of the sides AD and DB to Ad and db present. And thus the triangles ABC to Abc are finally in the triplicate ratio of the sides BC to bc. Truly I call the three on two ratio the square root of the cube, which is composed certainly from the simple cube and square root [ratios]. [For the areas of the triangles are as

3 32 2

2 2 3 3 3 3

: : :

: : : :

Also, : : : : : : ]

AD.DB Ad.db AD Ad BD bd

AD ad AB Ab AB Ab AD ad .

AD.DB Ad.db AD ad BD bd BD bd BD bd BD bd .

= ×

= × = =

= × = × =

Corol.5. And because DB to db finally are parallel and in the square ratio of AD to Ad : the final curvilinear areas ADB to Adb (from the nature of parabolas) are two thirds of the parts of the rectilinear triangles ADB to Adb; and the segments AB to Ab one third parts of the same triangles. And thence these areas and these segments will be in the



cubic ratio both of the tangents AD to Ad; as will as of the chords and of the arcs AB to Ab.

Scholium.

Moreover in all these we suppose the angle of contact neither to be infinitely greater to the angles of contact which circles maintain with their tangents, nor with the same infinitely small ; that is, the curvature at the point A, neither to be infinitely small nor infinitely great, or the interval AJ to be of finite magnitude. For DB can taken as AD3: [Thus far, as the angle DAB becomes very small, AB AD→ , and

2 2: : BD bd AD Ad→ .] as in that case no circle can be drawn through the point A between the tangent AD and the curve AB, and hence the angle of contact will be infinitely smaller than with circles. And by a similar argument, if DB becomes successively as

4 5 6 7 AD , AD , AD , AD ,&c. there will be had a series of contact angles going on to infinity, of which any of the latter is infinitely smaller then the first. And if DB is made successively as

6 73 4 53 5 62 42 , AD , AD ,AD , AD AD ,AD ,&c. [i.e. powers between 1 and 2]

another infinite series of contact angles will be had, the first of which is of the same kind as with circles, the second infinitely greater, and any later infinitely greater with the previous. But between any two from these angles a series of intermediate angles to be inserted can go off on both sides to infinity, any latter of which will be infinitely greater of smaller with the previous. As if between the terms 2 3 and AD AD the series

13 7 8 1711 9 6 5 14116 5 3 3 5 64 4 2 4 , , AD ,AD , AD AD ,AD ,AD ,AD , AD AD ,AD ,&c. is inserted. And again

between any two angles of this series a new series of intermediate angles can be inserted in turn with infinite intervals of differences. And nor by nature will it know a limit Everything which have been shown fully about curved lines and surfaces , easily may be applied to the surface curves of solids and the curves within. Truly I have presented these lemmas, so that I might escape the tedium of deducing long demonstrations ad absurdum, in the custom of the old geometers. For they are rendered more contracted by the method of indivisibles. But because the hypothesis of indivisibles is harder, and therefore that method is less recommended geometrically; I have preferred the demonstrations of the following things, the sums and ratios of the first arising and for the final of the vanishing quantities, that is, to deduce the limits of sums and ratios; and therefore demonstrations of these limits which I have been able to present briefly. Because indeed with these the same may be done better by the method of indivisibles ; and with the principles now demonstrated we may use them without risk. Hence in the following, if when I have considered quantities as it were from [being] small and unchangeable, or if I have used small curved lines for right lines ; I may not wish indivisibles to be understood, but vanishing divisibles, not the sums and ratios of parts but the limits of the sums and ratios of parts being determined always to be understood ; and the strength of such demonstrations always to be recalled to the method of the preceding lemmas. To the accusation, that the final proportion of vanishing quantities shall be nothing; obviously which, before they will have vanished, it is not the final, where they will have



vanished, and there is nothing. But by the same argument it may be equally contended the velocity is to be zero for a body at a certain place, when the motion may have finished, you may arrive at the final velocity : for this not to be the final, before the body has reached the place, and when it has reached there it is zero. And the easy response is : By the final velocity that is to be understood, by which the body is moving, and neither before it has reached the final place and the motion has ceased, nor afterwards, but then at the instant it touches that place; that is, that velocity itself by which the body reaches the final place and from which motion it ceases. And similarly by the ultimate ratio of vanishing quantities, the ratio of the quantities is required to be understood, not before they vanish, not after, but with which they vanish. And equally the first ratio arising is the ratio by which they are generated. And the first sum and final sum [in turn] is to be, by which they begin or cease (either to be increased or decreased). The limits stand out to which the velocity is able to reach at the end of the motion, but not to be transgressed. This is the final velocity. And in a like manner is the ratio of the limit of a quantity and of the proportions of all beginnings and endings. And since here the limit shall be certain and defined, the problem is truly the same geometrical one to be determined. Truly everything geometrical has been legitimately used in all the geometrical determinations and demonstrations. It may be contended also, that if the final vanishing ratios of vanishing quantities may be given, and the final magnitudes will be given: and thus a quantity will be constructed entirely from indivisibles, contrary to what Euclid demonstrated concerned with incommensurables, in Book X of the Elements. Truly this objection is supported by a false hypothesis. These final ratios actually vanishing from any quantities, are not the ratios of the final quantities, but the limits to which the ratios of quantities always approach by decreasing without bounds ; and which can be agreed they approach nearer than for whatever differences, at no time truly to be transgressed, nor at first to be considered as quantities being diminished indefinitely. The matter is understood more clearly with the infinitely great. If two quantities of which the difference has been given may be increased indefinitely, the final ratio of these is given, without doubt the ratio of equality, nor yet thus will the final quantities of this be given or the maxima of which that is the ratio. In the following, therefore if when I discuss quantities, in advising about things to be considered easily, either as minimas, vanishing or final ; you may understand the quantities are determined with great care, but always to be thought of as diminishing without limit. [In the sense that they are finite and getting smaller, and keep on doing so.] [Translator's Note: At this stage, we are made aware by Newton of the two methods of doing calculations, apart from the ponderous reducti ad absurdum type geometrical methods of the ancient Greeks, e.g. the approach of Archimedes to solving certain problems, and as used by Huygens in his Horologium in deriving the isochronous property of his cycloidal pendulum : The method of the first and last ratio, which appears to be none other than extracting a limit from first principles by seeking closer and closer upper and lower bounds indefinitely; and the method of indivisibles, which is akin to the modern calculus. At present, for the sake of economy and to avoid further controversy it would seem regarding vanishing quantities, only the first of the two has been used and shown in some detail.



Newton goes to great pains to construct a geometrical method which embodies the ideas both of integration and differentiation, but which avoids directly the forming of integrals and derivatives as we know them now; instead, the idea of a limit is set out initially, which incidentally demolishes Zeno' s Paradox in a sentence, and which in words is more or less the present definition of such, the difference between the limiting value and nearby values can be made as small as it pleases without end, without actually being made equal. In the method of first and last sums and ratios, a geometrical method is established for carrying out this process, and so for treating integration and differentiation. At first we look at slightly greater and slightly smaller rectangles enclosing a curve, which approach each other and the segment of the area under the curve closer and closer as their number is increased and the bases diminished, which on the whole seems highly credible. Thus a formulation of integration is obtained. In the second a parallelogram is presented in which a small arc of the curve lies near the diagonal, crossing at the ends, and one side of the parallelogram is a tangent at one vertex; as the parallelogram is diminished in size, the diagonal, the curve, and the tangent finally merge together closer and closer; on elaboration, we are to follow points on lines in diminishing similar triangles approaching an ultimate point at their intersection, where the two similar triangles vanish, but their finite ratios of corresponding sides is maintained by two other trustworthy similar triangles which remain finite all along, this method also seems intuitively ok, as we are told that the final ratio is to be evaluated from the finite similar triangles, at least in the case of the curve being a circle, which follows readily from elementary geometry, while of course Newton rightly asserts that the dreaded zero on zero is never used. Thus the objections of the doubters were allayed, or at least they had less to argue about. This approach is and was not enough for the pure mathematicans at the time, at least those on the continent; we must remember that Newton was (in my view) essentially a physicist doing mathematics, at which he was extraordinarily adept, as well as an experimentalist cum alchemist cum theologian, rather than a pure mathematician. The method considers points moving along lines in time, which we now would consider as a mere parameter, and we are asked to retrace these positions into the past to arrive at the starting point, or very close to it. Thus, Newton's calculus in the Principia is about the rates of change of quantities with time.]


Book I Section II. Translated and Annotated by Ian Bruce. Page 95

SECTION II.

On the finding of centripetal forces.

PROPOSITION I. THEOREM I.

The areas which bodies will describe driven in circles, with the radii drawn towards a

motionless centre of the forces, remain together in immoveable planes, and to be proportional to the times.

The time may be divided into equal parts, and in the first part of the time the body may describe the right line AB with the force applied [at B]. Likewise in the second part of the time, if nothing may hinder, the body may go on along the right line to c, (by Law I.) describing the line Bc itself equal to AB; thus so that from the radii AS, BS, cS acting towards the centre, the equal areas ASB, BSc may be completed. Truly, when the body comes to B, by a single but large impulse the centripetal force acts, and brings about that the body deflects from the line Bc and goes along in the line BC ; cC is acting parallel to BS itself, crossing BC in C; and with the second part of the time completed, the body (by the corollary to Law I.) may be found at C, in the same plane with the triangle ASB. Join SC; and the triangle SBC, on account of the parallels SB, Cc, will be equal to the triangle SBc, and thus also equal to the triangle SAB. By a similar argument if the centripetal force acts successively at C, D, E, &c. making it so that the body in particular small times will describe the individual lines CV, DE, EF, &c. all these will lie in the same plane; and the triangle SCD to the triangle SBC, and SDE to SCD itself, and SEF will be equal to SDE itself. Therefore in equal times equal areas are described in the motionless plane : and by adding together, the sums of any areas SADS, SAFS are between themselves, as the times of describing. Now the number may be increased and the width of the triangles diminished indefinitely; and finally the perimeter of these ADF, (by the fourth corollary of the third lemma) will be a curved line: and thus the centripetal force, by which the body is drawn perpetually from the tangent of this curve, may act incessantly ; truly any areas described SADS, SAFS always proportional to the times of description, in this case will be proportional to the same times. Q. E. D. Corol. I. The velocity of the body attracted towards the motionless centre, in intervals without resistance, is reciprocally as the perpendicular sent from that centre to the rectilinear tangent of the orbit. For the velocities at these locations A, B, C, D, E, are as the bases of equal [area] triangles AB, BC, CD, DE, EF, and these bases are reciprocally as the perpendiculars sent to themselves.



Corol. 2. If the chords AB, BC of two arcs described successively by the same body in equal times in non-resisting distances [or spaces] may be completed in the parallelogram ABCV; and of this diagonal, as it finally has that position when these arcs are diminished indefinitely, BV may be produced both ways, and will pass through the same centre of the forces. Corol. 3. If the chords of arcs described in equal times in distances without resistance AB, BC and DE, EF may be completed into the parallelograms ABCV, DEFZ; the forces at B and E are in turn in the ratio of the final diagonals BV, EZ, when the arcs themselves are diminished indefinitely. For the motions of the body BC and EF are composed (by the Law I corollary) from the motions Bc, BV and Ef, EZ: but yet BV and EZ are equal to Cc and Ff themselves, in the demonstration of this proposition they were being generated by the impulses of the centripetal force at B and E, and thus they are proportional to these impulses. Corol. 4. The forces by which any bodies in non-resisting intervals are drawn back from rectilinear motion and may be turned into curved orbits are between themselves as these versed sines of the arcs described in equal times which converge to the centre of forces, and they bisect the chords when these arcs are diminished indefinitely. For these versed sines are as the half diagonals, about which we have acted in the third corollary. Corol.5. And thus the same forces are as the force of gravity, as these versed sines are to the perpendicular versed sines to the horizontal of the arcs of parabolas, which projectiles describe in the same time. Corol. 6. All remains the same by Corollary V of the laws, when the planes, in which the bodies may be moving, together with the centres of forces, which in themselves have been placed, are not at rest, but may be moving uniformly in a direction.

[Some important features to note in these corollaries are : 1. The distances travelled by the body AB, BC, CD, etc., in successive constant time intervals tδ are related to the constant velocities in the intervals from the start of each interval by the simple relations :

; ; etcA BAB v t BC v t .δ δ= = ; since the areas of the triangles etcSAB, SCB, .Δ Δ described in the equal time intervals are equal, (and note how this has been achieved in diagram by means of the triangles of equal areas and etc.ScC CcB,Δ Δ ), then on calling the perpendicular distances from S to AB, BC, etc., A Bh ,h , etc.,

etc ; or, , etc.A B CSAB SBC SCD, . h AB h BC h CDΔ Δ Δ= = = = , and hence , etc.A A B B C Ch v h v h v= = ,



and thus the perpendicular distance from the centre of force varies inversely as the velocity of the body in that section of the motion. We would now interpret this result in terms of conservation of angular momentum of the orbiting body. 2.When the parallelgrams VBcC, etc. collapse as tδ tends to zero, the impelling force acts along the diagonal towards S. 3. The distances through which the centripetal force act, when applied, are given by the displacements BV or cC, EZ or fF, etc. By Lemma XI of Section I, these displacements tend towards the squares of the versed sines of the arcs AC and DF as tδ tends to zero. 4. These deliberations apply to parabolic arcs with uniform gravity acting uniformly, and also under the circumstances that the whole system is moving with a uniform motion in some direction. It should be noted that Newton, in the following propositions, has essentially let the number of sides of the polygon go to to infinity, and a geometrical analysis is performed on one of the small segments of the original polygon envelope of the resulting curve.]

PROPOSITION II. THEOREM II.

Any body, that is moving in some curved line described in a plane, and with the radius

drawn to some point, either motionless or progressing uniformly in a rectilinear motion, will describe areas about that point proportional to the times, and is urged on

by a centripetal force tending towards the same point. Case I. For any body, because it is moving in a curved line, is turned from rectilinear motion by some force acting on itself (by Law I.). And that force, by which the body is turned from a rectilinear course, and it is known that the minima triangles SAB, SBC, SCD, &c. are described equally in equal intervals of time about the fixed point S, acts in the location B along a line parallel to cC itself (by Prop.XL, Book. I. Elem. and Law II.) that is, along the line BS; and in the place C along a line parallel to dD itself, that is, along the line SC, &c. Therefore the force always acts along lines tending towards that fixed point S.

Q. E. D. Case:2. And, by the fifth corollary of the laws, it is likewise, either the surface is at rest in which the body will describe a curvilinear figure, or it may be moving together with the same body, and with its own point S moving uniformly in a direction, with the figure described. Corol. I. In intervals or in non-resisting mediums, if the areas are not proportional to the times, the forces do not tend to the meeting point of the radii, but thence are deviated as a consequence, or towards the direction in which the motion is made, but only if the description of the areas is accelerated: if it may be retarded ; they are deviated in the opposite way.



Corol. II. Also in mediums with resistance, if the description of the areas is accelerated, the directions of the forces deviate from the running together of the radii towards the direction in which there shall be motion.

Scholium. A body can be urged by a centripetal force composed from several forces. In this case the understanding of the proposition is, because that force which is composed from all the forces, tend towards the point S. Again if some force may act always following a line described perpendicular to a surface ; this will act so that the body is deflected from the plane of its motion: but the magnitude of the surface described neither will be increased or diminished, and therefore in the composition of the forces is required to be ignored.

PROPOSITION III. THEOREM III.

Every body, which with a radius drawn to the centre of some other moving body will describe areas about that centre proportional to the times, is acted on by a force

composed from the centripetal force tending towards that other body, and by the force arising from all the acceleration, by which that other body is acted on.

Let L be the first body, and T the other body : and (by corollary VI of the laws) if by a new force, which shall be equal and contrary to that, by which the other body T is driven, may urge each body along parallel lines ; the first body L goes on to describe the same areas as before about the other body T: but the force T, by which the other body was being, now is destroyed by the force equal and contrary to itself ; and therefore (by Law I) that other body T itself now left to itself, itself either will be at rest or it will be moving uniformly forwards : and the first body L by being pressed on by the difference of the forces, that is, by being urged by the remaining force, goes on to describe areas proportional to the times about the other body T. Therefore it tends (by theorem II.) by the difference of the forces to the other body T as centre.

Q. E. D. Corol. 1. Hence if the one body L, by a radius drawn to the other T, will describe areas proportional to the times ; and concerning the total force (either simple, or from several forces joined together composed according to the second corollary of the laws,) by which the first body L is urged, the total accelerative force is subtracted (by the same corollary of the laws), by which the other body is impressed : all the remaining force, by which the first body is impressed, tends towards the other body T as centre. Corol. 2. And, if these areas are approximately proportional to the times, the remaining force will tend approximately to the other body T. Corol. 3. And in turn, if the remaining force tends approximately towards the other body T, these areas will be approximately proportional to the times.



Corol. 4. If the body L with the radius drawn to the other body T will describe areas, which with the times are deduced to be very unequal ; and that other body T either is at rest, or is moving uniformly in a direction : the action of the centripetal force tending towards the body T either is zero, or it is a mixture and composed from the actions of other very strong forces : and the total force from all, if there are several forces added together, is directed towards another centre (either fixed or moving). The same will be obtained, when the other body is moved by some other motion ; but only if the centripetal force is assumed, which remains after subtracting the total force acting on that other body T .

Scholium.

Because the description of the equal areas is an indication of the centre, which that force considers, by which the body is influenced especially, and by which it is drawn back from rectilinear motion, and retained in its orbit ; why may we not take in the following the equal delineation of the areas as the sign of a centre, about which the motion of all circles in free intervals is carried out ?

PROPOSITION IV. THEOREM IV. The centripetal forces of bodies, which describe different circles with equal motion, tend towards the centres of these circles ; and to be between themselves, so that they are in the same time as the squares of the arcs described and as the radii of the circles. These forces tend towards the centres of the circles by Prop. II. and Corol.2, Prop. I. and are between themselves as with the smallest arcs in equal times, as without doubt of the versed sines described by Corol.4, Prop. I, that is , as the squares of the arcs themselves applied to the diameters of the circles by Lem. VII. and therefore, since these arcs shall be as the arc described in any equal times, and the diameters shall be as the radii of these ; the forces will be as the squares of any arcs described in the same time applied to the radii of the circles.

Q. E. D. Corol. I. Since these arcs shall be as the velocities of the bodies, [and inversely as the radii] the centripetal forces shall be composed from the square ratio of the velocities directly, and in the simple inverse ratio of the radii. Corol. 2. And, since in the periodic times they shall be in the ratio composed from the ratio of the radii directly, and in the ratio of the velocities inversely ; the centripetal forces are in the ratio composed from the ratio of the radii directly, and in the square ratio of the periodic times inversely. Corol. 3. From which if the periodic times may be equal, and therefore the velocities shall be as the radii; also the centripetal forces shall be as the radii; and vice versa.



Corol. 4. And if both the periodic times and the velocities shall be as the square roots of the radii ; the centripetal forces shall be equal among themselves ; and vice versa. Corol.5 . If the periodic times shall be as the radii, and therefore the velocities are equal; the centripetal forces shall be inversely as the radii : and vice versa. Corol. 6. If the periodic times shall be in the three on two power ratio of the radii, and therefore the velocities reciprocally in the square root ratio of the radii ; the centripetal forces shall be reciprocally as the square of the radii, and vice versa. Corol. 7. And universally, if the time of the period shall be as any power nR of R, and therefore the velocity inversely as the power 1nR − of the radius ; the centripetal force will be reciprocally as the power 2 1nR − of the radius: and vice versa. Corol. 8. All the same things described concerning times, velocities, and forces, by which similar bodies of any similar shape, and with centres having been put in place in these figures similarly, follow from the preceding demonstration and applied to these cases. But it is required for the description of the equality of areas to be substituted for the equality of motion, and the distances of the bodies from the centres to be taken for the radii. [This had been shown experimentally by Kepler in his second law of planetary motion.] Corol. 9. It also follows from the same demonstration ; that the arc, which a body will describe in a given circle by the centripetal force on rotating uniformly in some given time, is the mean proportional between the diameter of the circle, and the descent of the body on falling in the same time caused by the same given force. [These results are set out in the analytical manner by Routh and Brougham in their Analytical View of Sir Isaac Newton's Pricipia, p. 36 – 37; and also are dealt with conclusively by modern authors.]

Scholium.

The case of corollary six prevails with celestial bodies, (as also our Wren, Hook, and Halley have deduced separately) and therefore who consider the centripetal force decreasing in the square ratio of the distances from the centre, I have decided to explain further in the following. Again from the preceding propositions and from the benefit of the corollaries of this, also the proportion of the centripetal force to any known force is deduced, such as that of gravity. For if a body may be revolving in a circle concentric with the earth by its gravity, this weight is the centripetal force of that [motion]. But by Corol. IX of this proposition, from the descent of the weight and the time of one revolution, and the arc described in some given time is given. And from propositions of this kind Huygens in his exemplary



treatise de Horologio Oscillatorio [Concerning the Oscillatory (i.e. Pendulum) Clock] brought together the force of gravity with the centrifugal forces of revolution. Also the preceding can be demonstrated in this manner. In any circle a polygon is understood to be described of some number of sides. And if the body be required to move with a given velocity along the sides of the polygon, and is to be reflected by the circle, according to the individual angles of this; the force, with which the individual reflections impinge on the circle, will be as the velocity of this: and thus the sum of the forces in a given time will be as that velocity, and the number of reflections jointly : that is (if the kind of polygon may be given) as the distance described in that given time, and increased or decreased in the ratio of the length of the same to the aforementioned radius ; i.e. , as the square of this length applied [i.e. multiplied by] to the radius: and thus, if a polygon with an infinite number of sides may coincide with the circle, as the square of the arc described in the given time applicable to the radius. This is the centrifugal force, by which the body acts on the circle ; and to this the contrary force is equal, by which the circle continually repels the body towards the centre.

PROPOSITION V. PROBLEM I.

With the velocity given in some places, by which a body will describe some given figure with forces commonly tending towards some given centre, to find that centre. The three lines PT, TQV, VR [along which the velocity acts] touch the described figure in just as many points P, Q, R, concurring in T and V. To the tangents the perpendiculars PA, QB, RC are erected reciprocally proportional to the velocities of the body at these points P, Q, R, from which they have been raised ; that is, thus so that PA to QB shall be as the velocity at Q to the velocity at P, and QB to RC shall be as the velocity at R to the velocity at Q . Through the ends of the perpendiculars A, B, C are drawn AD, DBE, EC at right angles concurring in D and E [parallel to the respective tangents]: And TD and VE are drawn concurrent at the centre sought S. For the perpendiculars sent from the centre S to the tangents PT, QT (by Corol. I, Prop. I.) are inversely as the velocities of the body at the points P and Q; and thus by the construction [these perpendiculars from S to the tangents from T and V will be] directly as the perpendiculars AP and BQ , that is: as the perpendiculars are sent from the point D to the tangents. From which it is easily deduced that the points S, D, T are on a single right line. And by a similar argument the points S, E, V also are on a single right line; and therefore the body is turning about the centre S at the concurrence of the lines TD and VE.

Q. E. D.



[We give here the Le Seur and Janquier proof of Prop. V ; note 206, p. 78 : The points S, D, and T do lie on a single straight line. For from the centre S, with the perpendiculars SG, SF sent to the tangents TV, TF; and from the point D, with the perpendiculars DK, DH, it is apparent the angles FSG, HDK to be equal and contained betwee parallel lines, and on account of the sides SF, SG, DH, DK, in an analogous manner, the triangles FGS, HKD are similar ; and thus the angle SFG, DHK are equal; and hence there will be :

: : :

: : :

TH TF HK FG DH SF ,&TK TG HK FG DK SG.

= =

= =

On account of which TD, produced, will pass through the centre S.]

PROPOSITION VI. THEOREM V. If a body is revolving in some orbit in a non resisting space about an immobile centre, and some arc just arisen may be described in the minimum time, and the versed sine of the arc may be understood to be drawn, which bisects the chord, and which produced passes through the centre of forces : the centripetal force at the middle of the arc will be directly as the versed sine and inversely with the time squared. For the versed sine at the given time is as the force (Per Cor.4. Prop. I.), and with an increase in the time in some ratio, on account of the increase of the arc in the same ratio, the versed sine may be increased in that ratio squared (By Cor.2. & 3, Lem. XI.) and thus the versed sine is as the force to the first power and the time to the second power. The squared ratio of the time is rearranged on both sides of the proportionality, and the force becomes directly as the versed sine and inversely as the time squared.

Q. E. D. The same is easily shown also by Cor.4., Lem. X.



[i.e. in an obvious notation, vSin ~Fc ; t tΔ αΔ→ and the arc Δϑ αΔϑ→ ; the versed sine increases as 2α , and thus ( )2cvSin F tΔ→ . See note to Cor.1 below.] Corol. I. If the body P may describe the curved line APQ by revolving about the centre S; truly the right line ZPR may touch the curve at some point P, and to the tangent from some other point of the curve Q , QR is acting parallel to the length SP, and QT may be sent perpendicular to that distance SP: the centripetal force will be reciprocally as the

volume 2 2SP QTQR× ; but only if the magnitude of this volume is always taken as that,

which it shall be finally, when the points P and Q coalesce. For QR is equal to the versed sine of twice the arc QP, in the middle of which is P, and twice the triangle SQP or SP QT× is proportional to the time, in which the double of this arc will be described; and thus can be written for the time to be expressed.

Q. E. D. [Recall that the versed sine, or the turned sine, is the line segment DB ; see the added

diagram here; given by 22 2vSin r r cos r sinθ θ= − = , or for very small arcs, by 22vSin rθ= ; thus

2

2srvSin = in this case, and the elemental area

of the corresponding sector OCB is 2 12r rsθ = . Now, in these motions

equal areas are described in equal times, and thus the elemental [i.e. instantaneous] area is proportional to the elemental time and vice versa :

2 12t r rsΔ α θ = and thus the centripetal force or acceleration for unit mass, for areas

such as found above becomes

( ) ( )2

2 2vSinc tt

F r rΔθΔΔ

α α ω= ,

the modern form, which is inversely as 2 2SP QTQR× as required. We may also note that

the idea of angular velocity was not apparent at the time, it seem that Euler was responsible for this, so that Newton's derivation of the centripetal acceleration goes as far as the deviation from the tangent towards S, QR; this distance is divided by the area of the associated triangle SP QT× squared, proportional to the time squared, as one expects for an acceleration.] Corol. 2. By the same argument the centripetal force is reciprocally as the volume

2 2SY QPQR× , but only if the perpendicular ST shall be sent from the centre of forces to

the tangent PR of the orbit. For the rectangles andSY QP SP QT× × are equal. Corol. 3. If the orbit either is a circle, or either touches or cuts a circle concentrically : that is, the angle of contact, or which contains the smallest section with a circle, and having the same radius of curvature at the point P [as the circle]; and if PV shall be a chord of this circle for a body acted on by a centre of forces: the centripetal force [which



was previously inversely as 2 2SY QPQR× ,] shall be inversely as the volume 2SY PV× .

For PV is as 2QP

QR.

[ We give here the Le Seur and Janquier proof of Prop. VI, Cor. III ; note 211, p. 81 :

PV shall be as 2QP

QR. For let PQVF be the osculating

circle, and with the chord QM drawn, as well as the other chord PV drawn through the centre of forces S, the former may be bisected in K, and there will be (by Prop. 35, Book III, Euclid Elements),

2QK VK PK= × ; but with PK evanescent, QR PK= ,

and (by Cor. I, Lemma VII) QK QP= , hence 22 and QP

QRQP PV QR, VP .= × = ]

Corol. 4. With the same in place, the centripetal force is directly as the velocity squared , and inversely as that chord [PV]. For the velocity is reciprocally as the perpendicular SY by Corol. I. Prop. I. Corol. 5. Hence if some curvilinear figure APQ is given, and in that there may be given also a point S, towards which the centripetal force is always directed, the law of the centripetal force can be found, from which some body P drawn back from a straight line always will be retained in the perimeter of that figure, and by revolving describes it.

Without doubt either the volume 2 2SP QTQR× or the volume 2SY PV× is required to be

computed, for this is reciprocally proportional to the force. We will give examples of this matter in the following problems.

PROPOSITION VII. PROBLEM II.

A body may rotate in the circumference of a circle, the law of the attracting centripetal force towards some

given point is required. Let VQPA be the circumference of the circle; S the point given, to which the force as it were tends towards its centre ; P the body brought to the circumference ; Q the nearest point, into which it will be moved; and PRZ the tangent to the circle at the previous point. The chord PV



may be drawn through the point S; and with the diameter of the circle VA drawn, AP is joined; and to SP there may be sent the perpendicular QT, which produced may run to meet tangent PR in Z; and then LR may act through the point Q, which shall be parallel to SP itself, and then crosses to the circle at L, and then with the tangent PZ at R. And on account of the similar triangles ZQR, ZTP, VPA; there will be RP2 (that is RL RQ× ) to

QT2 as AV2 to PV2. And thus 2

2RL RQ PV

AV× × is equal to QT2. [Note the degenerate cyclic

quadrilateral LQPP.] This equality may be multiplied by 2SP

QR, and with the points P and

Q coalescing there may be written PV for RL. Thus 2 3

2SP PV

AV× becomes equal to

2 2SP QT .QR× . Therefore (by Corol.1. & 5. Prop. VI.) the centripetal force is inversely as

2 3

2SP PV

AV× as the square of the distance or of the height SP, and jointly as the cube of

the chord PV. Q. E. D.

The Same Otherwise.

To the tangent PR produced there may be sent the perpendicular SY: and on account of

the similar triangles SYP, VPA; there will be AV to PV as SP to SY : and thus SP PVAV×

equals SY, and 2 3

2SP PV

AV× equals 2SY PV× . And therefore (by Coral.3. & 5. Prop. VI.)

the centripetal force is reciprocally as 2 3

2SP PV

AV× , that is, on account of AV given,

reciprocally as 2 3SP PV .× Q. E. D. Corol. I. Hence if the given point S, towards which the centripetal force always tends, may be located on the circumference of this circle, for example at V, the centripetal force will be reciprocally as the fifth power of the height SP. Corol. 2. The force, [proportional to 2 3SP PV× ] by which the body P is revolving in a circle APTV about the centre of forces S, is to the force, by which the same body P in the same circle and in the same periodic time can be revolving about some other centre of forces R, as 2RP SP× to the cube of the right line SG, which is a right line drawn from the centre of the first force S to the tangent of the orbit PG, and which is parallel to the distance of the body from the second centre of forces. For from



the construction of this proposition, the first force is to the second force as 2 3 2 3toRP PT SP PV× × that is, as

3 32

3to SP PVSP RPPT×

× , or (on account of the

similar triangles PSG, TPV [for : SG SP PGPV TP TV

= = ]) to SG3.

Corol.3. The force, by which a body P is revolving in some orbit about a centre of forces S is that force, by which the same body P in the same orbit and in the same periodic time can revolve about some other centre of forces R, as 2SP RP× ; and certainly contained under the distance of the body from the first centre of forces S and with the square of the distance of this from the second centre of force R, to the cube of the right line SG, which is drawn from the centre of the first forces S to the tangent of the orbit PG, and is parallel to the distance RP of the body from the second centre of forces. For the forces in this orbit at some point P of this are in a circle of the same curvature.

PROPOSITION VIII. PROBLEM III.

A body may move in the semicircle PQA : towards effecting this, a law of the centripetal force is required thus tending towards a remote point S, so that all the lines PS, RS drawn towards that, will be had as parallel. From the semicircle with centre C, the radius CA is drawn, cutting such parallels in M and N, and CP may be joined. On account of the similar triangles CPM, PZT and RZQ there is CP2 to PM2 as PR2 to QT2, and from the nature of the circle PR2 is equal to the rectangle QR RN QN× + , or if with the points P and Q merged together to the rectangle

2QR PM× . Therefore CP2 to PM2 is as 2QR PM× to QT2 and thus 2QT

QR equals

3

22PMCP

, and [the ratio for the inverse of the centripetal force]

2 2 3 2

22equalsQT SP PM SP .

QR CP× ×

Therefore (by Corol.I.&5, Prop. VI.) the centripetal force is reciprocally as 3 2

22PM SP

CP× , that is (with the ratio determined

2

22SPCP

ignored) reciprocally as 3PM .

Q. E. D. The same may be deduced easily from the preceding proposition.



Scholium.

And from an argument not much dissimilar the body may be found to move in an ellipse, or also in a hyperbola or a parabola, by a central force which shall be reciprocally as the cube of the applied ordinate to a centre of forces acting at a great distance.

PROPOSITION IX. PROBLEM IV.

A body may be rotating on the [equiangular] spiral PQS with all the radii SP, SQ, &c. cut in a given angle : the law of the centripetal force is required tending towards the centre of the spiral. A small indefinite angle PSQ may be given, and on that account all the given angles will be given by a figure with the appearance SPRQT .

[Recall that we need to evaluate 2 2QT SPQR× which varies inversely as the centripetal

force.] Therefore the ratio QTQR

is given [i.e. constant], and 2QT

QR is as QT, that is (on

account of the kind of that given figure) as SP. Now the angle PSQ may be changed in some manner, and the right line QR subtending the contact angle QPR will be changed (by lemma XI) in the square ratio of PR itself or in the square ratio of QT. Therefore

2QTQR

will remain the same which it was first, that is as SP. Whereby 2 2QT SPQR× is as

SP3 and thus (by Corol.1 & 5, Prop. VI ), the centripetal force is reciprocally as the cube of the distance SP.

Q. E. D. The Same Otherwise.

The perpendicular SY sent to the tangent, and the chord PV of the circle concentrically

cutting the spiral are in given ratios to the altitude SP [i.e. and SY PVSP SP

are both

constant]; and thus 3SP is as 2SY PV× , that is (by Corol.3.& 5, Prop. VI.) reciprocally as the centripetal force.



LEMMA XII.

All the parallelograms drawn about any conjugate diameters of a given ellipse or hyperbola are equal to each other. Shown in works on conic sections. [Note that conjugate diameters for the ellipse are defined from the chords parallel to parallel tangents on either side of the ellipse; the one parallel chord passing through the centre is the one diameter, while the conjugate diameter bisects all these chords, beginning and ending on the contact points of the parallel tangents. ]

PROPOSITION X. PROBLEM V.

A body may be rotating in an ellipse : the law of the force is required tending towards

the centre of the ellipse. With the semi-axes CA, CB of the ellipse taken ; GP, DK the other conjugate diameters ; PF, QT the perpendiculars to the diameters ; Qv the applied ordinate to the diameter G P; and if the parallelogram QvPR may be completed, there will be (from works on conics ); [Newton defers proving the following theorem, which is not quite trivial : but which follows from a like theorem for the circle, as the ellipse can be regarded as a projected circle, for which some theorems apply with a little modification. It is perhaps the case that theorems used by Newton for the circle were thus extended to the other conic sections where appropriate. Using the diagram given here, this theorem states that 2 2 2: :Pv vG Qv PC CD× = , where PC and CD are semi-conjugate diameters and thus PR is parallel to CD ; now, in the case of the circle, the corresponding theorem is

2: 1:1Pv vG Qv× = , where QT and Cv are perpendicular to GP and 2 2=PC CD , a well-known result ; but the ratios based on parallel lines are not altered on projection, and the result follows with the others accounted for. This extension to the ellipse in the form given is shown, for example, in Theorem 9, § 136, Elements of Analytical Geometry, Gibson & Pinkerton, Macmillan & Co., 1911. A branch of geometry not studied much these days. To return to the translation : ] the rectangle PvG [i.e. Pv vG× ] is to 2Qv as 2PC is to 2CD and (on account of the

similar triangles QvT, PCF) 2Qv to 2QT as 2PC to 2PF ; and with the ratios taken

jointly, the rectangle PvG to 2QT as 2PC to 2CD and 2PC to 2PF that is, 2 2 2

22to as toQT CD PFvG PC .

Pv PC×



2 2 2

2 2 2 2

2 2

2 2 2

2

2 2 2 2

[ and ; or

or

= .]

Pv vG PC Qv PCi.e.Qv CD QT PF

Pv vG PC PC ,QT CD PF

vG PCQT / Pv CD PF / PC

×= =

×∴ = ×

×

Write QR for Pv, and (by Lemma XII.) for BC CA CD PF× × , also (with the points P and Q coinciding) 2PC for vG, and with the extremes and means multiplied in turn there

becomes 2 2 2 22equal to QT PC BC CAQR PC× × . Therefore there is, (by Coral.5, Prop. VI.)

the centripetal force varies reciprocally as 2 22BC CAPC× ; that is (on account of the given

2 22BC CA× ) reciprocally as 1PC

; that is, directly as the distance PC.

Q. E. I.

The same otherwise. In the right line PG from the one side of the point T there may be taken u so that Tu shall be equal to Tv itself; then take uV, which shall be to vG as 2DC is to 2PC . And since from the theory of conics there is 2 2 2to as to Qv Pv vG DC PC× , 2Qv is equal to PV uV× . With the rectangle uPv added to both sides, and the square of the chord of the arc PQ will be produced equal to the rectangle VPv ; and thus the circle, which touches the conic section at P and passed through the point Q, will also pass through the point V. The points P and Q run together, and the ratio uV to vG, which is the same as the ratio

2 2 to DC PC , becomes the ratio PV to PG or PV to 2PC; and thus PV will be equal to 22DC

PC. Therefore the force, by which the body P is revolving in an ellipse, will be

reciprocally as 22DC

PC into 2PF (by Carol.3, Prop. VI) that is (on account of 22DC

into 2PF given) directly as PC. Q. E. I.

Scholium

Corol. I. Therefore the force is as the distance of the body from the centre of the ellipse : and in turn, if the force shall be as the distance, the body will move in an ellipse having the centre at the centre of forces, or perhaps in a circle, into which certainly the ellipse can be transported.



Corol.2. And the periodic times of the revolutions made will be equal in all ellipses around the same centre. For these times are equal in similar ellipses (by Corol. 3. and 8. Prop. IV.) but in ellipses having a common major axis they are accordingly in turn as the whole total elliptic areas, and inversely likewise of the particular areas described ; that is, directly as the minor axes, and the velocities of the bodies inversely at the principal vertices ; that is, as the minor axes directly, and inversely as the ordinates at the same point of the common axis ; and therefore (on account of the equality of the direct and inverse ratios) in the ratio of equality.

Scholium.

If with the centre of the ellipse departing to infinity it may change into a parabola, the body will be moving in this parabola ; and a constant force now emerges at an infinite distance from the centre. This is Galileo's theorem. And if with a parabolic section of a cone (with the inclination of the plane to the section of the cone changed) may be changed into a hyperbola, the body will be moved in the perimeter of this with the centripetal force turned into a centrifugal force. And just as in a circle or ellipse, if the forces tending towards the centre of the figure placed on the abscissa [i.e. the x or ordinate axis]; these forces by augmenting or diminishing the ordinates in some given ratio, either by changing the angle of inclination of the ordinates to the abscissae, always may be augmented or diminished in some ratio of the distances from the centre, but only if the periodic times remain constant; thus also in general figures, if the ordinates be augmented or diminished in some given ratio, or the angle of the ordinates may be changed in some manner, with the periodic times remaining ; the forces tending towards some centre placed on the abscissa tending to particular ordinates may be augmented or diminished in the ratio of the distances from the centre.


Book I Section III. Translated and Annotated by Ian Bruce. Page 122

SECTION III.

Concerning the motion of bodies in eccentric conic sections.

[The Rectangle Theorem is a general theorem for Conic Sections with many applications: If two variable secants of a conic whose directions are fixed cut the conic in P, Q and P', Q', and intersect in O, then OP.OQ

OP' .OQ' is constant for all

positions of O. (For a coordinate geometry proof, see e.g. Elements of

Analytical Conics, Gibson & Pinkerton, p.434. (1911)) We will demonstrate this theorem for an elliptical section, following this reference. Thus, if the chords PQ and P'Q' intersect at O outside the ellipse shown here, then if the

tangents parallel to these chords in turn, op and op' , intersect at o, then 2

2OP.OQ op

OP' .OQ' op'= .

Similarly, if the semi-diameters CD and CD' parallel to the chords are considered, then 2 2

2 2OP.OQ op CD

OP' .OQ' op' CD'= = . A useful case occurs when O lies on the focus S of the ellipse, in

which case we have the added relations 2 2

2 2OP.OQ op CD SL.SK

OP' .OQ' SL' .SK 'op' CD'= = = . In this latter case,

there is added property that the harmonic mean of the two segments of a focal chord is equal to the semi-latus rectum l , where

2

2ba

l = for the standard ellipse: i.e. 1 1 2SL SK l+ = , or

2 ;KLSL.SK l= hence we have all these relations :

2 2

2 2OP.OQ op CD SL.SK KL

OP' .OQ' SL' .SK' K ' L'op' CD'= = = = .

Thus, we have Newton's Theorem : If VQ is an ordinate of the diameter PCP' of a conic, CP and CD are conjugate semi-diameters, then

2

2 2P'V .VP CP

VQ CD= .

From the given figure, which satisfies the requirements of the Rectangle Theorem, we have now in addition :

;P' C.CP P'V .VPDC.CD' QV .VQ'= giving

2

2 2P'V .VP CP

VQ CD= for conjugate axes, where C is the centre of the

ellipse or hyperbola . A similar result holds for the hyperbola, though care must be taken with signs.



To prove a similar theorem for the parabola, used below by Newton, i.e. that 2 4QV SP.PV= , we proceed as follows : . From the Rectangle Theorem applied to the

infinite parallel chords and R T∞ ∞ , and with the conjugate semi-diameter formed by the tangent TPR:

22 2 2 or QVRP TP TP

RQ.R TA.T PV TA∞ ∞= = , since =1 RT∞∞ as RQ

and TA are parallel. Therefore 22 PV .TP

TAQV = Also, PG is normal to the tangent, and PN is the ordinate of the point P. Now, from the geometry of the

figure, TA AN= and TS SP SG= = ; 2or 2 2TNTPTG TP TP TN .TG .TA. SP= = = . Therefore

22 4PV .TPTAQV SP.PV= = , as required.

In addition, 2 22 4 4 4.UA PN .UA PN TA.AS AN .AS ,= ∴ = = = giving the familiar formula 2 4y ax.= ]

PROPOSITION Xl. PROBLEM VI.

A body may be revolving in an ellipse: the law of the centripetal force is required attracting towards the focus of the ellipse.

[Note : This is one of the main results of the Principia, that was of great interset at the time, and was the proposition that Newton had mislaid when first visited by Halley enquiring about such; surely part of the folklore that now extends around Newton, and indicative of his disregard for the intellectual pursuits of others in the years following his rebuttals by Hooke. Thus, as always in these diagrams, we are looking for an expression for the area transcribed about the centre of force, here the focus F, in the element of time the body travels from P to Q, and this quantity squared is divided by a length corresponding to the versed sine in the limit, a distance proportional to and in the direction of the force within the arc of the motion (see Prop.I, Th.I, Sect. 2); thus the inverse of the acceleration is obtained as the limit is approached. Newton is careful enough to state in Sect. I that he does not take a limit, just approaches as near as you wish with smaller and greater magnitudes, effected by the longer sides of the parallelogram that encompass the arc that tends towards the long diagonal of the parallelogram. This incremental parallelogram is itself of some interest, as its 'long' and 'short' sides are determined by different causes : the long side PR corresponds to a small increment of the tangent line – following the First Law of Motion, and the corresponding side Qx is an increment of a diameter of the section, the conjugate diameter of which passes through the centre C of the conic; the short side QR – following the Second Law of Motion corresponds to the versed sine of the force, and acts towards the focal point F. Thus, from



the point P, one line is sent to S, and another to C; while from Q one line is Qv, while the other drawn is QT, perpendicular to SP. Hence the area traced out is proportional to PS QT× which in turn is proportional to the time, while QR is the distance moved under the action of the central force. Hence the centripetal acceleration can be found proportional to

( )2QR

PS QT× .This ratio must be identified with known properties of the curve

via proportionals, etc. derived from theorems of the conic section involved, one of which is :

2

2 2Gv.vP PCQv CD

= , shown above from elementary considerations . Thus Newton used what

might be called geometrical dynamics in his calculations.] Let S be a focus of the ellipse. SP may be drawn cutting both the diameter of the ellipse DK in E, and the applied ordinate Qv [i.e, a semi-chord of the axis PCG] in x, and the parallelogram QxPR may be completed. It is apparent that EP is equal to the major semi-axis AC : because there, with the line HI drawn from the other focus of the ellipse parallel to EC itself, on account of the equal lines CS, CH, the lines ES, and EI are made equal, thus so that EP shall be half the sum of PS, PI, that is (on account of the parallel lines HI, PR, and the equal angles IPR, HPZ) half the sum of PS, PH, which themselves jointly are equal to the total axis length 2AC. [Thus, in the customary equation for the ellipse

22

2 2 1yxa b+ = , and from the construction :

2SP PH a+ = ; from the reflection theorem for a ray travelling from one focus to the other, the angles IPR, HPZ are equal; then

; PH PI SE EI= = ; therefore, 2SP PI a+ = and ( )12 SP PI a+ = .]

The perpendicular QT may be sent to SP, and I call L the principal latus rectum of the

ellipse [a useful constant, the length of the vertical focal chord, equal to 22b

a ] (or22BC

AC),

there will be L QR× to L Pv× as QR to Pv, that is, as PE or AC to PC [from the similar

triangles Pvx and PFC]; and L Pv× to Gv.vP as L to Gv; and Gv.vP to 2Qv as 2PC to 2CD ,

[thus we have the ratios : L QR QR ACPE

L Pv Pv PC PC×× = = = ; L Pv L

Gv.vP Gv× = ;

2

2 2Gv.vP PCQv CD

= ],

and (per Corol.2, Lem. VII.) 2Qv to 2Qx with the points Q and P merged together is the

ratio of equality; 2Qx or 2Qv is to 2QT as 2EP to 2PF , that is, as 2CA to 2PF or (per

Lem. XII.) as 2CD to 2CB . [Note 261 (i) Leseur & Janquier: From the nature of conics,



the diameters of the parts made in terms of squares of the ordinates : as the square of the transverse diameter to the square of the conjugate of that : PF CD AC BC× = × , and thus 2 2 2 2PF CD AC BC× = × , and thus

2 2

2 2AC CDPF BC

= . Note also that the lines DK, IH, Qv,

and RP are all parallel; hence the angle FEP in the right angle triangle PEF is equal to the angle QxT in the small right angled triangle QxT, which triangles are hence similar.]

[i.e. 2 2 22

2 2 2 2Qx Qv CAEPQT QT PF PF

= = = .]

And with all these ratios taken together, L QR× becomes to 2QT as 2 2AC L PC CD× × × , or 2 2 22.CB PC CD× × to 2 2PC Gv CD CB× × × , or as 2.PC to Gv.

2 2 2 2 2

2 2 2 2 22 2[ .]L QR AC L PC CD .CB PC CD .PC

GvQT PC Gv CD CB PC Gv CD CBi.e. × × × × × ×

× × × × × ×= = =

But with the points Q and P coalescing, 2.PC and Gv are equal. Therefore from these

proportionalities L QR× and 2QT are equal. These equalities are multiplied by 2SP

QR,

and there becomes 2L SP× equals 2 2SP QTQR× . Therefore (by Corol.1 & 5, Prop.VI.) the

centripetal force is reciprocal as 2L SP× that is, reciprocally in the squared ratio of the distance SP.

Q E.I. [Note the desire to be rid of vanishing quantities, and to replace these with finite lengths ; the triangles QTx and PCE fulfil this role; the first has vanishing sides, and the latter does not; hence a ratio of vanishing sides in the first gives rise to a finite ratio in the second. This can be accomplished by setting one vanishing ratio equal to a finite ratio times by a vanishing ratio common to both the desired final quantities : considerable skill must be exercised to do this. Thus, no quantities are allowed to vanish, merely to cancel to a finite number. A crucial deduction in this matter is finding the length EP equal to AC. There is also the ratios of the sections of chords:

2

2 2Gv.vP PCQv CD

= ; thus it is necessary also to get rid

of Pv somehow, as well as Qx2 – the latter from the similar triangles mentioned; the former from the vanishing triangle Pxv which is similar to the triangle PFE, which gives the ratio Pv PC PC

Px PE AC= = . Thus, regarding the finite value of the vanishing quantities 2QR

QT,

we can set Pv PvPC PCQR Px PE AC= = × = × ; while

22

2QvPCGvCD

Pv = × ; thus, in place of QR,

we may write 2 22

2 2Qv QvAC PC PC

PC Gv GvCD CDAC× × = × × . Again, QT2 can be replaced by

2

22 PF

CAQx × . It then follows that ( )2 2

2 2 2 21QR QvPC AC

GvQT CD Qx PFAC= × × × × ; but from above,



2 2

2 2AC CDPF BC

= , and hence ( )2 2

2 2 2 2 2 21 1

2QR QvPC CD PC AC

Gv GvQT CD Qx BC BC BCAC AC= × × × × = × × = , since

Gv tends to 2PC in the limit; hence 2

22 1QR BC

ACQT× = , or 2 1QR

QTL× = , as required.]

The Same Otherwise.

Since the force attracting the body towards the centre of the ellipse, by which the body is able to rotate about that, shall be (by Corol.I, Prop. X.) as the distance CP from the centre of the ellipse C; CE is taken parallel to the tangent of the ellipse PR ; and the force, by which the same body P can revolve about some other point of the ellipse S, if

CE and PS are concurrent in E, will be as 3

2PESP

(by Corol.3, Prop.VII.) that is, if the

point S shall be the focus of the ellipse, and thus PE may be given, as reciprocally as 2SP . Q.E.1.

By the same brevity, with which we have extended the fifth problem to the parabola and the hyperbola, here it is allowed to be used likewise; truly on account of the worth of the problem, and the use of this in the following, it will be a pleasure to confirm the other propositions by demonstration.

PROPOSITION XII. PROBLEM VII.

A body may be moving in a hyperbola : the law of the centripetal force is required

tending towards the focus of the figure.

With the semi-axes CA, CB of the hyperbola taken; PG, KD are other conjugate diameters; PF is the perpendicular to the diameter KD; and Qv the applied ordinate to the diameter GP. Construct SP cutting with the diameter DK in E, and then with the applied ordinate Qv in x, and the parallelogram QRPx may be completed. It is apparent that EP is equal to the transverse semi-axis AC, because there, from the other focus of the hyperbola H the line HI parallel to EC is drawn, on account of which the equal quantities CS, CH may be equal to ES, E1 ; thus so that EP shall be the semi-difference of PS, PI, that is (on account of the parallel lines IH, PR and the equal angles IPR, HPZ) of PS, PH, which amounts to the total difference of the axis 2.AC . [This is a special case, where the chord SP is perpendicular to the axis, perhaps chosen to ease the diagram : from symmetry and from the construction, and CH CS EI ES= = ; for the hyperbola, 2PH – PS .AC= ; now PI = PH, i.e. the triangle PHI is isosceles : for if we assume the physical result that a ray from H striking at P appears to come from S on reflection, then the angles RPH and IPZ are equal, hence the normal line FP bisects the angle HPI, and so the triangle is isosceles, and PI PH= . Hence

2 or on dividing by 2.PI – PS .AC PE AC= = ]



To SP there may be sent the perpendicular QT. And I call L the principal latus rectum of

the hyperbola (or22BC

AC), there will be L QR× to L Pv× as QR to Pv, or Px to Pv, that is

: (on account of the similar triangles Pxv, PEC) as PE to PC, or AC to PC. Also there will be L Pv× to Gv Pv× as L to Gv; and (from the nature of conics) the rectangle Gv.vP to

2Qv as 2 2 to PC CD ; and (by Corol.2, Lem.VII.) 2 2 to Qv Qx with the points Q and P

coalescing shall become the ratio of equality ; and 2 2or Qx Qv is to 2 2 2 as to QT EP PF ,

that is, as 2 2 to CA PF , or (by Lem.XII.) as 2 2 to CD CB : and with all these ratios taken together L QR× shall be to 2QT as 2 2AC L PC CD× × × , or 2 2 22CB PC CD× × to

2 2PC Gv CD CB× × × , or as 2PC to Gv. [See below for modern notation.] But with the points P and Q coalescing 2PC and Gv are equal. And therefore from these proportions L QR× and 2QT are equal. These equalities are multiplied by

2SPQR , and there becomes

2L SP× equal to 2 2SP QTQR× . Therefore (by Corol.I & V, Prop.VI.) the centripetal force is

reciprocally as 2L SP× , that is, reciprocally in the ration of the square of the distance SP. Q. E. I. [The derivation and notation follow the ellipse above exactly.]



2

2 2

2 2 22

2 2 2 2

2 2 2 2 2

2 2 2 2 22 2

[ ; ; ;

as = = = .

Finally, = = = .]

L QR QR Px AC L Pv Gv.vP PCPE LL Pv Pv Pv PC PC Gv Pv Gv Qv CD

Qx CA CDEPQT PF PF CB

L QR AC L PC CD CB PC CD PCGvQT PC Gv CD CB PC Gv CD CB

Q P,

× ×× ×

× × × × × ×× × × × × ×

= = = = = =

→

The same otherwise.

The force may be found, by which the body tends from the centre of the hyperbola C. This will be produced proportional to the distance CP. Truly thence (by Carol.III,

Prop.VII.) the force tending towards the focus S will be as 3

2PESP

that is, on account of

PE given reciprocally as 2SP . Q.E.1.

In the same manner it may be shown, because the body by this force turned from centripetal to centrifugal will be moving in the above hyperbola. [The idea of free positive and bound negative energy orbits was not of course available at the time, whereby a body with total energy > 0 describes a hyperbola about an attracting source of force such as the sun, a body with total energy < 0 describes an ellipse, while the parabola corresponds to zero energy, such as a comet starting from rest at essentially an infinite distance from the sun. Thus, an attractive gravitational force can still give rise to this hyperbolic motion. Such comets with zero or positive energy have been observed occasionally. In the diagram, the attractive force may act from the one



focus, while a repulsive force may act from the other, and in this respect they have the same effect.]

LEMMA XIII. The latus rectum of a parabola pertaining to some vertex is four times the distance of that

vertex from the focus of the figure. This is apparent from conic sections. [For the modern reader, the delightful Book of Curves by E. H. Lockwood (CUP) can be consulted for the more common properties referring to conic sections, as well as for details about many other well-known curves.]

LEMMA XIV. The perpendicular, which is sent from the focus of the parabola to the tangent of this curve, is the mean proportional between the distance of the focus from the point of contact and from the principal vertex of the figure. For let AP be the parabola, S the focus of this, A the principle vertex, P the point of contact of the tangent, PO the applied ordinate to the principle diameter, PM the tangent crossing the principal axis M, and SN the perpendicular from the focus to the tangent. AN is joined and on account of the equal lines MS and SP, MN and NP, MA and AO , the right lines AN and OP are parallel ; and thence the triangle SAN will be right-angled at A, and similar to the equal triangles SNM and SNP: therefore PS is to SN as SN is to SA. Q.E.D.

[ PS SNSN SAi.e. = .]

[For the envelope of the parabola can be constructed from the tangents by rotating the right angle SAN about S while N moves progressively along the coordinate line AN from N to some N' (not shown here) whereby SNN'P is a cyclic quadrilateral and the angle ANS is the angle between the chord and the tangent, equal to the angle in the alternate segment NPS, on letting N' tend towards N. See Lockwood p.4 for the details.] Corol. 1. 2PS is to 2SN as PS to SA. [For 2SN PS.SA= and

2PS PSPS.SA SA= .]

Corol. 2. And on account of SA given there is 2SN as PS. Corol. 3. And the running together of any tangent PM with the right line SN, which is from the focus into the perpendicular itself, falls on the right line AN, which is a tangent to the parabola at the principle vertex.



PROPOSITION XIII. PROBLEM VIII.

A body may be moving in the perimeter of a parabola :the law of the centripetal force

tending towards the focus of this figure is to be acquired.

The construction of the lemma may remain, and let P be a body on the perimeter of the parabola, and from the closest place Q, into which the body may move, with the line QR drawn acting parallel to Sp itself and the perpendicular QT, and also Qv parallel to the tangent, and running to meet both with the diameter PG in v, as well as the interval SP in x. Now on account of the similar triangles Pxv, SPM, and the equality of the sides SM, SP, the other sides Px or QR and Pv are equal. But from the theory of conics [See note at start of this section.] the square of the ordinate Qv is equal to the rectangle under the latus rectum and the segment of the diameter Pv, that is (by Lem. XIII.) to the rectangle 4PS Pv× , or to 4PS QR× ; and with the points P and Q merging together, the ratio Qv to Qx (by Corol 2, Lem.VII.) shall be one of equality. Hence 2Qx in that case is equal to the rectangle 4PS QR× . But (on account of the similar

triangles QxT , SPN) 2 2 2 2 is to as to Qx QT PS SN , that is (by Corol.I, Lem.XIV.) as PS to SA, that is, as 4PS QR× to 4SA QR× , and thence (by Prop. IX, Lib. V. Elem.)

2QT and 4SA QR× are equal. This equality is multiplied by 2SP

QR , and there becomes 2 2SP QTQR× equal to 2 4SP SA× : and therefore (by Corol.I. and V, Prop. VI.) : the

centripetal force is reciprocally as 2 4SP SA× , that is, on account of 4SA given, reciprocally in the square ration of the distance SP.

Q.E.I.

[ 2 2 4Qv Qx PS QR→ = × ;2 2

2 244

Qx PS QRPS PSSA SA QRQT SN

××= = = ; 2 4QT SA QR= × .]

Corol. I. From the three latest propositions it follows, that if some body P may emerge from the position P and follows some line PR with whatever velocity, and with the centripetal force, which shall be reciprocally proportional to the square of the distance of the position from the centre, likewise may be acting ; this body will be moved in some conic sections having the focus in the centre of forces ; and vice versa. For from given focus and point of contact, and from the position of the tangent, it is possible to describe a conic section, which will have a given curvature at that point. But the curvature is given from the given centripetal force, and from the velocity of the body : and it is not possible to describe two orbits mutually touching each other with the same centripetal force and with the same velocity.



Corol 2. If the velocity, with which the body leaves from its position P, shall be that, by which the short line PR may be able to be described in some smallest part of time ; and the centripetal force shall be able in the same time to move through the distance QR: this body will be moved in some conic section, the principal latus rectum is that quantity

2QTQR

which finally is made when the small lines PR and QR are diminished indefinitely.

The circle I refer to the ellipse in these corollaries ; and I rule out the case, where the body descends to the centre along a straight line.

PROPOSITION XIV. THEOREM VI. If several bodies are rotating about a common centre, and the centripetal force shall be in the reciprocal square ratio of the distances of the places from the centre ; I say that the principal latera recta are in the square ratio of the areas, which the bodies describe by the radii to the centre in the same time. For (by Corol 2, Prop. XIII, [and Prop. XI]) the latus

rectum L is equal to the quantity 2QT

QR which finally comes

about, when the points P and Q coincide. But the minimum line QR in the given time is as the generating centripetal force, that is (by hypothesis) reciprocally as SP2. Therefore

2QTQR

is as 2 2QT SP× , that is, the latus rectum L is in the

square ratio of the area QT SP× . Q.E.D.

Corol. Hence the total area of the ellipse, and to which the rectangle under the axis is proportional, is composed from the square root ratio of the latus rectum, and from the ratio of the periodic time. For the total area is as the area QT SP× , which will be described in the given time, taken into the periodic time.

PROPOSITION XV. THEOREM VII.

With the same in position, I say that the periodic times for ellipses are in the ratio of the three on two power of the major axis.

For the minor axis is the mean proportional between the major axis and the latus rectum, and thus the rectangle under the axes is in the ratio of the major axis. But this rectangle (by Corol. Prop. XIV.) is in the ratio composed from the square root of the latus rectum and in the ratio of the periodic time. The ratio of the square root of the lengths of the lines are subtracted from both sides, and there will remain the three on two ratio of the major axis with the ratio of the periodic time.

Q.E.D.



Corol. Therefore the periodic times are the same in ellipses and circles, of which the diameters are equal to the major axes of ellipses.

PROPOSITION XVI. THEOREM VIII.

With the same in place, and with the right lines drawn to the bodies, which at that instant are tangents to the orbits, and with perpendiculars sent from the common focus to these tangents: I say that the velocities of the bodies are in a ratio composed from the inverse ratio of the perpendiculars, and directly with the square root ratio of the principal latera recta. Send the perpendicular SY from the focus S to the tangent PR, and the velocity of the body P will be reciprocally in the ratio of the square root of the

quantity 2SY

L. For that velocity is as the minimum

arc PQ described in the given moment of time, that is (by Lem.VII.) as the tangent PR, that is, on account of the proportionals PR to QT and SP to

SY, as SP QTSY× or as SY reciprocally and

SP QT× directly; and SP QT× shall be as the area described in the given time, that is (by Prop.XIV.) in the square root ratio of the latus rectum.

Q E.D. Corol.I. The principal latera recta are in a ratio composed from the square of the ratio of the perpendiculars and, and in the squared ratio of the velocities. Corol. 2. The velocities of bodies, at the maximum and minimum distances from the common focus, are in a ratio composed from the inverse ratio of the distances, and directly as the square root of the principal latera recta. For the perpendiculars now are these distances themselves. Corol.3. And thus the velocities in a conic section, at the maximum or minimum distances from the focus, is to the velocity in a circle at the same distance from the centre, in the square root ratio of the principal latus rectum to twice that distance. Corol. 4. The velocities of bodies gyrating in ellipses at their mean distances from the common focus are the same as of bodies gyrating in circles at the same distances; that is (by Corol.6, Prop. IV.) reciprocally in the square root ratio of the distances. For the perpendiculars now are the minor semi-axes, and these are as the mean proportionals between the distances and the latera recta. This ratio may be taken inversely with the



square root ratio of the latus rectums directly, and it becomes the ratio of the inverse square of the distances. Corol. 5. In the same figure, or even in different figures, of which the principal latera recta are equal, the velocity of the body is inversely as the perpendicular sent from the focus to the tangent. Corol. 6. In a parabola the velocity is reciprocally in the square root ratio of the distance of the body from the focus of the figure; in an ellipse it is changed more, in a hyperbola less than in this ratio. For (by Corol. 2, Lem. XIV.) the perpendicular sent from the focus to the tangent of the parabola is in the inverse square ratio of the distance. In a hyperbola the perpendicular is varied less, in an ellipse more. Corol. 7. In the parabola the velocity of a body at some distance from the focus is as the velocity of the body revolving in a circle at the same distance from the same centre in the square root ratio of two to one; in an ellipse it is less, in a hyperbola it is greater than in this ratio. For (by Corollary 2 of this Prop.)the velocity at the vertex of the parabola is in this ratio, and (by Corollary 6 of this Prop. and Proposition IV) the same proportion is maintained between all the distances. Hence also in the parabola the velocity everywhere is equal to the velocity of the body revolving in a circle at half the distance, in an ellipse it is less, greater in a hyperbola. Corol. 8. The velocity of gyration in some conic section is to the velocity of gyration in a circle at a distance of half the principle latus rectum of the section, as that distance to the perpendicular from the focus sent to the tangent of the section. This is apparent from corollary five. Corol. 9. From which when (by Corol 6. Prop. IV.) the velocity of gyration in this circle shall be to the velocity of gyration in some other circle reciprocally in the square root ratio of the distances ; there becomes from the equality the velocity of gyration in the conic section to the velocity of gyration in a circle at the same distance, as the mean proportional between that common distance and half the principle latus rectum of the section, to the perpendicular sent from the common focus to the tangent of the section.

PROPOSITION XVII. PROBLEM IX.

Because the centripetal force shall be reciprocally proportional to the square of the distance of the places put in place from the centre, and because the magnitude of that

force shall be known absolutely; the line is required, that the body will describe from that place progressing with a given velocity along a given right line.

The centripetal force tending towards the point S shall be that, by which the body p may gyrate in some given orbit pq, and the velocity of this may be known at the position p. From the place P the body P may emerge following the line PR with a given velocity, and thence soon, with the centripetal force acting, that may be deflected in the section of



a cone PQ. Therefore this right line PR is a tangent at P. Likewise pq may touch the other orbit in p, and if from S these perpendiculars are understood to be sent, (by Corol. I, Prop. XVI.) the principle latus rectum of the conic section will be to the principle latus rectum of the orbit in a ratio composed from the squared ratio of the perpendiculars and from the squared ratio of the velocities, from which thus it is given. Let L be the latus rectum of the conic section. Therefore the focus S of this conic section is given. The complement of the angle RPS to two right angles makes the angle RPH; and from the position given of the line PH, on which the other focus H may be located. With the perpendicular SK sent from PH, the conjugate semi-axis BC is understood to be erected, and there will be

2 2 2 2 2 2

2 2 2

2 4 4 4

2

SP KPH PH SH CH BH BC

SP PH L SP PH SP SPH PH L SP PH .

− + = = = − =

+ − × + = + + − × +

On both sides there may be added 2 22.KPH SP PH L SP PH− − + × + , and there becomes 2 2L SP PH SPH KPH ,× + = + or to as 2 2 to SP PH PH .SP .KP L+ + . From which PH is given both in length as well as in position. Without doubt if that shall be the velocity of the body at P, so that the latus rectum L were less than 2 2.SP .KP+ , PH will be added to the same direction of the tangent PR with the line PS; and thus the figure will be an ellipse, and it will be given from the given foci S, H, and the principle axis SP AH+ . But if the velocity of the body shall be so great, so that the latus rectum L were equal to 2 2.SP .KP+ , PH will be infinitely long; and therefore the figure will be a parabola having the axis SH parallel to the line PK, and thence it will be given. But if the body at this point emerges with a greater velocity from its place P , a length may be required to be taken PH at another direction of the tangent; and thus with the tangent arising between the two foci, the figure will be a hyperbola having the principle axis equal to the difference of the lines SP and PH, and thence it will be given. For if the body in these cases may revolve in a conic section thus found, it has been shown in Prop. XI, XII, and XIII, that the centripetal force will be inversely as the square of the distance from the centre of the forces S; and thus the line PQ is shown to be correct, as the body may describe by such a force, from some given place P, with a given velocity, emerging along the given right line PR in place. Q.E.F. Corol. I. Hence in every conic section from a given vertex D , with the latus rectum L, and with the focus S, another focus H is given on taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the proportion to SP PH PH+ is as 2 2SP KP+ to L in the case of this corollary, shall be DS DH+ to DH as 4DS to L, and separately DS to DH as 4DS L− to L.



Corol. 2. From which if the velocity of the body is given at the principal vertex D, the orbit may be found readily, clearly on taking the latus rectum of this to twice the distance DS, in the square ratio of the velocity of this given to the velocity of the body in a circle at a distance of rotation DS (by Corol.3, Prop. XVI. ;) then DH to DS shall be as the latus rectum to the difference between the latus rectum and 4DS. Corol. 3. Hence also if the body may be moving in some conic section, and it may be disturbed from it orbit by some external impulse ; it is possible to know, what course it will pursue afterwards. For on compounding the proper motion of the body with that motion, that the impulse alone may generate, the motion of the body will be had with which it will emerge along a given line in place with the given impulse in place,. Corol. 4. And if that body may be disturbed continually by some extrinsic impressed force, as the course may become know approximately, by requiring the changes to be deduced which the force induces on that body at some points, and from analogous series the continuous changes can be judged at the intermediate places.

Scholium.

If the body P tending towards some given point R by the centripetal force, may be moving on the perimeter of some given conic section, the centre of which shall be C; and the centripetal law of the force may be required : CG is drawn parallel to the radius RP , and crossing to the tangent of the orbit PG at G; and that force (by Corol.I. and Schol. Prop.

X. & Corol.3, Prop.VII.) will be as 3

2CGRP

.


Book I Section IV. Translated and Annotated by Ian Bruce. Page 142

SECTION IV.

Concerning the finding of elliptical, parabolic and hyperbolic orbits from a given

focus.

LEMMA XV. If the two right lines SV and HV are changed in direction at some third point V, with an ellipse or hyperbola for which the two foci are S, H, of which the one line HV shall be equal [in length] to the principle axis of the figure, that is, to the axis on which the foci are placed, and the other line SV is bisected by a perpendicular TR sent from T ; that perpendicular TR will be a tangent at some point [R] on the conic section: and vice versa, if it touches, then HV will be equal [in length] to the principle axis of the figure. For the perpendicular TR cuts the right line HV at R, produced if there were a need ; and SR may be joined. On account of the equal lines TS, TV, the angles TRS and TRV and the lines SR and VR will be equal. From which the point R will be on the conic section, and the perpendicular TR will touch the same : and vice-versa.

Q.E.V. [This section is concerned with the construction of conic sections satisfying various conditions of the focii and tangents, and so is not involved with mechanics directly; due to symmetry there will often be more than one point on the curve where a tangent with the same or a known gradient acts. The ellipse and hyperbola written in standard form are not of course functions as such, and are made up from the positive and negative square root functions which are symmetric. ]

PROPOSITION XVIII. PROBLEM X. With a focus and the principle axis given, to describe elliptic and hyperbolic trajectories, which will pass through a given point, and will be tangents to given lines in place. S shall be the common focus of the figures ; AB the length [i.e. 2a in modern notation] of the principle axis of any trajectory ; P the point through which the trajectory must pass ; and TR the right line that it must touch. With centre P, and with the interval [i.e. radius] AB – SP , if the orbit shall be an ellipse. or AB SP+ , if that shall be a hyperbola, a circle HG may be described. The perpendicular ST may be sent to the tangent TR, and the same may be produced to V , so that TV shall be equal to ST; and the circle FH is described with centre



V, and the interval [or radius] AB. By this method from the two circles either two points P, p may be given, or two tangents TR, tr, or the point P and a tangent, are required to be described. H shall be the common intersection of these, and from the foci S and H, with that axis given, the trajectory may be described. I say that this has been accomplished. For the trajectory described (because therefore PH SP+ is equal to the axis in the case of an ellipse, and PH SP− in the case of a hyperbola) will pass through the point P, and (by the above lemma) it touches the right line TR. And by the same argument the same will pass through two [given] points P and p, or touch two [given] right lines TR and tr. Q.E.F.

PROPOSITION XIX. PROBLEM XI. To describe a parabolic trajectory about a given focus, which will pass through a given point, and touch a given right line in position. S shall be the focus, P the point and TR the tangent of the trajectory to be described. With centre P, with the interval [i.e. radius] PS, describe the circle FG. Send the perpendicular ST from the focus to the tangent, and produce the same to V, in order that TV shall be equal to ST. In the same manner another circle fg is required to be described, if another point p is given; or finding another point v, if another tangent tr is given ; then the right line IF must be drawn which touches the two circles FG, fg if the two points P and p are given, or it may pass through the two points V and v, if the two tangents TR and tr are given, or touch the circle FG and pass through the point V, if the point P and the tangent TR are given. Send the perpendicular SI to FI, and bisect the same in K; and the parabola may be described with the principle axis SK and vertex K. I say the proposition has been accomplished. For the parabola, on account of the equal lines SK and IK, SP and FP, will pass through the point P; and (by Lem. XIV, Corol. 3.) on account of the equal lines ST and TV and the right angle STR, touches the right line TR.

Q E.F.

PROPOSITION XX. PROBLEM XII.

To describe a trajectory of some given kind about a given focus, which will pass through given points and touch a given right line in place.

Case I. With S the given focus, the trajectory ABC shall be described through the two points B and C. Because with the kind of trajectory given, the ratio of the principle axis to the separation of the focal points will be given. On that account take KB to BS, and LC to CS. From the centres B and C, with the distances BK, CL, describe



two circles, and to the right line KL, which may touch the same [circles] at K and L, send the perpendicular SG, and cut the same line at A and a, thus so that GA is to AS and Ga to aS as KB is to BS, and with the axis Aa, vertices A, a, the trajectory may be described. I say the construction is complete. For the other focus of the figure described shall be H, and since GA shall be to AS as Ga to aS, there will be separately Ga GA− or Aa to aS AS− or SH in the same ratio, and thus in the ratio that the principle axis of the figure described has to the separation of the foci of this figure; and therefore the figure described is of the same kind as that required to be described. And since KB to BS and LC to CS shall be in the same ratio, this figure will pass through the points B, C, as has been shown from the conics.

Q E.F. [Here the proof indicated relies on the focus directrix property between S and GL for a point on the ellipse, for which the ratios CS

CL , etc., are constant. Thus,

1 1; or giving or AS aS AG aG AG AS Aa HS Aa aGAG aG AS aS aG aS aG aS HS aS, ,− = − = = = = .]

Case 2. With S the given focus, the trajectory is to be described which may touch the two right lines TR and tr at some points. [Note : The original diagram in the 3rd edition has r at the wrong side of t, if we want to apply Lemma VI, assuming the trajectory is an ellipse.] Send the perpendiculars ST and St from the focus to the tangents and produce the same to V and v, so that TV and tv shall be equal to TS and tS. Bisect Vv in O, and erect the indefinite perpendicular OH, and cut the right line VS produced indefinitely in K and k, thus so that VK shall be to KS and Vk to kS as the principal axis of the described trajectory is to the separation of the foci. [Thus, TR is a tangent at the point R and VK Vk VH

KS kS SH= = .] With diameter Kk, a circle is described cutting OH in H; and with the foci S, H, with the principal axis itself made equal to VH, the trajectory is described. I say the construction is complete. For bisect Kk in X, and join HX, HS, HV, and Hv. Because VK to KS is as Vk to kS [i.e. =VK Vk

KS kS ]; and on adding together as VK Vk+ to KS kS+

[i.e. 22 ; or VK Vk KS kS VK Vk VX Vk

Vk kS KS kS KX kS+ + +

+= = = ]; and separately as Vk VK− to kS KS− , that is, as 2VX to 2KX and 2KX to 2SX

[i.e. 22 ; or Vk VK kS KS Vk VK VkKX

Vk kS kS KS SX kS− − −

−= = = ];



and thus, as VX to HX and HX to SX,[i.e. as andVX HXHX SX which are hence equal to Vk

kS ,] there will be the similar triangles VHX and HXS, and therefore VH will be to SH as VX to XH, and thus as VK to KS [i.e. =VH VX VK

SH XH KS= ]. Therefore the principal axis of the described trajectory VH has that ratio to the separation of the foci SH, and therefore is of the same kind. Since in addition VH and vH may be equal in length to the principal axis, and VS and vS may be bisected by the perpendicular lines TR and tr, it is clear (from Lem. XV.) these right lines touch the described trajectory.

Q E.F. Case. 3. With the focus S given, a trajectory shall be described which touches the right line TR at the given point R. Send the perpendicular ST to the right line TR, and produce the same to V, so that TV shall be equal to ST. Join VR and cut the right line VS produced indefinitely in K and k, thus so that VK to SK and Vk to Sk shall be as the principal axis of the ellipse required to be described to the separation of the foci [i.e. =VK Vk VH

SK Sk SH= ] ; and with a circle described on the diameter Kk, with the right line VR produced to be cut in H, and with the foci S, H, with the principal axis made equal to the right line VH, the trajectory will be described. I say that the construction is complete. For VH is to SH as VK to SK, and thus as the principal axis of the trajectory described to the separation of the foci of this, as may be apparent from the demonstration in the second case, and therefore the trajectory described to be of the same kind with that to be described, truly the right line TR by which the angle VRS may be bisected, to touch the trajectory in the point R, is apparent from the theory of conics.

Q E. F. Case 4. Now the trajectory APB shall be described about the focus S, which may touch the right line TR, and may pass through some point P beyond the given tangent, and which shall be similar to the figure apb, with the principal axis ab, and described with the foci s, h. Send the perpendicular ST to the tangent TR, & produce the same to V, so that TV shall equal ST. Moreover make the angles VSP, SVP equal to the angles hsq, shq ; and with the centre q and with an interval [i.e. radius] which shall be to ab as SP to VS describe a circle cutting the figure apb in p. Join sp and with SH acting which shall be to sh as SP is to sp, and which angle PSH may be put in place equal to the angle psh and the angle VSH equal to the angle psq. And then with the foci S, H, and with the principle axis distance AB equal to VH, the section of a cone may be described. I say that the construction is done. For if sv is acting which shall be to sp as sh is to sq, and which put in place the angle vsp equal to the angle hsq and the angle vsh equal to the angle psq, the triangles svh and spq will be similar, and therefore vh will be to pq as sh to sq, that is (on account of the similar triangles VSP, hsq ) so that VS is to SP or ab to pq. Therefore vh and ab are equal.



Again on account of the similar triangles VSH, vsh, VH is to SH as vh to sh, that is, the axis of the conic section now described is to the interval of separation of the foci, as the axis ab to the separation of the foci sh; and therefore the figure now described is similar to the figure apb. But this figure passes through the point P, on account of which the triangle PSH shall be similar to the triangle psh; and because VH is equal to the axis itself and VS may be bisected perpendicularly by the right line TR, it will touch the same right line TR.

Q.E.F. LEMMA XVI.

From three given points, to put in place three right lines to a fourth point which is not

given, of which the differences are given or are zero. Case I. Let these points be given A, B, C and let Z be the fourth point, that it is required to find; on account of the given difference of the lines AZ, BZ, the point Z will be found on a hyperbola of which the foci are A and B, and that given difference the principal axis. Let MN be that axis. Take PM to MA so that it is as MN to AB, [i.e. MNPM

MA AB= ; note that PR is the directrix of this branch of the hyperbola.] and with PR erected perpendicular to AB, and with the perpendicular ZR sent to PR; there will be, from the nature of this hyperbola, ZR to AZ as MN is to AB [i.e. MNZR

AZ AB= ; here Newton has used the inverse of the eccentricity to



define a constant ratio]. By similar reasoning the point Z will be located on another hyperbola, of which the foci are A and C and the principal axis the difference between AZ and CZ, and QS itself can be drawn perpendicular to AC, to which if from some point Z of this hyperbola the normal ZS may be sent, this will be to AZ as the difference is between AZ and CZ is to AC. Therefore the ratios of ZR and ZS to AZ are given, and on that account the same ratio of ZR and ZS in turn is given ; therefore if the right lines RP and SQ meet in T, and TZ and TA may be drawn, a figure of the kind TRZS will be given, and with the right line TZ on which the point Z will be given in place somewhere. Also there will be given the right line TA, and also the angle ATZ ; and on account of the given ratios of AZ and TZ to ZS the ratio of these will be given in turn; and thence the triangle ATZ will be given, the vertex of which is the point Z.

Q.E.I. Case 2. If two from the three lines such as AZ and B Z may be made equal, thus draw the right line TZ, so that it may bisect the angle AB ; then find the triangle ATZ as above. Case 3. If all three are equal, the point Z may be located in the centre of the circle passing through the points A, B, C.

Q.E.1. This lemma problem is solved also in the book of Appolonius on tangents restored by Vieta.

PROPOSITION XXI. PROBLEM XIII.

To describe a trajectory around a given focus, which will pass through given points and touch given right lines in place.

The focus S may be given, a point P, and touching TR, and it shall be required to find the other focus H. To the tangent send the perpendicular ST and produce the same to Y; so that TY shall be equal to ST, and YH will be equal to the principal axis. Join SP and HP, and SP will be the difference between HP and the principal axis. In this manner if several tangents TR may be given, or more points P, always just as many lines TH, or PH, drawn from the said points Y or P to the focus H, which either shall be equal to the axis, or to some given lengths SP different from the same, and thus which either are equal among themselves in turn, or have some given differences; and thence, by the above lemma, that other focus H is given. But with the foci in place together with the length of the axis (which either is YH; or, if the trajectory be an ellipse, PH SP+ ; or PH SP− for a hyperbola,) the trajectory may be had.

Q E.1.



Scholium. When the trajectory is a hyperbola, I cannot deal with the opposite hyperbola under the name of this trajectory. For a body by progressing in its motion cannot cross over into the opposite hyperbola. The case where three points are given can be solved expediently thus. The points B, C, D may be given. Join BC, CD produced to E, F, so that there shall be EB to EC as SB to SC, and FC to FD as SC to SD. To EF drawn and produced send the normals SG, BH, and on GS produced indefinitely take GA to AS and Ga to aS as HB is to BS; and A will be the vertex, and Aa the principal axis of the trajectory: which, just as Gd shall be greater, equal, or less than AS, will be an ellipse, parabola or hyperbola; the point a falling in the first case on the same part of the line GF with the point A; in the second case departing to infinity; in the third case on the opposite side of the line GF. For if the perpendiculars CI, DK may be sent to GF ; IC will be to HB as EC to EB, that is, as SC to SB; and in turn IC to SC as HB to SB or as GA to SA. And by a like argument it is approved that KD to SD be in the same ratio. Therefore place the points B, C, D in a conic section around the focus S thus described, so that all the right lines, drawn from the focus to the individual points of the section, shall be in that given ratio to the perpendiculars sent from the same points to the line GF. By a method not much different the solution of this problem has been treated most clearly in the geometry of de la Hire; Book VIII, Prop. XXV of his book on conic sections. [The New Elements of Conick Sections by Philip de la Hire was translated from Latin and French editions into English in 1724; it is available on microfilm.]


Book I Section V. Translated and Annotated by Ian Bruce. Page 155

SECTION V.

Finding the orbits where neither focus is given.

[A thorough investigation of the origin and use of these Lemmas is given by D.T.Whiteside in Vol. VI of his Mathematical Papers of Isaac Newton, CUP, p.238 onwards. In addition, a work can be reconstructed from Newton's Waste Book on the Solid Locus of the ancient Greek mathematicians, which was lightly modified for these Lemmas of the Principia (See Vol. IV Whiteside, p. 274 onwards for an account of this, which bears a close resemblance to the version in the Principia.) Mention should also be made of J. L. Coolidge's little book : A History of Conic Sections and Quartic Surfaces, available as a Dover reprint, especially Ch.'s 3 & 4. This book gives a modern impression on some of Newton's trail-blazing work, as he was unaware of the work done already by others into the projective nature of conics. Newton clearly had an eye towards an exhaustive survey of the construction of conic sections dating from antiquity, to which he added significantly, with regard to possible applications to the orbits of planets and comets; for in addition to the conventional treatment, he investigated the construction and properties of conic sections from points on the curve only; for the directrix and focus, relating to such curves given at a few points only, are unknown initially.]

LEMMA XVII. If from some point P of a given conic section to the four sides AB, CD, AC, DB of some trapezium ABDC produced indefinitely, and inscribed in that conic section, just as many right lines PQ, PR, PS, PT may be drawn at given angles, one line to each side : the rectangle PQ PR× drawn to the two opposite sides, will be in a given ratio to the rectangle PS PT× drawn to the other two opposite sides. Case 1. In the first place we may put the lines drawn to the opposite sides to be parallel to one of the remaining sides, e.g. PQ and PR [are parallel] to the side AC, PS and PT to the side AB. And in addition the two opposite sides [of the trapezium], e.g. AC and BD, themselves in turn shall be parallel. A right line, which may bisect those parallel sides, will be one of the diameters of the conic section, and it also will bisect RQ. Let O be the point in which RQ may be bisected, and PO will be the applied ordinate for that diameter. Produce PO to K, so that OK shall be equal to PO, and OK will be the applied ordinate for the other part of the diameter [Note: The use of the term applied ordinate by Apollonius for the distance from the centre of the conic along an oblique axis to the curve was a forerunner of the idea of a coordinate, developed by De Cartes some 1800 years later.]



Therefore since the points A, B, P and K shall be on the conic section, and PK may cut AB in a given angle, the rectangle PQ.QK will be (by Prop.17,19, 21 & 23. Book III. Apollonius Conics) in a given ratio to the rectangle AQ.QB. But QK & PR are equal, as from the equality of OK, OP, and their difference from OQ, OR, and thence also the rectangles PQ.QK and PQ PR× are equal; and thus the rectangle PQ PR× is to the rectangle AQ.QB, that is in the given ratio to the rectangle PS PT× .

Q.E.D. [ The initial theorems referring to Apollonius relate to the rectangles formed by chords of

a conic section IJ and HF intersecting at the point G, drawn through two random points on the section I and H, to the ratio of the tangents squared CA and CB from an external point C, which are parallel to the given chords and vice versa. Thus, in the diagram added, the letters of which bear no relation to those above, the red and blue chords are parallel to the tangents from some

external point C. The normals AE and BD also have been drawn and are part of a proof, which we do not give here, but the proposition shown by Apollonius is that

2

2=FG GH CAJG GI CB×× . We indicate here the ellipse drawn for these five points :

This Lemma can be extended to hyperbolic, circular and parabolic sections, and is further generalised below. In the following, we shall include the ellipse that the reader had to imagine drawn around the trapezium or quadrilateral; in general coloured lines have been added by this translator; I am sorry if they cause offense; the purpose is to improve the readability of the work.] Case 2. Now we may consider the opposite sides of the figure [trapezium] AC and BD not to be parallel. Bd acts parallel to AC and then crosses to the right line ST at t, and to the section of the cone at d. Join Cd cutting PQ in r, and PQ itself acts parallel to DM, cutting Cd in M and AB in N. Now on account of the similar triangles BTt, DBN; Bt or PQ is to Tt as DN to NB. Thus Rr is to AQ or PS as DM to AN.



[i.e. = =PQBt DNTt Tt NB and = =Rr Rr DM

AQ PS AN . ]

Hence, by taking antecedents multiplied into antecedents and consequents into consequents, so that the rectangle PQ Rr× is to the rectangle PS Tt× , thus as the rectangle ND.DM it to the rectangle AN.NB, and (by case I.) thus the rectangle PQ Pr× is to the rectangle PS Pt× , and dividing thus the rectangle PQ PR× is to the rectangle PS PT× .

Q.E.D. Case 3. And then we may put the four lines PQ, PR, PS, PT not to be parallel to the sides AC, AB but at some inclination to that. Of these in turn Pq, Pr act parallel to AC itself; Ps, Pt parallel to AB itself ; and therefore the given angles of the triangles PQq, PRr, PSs, PTt, will give the ratios PQ to Pq, PR to Pr, PS to Ps, and PT to Pt ; [i.e. , and PQ PSPR PT

Pq Pr Ps Pt, .]

and thus the composite ratios to and to PQ PR Pq Pr, PS PT Ps Pt× × × × . But, by the above demonstrations, the ratio

to Pq Pr Ps Pt× × has been given : and therefore the ratio PQ PR× to PS PT× also is given. Q.E.D.

LEMMA XVIII.

With the same in place ; if the rectangle drawn to the two opposite sides of the trapezium PQ PR× shall be in a given ratio to the rectangle drawn to the remaining two sides PS PT× ; the point P, from which the lines are drawn, will lie on the conic section described about the trapezium. Consider a conic section to be described through the points A, B, C, D, and any of the infinitude of points P, for example p: I say that the point P always lies on this section. If you deny this, join AP cutting this conical section elsewhere than at P, if it were possible, for example at b. Therefore if the lines pq, pr, ps, pt & bk, bn, bf, bd may be drawn from these points p & b at given angles to the sides of in the right trapezium ; so that

will be to bk bn bf bd× × as (by Lem. XVII.) to pq pr ps pt× × , and thus (by hypothesis) to PQ PR PS PT× × . And on account of the similitude

of the trapeziums bkAf, PQAS, so that bk is to bf thus as PQ to PS. Whereby, on applying the terms of the first proportions to the corresponding terms of this, there will be bn to bd as PR to PT. Therefore the equal angled trapeziums Dnbd and DRPT are similar, and the diagonals of these, Db and DP are similar on that account. And thus b lies at the



intersection of the lines AP,DP and thus it coincides with the point P. Whereby the point P, where ever it is taken, to be inscribed on the designated conic section.

Q E.D. Corol. Hence if the three right lines PQ, PR, PS are drawn at given angles from a common point P to just as many given right lines in position AB, CD, AC, each to each in turn, and let the rectangle under the two drawn PQ PR× to the square of the third PS be in a given ratio: the point P, from which the right lines are drawn, will be located in the section of a cone which touches the lines AB, CD in A and C; and vice versa. For the line BD may fit together with the line AC, with the position of the three lines AB, CD, AC remaining in place; then also the line PT fits with the line PS: and the rectangle PS PT× becomes PS squared and the right lines AB, CD, which cut the curve in the points A and B, C and D, now are no longer able to cut the curve in these points taken together, but only touch. [Thus, the lines AB and CD are now tangents to the conic. Apollonius derived the classical three-line locus as a special case of the four-line locus for generating a conic : See Conics III, Prop. 54-56.]

Scholium. The name of the conic section in this lemma is taken generally, thus so that both a section passing through a vertex of the cone as well as a circle parallel to the base may be included . For if the point p falls on the line, by which the points A and D or C and B are joined together, the conic section is changed into two right lines, of which one is that right line on which the point p falls, and the other is a right line from which the two others from the four points are joined together. If the two opposite angles of the trapezium likewise may be taken as two right angles, and the four lines PQ, PR, PS, PT may be drawn to the sides of this either perpendicularly or at some equal angles, and let the rectangle drawn under the two PQ PR× be equal to the rectangle under the other two PS PT× , so that the rectangle under the sines of the angles S, T, in which the two final PS, PT are drawn, to the rectangle under the sines of the angles Q, R, in which the first two PQ, PR are drawn. In the rest of the cases the position of the point P will be from the other three figures, which commonly are called conic sections. But in place of the trapezium ABCD it is possible to substitute a quadrilateral, the two opposite sides of which cross each other mutually like diagonals. But from the four points A, B, C, D one or two are able to go off to infinity, and in that case the sides of the figure, which converge to these points, emerge parallel: in which case the section of the cone will be crossed by the other points, and will go off to infinity as parallel lines. [A full solution of this problem can be found as a note in Whiteside, Vol. VI, p. 275. ]



LEMMA XIX. To find a point P, from which if four right lines PQ, PR, PS, PT may be drawn to just as many other right lines AB, CD, AC, BD, given in position, from one to the other in turn, at given angles, the rectangle drawn under the two, PQ PR× , will be in a given ratio to the rectangle under the other two, PS PT× . The lines AB, CD, to which the two right lines PQ, PR are drawn containing one of rectangles, come together with the other two lines given at the points A, B, C, D. From any of these points A some right line AH may be drawn, in which the point you wish P may be found. That line cuts the opposite lines BD, CD, without doubt BD in H and CD in I, and on account of all the given angles of the figure, the ratios PQ to PA and PA to PS are given, and thus the ratio PQ to PS is given. By taking [i.e. by dividing] this ratio from the given ratio PQ PR× to PS PT× , the ratio PR to PT will be given, and by adding [i.e. multiplying by] the given ratios PI to PR, and PT to PH the ratio PI to PH will be given, and thus the point P.

Q.E.1. Corol. I. Hence also it is possible to draw the tangent at some point D of the infinite numbers of locations of the points P. For the chord PD, when the points P and D meet, that is, where AH is drawn through the point D, becomes the tangent. In which case, the final vanishing ratio of the lines IP and PH may be found as above. Therefore draw CF parallel to AD itself, crossing BD in F, and cut at E in the same final ratio, and DE will be the tangent, because therefore CF and the vanishing IH are parallel, and similarly cut in E and P. Corol. 2. Hence it is apparent also that the position of all the points P can be defined. Through any of the points A, B, C, D, e.g. A, draw the tangent AE of the locus and through some other point B draw the parallel of the tangent BF meeting the curve [or locus] at the position F. But the point F may be found by Lem. XIX. With BF bisected in G, and AG produced indefinitely, this will be the position of the diameter to which the ordinates BG and FG may be applied. This line AG may meet the curve in H, and AH will be a diameter or a transverse width to which the latus rectum will be as BG2 to AG GH× . If AG never meets the curve, the AH proves to be infinite, the locus will be a parabola, and the

latus rectum of this pertaining to the diameter AG will be 2BG

AG. But if that meets

somewhere, the locus will be a hyperbola, where the points A and H are placed on the same side of G: and an ellipse, when G lies between, unless perhaps the angle AGB shall



be right, and the above BG2 is equal to the rectangle AGH, in which case a circle will be had. And thus [a solution] of the problem of the ancients concerning the four lines, started by Euclid and continued by Apollonius and such as the ancients sought, not from a calculation but composed geometrically, is shown in this corollary. [There is next presented an important Lemma that is fundamental to the applications that follow.]

LEMMA XX.

If in some parallelogram ASPQ, the two opposite angles A and P touch the section of a cone at the points A and P ; and with the sides of one of the angles AQ and AS produced indefinitely, meeting the same section of the cone at B and C ; moreover from the meeting points B and C to some fifth point D of the conic section, the two right lines BD and CD are drawn meeting the other two sides of the parallelogram PS and PQ produced indefinitely at T & R: the parts PR and PT of the sides [ of the parallelogram] will always be cut in turn in a given ratio. And conversely, if these cut parts are in turn in a given ratio, the point D touches the section of the cone passing through the four points A, B, C, P.

Case I. BP and CP are joined together and from the point D the two right lines DG and DE are acting , the first of which DG shall be parallel to AB itself and meets PB and PQ and CA in H, I and G; the other shall be DE parallel to AC itself and meeting PC and PS and AB in F, K and E: and the rectangle DE DF× will be (by Lem. XVII.) in a given ratio to the rectangle DG DH× . But PQ to DE (or IQ) shall be as PB to HB, and thus as PT to DH; and in turn PQ to PT as DE to DH. And there is PR to DF as RC to DC, thus as (IG or) PS to DG, and in turn PR to PS as VF to DG; and with the ratios joined the rectangle PQ PR× shall be to the rectangle PS PT× as the rectangle DE DF× to the rectangle DG DH× , and thus in a given ratio. But PQ and PS are given, and therefore the ratio PR to PT is given. Q E.D.



Case 2. Because if PR and PT may be put in place in a given ratio in turn, then by retracing the reasoning, it follows that the rectangle DE DF× to be in a given ratio to the rectangle DG DH× , and thus the point D (by Lem. XVIII.) touches the conic section passing through the points A, B, C and P.

Q. E. D. Corol. I. Hence if BC acts cutting PQ in r, & on PT there may be taken Pt in the ratio to Pr that PT has to PR, Bt will be a tangent of the conic section at the point B. For consider the point D to coalesce with the point B, thus so that, as with the chord BD vanishing, BT may become a tangent; and CD and BT coincide with CB and Bt. Corol. 2. And in turn if Bt shall be a tangent, and at some point D of the conic section BD and CD may come together; R will be to PT as Pr to Pt. And counter wise, if there shall be PR to PT as Pr to Pt: BD and CD may come together at some point D of the conic section. Corol. 3. A conic section does not cut a conic section in more than four points. For, it were possible to happen, the two conic sections may pass through each other in the five points A, B, C, P, O; and these may cut the right line BD in the points D, d, and PQ itself may cut the right line Cd in q. Hence PR is to PT as Pq to PT; from which PR and Pq in turn themselves may be equal, contrary to the hypothesis. [The following lemma, related to the above, shows how to describe a branch of a hyperbola without making use of the focus, using points on the curve only, as well as a reference line on which related points and angles may be defined. Note the positions of the points A, B, C, D and P in the diagrams relating to these lemmas, where the hyperbola in the latter can be viewed as an inverted form of the ellipse in the former. Newton has not followed with a like proof, but has introduced a new way of drawing a conic section.]



LEMMA XXI.

If two moveable and indefinite right lines BM and CM drawn through the given points or poles B and C, a given line MN may be described from their meeting position M;

and two other indefinite right lines BD and CD may be drawn making given angles MBD and MCD with the first two lines at these given points B and C : I say that these two lines BD and CD, by their meeting at D, describe the section of a cone passing through the points B and C. And vice versa, if the right lines BD and CD by their meeting at D describe the section of a cone passing through B, C, and A, and the angle DBM shall always be equal to the given angle ABC, and the angle DCM always shall be equal to the given angle ACB: then the point M remains in place on the given line. For a [fixed] point N may be given on the line MN, and when the mobile point M falls on the motionless point N, the mobile point D may fall on the motionless [i.e. fixed] point P. Join CN, BN, CP, BP, and from the point P direct the lines PT and PR crossing with BD and CD themselves in T and R, and making the angle BPT equal to the given angle BNM, and the angle CPR equal to the given angle CNM. Therefore since (from the hypothesis) the angles MBD and NBP shall be equal, and also the angles MCD and NCP; take away the common angles NBD and NCD, and the equal angles NBM and PBT , NCM and PCR remain: and thus the triangles NBM and PBT are similar, and also the triangles NCM, PCR. Whereby PT is to NM as PB to NB, and PR to NM as PC to NC. But the points B, C, N, P are fixed. Therefore PT and PR have a given ratio to NM, and therefore a given ratio between themselves; and thus (by Lem. XX.) the point D, always the meeting point of the mobile right lines BT and CR , lies on a conic section passing through the points B, C, P. [The triangles NBM , PBT , and NCM, PCR are similar ; and NM NB NM NC MCMB

PT PB TB PR PC CR∴ = = = = ; hence a definite ratio is formed for the lines PT and PR , as in the above lemma.]

Q E.D.



And conversely, if the moveable point D may lie on a conic section passing through the given points B, C, A, and the angle DBM always shall be equal to the given angle ABC, and the angle DCM always equal to the given angle ACB, and when the point V falls successively on some two immoveable points of the section p, P, the moveable point M falls successively on two immoveable points n, N: through the same n and N the right line nN acts, and this will be the perpetual locus of that mobile point M. For, if it

should happen that the point M can move along some curved line. Therefore, the point D will touch the conic section passing through the five points B, C, A, p, P, where the point M always lies on a curved line. But also, from the demonstration now made, the point D also lies on the conic section passing through the five points B, C, A, p, P, where the point M always lies on a right line. Therefore the two conic sections will pass through the same five points, contrary to Corol. 3, Lemma. XX. Therefore is absurd for the point M to be moving on some curved line. Q. E. D.

PROPOSITION XXII. PROBLEM XIV.

To describe a trajectory through five given points. Five points A, B, C, P and D may be given. From any one of these points A to some other two, which may be called the poles B and C, draw the right lines AB and AC, and from these draw the parallel lines TPS, PRQ through the fourth point P. Then from the two poles B and C, draw the two indefinite lines BDT, CRD through the fifth point D, crossing the most recently drawn lines TPS and PRQ at T and R (the first to the first and the second to the second). And then from the right lines PT and PR, with the right line drawn tr parallel to TR itself, cut some proportion Pt and Pr of PT and PR; and if through the ends t and r of these and the poles B and C, Bt and Cr are drawn concurrent in d, that point d will be located in the trajectory sought. For that point d (by Lem. XX) may be placed in a conic section crossed over by the four points A, B, C, P ; and with the lines Rr and Tt



vanishing, the point d coincides with the point D. Therefore the five points A, B, C, P, D will pass through the conic section.

Q.E.D. The same otherwise.

From the given points join any three A, B, C; and around two of these B, C, or the poles, by rotating the given angles with magnitude ABC and ACB, the sides BA and CA may be applied first to the point D, then to the point P, and the points M and N may be noted with which the other sides BL and CL, themselves cross over in each case. The indefinite line MN may be drawn and these mobile angles may be rotated around their poles B, C, from that rule so that the intersection of the legs BL, CL or BM, CM, which now shall be m, always lies on that infinite line MN; and the intersection of the legs BA, CA, or BD, CD, which now shall be d, will delineate the trajectory sought PADdB. For the point d (by Lem. XXI.) contains the section of the cone passing through the points B and C; and when the point m approaches towards the points L, M, N, the point d (by construction) will approach towards the points A, D, P. And therefore the conical section passes through the five points A, B, C, P, D.

Q.E.F. Corol. 1. Hence the right line can be drawn readily, which touches the trajectory at some given point B. The point d may approach the point B, and the line Bd emerges as the tangent sought. Corol.2. From which also the centres of the trajectories, the diameters and the latera recta can be found, as in the second corollary of Lemma XIX.

Scholium. The first construction arose a little simpler by joining BP, and in that, if there was a need, produced by requiring that Bp to BP is as PR ad PT; and by drawing an infinite right line pe through p parallel to SPT itself, and on that always by taking pe equal to Pr; and with the right lines Be , Cr drawn concurrent in d. For since there shall be Pr to Pt, PR to PT, pB to PB, pe to Pt in the same ratio ; pe and Pr always will be in the same ratio. By this method the points of a trajectory can be found most expeditiously, unless you prefer a curve, as in the following construction, to be described mechanically. [More information on this and related topics can be found in the book by J.L. Coolidge : A History of Conic and Quartic Sections, originally published by OUP (1945), and later as a paperback by Dover Books. The connection to Newton's ongoing research activities can be found in Vol. IV of Whiteside's Mathematical Papers......, p.299, and in Vol. VI, p.258 of the same. The entire writings of Greek geometry and many other things can be found at the wilbourhall.org website, in Greek and Latin; these are corrected versions of the ham-fisted efforts of Google in scanning old texts. Of particular interest is the



monumental translation of the works of Apollonius by Edward Halley in 1712 from Greek and Arabic sources into Latin; this was published about the same time as the second edition of the Principia.]

PROPOSITION XXIII. PROBLEM XV.

To describe the trajectory, which will pass through four given points,

and which will touch a given right line in place. Case 1. The tangent HB may be given, the point of contact B, and three other points C, D, P. Join BC, and with PS acting parallel to the right line BH, and PQ parallel to the right line BC, complete the parallelogram BSPQ. Draw BD cutting SP in T; and CD cutting PQ in R. And then, with some line tr parallel to TR , from PQ, PS cut Pr, Pt proportional to PR, PT themselves respectively; and the meeting point d of the lines drawn Cr, Bt (by Lem. XX.) always lies on the described trajectory. [Thus, the two methods of defining the conic section are shown, the first above using the parallelogram method, while the second below uses the idea of poles with an angle rotating about one pole and chords passing through the other pole from a variable point on a line.]

The same otherwise. While the angle with given magnitude CBH may rotate about the pole B, then also some rectilinear radius DC has been produced at both ends about the pole C. The points M, N may be noted, in which the leg BC of the angle may cut that radius, when the other leg BH meets the same radius at the points P and D. Then for MN drawn indefinitely always meeting that radius CP or CD, and the leg BC of the angle, the join of the other leg BH with the radius will delineate the trajectory sought. For if in the constructions of the above problems the point A may fall on the point B, the lines CA and CB coincide, and the line AB in its ultimate position becomes the tangent BH; and thus the constructions put in place there become the same as the constructions described here. Therefore the meeting of the leg BH with the radius passing through the points C, D, P will delineate the section of the cone, and the right line BH tangent at the point B.

Q E.F. Case 2. Four points may be given B, C, D, P , the tangent HI placed outside. With the two lines BD, CP joined meeting in G, and with these lines crossing the tangent line in H



and I. The tangent may be cut at A, thus so that HA shall be to IA, as the rectangle under the mean proportion between CG and GP and the mean proportion between BH and HD, to the rectangle under the mean proportion between DG and GB and the mean proportion between PI and IC; and A will be the point of contact. [Thus,

2

2CG PGHA BH DHDG BG PI CIAI

× ×× ×= × , which is a constant ratio.

This can be viewed expediently as an application in analytic geometry relating to the Rectangle Theorem mentioned earlier; Whiteside has related this and the following lemmas to invariant cross ratios in projective geometry, which of course did not exist as a theory at the time; we have to take the Proposition of Apollonius (Book III, Prop.17) mentioned above as the basis of this lemma and the following.] For if HX parallel to the right line PI may cut the trajectory at some points X and T: the point A thus will be located (from the theory of conics [Apollonius Conics III, 17&18. ]), so that HA2 will be to AI2 in the ratio composed from the ratio of the rectangle XHT to the rectangle BHD, or of the rectangle CGP to the rectangle DGB, and from the ratio of the rectangle BHD to the rectangle PIC. Moreover with the point of contact found A, the trajectory may be described as in the first case.

Q.E.F. But the point A can be taken either between the points H & I, or beyond ; and likewise a twofold trajectory can be described.

PROPOSITION XXIV PROBLEM XVI.

To described a trajectory, which will pass through three given points and which may touch two given right lines in place.

The tangents HI, KL and the points B, C, D may be given. Through any two points B, D draw the indefinite right line BD meeting the tangents in the points H, K. Then also through any two of the other points C, D draw the indefinite line CD crossing the tangent lines at the points I, L. Thus with the drawn lines cut these in R and S, so that HR shall be to KR as the mean proportional between BH and HD is to the mean proportional between BK and KD; and IS to LS as the mean proportional is between CI and ID to the mean proportional between CL and LD. Moreover cut as it pleases either between the points K and H, I and L, or beyond the same ; then draw RS cutting the tangents at A and P, and A and P will be the points of contact. For if A and P may be supposed to be the points of contact situated somewhere on the tangents ; and through



some of the points H, I, K, L some I, placed in either tangent HI, the right line IT is drawn parallel to the other tangent KL, which meet the curve at X and Y, and on that IZ may be taken the mean proportional between IX and IY: there will be, from the theory of conics, the rectangle XIY or IZ2 to LP2 as the rectangle CID to the rectangle CTD, that is (by the construction) as SI2. ad SL2 and thus IZ to LP as SI to SL. Therefore the points S, P, Z lie on one right line. Again with the tangents meeting at G, there will be (from the theory of conics), the rectangle XIY or IZ2 to IA2 as GP2 to GA2 and thus IZ to IA as GP to GA. Therefore the points P, Z and A lie on a right line, and thus the points S, P and A are on one right line. And by the same argument it will be approved that the points R, P and A are on one right line. Therefore the points of contact A and P lie on the right line RS. But with these found, the trajectory may be described as in the first case of the above problem.

Q.E.F. In this proposition, and in the following case of the above proposition the constructions are the same, whither or not the right line XY may cut the trajectory at X and Y ; and these may not depend on that section. But from the demonstrated constructions where that right line may cut the trajectory, the constructions may be known, where it is not cut ; I shall not linger with further demonstrations for the sake of brevity. [According to Whiteside, Pemberton, the editor of the 3rd and final edition, tried to induce Newton to make some corrections to indicate the existence of two real solutions : see note 60 p.243, Vol.6 Math. Papers......]

LEMMA XXII.

To change figures into others of the same kind.

Some figure HGI shall be required to be changed. Two parallel lines may be drawn in some manner AO, BL cutting some third given line AB in place at A and B, and from some point G of the figure, some line GD may be drawn to the line AB, parallel to OA itself. [The initial skew axis can be taken as BDI of the abscissa or ordinate x with origin A, and AO as the applied line or coordinate y; this general one to one degree preserving transformation, the product of a simple affine transformation and a plane perspectivity, or a simple translation and rotation, and rescaling, had been published originally by de la Hire in his Conic Sections, (Note 67 Whiteside); subsequently used here to convert converging lines into parallel lines.].Then from some point O, given on the line OA, the right line OD is drawn to the point D, crossing BL itself in d, and from the crossing point there is raised the given line dg containing some angle with the right line BL, and having that



ratio to Od which DG has to OD; and g will be the point in the new figure hgi corresponding to the point G. [Thus the first axis are translated and rotated and rescaled to become th new axis; the new abscissa of the original point G is ad.] By the same method the individual points of the first figure will give just as many points in the new figure. Therefore consider the point G by moving continually to run through all the points of the first figure, and likewise the point g by moving continually will run through all the points of the new figure and describe the same. For the sake of distinction we may call the DG the first order, and dg the new order; AD the first abscissa, ad the new abscissa; O the pole, OD the cutting radius, OA the first order radius, and Oa (from which the parallelogram OABa is completed) the new order radius. Now I say that, if the point G touches the right line in the given position, the point g will also touch the same right line in the position given. If the point G touches a conic section, the point g will also touch a conic section. Here I count the circle with the conic sections. Again if the point G touches a line of the third analytical order, the point g touches a line of the third analytical order ; and thus with curved lines of higher order. The two lines which the points G, g touch will always be of the same analytical order. And indeed as ad is to OA thus are Od to OD, dg to DG, and AB to AD; and thus AD is

equal to OA ABad× , and DG is equal to OA dg

ad× . Now if the point G touches a right line,

and thus in some equation, in which a relation may be had between the abscissa AD and the ordinate DG, these indeterminate lines AD and DG rise to a single dimension only, by

writing OA ABad× for AD in this equation, and OA dg

ad× for DG, a new equation will be

produced, in which the new abscissa ad and the new ordinate dg rise to single dimension only, and thus which designate a right line. But if AD and DG, or either of these, will rise to two dimensions in the first equation, likewise ad and dg will rise to two in the second equation. And thus with three or more dimensions. The indeterminates ad, dg in the second equation, and AD, DG in the first always rise to the same number of the dimensions, and therefore the lines, which touch the points G, g, are of the same analytical order. I say besides, that if some right line may touch a curved line in the first figure ; this right line in the same manner with the transposed curve in the new figure will touch that curved line in the new figure ; and conversely. For if some points of the curve approach to two and join in the first figure, the same transposed points will approach in turn and unite in the new figure; and thus the right lines, by which these points are joined, at the same time emerge as tangents in tangents of the curves in each figure. The demonstrations of these assertions may be put together in a more customary manner by geometry. But I counsel brevity. Therefore if a rectilinear figure is to be transformed into another, if it is constructed from right lines, it will suffice to transfer intersections, and through the same to draw right lines in the new figures. But if it may be required to transform curvilinear figures, points, tangents and other right lines are to be transferred, with the aid of which a curved line may be defined. But this lemma is of assistance in the solution of more difficult problems, by transforming the proposed figures into simpler ones. For any converging right lines are transformed into parallel lines, by requiring to take some right line for the



first order radius, which passes through the meeting point of convergent lines; and thus because that meeting point with this agreed upon will go to infinity ; since they are parallel lines, which never meet. But after the problem is solved in the new figure; if by inverse operations this figure may be changed into the first figure, the solution sought will be had. Also this lemma is useful in the solution of solid problems. For as often as two sections of cones are come upon, of which a problem is required to be solved by the intersection, it is possible to change either of these, if it shall be either a hyperbola or a parabola into an ellipse: then it may be easily changed into a circle. Likewise a right line and a conic section, in the construction of plane problems, may be turned into a right line and a circle. [The interested reader may like to know that Edmond Halley the first editor, according to Note 71 on p. 272 of Vol. VI of the Math. Works....., wished further explanation from Newton on this Lemma; this he was given in a letter from Newton, but which never made it translated into Latin into the Principia. Coolidge, using coordinates, derives a transformation of the kind indicated on p. 46 of his history of sections.]

PROPOSITION XXV. PROBLEM XVII.

To describe a trajectory, which will pass through two given points, and touch three given lines in place.

Through the meeting of any two tangents with each other in turn, and the meeting of the third tangent with that line, which passes through the two given points, draw an indefinite line; and with that taken for the first order radius, the figure may be changed by the above lemma, into a new figure. In that new figure these two tangents themselves emerge parallel in turn to each other, and the third tangent becomes parallel to the right line passing through the two given points. Let hi, kl be these two parallel tangents, ik the third tangent, and hl a right line parallel to this passing through these points a, b, through which the conic section in this new figure must pass through, and completing the parallelogram hikl. The right lines hi, ik, kl may be cut in c, d, e, thus so that the side hc is to the square root of the rectangle ah.hb, [i.e. hc squared shall be to the rectangle ah.hb], as ic to id, and ke to kd as the sum of the rectangles hi and kl is to the sum of the three lines, the first of which is the right line ik, and the other two are as the squared sides of the rectangles ah.hb and al.lb: and c, d, e will be the points of contact. For indeed; from the conics, hc2 is to the rectangle ah.hb, as ic2 to id2, and ke2 to kd2, and in the same ratio el2 to the rectangle al.lb; and therefore hc is to the square root of ahb, ic to id, ke to kd,



and el to the square root of al.lb are in that square root ratio, and on adding, in the given ratio of all the preceding hi and kt to all the following, which are to the square root of the rectangle ah.hb, and to the rectangle ik, and the square root of the rectangle al.lb. Therefore the points of contact c, d, e may be had from that given ratio in the new figure. By the inverse operations of the newest lemmas these points may be transferred to the first figure, and there (by Prob. XIV.) the trajectory may be described. Q. E. F. [From the Rectangle Theorem :

2 2 2 2 2 2

2 2 2 and hence hc ic ke ic le hcha hb lb la ha hbid kd id× × ×= = = = ; or

hc ic ke leid kdha hb lb la× ×

= = = . Hence given ratio.hc ci ke el hi klid dk ha hb lb la ik ha hb lb la

+ + + ++ + × + × + × + ×

= = ]

Moreover thence so that the points a,b lie either between the points h, l, or beyond, the points c, d, e must lie between the points taken h, i, k, l , or beyond. If either of the points a, b fall between the points h, l, and the other beyond, the problem is impossible.

PROPOSITION XXVI. PROBLEM XVIII.

To describe a trajectory, which will pass through a given point, and will touch four

given lines in place. From the common intersection of any two tangents to the common intersection of the remaining two an indefinite right line is drawn, and the same taken for the first order radius, the figure may be transformed (by Lem. XXII.) into a new figure, and the two tangents, which met at the first order radius now emerge parallel. Let these be hi and kl ; ik and hl containing the parallelogram hikl. And let p be the point in this new figure corresponding to a given point in the first figure. Through the centre of the figure Opq is drawn, and putting Oq to equal Op, q will be another point through which the conic section in this new figure must pass. By the operation of the inverse of Lemma XXII this point may be transferred into the first figure, and here two points will be had through which the trajectory is to be described. Truly that same trajectory can be described by Problem XVII.

Q.E.F. [On p. 272 of Vol. VI of the Math. Papers...., Whiteside has drawn a hyperbola for the external case, where the curve and the points p and q lie outside the parallelogram.]



LEMMA XXIII.

If two given right lines in place AC, BD may be terminated in two given points A, B, and they may have a given ratio in turn, and the line CD, by which the indeterminate points C, D are joined together, may be cut in the given ratio at K : I say that the point K will be located on a given fixed right line. For the right lines AC and BD meet in E, and on BE there may be taken BG to AE as BD is to AC, and FD always shall be equal to the given EG ; and from the construction there will be EC to GD, that is, to EF as AC to BD, and thus in the given ratio, and therefore a kind of triangle EFC is given. CF may be cut in L so that CL to CF shall be in the ratio CK to CD; and on account of that given ratio, a kind of triangle EFL will also be given ; and thence the point L will be place in a given position on the line EL. Join LK, and CLK, CFD will be similar triangles; and on account of FD given and the given ratio LK to FD, LK will be given. This may be taken equal to EH, and ELKH always will be a parallelogram. Therefore the point K is located on the side HK of this parallelogram in place.

Q.E.D. Corol. On account of the given kind of figure EFLC, the three right lines EF, EL and EC, that is, GD, HK and EC, in turn have given ratios.

LEMMA XXIV.

If three right lines may touch some conic section, two of which shall be parallel and may be given in position; I say that the semi-diameter of the section parallel to these two lines, shall be the mean proportional between the segments of these, from the points of contact and to the third interposed tangent. Let AF and GB be the two parallel tangents of the conic section ADB touching at A and B; EF the third tangent of the conic section touching at I, and crossing with the first tangents at F and G; and let CD be the semi-diameter of the figure parallel to the tangents : I say that AF, CD, BG are continued proportionals. For if the conjugate diameters AB and DM cross the tangent FG at E and H, and mutually cut each other at C and the parallelogram IKCL may be completed; from the nature of the conic section as EC is to CA thus CA is to CL, [by Apoll. Book III, Prop. 42] and thus by division EC–CA to CA–CL, or EA to AL, and from adding EA to +EA AL or EL as EC to EC CA+ or EB; and thus, on account of the similar triangles EAF, ELI, ECH, EBG, AF to LI as CH to BG. Likewise, from the nature of conic sections, LI or CK is to CD as CD is to CH; and thus from the rearranged equation, AF to CH as CD to BG.

Q.E.D.



Corol. I. Hence if the two tangents FG, PQ with the parallel tangents AF, BG cross at F and G, P and Q and mutually cut each other at O; from the rearranged equation there will be AF to BQ as AP to BG, and on dividing as FP to GQ, and thus as FO to OG. Corol. 2. From which also the two right lines PG, FQ, drawn through the points P and G, F and Q, concur at the right line ACB through the centre of the figure and passing through the points of contact A, B .

LEMMA XXV. If the four sides of a parallelogram produced indefinitely may touch some conic section, and are cut by some fifth tangent; moreover the ends of any two neighbouring sides [sections] cut off opposite the angles of the parallelogram may be taken : I say that each section shall be to its side, as the part of the other neighbouring side between the point of contact and the third side, is to the section of this other side. The four sides ML, IK, KL, MI of the parallelogram MLIK may touch the conic section at A, B, C, D, and the fifth tangent FQ cuts these sides at F, Q, H and E; moreover the sections of the sides MI, KI may be taken ME and KQ, or of the sides KL, ML the sections KH, MF : I say that ME to MI shall be as BK to KQ; and KH to KL as AM to MF. For by the first corollary of the above lemma ME is to EI as AM or BK to BQ, and by taking ME to MI as BK to KQ. Q. E. D. Likewise KH to HL as BK or M to AF, and on dividing KH to KL as AM to MF. Q E.D. Corol. I. Hence if the parallelogram IKLM is given, described about some given conic section, the rectangle KQ ME× will be given, and also as equally to that the rectangle KH MF× . For the rectangles are equal on account of the similarity of the triangles KQH and MFE. Corol. 2. And if a sixth tangent eq is drawn crossing with the tangents KI, MI at q and e; the rectangle KQ ME× will be equal to the rectangle Kq Me× ; and there will be KQ to Me as Kq to ME, and by division as Qq to Ee. Corol. 3. From which also if Eq, eQ may be joined and bisected, and a right line is drawn through the point of bisection, this will pass through the centre of the conic section. For since there shall be Qq to Ee as KQ to Me, the same right line will pass through the midpoints of every EQ, eQ, M K (by Lem. XXIII.), and the midpoint of the line MK is the centre of the section.



[See Whiteside note 80, Vol. VI p.280 Math. Papers... for some of the interesting history on this Lemma]

PROPOSITION XXVII. PROBLEM XIX.

To describe a trajectory, which touches five given lines in position.

The tangents ABG, BCF, GCD, FDE, EA may be given in position. For the quadrilateral figure ABFE contained by any four, bisect the diagonals AF, BE in M and N, and (by Corol.3. Lem. XXV.) the right line MN drawn through the point of bisection will pass through the centre of the trajectory . Again for the figure of the quadrilateral BGDF, contained by any other four tangents, the diagonals (as thus I may say) BD, GF bisected in P and Q: and the right line PQ drawn through the point of bisection will pass

through the centre of the trajectory. Therefore the centre will be given at the meeting point of the bisectors. Let that be O. Draw KL parallel to any tangent BC of this [trajectory], to that distance so that the centre O may be located at the mid-point between the parallel lines ; and the line KL drawn will touch the trajectory to be described [note : the diagram is misleading: K does not lie on the ellipse] . It will cut these other two tangents in GCD and FDE in L and K. Through the meeting points of the non-parallel tangents CL, FK with the parallel tangents CF, KL, C and K, F and L draw CK, FL meeting in R, and the right line OR drawn and produced will cut the parallel tangents CF, KL in the points of contact. This is apparent by Corol.2, Lem. XXIV. By the same method it will be possible to find the other points of contact, and then finally by the construction of Prob. XIV. to describe the trajectory. Q. E .F.

Scholium. The problems, where either the centres or asymptotes of the trajectory are given, are included in the proceeding. For with the points and tangents given together with the centre, just as many other points and tangents are given from the other parts of this trajectory equally distant from the centre. But an asymptote may be taken as a tangent, and the end of this will be considered for the point of contact at an infinite distance (if thus it shall be spoken of). Consider the contact point of any tangent to go off to infinity,



and the tangent will change into an asymptote, and the constructions of the preceding problems [XIV and Case 1 of XV] will be changed into constructions where the asymptote is given. After the trajectory has been described, we are free to find the axes and the foci of this curve by this following method. In the construction and in the figure of Lemma XXI,

made so that the legs BP, CP of the moveable angles PBN, PCN, by the meeting of which the trajectory was described, in turn themselves shall become parallel, and then maintaining that position [the angles] may be rotated about their poles B and C in that figure. Meanwhile truly the other legs CN, BN of these angles may describe the circle BGKC, by their meeting at K or k,. Let O be the centre of this circle. From this centre to the ruler MN, to which these other legs CN and BN meanwhile will concur, while the trajectory may be described, send the normal OH from the centre crossing at K and L. And where these other legs CK and BK meet at that point K which is closer to the ruler, the first legs CP and BP will be parallel to the major axis, and perpendicular to the minor; and the opposite comes about, if the same legs meet at the more distant point L. From which if the centre of the trajectory may be given, the axes are given. But from these given, the foci are evident. [A detailed explanation of these results is given by Whiteside in the notes 92 – 95 Vol. VI of the Math. Papers....p.285 ; Whiteside also considers the hyperbolic case with a splendid diagram.] Truly the squares of the axes are one to the other as KH to LH, and from this the kind of the trajectory given by the given four points is easily described. For if from the two points given [on the curve], the poles C and B may be put in place, the third will give the mobile angles PCK and PBK; moreover with these given the circle BGKC can be described. Then on account of the given kind of trajectory, the ratio OH to OK will be given, and thus OH itself, With centre O and with the interval OH describe a circle, and the right line, which touches this circle, and passes through the meeting point of the legs CK and BK, where the first legs CP and BP concur at the fourth given point, will be that ruler MN with the aid of which the trajectory may be described. From which also in turn the kind of trapezium [i.e. quadrilateral] given (if indeed certain impossible cases are excepted) in which some given conic section can be described. Also there are other lemmas with the aid of which given kinds of trajectories, with given points and tangents, are able to be described. That is of this kind, if a right line may be drawn through some given point in place, which may intersect the given conic section



in two points, and the interval of the intersection may be bisected, the point of bisection may touch another conic section of the same kind as the first, and having the axes parallel with those of the former. But I will hurry on to more useful matters. [This question is examined by Whiteside in note 98 of the above.]

LEMMA XXVI.

The three angles of a triangle given in kind and magnitude are put in place one to one to as many given right lines in place, which are not all parallel. Three right lines AB, AC, BC are given in place and it is required thus to locate the triangle DEF, so that the angle of this D may touch the line AB, likewise the angle E the line AC, and the angle F the line BC. Upon the sides DE, DF & EF describe three segments of circles DRE, DGF, EMF, which take angles equal to the angles BAC, ABC, ACB respectively. But these segments may be described to these parts of the lines DE, DF, EF, so that the letters DRED may be returned in the same cyclic order with the letters BACB, the letters DGFD with the same letters ABCA, and the letters EMFE with the letters ACBA; then these segments may be completed in whole circles. The first two circles cut each other mutually in G, and let P and Q be the centres of these. With GP and PQ joined, take Ga to AB as GP is to PQ , and with the centre G, with the interval Ga describe the circle, which will cut the first circle DGE in a. Then aD is joined cutting the second circle DFG in b, then aE cutting the third circle EMF in c. And now the figure ABCdef can be set up similar and equal to the figure abcDEF. With which done the problem is completed. For Fc itself may be drawn crossing aD in n, and aG, bG, QG, QD, PD may be joined. From the construction the angle EaD is equal to the angle CAB, and the angle acF is equal to the angle ACB, and thus the triangle anc is equiangular to the triangle ABC . Hence the angle anc or FnD is equal to the angle ABC, and thus is equal to the angle FbD ; and therefore the point n falls on the point b. Again the angle GPQ, which is half of the angle at the centre GPD, is equal to the angle at the circumference GaD; and the angle GQP, which is half the angle at the centre GQD, is equal to the complement of two right angles at the circumference GbD, and thus equal to the angle Gba; and thus the two triangles GPQ and Gab are similar; and Ga is to ab as GP to PQ ; that is (from the construction) as Ga to AB. And thus ab and A B are equal; and therefore the triangles abc and ABC,



which we have approved in a similar manner, are also equal. From which, since the sides ab, ac, bc respectively may touch the above angles D, E, F of the triangle DEF , the angles of the triangle abc, the figure ABCdef can be completed similar to the similar and equal figure abcDEF, and that on completion solves the problem. Q E F. Corol. Hence a right line can be drawn the parts of which given in length will lie between three given right lines in place. Consider the triangle DEF, with the point D approaching the side EF, and with the sides DE, DF placed along a line, to be changed into a right line, the given part of which DE must be placed between the right lines given in place AB, AC, and the preceding part DF between the given right lines AB, BC in place ; and by applying the preceding construction to this case the problem may be solved.

PROPOSITION XXVIII. PROBLEM XX.

To describe a trajectory given in kind and magnitude, the given parts of which will lie in position between three given lines.

The trajectory shall be required to be described, which shall be similar and equal to the curved line DEF, and which with the three right lines AB, AC, BC given in position, will be cut into the given parts of this by the similar and equal parts of this DE and EF. Draw the right lines DE, EF, DF, and to the triangle of this DEF place the angles D, E, F to these given right lines in place (by Lem. XXVI), then describe a similar and equal trajectory of the curve DEF about the triangle. Q.E.F.

LEMMA XXVII.

To describe a given kind of trapezium [i.e. quadrilateral ], the angles of which are given to four right lines in position, one to one in position, and which are not all parallel, nor converge to a common point,.



The four right lines may be given in position ABC, AD, BD, CE; the first of which may cut the second at A, the third at B, and the fourth at C: and the trapezium fghi shall be required to be described, which shall be similar to the trapezium FGHI ; and the angle f of which shall be equal to the given angle F, may touch the right line ABC; and the other angles g, h, i, shall be equal to the other given angles G, H, I , may touch the lines AD, BD, CE respectively. FH and the above FG may be joined, FH and F1 may describe as many sections of the circle FSG, FTH, FVI; of which the first FSG may take an angle equal to the angle BAD, the second FTH may take an angle equal to the angle CBD, and the third FVI may take an angle equal to the angle ACB. But the segments must be described according to these parts of the lines FG, FH, F1, so that the letters FSGF shall be in the same cyclic order as the letters BADB, and so that the letters FTHF shall be returned in the same circle with the letters CBDC, and the letters FVIF with the letters ACEA. The segments may be completed into whole circles, and let P be the centre of the first circle FSG, and Q the centre of the second FTH. Also PQ may be joined and produced in each direction and in that QR may be taken in the same ratio to PQ as BC has

to AB. But QR may be taken on the side of the point Q so that the order P, Q, R of the letters shall be the same as of the letters A, B, C: and with centre R and with the interval RP the fourth circle may be described FNc cutting the third circle FVI in c. Fc may be joined cutting the first circle a, and the second in b. aG, bH, and cI are constructed and the figure abcFGHI can be put in place similar to the figure ABCfghi. With which done the trapezium fghi will be that itself, which it was required to construct. For the two first circles FSG and FTH mutually cut each other in K. PK, QK, RK, aK, bK, and cK may be joined and QP may be produced to L. The angles to the circumferences FaK, FbK, FcK are half of the angles FPK, FQK, FRK at the centres, and thus equal to the halves of these angles LPK, LQK, LRK. Therefore the figure PQRK is equiangular and similar to the figure abcK, and therefore ab is to bc as PQ to QR, that is, as AB to BC. By construction, to the above FaG, FbH, FcI the angles fAg, fBh, fCi are equal. Therefore to the figure abcFGHI the similar figure ABCfghi is able to be completed. With which done the trapezium fghi may be constructed similar to the trapezium FGHI, and with its angles f, g, h, i touching the right lines ABC, AD, BD, CE.

Q. E. F.



Corol. Hence a right line can be drawn whose parts, with four given lines in position intercepted in a given order, will have a given proportion to each other. The angles FGH and GH1 may be augmented as far as that, so that the right lines FG, GH, HI may be placed in a direction, and these in this case by constructing the problem will lead to the right line fghi, the parts of which fg, gh, hi, intersected by the four right lines given in position AB and AD, AD and BD, BD and CE, are to one another as the lines FG, GH, HI, and they will maintain the same order among themselves. Truly the same shall be expedited thus more readily. AB may be produced to K, and BD to L, so that B K shall be to AB as HI to GH; and DL to BD as GI to FG; and KL may be joined crossing the right line CE in i. There may be produced iL to M, so that there shall be LM to iL as GH to HI, and then there may be drawn MQ parallel to LB itself, and crossing the right line AD in g, then gi cuts AB, BD in f, h. I say that it has been done. [See note 110 of Whiteside for an explanation.] For Mg may cut the right line AB in Q, and AD the right line KL in S, and AP is drawn which shall be parallel to BD itself and may cross iL in P, and there will be gM to Lh (gi to bi, Mi to Li, GI to HI, AX to BK) and AP to BL in the same ratio. DL may be cut in R so that DL shall be to RL in that same ratio, and on account of the proportionals gS to gM, AS to AP, and DS to DL; there will be, from the equation, as gS to Lh thus AS to BL and DS to RL; and on mixing [the ratios], BL–RL to Lh–BL as AS–DS to gS–AS. That is, as BR to Bh so AD to Ag, and thus as BD to gQ. And in turn BR to BD as Bh to gQ, or fh to fg. But by construction the line BL will be cut in the same ratio in D and R and the line FI in G and H: and thus BR is to BD as FH to FG. Hence fh is to fg as FH to FG. Therefore since also there shall be gi to hi as Mi to Li, that is, as GI to HI, it is apparent the lines FI, fi similarly are cut in g and h, G and H.

Q. E.F. In the construction of this corollary after LK is drawn cutting CE in i, it is allowed to produce iE to V, so that there shall be EV to Ei as FH to HI, and to draw Vf parallel to BD itself. The same is returned if from the centre i, with the radius IH, a circle may be described cutting BD in X, and iX may be produced to r, so that iT shall be equal to IF, and Tf may be drawn parallel to BD. Other solutions of this problem were devised formerly by Wren and Wallis.



PROPOSITION XXIX PROBLEM XXI.

To describe a trajectory of a given kind, which from four given right lines will be cut

into parts, in order, in given kind and proportion. A trajectory shall be required to be described, which shall be similar to the curved line FGHI, and the parts of which, similar and proporional to the parts of this FG, GH, HI, with the right lines given in position AB and AD, AD and BD, BD and CE, the first may lie between the first, the second with the second, and the third with the third. With the right lines drawn FG, GH, HI, FI, the trapezium [read as quadrilateral] fghi may be described (by Lem. XXVII.) which shall be similar to the trapezium FGHI, and the angles of which f, g, h, i may touch these right lines given in position AB, AD, BD, CE, one to one in the said order. Then about this trapezium the trajectory of the like curved line FGHI may be described.

Scholium.

It is possible for this problem to be constructed as follows. With FG, GH, HI, FI joined produce GF to P, and join FH, 1G, and with the angles FGH, PFH make the angles CAK, DAL equal. A K and AL are concurrent with the right line BD in K and L, and thence KM and LN are drawn, of which KM may make the angle AKM equal to the angle GHI, and it shall be to AK as HI is to GH ; and LN may make the angle ALN equal

to the angle FHI, and it shall be to AL as HI to FH. Moreover AK, KM, KM, AL, LN may be drawn to these parts of the lines AD, AK, AL, so that the letters CAKMC, ALKA, DALND may be returned in the same order with the letters FGHIF in the orbit; and with MN drawn it may cross the right line CE in i. Make the angle iEP equal to the angle 1GF, and PE shall be to Ei as FG to G1; and through P there is drawn PQf, which with the right line ADE may contain the angle PQE equal to the angle FIG, and crosses the right line AB in h and fi may be joined. But PE and PQ may be drawn to these sections of the lines CE and PE, so that the cyclic order of the lines PEiP and PEQp shall be the same as of the letters FGHIF, and if above on the line fi also the same order of the letters may be put in place, the trapezium fghi will be similar to the trapezium FGHI, and the given kind of trajectory may be circumscribed, the problem may be solved. Up to this point concerned with the finding of orbits. It remains that we may determine the motion of bodies in the orbits found.


Book I Section VI. Translated and Annotated by Ian Bruce. Page 201

SECTION VI.

Concerning the finding of motions in given orbits. (s)

(t) Thus, according to the annotated edition of the Principia by Fathers Le Seur and Jacquier, the last edition of which was pub. in 1833 in Glasgow, the so-called 'Jesuit Edition' , we have this summary :- Note 338 (s), p. 202: Newton in this whole section supposes the body thus to be moving in a given conic trajectory, so that with the radii drawn to the focus of the trajectory, areas or sectors will be described proportional to the times; for by that law in Book 3, all the planets have been shown by the phenomena to revolve in conic section orbits. In addition the time is required to be noted, in which the body will arrive at some given point of the trajectory of the orbit from some given point of the trajectory, e.g. from the principal vertex point of that to some other point on the same given trajectory ; and for the area to be given or the sector of the trajectory corresponding to this given time ; and from these given, the moving position on the trajectory may be sought at some other given time; or on the other hand the time may be sought in which the moving point will reach some given point in the trajectory; for since the areas shall be proportional to the times, in some given time, the area described in this time may be found, and in turn with the area described given, the time may be found in which the motion was described.

PROPOSITION XXX. PROBLEM XXII. To find the position of a body in a given parabolic trajectory at a designated time. Let S be the focus and A the principle vertex of the parabola, and let 4AS M× be equal to the area of the parabola APS cut off, by which the radius SP, either after the height of the body had been described from the vertex, or before the approach of this to the vertex has been described. The magnitude of this area cut off is known to be proportional to the time itself. Bisect AS in G, and raise a perpendicular GH equal to 3M, and the circle described with centre H, radius HS will cut the parabola at the position sought P. For, with the perpendicular PO sent to the axis and with PH drawn, there is



2 2 2 2 2

2 2 2 2

( )

2 2

AG GH HP AO AG PO GH

AO PO AG AO GH PO AG GH .

+ = = − + −

= + − × − × + +

From which 2 2 2 23

42 ( 2 )GH PO AO PO AG AO AO PO .× = + − × = + [Since 2

4OP AS AO= × , ]

For 2AO write 2

4PO

ASAO× [i.e. 2 23

4 42 POASGH PO AO PO× = × + ] ; and with all the applied

terms divided by 3PO and multiplied by 2AS, there becomes

4 1 13 6 2

3 4 36 6

(

area ) area AO AS AO SO

GH AS AO PO AS PO

PO PO APO SPO APS.+ −

× = × + ×

= × = × = − =

[For the area under the parabola is 23 AO OP× (either by integration, or from Archimedes

Prop. 17, quadr. Parab. sup. Theor. IV, de Parabola).] But GH was equal to 3M, and thence 4

3 GH AS × is 4AS M× . Therefore the area cut APS is equal to the area 4AS M× , that was required to be cut.

Q. E. D. Corol.1. Hence GH is to AS, as the time in which the body has described the arc AP to the time in which the body has described the arc between the vertex A and the perpendicular to the axis erected from the focus S. [Let P'S be the semi-latus rectum, for which

2P' O AS= , then the area APS : area AP'S = 4 23 3: 2 :GH AS AS AS GH AS× × = .]

Corol. 2. And with the circle ASP always passing through the moving body at P , the velocity of the point H is to the velocity which the body had at the vertex A as 3 to 8; and thus also in that ratio is the line GH to the right line that the body in the time of its motion from A to P, that it may describe with the velocity it had at the vertex A. [See Note 339 (b) ]

338 (t) : Let S be the focus, and A, the principal vertex of the parabola, and the time shall be given in which the body in moving on the parabola, as we have established (358.), from the vertex A to the point P, or arrives at the vertex A from the point P , or the time shall be given in which the sector APS is described. Note 339 (b): Join AP, and at the mid-point q of this, raise the perpendicular L, and since (from the demonstration) always HP HA= , and thus AP is the chord of this circle whose centre is H. And thus (by Book I, Sect. I Euclid's Elements) that perpendicular qL cuts the right line GH in H ; and on account of the similar triangles LGH, LqA there is

12: or :GH qA AP LG Lq= . There may be



taken 2AC AS= clearly half of the latus rectum of the parabola, and with centre C, and with the radius CA, the circle AN may be described, this is a tangent to the parabola at A (note 241); truly with the points P and A, H and G coincident, and thus also the points L and C coincide. Let 2 4Lq LA CA AS GS= = = = and 3LG CG GS= = , and with the arc AP equal to the chord AP, (Lem. VII) ; from which since in the above proportion there shall be 1

2: :GH AP LG Lq= , in this case there shall be 12: 3 : 4GH AP GS GS= ; that is,

: 3: 8GH AP = . Truly on account of the uniformity of the motion and equally by continuing in its state over the time through the points A, P, G, H, the velocity of the point H at G, is to the velocity of the body P at A as GH to AP, and because (from the Dem.) 4

3 AS GH× is always equal to the area APS, and 43 AS is a constant quantity, GH

will always be as the area APS, that is as the time in which the point H, has run through GH, and is hence whose motion is uniform and the same everywhere. Whereby the velocity of the point H is everywhere to the velocity that the body P has at A, as arising at GH , to that arising at AP, that is, as 3 to 8. Q.e.d.

Corol. 3. Hence also in turn it is possible to find the time in which the body has described some designated arc AP. Join AP and to the mid-point of this erect the perpendicular GH to the line crossing in H. [i.e. the same argument for the time can be applied to any point.]

LEMMA XXVIII.

No figure is extant, cut by right lines as you please, of which the oval area may be able to be found generally by equations with a finite number of terms and dimensions.

Within an oval some point may be given, about which or pole a right line may rotate perpetually, with a uniform motion, and meanwhile on that right line a point may emerge moveable from the pole, and it may always go forwards with that velocity, which shall be as the square of that right line within the oval. [See note 359.] By this motion the point will describe a spiral with infinite rotations. Now if a part of the oval area can be found cut from that right line by a finite equation, also the distance may be found by the same equation of the point from the pole, which is proportional to this area, [See note 360 below.] and thus all the points of the spiral can be found by a finite equation: and therefore from any position of this right line the intersection with a given position with the spiral also can be found by a finite equation. And every right line produced infinitely cuts the spiral in an infinite number of points, and the equation, [See note 361 below.] by which some intersection of the two lines may be found, shows all the intersections of these with just as many roots, and thus it rises to just as many dimensions as there are intersections. Because two circles cut each other mutually in two points, a single intersection may not be found except by an equation of two dimensions, from which the other intersection also may be found. Because there can be four intersections of the two conic sections [See note 362 below.], it is not possible to find any of these generally except by an equation of four dimensions, from which all may be found at the same time.



For if these intersections themselves may be sought, since the law and the condition of all is the same, the calculation will be the same in each case, and therefore always the same conclusion, which therefore must include all like intersections and shown indifferently. From which also the intersections of conic sections and of curves of the third order, because from that there can be six, likewise they may be produced by equations of six dimensions, and six intersections of the two curves of the third power, because nine can be possible, likewise they may arise from equations of nine dimensions. Unless that by necessity may come about, all solid [i.e. volume or 3 dimensional] problems may be allowed to be reduced to the plane, and greater than solid [three dimensions] to three dimensions. I am concerned here with curves that are irreducible in power. For if the equation, by which a curve is defined, can be reduced to a lower power : the curve will not be single, but composed from two or more equations, whose intersections can be found separately by different calculations. In the same manner the intersections of two right lines and the sections of cones will always be produced by equations of two dimensions, of three right lines and of irreducible curves of the third power by equations of the third dimension, of four right lines and of irreducible curves of the fourth power by equations of the fourth dimension, and thus indefinitely. Therefore the intersections of right lines and spirals will require equations with an infinite number of dimensions, since this curve shall be simple and irreducible into many curves, and an infinitude of roots, by which all the intersections can be likewise shown. For this is the same law and the same calculation of everything. For if from the pole a perpendicular may be sent through that intersecting right line, and that perpendicular may be rotated together with the intersecting line about the pole, the intersections of the spiral will cross mutually from one into another, whatever shall be first or nearest, after one revolution will be the second, after two the third, and thus henceforth: nor meanwhile will the equation be changed except for the change in the magnitude of the quantities by which the position of the cuts may be determined. From which since these quantities after individual rotations may return to the first magnitudes, the equation will be returned to the first form, and thus one and the same will show all the intersections and therefore an infinite number of roots will be had, from which everything is able to be shown. Therefore the intersection of a right line and a spiral will be unable to be shown generally by a finite equation, and therefore nothing may be shown by such an equation generally with the area of which oval, cut by designated right lines. By the same argument, if the distance between the pole and the point, by which the spiral may be described, should be taken proportional to the perimeter of the oval cut, it cannot be proved because the length of the perimeter cannot be shown generally by a finite equation. But here I talk about ovals which are not touched by conjugate figures going off to infinity.

Further notes from the Le Seur and Jacquier edition:



Note 359: Within the oval ACBA some point P may be given, or about which pole the right line PS may rotate perpetually, with a uniform motion, thus so that the given point A of this line may describe equal arcs of circles AamX in equal intervals of time, and meanwhile on that right line PS, the moving point p may emerge from the pole P, and will always go on the same right line PS with the velocity which shall [note the error here!] be as the square of that right line within the oval, that is, when the line PS arrives at the place Ps, and the mobile point P at p, the velocity of the point p shall be as the square of the right line PQ contained between the pole P and the oval AQCP, in this motion that point p, describes the spiral PpnZ, with infinite rotations. Note 360. With these in place the right line Pp will always be as the area PAQP; for the circle AamX may be understood to be divided into innumerable equal arcs as am, and with the radii of the spiral PQ, Pq drawn, the perpendiculars Qr, pL sent from the circle to the oval crossing at p and n, a and m, Q and q, to Pq, and in the same time in which the point a, will traverse the arc am, the point p may run through the arc am, the point p may run through the line Ln ; on which account with the arc am arising, Ln will be as the velocity of the point p on the right line Ps, that is, (by hyp.) as the square of the line PQ; again on account of the similar triangles Pam, PQr : : PQ am

PaPa PQ am Qr ×= = , and

hence the area of the sector arising PQq, 2

12 2

PQ amPaQr PQ ×× = . Therefore since am and

2Pa shall be constant quantities (by hyp.), the area PQq arising or the flux of the area PAQ will be as PQ2, and thus as the emerging Ln, or as the flux of the right line Pp, and hence the total area of the fluent PAQ, will be as the total right line of the flux Pp , (Cor. Lem. IV) Q.e.d. Note 361. The points p and Q may be referred to the right line AB, in the given position with the perpendiculars Q, H, pF sent to AB and the area PAQ shall be equal to a finite quantity E from the variable lines PH, QH and composed from other constants as desired, and since the line Pp is proportional to the area PAQ or to the finite quantity E (360), that line will be able to be expressed by a factor from the quantity E into a constant quantity B, and Pp E B= × will be a finite equation. Truly from the similar triangles PFp, PHQ



and the right angle to H, :Pp pF PQ= , or 2 2 :PH QH QH+ , and :Pp PF PQ= , or 2 2 :PH QH PH+ , and besides from the nature of the oval AQCB, another equation

may be given between PH and QH, four finite equations therefore may be found, which likewise contain five variables, to wit Pp, PF, pF, PH, QH, and which hence will be able to be reduced to a single finite equation in which only two variables PF, pF may be found, and thus by this finite equation all the points of the spiral will be able to be found, and therefore with the position given of any right line Sp, the intersection p of any given right line Sp with the spiral also can be found from a finite equation; for since two right lines Sp, SB shall be given in position, the magnitude of the line SP and the nature of the triangle SpF may be given, and hence the ratio of the lines SF or SP PF∓ to Fp, and a new equation may be found between PF and Fp; therefore by this equation and by the other which is to the spiral, PF may be determined, and Fp, and the point of intersection p may be found by a finite equation. Note 362. The two lines AMS, Sms may be referred to the same right line AQ given in place, mutually intersecting in the points S, s , and let AQ, AP be the common abscissae, and QS, PM, Pm the ordinates accorded to these; because with the common intersections of the lines SMs, Sms, the ordinates PM, Pm are equal, if in the two equations for the lines SMs, Sms, with the abscissae remaining common, in place of the ordinates PM, Pm, the same letters may be written, such as y, and then from these equations the letter which expressed the common abscissa may be eliminated, an equation will be obtained composed from y and constants only. Again this final equation no more will determine the first common ordinate SQ, or the first intersection S, than the third of the fourth, etc, since there shall be the same law for everything and likewise the same condition in the calculation; therefore this equation must show completed and indifferently all the common coordinates QS, and likewise all the intersections S, and thus just as many roots or the values of y to be returned as there are common ordinates or intersections, but the equation has just as many dimensions as the number of roots; and thus if the intersections S, s of the lines SMs, Sms, shall be finite in number, also the equation which may determine those is finite ; but if the intersections were infinite in number, the equation will be of infinite dimensions and with an infinitude of roots. See also Chandrasekbar, p. 133. Here the point is made that 'smooth' curves are geometrically rational or algebraic, and thus their area can be found; curves with points of discontinuity, called 'geometrically irrational' by Newton, such as a sector of the ellipse, are not smooth, and do not satisfy an algebraic equation of a finite number of dimensions, and therefore cannot be integrated exactly. However, we find in what follows that such sectors can be related to the area of the corresponding circular circle, and approximate solutions can be found for the angle, if we know the area : which is of course the aim of this section – the area is known and the angle or position in the orbit is required. Open curves such as the parabola and hyperbola do not suffer from this condition.



Corollary.

Hence the area of [the sector] of an ellipse, which will be described by a radius drawn from the focus to a moving body, will not be produced by a finite equation for the time given; and therefore cannot be determined geometrically from the drawing of rational curves . I call curves geometrically rational [i.e. algebraic], all of the points of which are defined by lengths, that is, they are able to be determined by complicated ratios of lengths ; and the others (as spirals, quadratrixes, trochoides) I call geometrically irrational. For the lengths which are or are not as number to number (just as in the tenth Book of the Elements) are arithmetically rationals or irrationals. [Whiteside notes that it was Barrow who made this connection on innumerability, and not Euclid, which Newton corrected.] Therefore I cut off an area of an ellipse proportional to the time as follows, by a geometrically irrational curve [i.e. a cycloid].

PROPOSITION XXXI. PROBLEM XXIII. To find the position of a body in a given elliptical trajectory at a given designated time.

Let A be the principle vertex of an ellipse APB; S the focus, and O the centre, and let P be the position of the body required to be found. Produce OA to G, so that OG shall be to OA as OA to OS [In modern terms, if we let the eccentricity be e, than OS ae= and OG a / e= , whereOA a= .] Erect the perpendicular GH, and with centre O and radius OG describe the circle GEF, and upon the ruler or established right line GH, the wheel GEF may progress by rotating about its axis, meanwhile with its own point A tracing out the trochoid ALI. With which done, take GK in the ratio to the perimeter of the wheel GEG, so that the time, in which the body by progressing from A will describe the arc AP, is to the time of one revolution of the ellipse. The perpendicular KL can be erected crossing the trochoid [or prolate cycloid] at L, and with LP itself drawn parallel to KG will meet the ellipse in the position of the body sought P.



[Note that the ellipse does not itself rotate, just the point P on its curve, along with the imaginary wheel , and the diagram is counter intuitive, as the circle must rotate clockwise while it moves to the left to produce the curve shown so that P falls on L at the position shown; this appears to be rather strange; the diagram by Wren who originally constructed this solution, had the cycloid going to the right ; thus we may consider the point P to coincide with A when 0θ = , which is also a point on the cycloid, and the wheel to be rolling to the left, so that P later actually coincides with L in the position shown. In this case, in rotating through the positive angle θ , the point P moves to the left a distance GK or a

eθ , while in the same time shortening this distance by the amount a sinθ ; hence

relative to the origin O fixed in space in the position shown, we have the point L situated at ( ) ;a

ex e sinθ θ= − the position of the y coordinate is given by (1 )aey ecosθ= − . Thus,

this point is assumed given, from which we may deduce the area swept out by P in the orbit.] For with centre O, with the radius OA the semicircle AQB may be described, and for LP to meet the arc AQ if there is a need by producing Q, and SQ, OQ may be joined. For the arc EFG may cross OQ in F, and to the same OQ there may be sent the perpendicular SR. The area APS is as the area AQS, that is as the difference between the sector area OQA and the triangle area OQS, or, as the difference of the products 1 12 2andOQ arcAQ OQ SR× × , [recall that for different ellipses or for an ellipse and a circle, on the same diameter and with the ordinates in a given ratio, then the corresponding areas of elliptic (and circular) sectors are in the same ratio b/a.] that is, on account of the given 1

2 OQ , as the difference between the arc AQ and the right line SR,

[i.e. the circular sector AQS = ( )12 OQ arcAQ SR− , and the elliptic sector b

aAPS AQS= ], and thus (since the equal ratios shall be given, SR to the sine of the arc AQ, OS to OA, OA to OG, and the arc AQ to the arc GF, and on separating, to arcAQ SR GF− − sin arcAQ), as GK to the difference between the arc GF and the sine of arc AQ. [i.e. ( )2

aAPS arcAQ SR= − giving SR OS sinarcAQ ae sinθ= = , and hence the area of

the elliptic sector APS = ( )2ab e sinθ θ− , which relates to the known x coordinate, which

is moving along the x axis at a constant speed. There is still the problem of finding the angle from the coordinate x analytically, if we are not content with simply measuring it from a mechanical model.]

Q.E.D. [Whiteside, in VI note134, points put that this mechanical solution of the problem of determining the time to travel along the arc of the ellipse from the apse A is essentially the same as that found by Wren in 1658, and published by Wallis in his tract De Cycloide the following year. Note that this mechanical solution uses arcs rather than areas. Further information can be found in the notes provided by Whiteside, especially note 138. Part of the confusion about the following method was due to a page being mislaid by Humphrey Newton, Newton's amanuensis, which resulted in several pages being deleted, as Newton was unable in the short time availabe due to the printing in process, to reproduce that page. The ellipse is now put upright, and a numerical method is used to solve Kepler's equation.]



Scholium.

The remaining description of this curve shall be had with difficulty, it is better to give an approximate solution. Then a certain angle B may be found, which shall be to the angle of the degrees 57.29578, that an arc subtends equal to the radius, as it is the distance SH of the focus to the diameter of the ellipse AB; then also a certain length L, which shall be to the radius [r] in the same ratio inverted. [Thus, in modern terms, we set

the eccentricity equal to the size of the angle B in radians or HS SO rAB AO LB e= = = = , the

eccentricity.] With which once found, the problem then may be put together by the following analysis. By some construction, or in some manner by making a guess, the position of the body P may be known near to the true position p. And with the applied ordinate PR sent to the axis of the ellipse, from the proportion of the diameters of the ellipse, the ordinate RQ of the circumscribed circle AQB is given, which is the sine of the angle AOQ with the radius being AO [by def.], and which cuts the ellipse at P. That will suffice in a rough calculation to find the angle in approximate numbers. [That is,

SHPRQR AB e= = .] Also the angle may be known to be proportional to the time, that is, which

may be to four right angles, as the time is in which the body will describe the arc AP, to the time of one revolution in the ellipse. Let this angle be N. [Thus, 2 2

arcApt NT rπ π= = .]

[Clearly, the solution is required to the problem: area of sector of ellipse ApS : area of sector of circle AqS = b/a. This may be developed as above, and we come upon an equation similar to that above : the elliptic sector ( )2

abApS e sinθ θ= − ; or N e sinθ θ= − , and the value of θ is required to satisfy this equaton. Following Whiteside, an approximate solution is set up 2 1 1θ θ ε= + , where iε is small; from which successive approximations follow. The initial value 1θ it taken as the angle AOQ, while the values for 1 2 etc, , .,ε ε are successively E, G... We may, for clarification, invoke Newton's Method for finding a better approximation to the root of an equation : if x0 is an approximate root of the equation ( ) 0f x = , then a better approximation is

1

1

( )2 1 ( )

f xf ' xx x= − , and so on; here ( )f N e sinθ θ θ= − + , in which case



( ) 1f ' ecosθ θ= − + , and ( )2

N AOQ L r sin AOQN AOQ B sin AOQB cos AOQ Lcos AOQAOQ AOQθ − +− += − = − ;

thus, the correction is the angle ( ) ( )N AOQ L r sin AOQ N AOQ L r sin AOQLcos AOQ Lcos AOQ Lcos AOQ

− + −− = + ; Newton sets

the correction into two parts, r sin AOQLcos AOQD = ; and applies the correction D to the other part,

giving : ( )L N AOQ DLcos AOQE − += .] Then an angle D may be taken to the angle B, as the sine of

the angle AOQ to the radius itself, and an angle E to the angle N AOQ D− + , as the length L to the same length L diminished by the cosine of the angle AOQ [this statement applies to both parts above], where that angle is less than a right angle, but increased when it becomes greater. Afterwards then the angle F may be taken to the angle B, so that the sine of the angle AOQ E+ is to the radius, and the angle G to the angle N AOQ E F− − + as the length L to the same length diminished by the cosine of the angle AOQ + E , when that is less than a right angle, increased when greater. In the third place in turn, the angle H may be taken to the angle B, as the sine of the angle AOQ E G+ + is to the radius; and the angle I to the angle N AOQ E G H− − − + , and the length L to the same length diminished by the cosine of the angle AOQ E G+ + , when this is less than a right angle, increased when greater. And thus it is permitted to go on indefinitely. Finally the angle AOQ may be taken equal to the angle AOQ E G I &c+ + + + . And from the cosine of this Or and with the ordinate pr, which is to the sine of this qr as the minor axis of the ellipse to the major axis, the correct position p of the body is obtained. If when the angle N AOQ D− + is negative, the + sign of E everywhere is changed into – , and the – sign into +. Likewise it is to be understood concerning the signs of G and I, where the angles

and N AOQ E F , N AOQ E G H− − + − − − + may be produced negative. But as the infinite series etcAOQ E G I+ + + + . converges rapidly, thus so that scarcely will there be any need to progress further than to the second term E. And the calculation may be based on this theorem, that the area APS shall be as the difference between the arc AQ and the right line from the focus S sent perpendicularly to the radius OQ. A similar calculation is put in place with the hyperbola problem. Let O be the centre of this, the vertex A, the focus S and the asymptote OK. The magnitude of the area cut off is requiring to be proportional to the time. Let that be A, and a guess is made concerning the position of the right line SP, which shall be cut approximate to the true area APS. OP may be joined, and from A and P to the asymptote draw AI and PK parallel to the other asymptote, and the area AIKP will be given by a table of logarithms, and from the area equal to OPA, which taken from the triangle OPS, the area cut APS will be left. From the difference of the area requiring to be cut off A and of the area cut off APS the double 2 2APS A− or



2 2A APS− to the line SN, which is perpendicular from the focus S into the tangent TP, the length of the chord PQ may arise. But that chord PQ may be inscribed between A and P, if the area cut APS shall be greater than the area required to be cut A, otherwise cut at the opposite side of the point P: and the point Q will be a more accurate position of the body. And with the computation repeated the same may be found more accurately indefinitely. And from these calculations the problem generally can be put together analytically. Truly the calculation which follows is more fitted to astronomical uses. With the semi-axis of the ellipse AO, OB, OD present, and L its latus rectum, and D the difference between the semi-minor axis OD and the semi-latus rectum 1

2 L ; whereby then the angle Y, the sine of which shall be to the radius as the rectangle formed from that difference D, and the half sum of the axes AO OD+ to the square of the major axis AB; then the angle Z, the sine of which shall be to the radius as twice the rectangle formed by the distance between the foci SH and that difference D to the triple of the square of the major semi-axis AD. With these angles thus found ; the position of the body thus can be determined. Take the angle T proportional to the time in which the arc BP has been described, or equal to the mean motion (as they say) ; and the angle V, as it is the first equation of the mean motion, to the angle Y, the first maximum equation, as the sine of double the angle T to the radius ; and the angle X, the second equation, to the angle Z, the second maximum equation, as the cube of the sine of the angle T is to the cube of the radius. [See Whiteside note 159 Vol. VI etc.] Take the sum of the angles T, V, X or the sum T X V+ + , if the angle T is less than a right angle , or the difference T X V+ − , if this greater than a right angle and less than two right angles, equal to the angle BHP, the mean motion equation ; and if HP should cut the ellipse at P, with SP drawn it will cut the area BSP approximately in proportion to the time. This practice may be seen to be work well enough, because of the very small angles V and X therefore arising, in seconds of arc, if it pleases to be sufficient to find two or three figures of the places. And also this is accurate enough for the theory of the planets. For in the orbit of Mars itself, the equation of the centre of which is ten degrees, the error scarcely will exceed a second of arc. But when the equated mean motion has been found (the angle BHP), then the true motion (the angle BSP) and the distance (SP)are had readily by the well-known method [of Seth-Ward; this useful reference is missing in the 3rd edition, in place in the 1st and 2nd edition]. Up to the present we have been concerned with the motion of bodies on curved lines. But it may also happen that the body ascends or descend by a right line, and I now go on to to set out matters relating to motions of this kind.


Book I Section VII. Translated and Annotated by Ian Bruce. Page 217

SECTION VII.

Concerning the rectilinear ascent and descent of bodies.

[Newton treats this problem as a limiting case of orbital motion, and there are three cases to consider: elliptic, parabolic, and hyperbolic orbits. It is easily shown in modern terms that in the elliptic case, the total energy of the body is given by 21

2 0rv μ− < , while in the second and third cases the total energy is zero, and > 0 respectively, where μ relates to the gravitational constant. Newton of course does not follow this approach. The task is to find the time to fall a given distance in a straight line towards the focus, or to be projected away likewise, with some given initial velocity and position. The method depends on finding the equivalent circular motion relating the areas, which are in proportion to the times as previously. The problem is then essentially a special case of Kepler's problem; arcs are related to areas.]

PROPOSITION XXXII. PROBLEM XXIV.

Because the centripetal force shall be inversely proportional to the square of the distance with the position from the centre, to define the intervals which a body by falling in a

straight line will describe in given times. Case. I. If the body does not fall perpendicularly, it will describe some conic section (by Corol. I, Prop. XIII.) the focus of which agrees with the centre of forces. Let that conic section be ARPB, and S the focus of this . And initially, if the figure is an ellipse, upon the major axis AB of this the semicircle ADB may be described, and by falling the body may cross the right line DPC perpendicular to the axes; and with DS and PS drawn, the area ASD will be [proportional] to the area ASP, and thus also proportional to the time. With the axes AB remaining, the width of the ellipse may be continually become less, and always the area ASD will remain proportional to the time. That width may be decreased indefinitely : and with the orbit APB now coincident with the axes AB and the focus S with the end of the axis B, the body will descend on the right line AC, and the area ABD becomes proportional to the time. And thus there will be given the interval AC, which the body will describe by falling perpendicularly from the position A in the given time, but only if the area ABD may be taken proportional to the time, and the perpendicular DC may be sent from the point D to the right line AB

Q.E.I. [See Chandrasekhar p.143 and beyond for a modern treatment, Routh & Brougham, p. 77 ; Whiteside Vol. VI, p. 325 onwards. In this case, the length of the latus rectum, given by

22 1a e− , can approach zero as the eccentricity e approaches 1, making the ellipse more narrow and keeping the transverse length fixed, while the focus tends towards B, and the



time is proportional to the area intercepted by a radius SP on the circle with diameter DB, where S coincides with B in the limiting case.] Case 2. If that figure RPB is a hyperbola [second diagram], the rectangular hyperbola BED may be described according to the same principal diameter AB : and because the areas CSP, CBfP, SPfB are in proportion to the areas CSD, CBED, SDEB, one to one, in the given ratio of the heights CP, CD; and the area SPfB is proportional to the time in which the body P will be moved through the arc PfB; the area SDEB will also be proportional to the same time. [Note that the here the rectangular hyperbola is the regular figure equivalent to the auxiliary circle for the ellipse, used in finding the area corresponding to the time.]The latus rectum of the hyperbola RPB may be diminished indefinitely with the transverse width remaining fixed, and the arc PB will coincide with the right line CB and the focus S with the vertex B and the right line SD with the right line BD. Therefore the area BDEB will be proportional to the time in which the body C by falling in a straight line will describe the line CB.

Q.E.I. [In this case, the latus rectum or the focal chord, is given by 22 1a e − , and as e approaches 1, the orbit becomes narrower, maintaining the same separation of the foci and centre.] Case 3. And by a similar argument if the figure RPB is a parabola, and with the same principal vertex B another parabola BED may be described, which always may be given, then meanwhile the first parabola, on the perimeter of which the body P may be moving, and with the latus rectum of this reduced to zero, it may agree with the right line CB; the segment of the parabola BDEB will be proportional to the time in which both P or C will descend to the centre.


Book I Section VII. Translated and Annotated by Ian Bruce. Page 219 PROPOSITION XXXIII. THEOREM IX.

With the positions now found, I say that the velocity of the falling body at some place C is to the velocity of a body describing a circle with the centre B and radius BC, in the square root ratio that AC, the distance of the body from the circle or from the more distant vertex of the rectangular hyperbola A, has to the principal semi-diameter of the figure 1

2AB.

AB may be bisected in O, the diameter of each common figure RPB, DEB; and the right line PT may be drawn, which may touch the figure RPB in P, and also cuts that common diameter AB (if there is a need for extending) in T; and let SY be perpendicular to that right line, and let BQ be perpendicular to this diameter, and L may be put as the latus rectum of the figure RPB . It may be agreed by Corol. IX, Prop. XVI, that the velocity at some place P of the body moving on the line RPB about the centre S, shall be to the velocity of the body being described about the circle with the same centre and radius SP as the square root of the ratio of the rectangle 1

2 L SP× to SY2.



[Recall that by Corol. IX, Prop. XVI, the velocity at P for the conic is as 12 L

SY, and for

the circle with radius SP, for which the velocity is as 1SP

[i.e., in modern terms, from

the force equation for motion in a circle, we have 2

21 1 or v

r r rvα α ], we have

2 12

2 2 2conicAPB

cir .rad . SP

L SPv Lv SY SY

SP=

×= × = , where we note that the circle is an ellipse with equal semi-

major and minor axes, and the latus rectum is the diameter, while SY becomes SP.] But from the theory of conics there is AC.CB to CP2 as 2AO to L, and thus

22CP AOACB×

equals L. [To show this analytically for an ellipse, note initially from

22

2 2 1yxa b+ =

that ( ) ( )( )2 2

2 22 2 1 x b

a ay b a x a x= − = + − ; hence

2

2 22 2AC.CB a a AOL LCP b

= = = , hence 2 2CP AO

AC.CBL ×= , as 22b

aL = .]

Therefore these velocities in turn are in the square root ratio 2CP AO SPAC.CB× × to SY2

[ or 2 2

2 2conic

circle

v CP AO SPv AC.CB SY

× ××

= ].

Again from the theory of conics CO is to BO as BO to TO, and by adding or taking from each other as CB to BT. From which either by taking or adding there shall be BO – or + CO to BO as CT to BT, that is, AC to AO as CP to BQ [i.e. =CO BO BO CO BC

BO TO TO BO BT++= = ; then 1 BC TC

BT BT− = , while 1 1OC OC ACOB OA OA− = − = because of

the similar triangles CPT and BQT, BQ.ACOACP = ];

and thence 2CP AO SPAC.CB× × is equal to

2BQ AC SPAO BC× ×× . Now with the width CP of the figure

RPB diminished indefinitely, thus so that the point P coincides with the point C; and the point S with the point B, and the line SP with the line BC, and the line ST with the line BQ ; and the velocity of the body now descending on the right line CB becomes to the velocity of the body describing the circle BC with the radius B, in the square root ratio of

2BQ AC SPAO BC× ×× to SY2, that is (with the equal ratios SP to BC and BQ2 to SY2 ignored), in the

square root ratio AC to AO or 12 AB , [i.e. C conic

circle

v ACv AO= .]

Q. E.D. Corol. I. With the points B and S coinciding, TC shall be to TS as AC to AO. Corol. 2. With the body rotating in some circle at a given distance from the centre of the circle, it may rise up by its own motion to twice is distance from the centre.



PROPOSITION XXXIV. THEOREM X.

If the figure BED is a parabola, I say that the velocity of the falling body at some place C is equal to the velocity by which the body can describe a circle uniformly with centre B and with radius half of its interval BC. For the velocity of the body describing a parabola RPB about a centre S at some place P (by Corol. Prop. XVI.) is equal to the velocity of the body describing a circle uniformly about the same centre S, with a radius half of the interval SP. There the width of the parabola CP may be diminished indefinitely, so that the arc of the parabola PfB may coincide with the right line CB, the centre S with the interval B, and the radius SP with the interval BC, and the proposition will be agreed upon. [Recall

2 2

2 2conic

circle

v CP AO SPv AC.CB SY

× ××

= ; in this case2

2 22 1

2AC.CB a a

LCP b= = = , and hence

2

2 22 1conic

circle

v AO SPv SY

× ×= = when 12AO BC= and SP SY BC= = .]

PROPOSITION XXXV. THEOREM XI.

With the same in place, I say that the area of the figure, described by the indefinite radius SD, shall be equal to the area that the body can describe in the same time, with a radius equal to half of the latus rectum of the rectilinear figure DES, by rotating uniformly about the centre S. For consider the body C as falling in the shortest interval of time to describe the element of length Cc, and meanwhile another body K, by rotating uniformly in a circle OKk about the centre S, to describe the arc Kk. The perpendiculars CD and cd may be erected meeting the figure DES in D and d. SD, Sd, SK, Sk may be joined and Dd may be drawn meeting the axis AS in T, and to that the perpendicular SY may be sent. Case. 1. Now if the figure DES is a circle or a rectangular hyperbola, the diameter AS may be the transverse bisector of this at O, and SO will be half of the latus rectum. And because TC is to TD as Cc to Dd, and TD to TS as CD to SY, so that from the equation there will be TC to TS as CD Cc× to SY Dd× .

[i.e. = and =TC Cc CDTDTD Dd TS ST, ; TC CD Cc

TS SY Dd××= ]

But (by Coroll. Prop. XXXIII.) TC is to TS as AC to AO, for example if the final ratios of the lines may be taken on placing the points D and d together. Therefore AC is to AO or SK as CD Cc× to SY Dd× . Again the velocity of the descending body at C is to the



velocity of the body described around the circle with radius SC and centre S in the square root AC to AD or SK (by Prop. XXXIII.) And this velocity to the velocity of the body describing the circle OKk is in the square root ratio SK to SC (by Corol. VI. Prop. IV.) and from that equation the first velocity to the final, that is the line element Cc to the arc Kk is in the square root ratio AC ad SC, that is in the ratio AC to CD. Whereby CD Cc× is equal to AC Kk× , and therefore AC to SK as AC Kk× to SY Dd× , and thus SK Kk× equals SY Dd× , and 1

2 SK Kk× equals 12 SY Dd× , that is the area KSk is

equal to the area SDd. Therefore in the individual small increments of time

the small increments of the two areas KSk and SDd may be generated, which, if the magnitude of these may be diminished and the number increased indefinitely, maintain a ratio of equality, and therefore (by the Corollaries of Lemma IV.) the whole areas generated likewise are always equal. Q.E.D. Case. 2. But if the figure DES shall be a parabola, there may be found to be as above CD Cc× is to SY Dd× as TC to TS, that is as 2 to 1, and thus 14 CD Cc× is equal as above 1

2 SY Dd× . But the velocity of the falling body at C is equal to the velocity by which the circle with radius 1

2 SC may be able to be described uniformly (by Prop. XXXIV.) And this velocity to the velocity by which the circle with radius SK may be able to be described, that is, the element Cc to the arc Kk (by Corol. VI., Prop. IV.) is in the square root ratio SK to 1

2 SC , that is, in



the ratio SK to 12 CD . Whereby 1

2 SK Kk× is equal to 14 CD Cc× and thus equal to

12 SY Dd× , that is, the area KSk is equal to the area SDd as above. Q.E.D.

PROPOSITION XXXVI. PROBLEM XXV.

With the place of the falling body A given to determine the descent times. Upon the diameter AS, describe the semi-circle ADS, the distance of the body at the start, and so that the semicircle OKH about the centre S is equal to this. From some position of the body C erect the applied ordinate CD. Join SD, and put in place the sector OSK equal to the area ASD. It is apparent by Prop. XXXV that the body on falling describes the distance AC in the same time that the other body, by rotating uniformly about the centre S, can describe the arc OK in the same time. Q. E. F.

PROPOSITION XXXVII. PROBLEM XXVI.

To find the times of ascent or descent of a body projected from some given place. [There is as need to classify the motion of a body, projected either up or down, into one of the three types, corresponding to degenerate motion on and ellipse, hyperbola, or a parabola. ] The body may emerge from a given place G along the line GS with some velocity. In the square ratio of this velocity to the uniform velocity in a circle, by which a body may be able to rotate about the centre S for a given radius SG,

take GA to 12 AS . If that ratio is of the number two to one, the point A infinitely far away,

in which case a parabola is being described with vertex S, axes SG, with some latus



rectum. This is apparent from Prop. XXXIV. But if that ratio were greater or smaller than 2 to 1, in the first case a circle, in the latter a rectangular hyperbola, must be described on the diameter SA. It is apparent by Prop. XXXIII. Then with centre S, with a radius equal to half the latus rectum, the circle HkK may be described, and at the position of the body G either descending or ascending, and at some other place C, the perpendiculars GI and CD meeting the conic section or the circle in I and D. Then with SI and SD joined, the sectors HSK and HSk are made equal to the segments SEIS, SEDS, and by Prop. XXXV the body G describes the interval GC in the same time in which the body K can describe the arc Kk. Q.E.F.

PROPOSITION XXXVIII. THEOREM XII. Because the centripetal force may be put proportional to the height or distance of the places from the centre, I say, that the times of falling, the velocities and the distances described, are proportional to the arcs, to the sines of the arcs and to the versed sines respectively. [Recall that an inverse square law of force acting on a body in orbit from the focus of the ellipse may be replaced by one of proportionality acting from the centre (Prop. IV) ; thus there is proportional motion between uniform rotation on the auxiliary circle, on any ellipse with the same semi-major axis, and the S.H.M. on the vertical line AS in the limiting case. Here the forces are in proportion to the distances SC, the speeds to the chords CD, and the distances fallen to the versed sines AC.] A body may fall from some position A along the right line AS; and from the centre of forces S, with a radius AS, the quadrant of a circle may be described AE, and let CD be the [right] sine of any arc AD; and the body A, in the time AD, by falling will describe the interval AC, and at the place C it will acquire the velocity CD. In the same manner it may be shown from Proposition X, as by which Proposition XXXII was demonstrated from Proposition XI. Corol. I. Hence the times are equal, in which a single body by falling from the place A arrives at the centre S, and another body by rotating will describe the fourth part of the arc ADE. Corol. 2. Hence all the times are equal in which bodies fall from any places as far as the centre. For all the periodic times of revolution may be equal (by Corol. III. Prop. IV.).



PROPOSITION XXXIX. PROBLEM XXVII. With centripetal forces of any kind put in place, and with the quadratures of the curvilinear figures agreed upon, the straight rise or fall of a body is required, as well as the velocity at individual places , and the time in which the body may arrive at some place: And conversely. The body E may fall from some place A on the right line ADEC, and from its place E a perpendicular line EG may always be put in place, proportional to the centripetal force at that place tending towards the centre C: And let BFG be the curved line that point G always touches [i.e. the curve traced out by the centripetal force or acceleration]. Moreover, at the start the line EG may coincide with the perpendicular AB itself, and the velocity of the body at some place E will be as the right line, [the square of which] can be as the curvilinear area ABGE. Q.E.I. [This is essentially an exercise in integration; the first curve BFG is integrated w.r.t. z from 0z = at A to some general point, giving the area under the force vs. distance, or the acceleration vs. distance curve, which we can interpret as the kinetic energy acquired, or as the work done by the force; we can show this readily starting from ( )2

ddzdt

F z= − ,

which can be written in the form ( )dv dz dvdz dt dz. v F z= = − , or as the indefinite integral

( )212 v F z dz= −∫ . Clearly Newton was familiar

with, and indeed was the originator of this wonderful new approach to solving problems, but chose not to divulge his method.] On EG, EM may be taken for the right line, which can be inversely proportional to [the square root of] the area ABGE, and VLM shall be the curved line, that the point N always touches, and the asymptote of this is the right line AB produced; and the time will be, in which the body by falling describes the line AE, as the area under the curve ABTVME.

Q. E. I. [Following on from the last note, we now have

( )122 1:dt

dz z EF z dz area ABGE

EM .

α−

=⎡ ⎤⎡ ⎤ =⎣ ⎦ ⎣ ⎦

=∫

If we call EM the function ( )T z , then

( )t T z dz= ∫ ; the limits of integration are chosen to

fit the circumstances, from A to E. There now



follows the verification of these integrations by the inverse process of differentiation.] And indeed on the right line AE there may be taken that line of the shortest length DE, and DLF shall be the locus of the line EMG, when the body will be moving through D; and if this shall be the centripetal force, as the right line, which can be the area ABGE, it shall be as the velocity of the descent: that area will be in the square ratio of the speed, that is, if for the velocities at D and E, there may be written V and V I+ , the area ABFD will be as VV, and the area ABGE as 2VV .VI II+ + , and separately the area DFGE as 2.VI II+ , and thus DFGE

DE as 2VI IIDE+ , that is, if the ratios are taken of the first vanishing

quantities, the length DF shall be as the quantity 2VIDE , and thus also as the quantity of half

of this I VDE× . But the time, in which the body falling will describe the element of line DE,

is as that element directly and as the velocity V inversely, and the force is as the velocity increment I directly and the time inversely, and thus if the first vanishing ratios are taken, as I V

DE× , that is, as the length DF. Therefore DF or EG becomes proportional to the force

itself so that the body may descend with that velocity, which shall be as the right line which can be as the [square root] of the area ABGE.

Q.E.D. Again since the time, in which, in which any line element DE of a given length may be described, shall be inversely as the velocity, and thus inversely as the right line which can become the area ABFV; and let it be DL, and thus the area arising DLME, as the same right line inversely: the time will be as the area DLME, and the sum of all the times as the sum of all the areas, that is (by Corol. Lem. IV.) the total time in which the line AE is described will be as the total area ATVME.

Q.E.D. Corol. 1. If P shall be the place, from which the body must fall, as urged by some uniform known centripetal force (such as gravity generally is supposed) it may acquire a velocity at the place D equal to the velocity, that another body falling by some other force has acquired at the same place D, and on the perpendicular DF, DR may be taken, which shall be to DF as that uniform force [PQ] to the other force at the place D; and the rectangle PDRQ may be completed, and an area ABFD equal to this may be cut off;

[Thus, ( )PQ PD F z dz× = ∫ with suitable limits chosen.]

A will be the place from which the other body has fallen. For with the rectangle DRSE completed, since there shall be the area ABFD to the area DFGE as VV to 2.VI, and thus as 1

2V to I, that is, as of half of the whole velocity to the increment of the velocity of the body falling by the unequal force; and likewise the area PQRD to the area DRSE as of half of the whole velocity to the increment of the velocity of the body falling under the uniform force; and these increments shall be (on account of the equality of the increments of time arising) as the generating forces, that is, as the applied lines DF, DR in order, and thus as the areas arising DFGE and DRSE; from the equality of the total area, ABFD and PQRD are as the half of the total speeds, and therefore, on account of the equal speeds, equal in turn.



Corol. 2. From which if some body may be projected from some place D with some given velocity either up or down, and the law of the centripetal force may be given, the velocity of this will be found at any other place e, on erecting the ordinate e.g., and by taking that velocity at e to the velocity at the place D as the right line, which can become [on squaring] the rectangular area PQRD either increase by the curved area DFge, if the place e is below the place D, or decreased by it, if this is above, to the right line which can become [on squaring] the area PQRD only.

[ 2 21 2Thus ( ),vel vel F z dz= ± ∫ .]

Corol. 3. The time too will become known by erecting the ordinate em inversely proportional to the square root of the side from PQRD + or – DFge, and by taking the time in which the body has described the line De to the time in which the other body fell with a uniform force from P and on falling arrives at D, as the curvilinear area DLme to the rectangle 2.PD DL× . For the time in which the body falling under the uniform force has described the line PD, is to the time in which likewise the body has described the line PE in the square root ratio PD to PE, that is (with the element of the line now arising) in the ratio PD to 1

2PD DE+ or 2.PD to 2PD DE+ , and separately, to the time in which the same body has described the line element DE as 2PD to DE, and thus as the rectangle 2PD DL× to the area DLME; and the time in which the one body has described the line element DE to the time in which the other body in the non uniform motion has described the line De, as the area DLME to the area DLme, and from the equality the first time to the final time as the rectangle 2PD DL× to the area DLme.


Book I Section VIII. Translated and Annotated by Ian Bruce. Page 236

SECTION VIII.

Concerning the finding of orbits in which rotating bodies are acted on by any centripetal forces.

PROPOSITION XL. THEOREM. XIII.

If a body may be moving in some manner, under the action of some centripetal force, and another body may ascend or descend, and the velocities of these in some case are equal

at some altitude, then the velocities of these shall be equal at all equal altitudes. [Although Newton of course does not use these exact words, this Proposition embodies the genesis of the idea of equipotential surfaces surrounding the source of a force field of some kind; in the diagram DE and EK are parts of such surfaces in the case of gravity, and a body falling or orbiting in some manner acquires an acceleration in moving from one surface to another, separated normally from it by an infinitesimal distance, and thus a change in velocity is produced; thus, the distances considered which Newton calls minimal are our infinitesimals, and likewise with the times involved. On the other hand, an orbiting body as well as a dropped body have to conserve the equal areas in equal times law, which amounts to the conservation of angular momentum, zero in the second case; all of these matters are attended to here. The same arguments are true of course for bodies projected upwards.] Some body may descend from A through D and E to the centre C, and another body may be moving from V on a curve VIKk. With the centre C, for some radii the concentric circles DI and EK may be described meeting with the right line AC in D and E, and with the curve VIK in I and K. IC may be joined crossing KE at N itself; and onto IK there may be send the perpendicular NT; and the separation DE or IN of the circumferences of the circles shall be as a minimum, and the bodies at D and I may have equal velocities. Because the distances CD and CI are equal, the centripetal forces at D and I are equal. These forces may be expressed by the equal line elements DE and IN; and if the one force IN may be resolved into the two forces NT and IT (by Corol. 2 of the Laws.); the force NT, by acting along the line NT perpendicular to the course ITK of the body, will not change the velocity of the body in that course, but will only draw the body away from moving in straight line, and it will always act to deflect the body from the tangent of the orbit itself, and the body to be progressing along the way of the curved line ITKk. That force will be completely used up in producing this effect: but the other force IT, by acting along the course of the body, the whole force will accelerate that body, and in the given time taken as the minimum possible will generate an acceleration proportional to that time itself. Hence the accelerations of the bodies at D and I



made in equal times (if the first ratios of the line elements arising are taken DE, IN, IK, IT, NT ) become as the lines DE and IT: but with unequal times as these lengths and times jointly. But the times in which DE & IK are described, on account of the equality of the velocities are as the paths described DE and IK, and thus the accelerations, in the described paths along the lines DE & IK, are as DE and IT, DE and IK jointly, that is as DE2 and the rectangle IT IK× . [If we let v be the common velocity at D and I initially, the first straight down, and the second along the curve, then after an increment in time DE

DE vtΔ = , the first body has

acquired an extra velocity 2DEDE vg t g DEΔ α= ; as the acceleration g (or force per unit

mass) is expressed by the line DE : (A lingering source of confusion in force diagrams at this time was that a line segment could represent a displacement, a velocity, a force, etc; only vector notation eventually removed some of this confusion.); hence in this case, what we may call now the impulse or the change in velocity, F tΔ , is proportional to DE2. For the other body on the curve, it has to travel a longer increment IK, but with the same initial velocity v as D (from the hypothesis) ; hence in this case IK

IK vtΔ = ; in addition,

the actual component of the force along the curve is diminished by the cosine factor ITIN ;

or the impulse above, to which this must be equal, becomes

IT IK ITIK IN v INg t IN IK ITΔ α× = × × × .

Chandrasekhar shows this on p. 167 ; he also talks about energy conservation, and it is quite wrong to do so, as there is no hint of conservation laws in the Principia, at least up to this point, apart from Kepler's 2nd Law, which Newton uses implicitly without naming it as such.] But the rectangle IT IK× is equal to IN2, that is, equal to DE2 [from the similar triangles ITN and IKN] and therefore the accelerations in the transition of the bodies from D and I to E and K are produced equal. Therefore the velocities of the bodies are equal at E and K : and will always be found equal by the same argument in the subsequent equal distances. Q.E.D. In addition by the same argument bodies both equidistant from the centre and with the same velocity, in ascending to equal distances, are equally retarded. Q.E.D. Cor. 1. Hence if a body may be oscillating hanging from a thread, or restrained to be moving on some impediment perfectly lubricated and without friction, and another body may ascend or descent directly, the velocities of these at the same height shall be equal : the velocities of these will be equal to any others at all heights. For that same transverse force NT shall be presented either by the thread or the impediment of the completely



slippery vessel. The body may not be retarded nor accelerated by that, but only constrained by the curve to cease being linear. Cor. 2. Hence also if the quantity P shall be the maxima distance from the centre to which the body either oscillating or rotating in some trajectory may ascend, from some lower point of the trajectory, as here it may be able to rise projected upwards with a velocity ; and let the quantity A be the distance of the body from the centre at some other point of the orbit, and the centripetal force always shall be as some power 1nA − of A, the index 1n − of which is any number n diminished by one ; the velocity of the body at any

height A will be as n nP A− , and therefore is given. For the velocity of the body ascending and descending on a straight line (by Prop. XXXIX) is in this ratio itself.

PROPOSITION XLI. PROBLEM XXVIII.

With any kind of centripetal force in place, and the quadratures of the figures granted, then both the trajectories in which bodies are moving are required, as well as the times of

the motions found in the trajectories. Any force may be drawing [a body] towards the centre C and the trajectory VIKk shall be required to be found. The circle VR with centre C may be given described with some radius CV, and from the same centre some other circle [arcs] ID and KE are described cutting the trajectory in I and K and the right line CV in D and E. Then draw the right line CNIX cutting the circles KE and VR in N and X, and also the right line CKY meeting with the circle VR in Y. Moreover let the points I and K themselves in turn be the closest possible together, and the body may go from V by I and K to k; and the point A shall be that place from which the other body must fall, so that at the place D it will acquire a velocity equal to the velocity of the first body at I. And with the matters in place from Proposition XXXIX, the line element IK, as given described in the shortest time, will be as the velocity [In modern terms we may write this velocity using polar coordinates ( )r,ϕ corresponding

to CN and the angle NCK, as ( ) ( )122 2 2ds r, dr r dϕ ϕ= + and ( ) ( )

122 2 2v r, r rϕ φ= + ],

and thus as the right line which can become [on integration] the area ABFD, [Thus, the area ABFD vds= ∫ ; and which now we call the energy integral, corresponding

to the area is : 2

221

2 2( )h

rr F r dr+ = −∫ , where F(r) is the attracting force on the body. This

equation arises from the force equation : 2

32 2( ) since h

rr r F r , r hϕ ϕ− = − + = , the

angular momentum equation, which Newton understood to be the 'equal area in equal



times' law (of Kepler.) Thus, the radial velocity below becomes

( )1

2 2

22 ( ) hr

r F r dr ABFD ZZ⎡ ⎤= − − = −⎣ ⎦∫ .]

and the triangle ICK will be given proportional to the time, and thus KN will be inversely as the altitude IC, that is, if some [constant] quantity Q may be given, and the altitude IC may be called A, [not to be confused with the vertex A, then the length KN varies ] as Q

A .

Hence we may use the name Z for the quantity QA

[i.e. a length proportional to KN, normal to the radius at that point, so that Z IC× is proportional to the rate of change of area ; this quantity Z, which is just h

r , was introduced by Halley in editing the work, to ease the typesetting, according to Whiteside p.347 of Vol. VI], and we may put the magnitude of Q to be that such that in some case there shall be ABFD to Z, as IK is to KN, and in every case there will be ABFD to Z as IK to KN, and ABFD to ZZ as IK2 to KN2, and separately ABFD –ZZ to ZZ as IN2 to KN2 [i.e.

2 2 2

2 2IK KN INABFD–ZZ

ZZ KN KN−= = . ]

[In modern notation this becomes ( ) ( ) ( )22

22

22 2hr r

hr

ABFD vdr INrd v KN .

ϕϕ−

= = = ]

and thus ABFD ZZ− is to Z (or QA ) is as IN to KN, and therefore A KN× equals

Q IN

ABFD ZZ×−

. From which since YX XC× shall be to A KN× as CX2 to AA, XY XC×

the rectangle will equal 2Q IN CX

AA ABFD ZZ× ×

−·

[i.e. 2 2 2

2 22

drdt

Q IN CX h dr CX h dt CXr rAA ABFD ZZ

r d hdt.ϕ × × × × × ×−

= = = = ],

Therefore if Db and Dc may be taken on the perpendicular DF always equal respectively

to 2

QABFD ZZ−

and2

2 Q CX

AA ABFD ZZ×

−,

[These are in turn 2 22 Q h hdt

r drABFD ZZ−= = and

2 2 22 2

2 2 22 22 r

Q CX r CX dh CX CXdrr v r rAA ABFD ZZ

.ϕ ϕ× ××−

= = =

The first quadrature gives an area proportional to the time of descent, while the second gives an area proportional to the sector angle, and thus locates the position of the body on the curve.] and the curved lines may be described ab and ac which always touch the points b and c; and from the point V to the line AC the perpendicular Va may be drawn cutting the curved areas VDba and VDca, and also the ordinates may be erected Ez, Ex: because the rectangle Db IN× or DbzE is equal to half of the rectangle A KN× , or to the triangle ICK; and the rectangle Dc IN× or DcxE is equal to half of the rectangle TX XC× or to



the triangle XCT; that is, since the small parts DbzE and ICK always equal the small parts arising of the areas VDba and VIC, and DcxE and XCY are always equal to the small parts arising of the areas Dca and VCX, the area generated VDba will always be an area equal to the area VIC, and thus proportional to the time, and the area produced VDca equal to the area generated VCX. Therefore with some time given from which the body has departed from the place V, the area will be given itself proportional to VDba, and thence the height of the body will be given CD or CI; and the area BDca, and for that the equal VCX together with the angle of this VCI. But with the angle VCI and the height CI given, the place I may be given, in which the body will be found in that time completed. Q. E. I. [This would have been a very difficult proposition for those of Newton's contemporaries to follow, who were not conversant with Newton's calculus, and even now it is a little difficult in the Latin until one knows what is going on; Chandrasekhar sets this out in more detail on p. 170, where it becomes quite straight forwards as the double integration derived from the original differential equation, while Whiteside provides a similar historical enlightenment on p.347 of Vol. VI . Chandrasekhar admits to solving the problems himself, those he tackles, and then relating his solution to that of Newton, which has raised questions of anachronism, but these can be taken in one's stride; while Whiteside digs deep into Newton's methods from a historical standpoint; clearly the latter is more satisfactory, though the former has a lot to recommend it from the immediate nature of the solutions provided for the common reader; in note (209) Whiteside sets out the integrals for the angle and time of the orbiting body under a general force law; unfortunately for Newton, his solution was poorly received or even understood by his contemporaries, who went to great lengths subsequently to prove the same results using Leibniz's notation, which later caused Newton much unhappiness : see Whiteside's notes for further details on this. ] Corol. I. Hence the maximum and minimum heights of bodies, that is, the apses of trajectories may be found conveniently. For the apses are the points in that trajectory in which the right line IC drawn through the centre falls perpendicularly on the trajectory VIK : because that will be where the right lines 1K and NK are equal [i.e. 0r = ], and thus the area ABFD is equal to ZZ. Corol. 2. But also the angle KIN, in which the trajectory cuts that line IC somewhere, may be found conveniently from the height IC of the body; without doubt by taking the sine of this to the radius as KN to IK, that is, as Z to the square root of the area ABFD.



Corol. 3. If from the centre C and from the principal vertex V some conic section VRS may be described, and from some point R of this the tangent RT may be drawn meeting the axis CV produced indefinitely at the point T; then with CR joined there may be drawn the right line CP, which shall be equal to the abscissa CT, and the angle VCP is put in place proportional to the sector VCR may be put in place ; moreover a centripetal force may draw towards the centre C inversely proportional to the cube of the distance of the place [of the body], and the body will emerge from the place V with the just velocity along the perpendicular right line CV : that body may be progressing in the trajectory VPQ which the point P always touches ; and thus if the conic section VRS shall be a hyperbola, the same may fall to the centre. If that conic were an ellipse, that body will always ascend and depart to infinity. And conversely, if some body may emerge from the place V with a velocity, and likewise so that it has began either to descend obliquely to the centre, or to ascend obliquely from that, the figure VRS shall be either a hyperbola or an ellipse, the trajectory can be found either by increasing or diminishing the angle VCP in some given ratio. But also, with the centripetal force changed into a centrifugal force, the body will ascend obliquely in the trajectory VRC which is found by taking the angle VCP proportional to the elliptic sector VRC, and the length CP equal to the length CT as above. All these follow from the preceding proposition, by the quadrature of some curve, the discovery of which, as that may be done readily enough, I omit for the sake of brevity. [Chandrasekhar, circa p.180, is not satisfied with this corollary , as it appears to contain uncorrected misprints, which do not entirely agree with his own derivations, and the diagrams have been altered in the third edition from the first two editions. Whiteside goes to some length to try to remove the confusion. Clearly this is a point in Principia still in need of complete final elucidation.]

PROPOSITION XLLI. PROBLEM XXIX. With the law of the centripetal force given, the motion of the body from a given place is required with a given velocity, arising along a given right line. With the matters remaining in the three preceding propositions : the body may arise from the place I along the line element IK, with that velocity which another body may acquire at D by falling from the place P, acted on by some uniform centripetal force : and this uniform force shall be to the first force by which the first body is acted on at I, as DR to DF. But the body may go towards k ; and with centre C and with the radius Ck it may



describe the circle ke, meeting the right line PD in e, and there may be erected the ordinates eg, ev, ew of the curves BFg, abv, acw. From the given rectangle PDRQ, and with the given law of the centripetal force by which the first body is disturbed, the curved line BFg is given, by the construction of problem XXVII, and Corol. I of this. Then from the given angle CIK there is given

the arising proportion IK, KN is given, and thence, by the given construction of Prob. XXVIII. the quantity Q is given, together with the curved lines abv and acw: and thus, in some completed time Dbve, both the height of the body Ce or Ck is given, together with the the area Dcwe, and the sector XCy is equal to that, and the angle ICk, and the place k at which the body now will be moving. Q. E.I. Moreover we may suppose the centripetal force in these propositions in receding from the centre to be varied in some manner according to some law, as with which able to be imagined at equal distances from the centre to be the same on all sides [i.e. symmetrical]. And up to this stage we have considered the motion of bodies in immoveable orbits. So that there remains a few things we may add concerning the motion of bodies in orbits, which are rotating around a centre of force. [This is the complete problem, given with the starting conditions of the body. The above integrations, which were indefinite, are now made definite.]


Book I Section IX. Translated and Annotated by Ian Bruce. Page 248

SECTION IX.

Concerning the motion of bodies in moving orbits, and from that the motion of the apses.

PROPOSITION XLIII. PROBLEM XXX.

It is required to bring about, that a body shall be able to move in a trajectory that likewise rotates about the centre of forces, and another body to remain [moving] in the

same stationary orbit. In the orbit VPK given stationary, the body P may be revolving by going from V towards K. From the centre C there may always be drawn Cp, which shall be equal to CP itself, and the angle VCp may be put in place proportional to the angle VCP; and the area, that the line Cp will describe, will be to the area PCP, that the line CP will describe likewise, as the velocity of the line describing Cp to the velocity of the line describing CP; that is, thus so that the angle VCp [will be] in a given ratio to the angle VCP, and therefore proportional to the time. Since the area that the line Cp will describe in the motionless plane shall be proportional to the time, it is evident that the body, urged by a centripetal force of the right size, together with the point p may be able to revolve in that curved line as the same point p now may be described in the fixed plane in the account established. The angle VCu is made equal to the angle PCp, and the Cu equal to the line CV, and the figure uCp equal to the figure VCP, and the body always present at p will be moving on the perimeter of the revolving figure uCp, and in the same time describes an arc of this up by which the body P can describe another arc similar and equal to VP in the stationary figure VPK. Therefore the centripetal force may be sought, by Corollary V of Proposition VI, by which a body may be able to revolve on that curved line that the point p will describe in the fixed plane, and the problem will be solved. Q. E.F. [A number of ideas are to be presented here in succession, that perhaps need to be examined in more detail; Newton's understanding of a physical system is set out in the continued evolution of the positions, velocities, etc., of the bodies involved in time; indeed time is the customary free variable in Newton's differential calculus. Here he considers initially a stationary orbit, the curve traced out by a point attracted towards the focus by some force; any point on which line can be described using polar coordinates ( )r,θ , with the focus C as origin; such a body sweeps out equal areas in equal times. Another version of the same orbit is desired, that rotates about the first stationary orbit; since it is similar to the first orbit at any instant, the only difference is a rotation in the



coordinates, and hence the corresponding point may be given by the coordinates ( )r,αθ , for some value of α ; in this way points in the stationary orbit are imaged onto points in the other rotating orbit : thus, the vertex V becomes u, a general point P becomes p, while the polar angle is increased in the ratio α , all at the same instant of time ; the identities involving the areas of the elliptic sectors correspond to the 'equal areas in equal times' rule for Keplerian orbits (assumed to apply in this case, which we now know to be so from the conservation of angular momentum), are related by a constant α > 1 for an anti-clockwise motion as defined in the diagrams, and clockwise for α < 1; clearly in the moving case the body in the moving orbit sweeps out a larger area in the same time than the body in the stationary orbit.]

PROPOSITION XLIV. THEOREM XIV. The difference of the forces, by which a body in a stationary orbit, and another body in a revolving orbit are able to be moving equally, is in the triplicate ratio [i.e. cube] of the common inverse altitude. Take similar and equal parts vp, pk of the revolving orbit with the parts VP and PK of the stationary orbit; and the separation of the points P and K may be understood to be the smallest [i.e. increments]. [At this point we note in the diagram, that ; CV Cv CP Cp;= = while the angle between a stationary orbit line and its rotating image line are rotated through the same angle, so that

VCv PCp∠ = ∠ , etc. ] Send the perpendicular kr from the point k to the right line pC, and produce the same to m, so that mr shall be to kr as the angle VCp to the angle VCP. [Note that the infinitesimal rk is the perpendicular from k to pC, and hence also is the perpendicular distance of K from PC, and by hyp. kC KC= , so that K and k lie on a circle with centre C; however, in the time tΔ in which P has advanced to K, and p would have advanced to k in the other orbit – if stationary, the rotating orbit has moved forwards to a new position m, following the angle magnification α , according to which rm rkα= . This becomes clear if we understand that at any instant, a point p on the moving orbit also at that instant is rotating in a circle with radius



Cp and centre C; thus, if we could 'freeze' the position of the body in the rotating orbit, we would be left with the pure circular motion of the body at p, which would present itself after the time increment at n ; thus the length mn corresponds to the incremental extra distance due to the effect of the motion of the body in the rotating ellipse, which Newton explains in detail. Another point that emerges, concerning the angle amplification factor α , is the area increase or decrease ratio per unit time : for if h is the rate of change of the area in the stationary orbit, so that the increment in the area swept out by the body in the time increment tΔ is h tΔ , then likewise in the rotating orbit, the area swept out per unit time is h', then the ratio of the areas swept out in the two orbits is h' t

h tΔΔ α= , which is the ratio

of the areas of the elliptic sectors in the case of elliptic orbits.] Because the altitudes of the bodies PC and pC, KC and kC always shall be equal, it is evident that the increments or decrements of the lines PC and pC always shall be equal, and thus if at the present places of the bodies P and p, the individual motions may be separated (by Corol. 2 of the Laws.) into two [motions] : these towards the centre, or along the lines PC and pC may be determined, and the others which shall be transverse to the former, and may have a direction perpendicular to the lines PC and pC themselves may also be determined. The motions towards the centres will be equal [as CP Cp= ], and the transverse motions of the body p will be to the transverse motion of the body P, as the angular motion of the line pC to the angular motion of the line PC, that is, as the angle VCp to the angle VCP. Therefore, in the same time in which the body P arrives at the point K by its own motion in both directions, the body p will be moved by an equal amount towards the centre from p towards C equally, and thus in that completed time it will be found somewhere on the line mkr, which is perpendicular to the line pC through the point k . From the transverse motion, it will acquire a distance from the line pC, which shall be to the distance that the other body P acquires from the line PC, as the transverse motion of the body p is to the transverse motion of the other body P. Whereby since kr shall be equal to the distance that the body P acquires from the line PC, and mr shall be to kr as the angle VCp to the angle VCP, that is, as the transverse motion of the body p to the transverse motion of the body P, it is evident that the body p will be found in that completed time at the place m. Thus these [motions] themselves will be in place when the bodies p and P are moving equably along the lines pC and PC, and thus the bodies are acted on by equal forces along these lines, [i.e. if the force on the body p in the mobile ellipse is equal to the force on the other body P in the stationary ellipse, due to the equal distances.] But if the angle pCn may be taken to the angle pCk as the angle VCp is to the angle VCP, and thus nC equals kC, and the body p in that time completed may actually be found at n . Thus [since this is not the case] it [ i.e. p] shall be acted on by a greater force than the body P, but only if the angle pCn is greater than the angle pCk, that is if the orbit upk either is advancing, or is moving with a greater speed that shall be double of that by which the line CP is carried in regression ; and with a smaller force if it is moving slower in the orbit. And the difference of the forces, so that the interval of the places mn, through which that body p from the action of this [extra force], in that given interval of time, must be



transferred. With the centre C and radius Cn or Ck, a circle may be described cutting the lines mr and ms product to s and t, and the rectangle mn mt× will be described equal to the rectangle mk ms× , and thus mn equals mk ms

mt× . But since the triangles pCk and pCn

may be given in magnitude at a given time, kr and mr and so their sum and difference mk and ms are inversely as the altitude pC [these triangles correspond to the areas swept out by the body in the mobile trajectory relative to that frame, and the corresponding area swept out by the body in the mobile trajectory relative to the stationary frame, corresponding to the their respective lengths by the common altitude Cp], and thus the rectangle mk ms× is inversely as the square of the altitude pC. And mt is directly as 1

2 mt , that is, as the altitude pC. These are the first ratios of the nascent lengths [Note that Newton calls differentials arising in pending integrations nascent, or the smallest of magnitudes coming into being; while the differentials associated with differentiation are called evanescent, or vanishing ones.]; and hence mk ms

mt× shall be, that is, the total increment arising mn, proportional to the

inverse difference of the forces, as the cube of the altitude pC. Q. E. D. [Thus, an extra force in addition to the central force responsible for the stationary trajectory, and varying as the inverse cube of the distance from the focus, is introduced in these circumstances, to produce the rotating orbit. For an analytical solution see, e.g. Whittaker's Analytical Dynamics, on the solvable problems of particle dynamics, p.83 ; and also of course Chandrasekhar, circa p.187; Brougham & Routh also treat this problem, p. 88, known as Newton's theorem of revolving orbits. In the following Corollaries we find the ratio of the difference of the forces to a circular force derived from the versed sine is given by mk ms

mt× to

2

2rkkC . Note that

( ) or 1ms mk rm rs rm rk rkα= ± = ± = ± ; Again, rk is the altitude of the elemental sector

pkC, and the area of this can be related to h' tΔ by being equal to 12 pC rk× ; hence

( ) ( )11 hPCms rm rs rk tαα Δ+= + = + = and ( ) ( )11 h

PCmk rm rs rk tαα Δ−= − = − = ;

and therefore ( ) ( )2 22

3

12 22

1hmk ms rk

mt mt PCt

αα Δ

−× = − = , and the result ( )2 2

3

1 22

h

PCmn t

αΔ

−= follows

on taking mt as almost a diameter equal to 2kC. These results imply that the orbits are almost circular, for the centre of curvature is taken as C. This is the total difference in the centripetal forces acting on the bodies in the two orbits, at the same distance from C, and is established in this manner by Chandrasekhar.] Corol. I. Hence the difference of the forces at the places P and p, or K and k, to the force by which the body shall be able to revolve in a circular motion from R to K in the same time in which the body P will describe the arc PK in the stationary arc, is as the line element arising mn, to the versed sine of the arc RK arising, that is as mk ms

mt× to

2

2rkkC , or



as mk ms× to rk squared [as the latter trajectory is circular.]; this is, if the given quantities may be taken F, G in that ratio in turn as the angle VCP may have to the angle VCp, as GG–FF to FF. [These results follow as a simplification above, on taking the ratio G to F as h' to h and in turn equal to α . Again, Chandrasekhar elaborates, and comes to the same conclusion.] And therefore, if from the centre C with some radius CP or Cp a sector of the circle may be described equal to the total area VPC, that the body P has described revolving at some time in the stationary orbit with the radius drawn to the centre of the circle: the difference of the forces, by which the body P in the stationary orbit and the body p in the mobile orbit are revolving, will be to the centripetal force, by which some body, with the radius drawn to the centre, may be able to describe that sector, by which the area VPC will be described uniformly, as GG–FF to FF. And if that sector and area pCk are in turn as the times in which they are described. [If we start from Kepler's first and second laws, then we can assume a stationary ellipse of the form : 1l

r ecosϕ= + . Now for an ellipse, the semi-latus rectum is given by 2b

al = the radial acceleration is given by 2r rϕ− , while the angular momentum or area is

changing at a constant rate : 212

dAdtr hϕ = = . The radial acceleration is readily shown to

be 2

2hlr

r = − ; this result is needed for the next Corollary.]

Corol.2. If the orbit VPK shall be an ellipse having the focus C and the higher apse V; and similar and equal to that there may be put the ellipse upk, thus so that pC equals PC always, and the angle VCp shall be in the given ratio G to F to the angle VCP ; but for the height PC or pC there may be written A, and for the ellipse 2R may be put in place for the latus rectum: the force will be, by which the body can revolve in the rotating ellipse, as 3

RGG RFFFFAA A

−+ and conversely.

[Note that R has been absorbed into the proportionality, as the force should be : 2 22

2 31 RG RFFR A A

−⎡ ⎤+⎣ ⎦ .]

For the force by which the body may revolve in the stationary ellipse may be set out by the amount FF

AA [from Prop. XI, Section III], and the force at V will be 2FFCA

. But the

force by which the body may be able to revolve in a circle at the distance CV with that velocity that the body may have in the ellipse at V, is to the force by which the body revolving in the ellipse may be acted on from the apse V, as half of the latus rectum of the ellipse to the semi diameter of the circle CV, and thus there arises 3

RFFCV

: and the force

which shall be to that as GG–FF as FF, becomes 3RGG RFF

CV− : and this force (by Corol. I.

of this section) is the difference of the forces at V by which the body P in the stationary ellipse VPK, and the body p in the moving ellipse upk are revolving. From which since



(by this proposition) that difference at some other height A shall be to the that written height CV as 3

1A

to 31

CV, the same difference 3

RGG RFFCV− will prevail at any height A.

Therefore to the force FFAA , by which the body can revolve in the stationary ellipse VPK,

the extra force must be added 3RGG RFF

A− ; and the total force may be put together

3RGG RFFFF

AA A−+ , by which the body shall be able to rotate in the moving ellipse upk.

[It appears that someone misread the diagram at some stage, for Whiteside refers to the vertex of the mobile ellipse as v from Newton's original notes, and it appears as such in the first 1687 edition, but is taken as u in the two later editions of the Principia.] Corol.3. It may be deduced in the same manner, if the stationary orbit VPK shall be an ellipse having the centre of forces at C; and to this there may be put in place the similar, equal, and concentric mobile ellipse upk; and let 2R be the principle latus rectum of this ellipse, and 2T transverse side or the major, and the angle VCp always shall be to the angle VCP as G to F; the forces, by which the bodies in the stationary ellipse and in the mobile ellipse are able to revolve in equal times, will be as 3 3 3 and RGG RFFFFA FFA

T T A−+

respectively. Corol. 4. And generally, if the maximum height CV of some body may be called T, and the radius of curvature that the orbit VPK has at V, that is the radius of the circle equal to the curvature, may be called R, and the centripetal force, by which the body can revolve in some stationary trajectory VPK at the place V, may be called

2

2F

RT,

[all the editions of the Principia have the nonsensical 2

2F VT

here, which both Whiteside

and Chandrasekhar discretely point out as being in error], and for the other mobile ellipses may be called indefinitely X at the positions P, with the altitude CP called A, and G to F may be taken in the given ratio of the angle VCp to the angle VCP: the centripetal force will be as the sum of the forces

2 2

3VRG VRF

AX −+ , by

which the body can move circularly in the same motion in the same trajectory upk in the same times with the same motions. Corol. 5. Therefore in some given motion of the body in the stationary orbit, the motion of this about the centre of forces can be increased or decreased in a given ratio, and thence new stationary orbits can be found in which bodies may be rotated by new centripetal forces.



Corol. 6. Therefore a perpendicular VP of indefinite length is erected to the given right line in place CV, and Cp may be drawn equal to CP itself, making the angle VCp, which shall be in a given ratio to the angle VCP; the force by which the body can revolve on that curve Vpk that the point p always touches, [or traces out] will be reciprocally as the cube of the altitude Cp. For the body P by the force of inertia, with no other force acting, can progress uniformly along the line VP. A force may be added at the centre C, inversely proportional to the cube of the altitude CP or Cp, and (by the present demonstration) that rectilinear motion will be turned away into the curved line Vpk. But this curve Vpk is that same curve as that curve VPQ found in Corol. 3. Prop. XLI , in which we have said there bodies attracted by forces of this kind ascend obliquely.

PROPOSITION XLV. PROBLEM XXXI.

The motions of the apses are required of orbits that are especially close to circles. The problem is required to be solved by using arithmetic because the orbit, that the body revolving in the mobile ellipse (as in Corol. 2. or 3. of the above proposition) will describe in the stationary place, may approach to the form of this orbit of which the apses are required, and by seeking the apses of the orbit that the body will describe in that stationary plane. But the orbits may acquire the same form, if the centripetal forces by which they are described, gathered together, may return proportionals with equal altitudes. Let the point V be the upper apse, and there may be written, T for the maximum altitude CV, A for some other altitude CP or Cp, and X for the difference of the altitudes CV– CP; and the force, by which the body is moving in an ellipse about its focus C (as in Corol. 2.), and which in Corol. 2. was as

2 22

2 3RG RFF

A A−+ , that is as

2 2 2

3F A RG RF

A+ − , by

substituting T X− for A, will be as 2 2 2 2

3RG RF TF F X

A− + − . Any other centripetal force may

be reduced similarly to a fraction whose denominator shall be 3A and the numerators, made from the gathering of the homogeneous terms, may be put in place in an analogous manner. The matter will be apparent from examples. Example 1. We may put the centripetal force to be uniform [in the two cases, some rotating apse u is compared with the upper apse V], and thus the force to be as

3

3AA

, or (by

writing T X− for the height A in the numerator), as 3 2 2 3

33 3T T X TX X

A− + − ;



[The use of T X− is just a computational convenience for the distance A, which is to be expanded out in the numerator of

3

3AA

, while the denominator is taken as constant, so that

small differences from the maximum value T can be found easily; here : T denotes the max. height of the body in the stationary ellipse; A denotes the altitude CP or Cp at some time; and X denotes the differenceCP – CV A T= − , which can be made infinitesimally small.] and with the corresponding terms of the numerators gathered together [for this particular arrangement in comparison with the general above], without doubt 'the given' with 'the given' and the 'not given' with 'not given', the expression becomes

2 2 2RG RF TF− + to 3T ; as 2F X− to 2 2 33 3T X TX X− + − or, as 2F− to 2 23 3T TX X− + − .

Now since the orbit may be put as especially close to being circular, the orbit may begin with a circle and on account of R and T made equal, and X diminished indefinitely, the final ratios become RG2 to T3 as 2F− to 23T− , or G2 to T2 as F2 to 3T2, and in turn G2

to F2 as T2 to 3T2, that is, as 1 to 3; and thus G to F, that is the angle VCp is to the angle VCP, as 1 to 3 . [i.e.

2 2

2 23=G F

T T or

2 2

2 2133

=G TF T

= . Also, the rate of change of the respective areas is CVph'

h CVPα ∠∠= = .]

Therefore since the body in the stationary ellipse, by descending from the highest apse to the lowest may make the angle VCP (thus as I may say) 180 degrees ; the other body in the mobile ellipse, and thus in the stationary orbit on which we have acted, from the upper apse to the lower apse on descending may make an angle VCp of 180

3 that therefore

on account of the similitude of which orbit, that the body will describe with a uniform centripetal force acting, and will be described performed in the plane at rest by rotating in the revolving ellipse. Through the above collection of the similar terms these orbits are returned, not universally but on that occasion since they may approach especially to the circular form. Therefore a body revolving with a uniform centripetal force in an almost circular orbit, between the upper and lower apses may always make an angle of 180

3degrees, or 103 gr. 55 m. 23 sec. to the centre ; arriving from the upper apse to the

lower apse once this angle has been made, and thence returning to the upper apse when again it has made the same angle ; and thus henceforth indefinitely. Example 2. We may put the centripetal force 3nA − to be as some power of the altitude A or as 3

nAA

where 3n − and n indicate some whole numbers or fractional indices of the

powers, either rational or irrational, positive or negative. That number nA or nT X− may be reduced into a converging infinite series, and the series



1 22 etcn n nnn nT nXT XXT .− −−− + emerges. And with the terms of this collected together

with the terms of the other numerator RGG RFF TFF FFX− + − , there shall be RGG RFF TFF− + to nT as FF− to 1 2

2 etcn nnn nnXT XXT .− −−− + And with the final

ratios taken when the orbits approach to a circular form, there shall be RGG to nT as – FF to 1nnXT −− , or GG to 1nT − as FF to 1nnT − , and in turn GG to FF as 1nT − to 1nnT − ; that is as I to n; and thus G to F, that is the angle VCp to the angle VCP, as 1 to n . Whereby the angle VCP, in the falling of the body performed in the ellipse from the upper apse to the lower apse, shall be 180 degrees; the angle VCp will be made, in the descent of the body from the upper apse to the lower apse, that the body will describe in an almost circular orbit proportional to some centripetal force of the power 3nA − , equal to an angle of 180

n; and with this angle repeated the body will return from the lower apse to the

higher apse, and thus henceforth indefinitely. So that if the centripetal force shall be as the distance of the body from the centre, that is, as A or

4

3AA

, n will be equal to 4 and n

equals 2; and thus the angle between the upper apse and the lower apse equals 1802 or

90 degrees. Therefore with a fourth part of one revolution completed the body will arrive at the lower apse, and with another quarter part at the upper apse, and thus henceforth in turn indefinitely. That which also had been shown from proposition X : for the body may be revolving in the stationary ellipse, acted on by this centripetal force, whose centre is at the centre of the forces. But if the centripetal force shall be reciprocally as the distance, that is directly as 1

A or 2

3AA

, n will be equal to 2, and thus between the upper apse and the

lower apse the angle will be 1802

degrees, or 127 degrees, 16m., 45 sec. and therefore

will be revolving by such a force, always by the repetition of this angle, in turns with the others, from the upper apse to the lower apse, and eternally may arrive at the lower apse from. Again if the centripetal force shall be reciprocally as the square root of the square root of the eleventh power of the altitude, that is inversely as

114A , and thus directly as

114

1

A, or as

143

AA

n will be equal to 14 and 180

n degrees equals 360 degrees and therefore the

body descending from the upper apse and thus perpetually descending, it will arrive at the lower apse when it will have completed a whole revolution, then by completing another revolution by ascending always, it will return to the upper apse : and thus eternally in turn. Example 3. We may take m and n for any indices of the powers of the altitude, and taking b and c for any given numbers, we may put the centripetal force to be as

3 m nbA cAA+ , that is as 3

m n

b T X c T XA

× − + × − , or (by the same method of our converging

series) as



1 1 2 22 2

3 + etc.m n m n m nmm m nn nbT cT mbXT ncXT bXXT cXXT

A

− − − −− −+ − − +

and with the terms of the numerator gathered together, it becomes RGG RFF TFF− + to

m nbT cT+ , as –FF to 1 1 2 2

2 2 + etc.m n m nmm m nn nmbT ncT bXT cXT− − − −− −− − + And on taking the final ratios which arise, here the orbits may approach to a circular form , let G2

be to 1 1m nbT cT− −+ , as F2 to 1 1m nmbT ncT− −+ , and in turn G2 to F2 as

1 1m nbT cT− −+ to 1 1m nmbT ncT− −+ . Which proportion, by setting the maximum altitude CP or T

arithmetically to one, shall be G2 to F2 as b c+ to mb+nc, and thus as 1 to mb ncb c++ . From

which G shall be to F, which shall be the angle VCp to the angle VCP, as 1 to mb ncb c++ .

And therefore since the angle VCP between the upper apse and the lower apse of the stationary ellipse shall be 180 degrees, the angle VCp between the same apses, in the orbit that the body described by a centripetal force proportional to the quantity

m nbA cAA cub.+ ,

is equal to an angle of 180 b cmb nc

++ degrees. And by the same argument if the centripetal

force shall be as 3

m nbA cAA− , an angle of 180 b c

mb nc−− may be found between the apses. Nor

will problems be resolved otherwise in difficult cases. A quantity, to which the centripetal force is proportional, must always be resolved into a converging series having the denominator A3. Then the given part of the numerator which arises from that operation to the other part of this which is not given, and the part given of the numerator of this 2 2 2 2RG RF TF F X− + − to the other part of this not given, are to be put in the same ratio: And by deleting the superfluous quantities, and on writing one for T, the proportion G to F will be obtained. Corol. I. Hence if the centripetal force shall be as some power of the altitude, it is possible to find that power from the motion of the apses; and conversely. Without doubt if the whole of the angular motion, by which a body returns to the same apse, shall be to the angular motion of one revolution, or of 360 degrees, as some number m to another number n, and the altitude may be called A : the force will be as that power of the altitude

22 3n

mA−

, the index of which is 22 3n

m− . That which has been shown by the second

example. From which it is clear that that force, in receding from the centre, cannot decrease in a ratio greater than the cube of the altitude. A body acted on by such a force revolving and descending from that apse, if it begins to descent, at no time will arrive at the lower apse or the minimum altitude, but will descend as far as the centre, describing that curved line which we have discussed in Corol. 3. Prop. XII. But if that body should have started from the descending [i.e. lower] apse, or to ascend minimally, it will ascend indefinitely, nor at any time will it arrive at the upper apse. For it describes that curved line which have been discussed in the same Corol. and in Corol. VI. of Prop. XLIV. And thus where the force, in receding from the centre, decreases in a ratio greater than the cube of the altitude, the body descending from the lower apse, likewise as it may have



started either to descend or ascend, descends to the centre or ascend to infinity. But if the force, in receding from the centre either may decrease to be less than the cube of the altitude, or increase in some ratio of the altitude ; the body at no time may descend as far as the centre, but will arrive at some lower apse: and conversely, if a body is ascending and descending from one apse to the other in turn, then at no time will it be called to the centre; the force in receding from the centre either will be greater, or may decrease in a ratio less than the cube of the altitude: and from which the body will have returned quicker from one apse to the other, there the further the ratio of the forces recedes from that cubic ratio. So that if the body may be returned in revolving either 8 , 4 , 2 or

121 times from the upper apse to the upper apse by ascending and descending ; that is, if m

to n were as 8, 4, 2 or 121 to 1, and thus 3nn

mm − may give rise to 1

64 3− , 116 3− , 1

4 3− , or 49 3− : the force will be as

164 3A − ,

116 3A − ,

14 3A − , or

49 3A − , that is,

reciprocally as 1

64 3A − ,1

163A − ,143A − or

493A − . If the body in individual rotations may have

returned to the same stationary apse; there will be m to n as 1 to 1, and thus 3nnmmA − equals

2A − or 21A

; and therefore the decrease of the forces are in the square ratio of the

altitudes, as has been shown in the preceding. If the body may return to the same apse in the parts of a revolution, either in three quarters, or two thirds, or one third, or in one quarter ; m to n will be as 1

4 or 23 or 1

3 or 14 to 1, and thus 3nn

mmA − equals either 169 3A − ,

94 3A − , 9 3A − , or 16 3A − ; and therefore the force varies reciprocally either as

119A ,

34A , or

directly as 6A , 13A . And then if the body on progressing from the upper apse to the same upper apse has completed a whole number of revolutions, and three degrees beyond, and therefore that apse by a single revolution of the body will have been completed as a consequence in three degrees; m to n will be as 363 degrees to 360 degrees or as 121 to I20, and thus the force 3nn

mmA − will be equal to 2952314641A− ; and therefore

the centripetal force varies reciprocally as 2952314641A or reciprocally as

42432A approximately.

Therefore the centripetal force decreases in a ratio a little greater than the square, but which in turn approaches 3

459 closer to the square than to the cube. Corol. 2. Hence also if the body, by a centripetal force which shall be reciprocally as the square of the altitude, may be rotating in an ellipse having a focus at the centre of forces, and to this centripetal force there may be added or taken away some other external force ; the motion of the apses can be known (by example three) which may arise from that external force: and conversely. So that if the force by which the body shall be revolving in an ellipse shall be as 2

1A

and the external force taken away shall be as cA, and thus the

force remaining shall be as 4A cA

A cub.− ; there will be (in the third example) b equals 1, m

equals 1, and n equals 4, and thus the angle of rotation between the apses equals an angle of 1

1 4180 cc

−− . We may put that external force to be 357.45 less in parts than the other



force by which the body is rotating in the ellipse, that is c becomes 10035745 , with A or T

present equaling 1, and 11 4180 c

c−− may become 35645

35345180 , or 180.7623, that is, 180 degrees, 45m. 44sec. Therefore the body descending from the upper apse, by moving through an angle of 180 degrees, 45m. 44sec , will reach the lower apse, and with this motion doubled it will return to the upper apse : and thus the upper apse by progressing will make 1 degree. 31 m. 28 sec. in individual rotations The apses of the moon progress around twice as fast. Up until now we have been concerned with the motion of bodies in orbits of which the planes pass through the centre of forces. It remains that the motion of bodies also may be determined in eccentric planes. For the writers who treat the motion of weights, are accustomed to consider the oblique ascent and descent of weights in any planes given, as well as straight up and down: and equally to judge the motions of bodies with any forces for whatever centres desired, and of the dependence on eccentric planes this comes to be considered. Moreover we may suppose the planes to be the most polished and completely slippery lest they may retard bodies. Certainly, in these demonstrations, on whatever of these planes on which in turn the bodies touch by pressing on, we may take the planes parallel to these, in which the centres of the bodies are moving and the orbits they may describe by moving. And at once we may determine by the same law the motions of bodies carried out on curves surfaces.


Book I Section X. Translated and Annotated by Ian Bruce. Page 267

SECTION X.

Concerning the motion of bodies on given surfaces, and from that the repeating motions of string pendulums.

PROPOSITION XLVI. PROBLEM XXXII.

With some general kind of centripetal force in place, and with both a given centre of forces as well as some plane on which the body is revolving, and with the quadratures of curvilinear figures granted : the motion of a body is required starting out along a right line in that given plane from some given place, and with some given velocity,. S shall be the centre of forces, SC the shortest distance of this centre from a given plane, P the body setting out from some place P along the right line PZ, Q the same body revolving in its trajectory, and PQR that trajectory described in the given plane, that it is required to find. CQ and QS are joined, and if on QS, SV may be taken proportional to the centripetal force by which the body is drawn towards the centre S, and VT may be drawn which shall be parallel to CQ and meeting SC in T. The force SV may be resolved (by Corol 2. of the laws) into the forces ST, TV; of which ST by acting on the body along a line perpendicular to the plane, will not change the motion of that body in this plane. But the other force TV, by acting the position of the plane, draws the body directly towards the point C in the given plane, and likewise it comes about, so that the body may be moving in this plane in the same manner, and if the force ST may be removed, and the body may be revolving about the centre C in free space acted on by the force TV alone. Moreover with the given centripetal force TV by which the body Q is revolving in free space about the given centre C, then both the trajectory PQR is given (by Prop. XLII.) that the body will describe, the place Q, in which the body will be rotating at some given time, as well as the velocity of the body at that place Q; and conversely. Q E.1. [We should bear in mind here, that if the body travels in an elliptic orbit in a gravitational field, then the length CS cannot remain constant, and when the body is at the maximum distance Q from C, it has risen to its greatest height, being lowest at the minimum distance, assuming the length of the string remains unchanged. Thus Kepler's criterion of the motion of the body always being in the same plane is not satisfied, and hence one cannot use the Kepler criterion of equal areas in equal times ; or, there is an unbalanced torque acting which changes the angular momentum during the course of the orbit. Thus, Newton's Proposition XLVI relates to a zero gravity situation, or, as Newton states, C is the only centre of force present. Hence we are not looking at a conical pendulum, which only applies to circular motions.]



PROPOSITION XLVII. THEOREM XV.

Because a centripetal force may be put in place proportional to the distance of the body from the centre; all bodies in any planes revolving in some manner describe ellipses, and the ellipses are performed in equal times ; and those moving on right lines, and also running to and fro, may complete the individual coming and going motions in the same periods of time. For, with which things in place from the above proposition, the force SV, by which the body Q rotating in some plane PQR is drawn towards the centre S, is as the distance SQ; and thus on account of the proportionals SV and SQ , TV and CQ the force TV, by which the body is drawn towards the point C given in the plane of the orbit, is as the distance CQ. Therefore the forces, by which bodies turning in the plane PQR are drawn towards the point C, on account of the distances, are equal to the forces by which any bodies are drawn in some manner towards the centre S; and therefore bodies are moving in the same times, in the same figures, in some plane PQR about the point Q and in the free space about the centre S; and thus (by Corol. 2, Prop. X. and Corol. 2, Prop. XXXVIII.) in equal times always they describe either ellipses in that plane about the centre C, or besides they will furnish periods by moving to and fro in right lines through the centre C drawn in that plane. Q.E.D.

Scholium. The ascent and descent of bodies on curved surfaces are related to these. Consider curved lines described in a plane, then to be revolved around some axis given passing through the centre of the forces, and from that revolution to describe curved surfaces ; then bodies thus can move so that the centres of these may always be found on these surfaces. If bodies by ascending and descending these obliquely besides can run to and fro, the motions of these will be carried out in planes crossing the axis, and thus on curved lines, by the rotation of which these curves surfaces have arisen. Therefore for these it will suffice to consider the motion in these curved line cases. [ The two following propositions are handled by similar reasoning, on separate diagrams, in what follows. Newton calls all his curves cycloids or epicycloids (the evolute or epicycloid of any cycloid is a similar equal figure with its cusps translated through half the arc of the original curve). According to Proctor, in his interesting book: A Treatise on Cycloids, (1878), which touches on some of the material in this sections, the best way to define such curves is as follows : The epicycloid/hypocycloid is the curve traced out by a point on the circumference of a circle which rolls without sliding on a fixed circle in the same plane, the rolling circle touching the outside/inside of the fixed circle. Different values of the two radii give rise to different curves, some of which are well-known. Full descriptions of such curves can be found, e.g. in the CRC Handbook of Mathematics, and of course on the web.



We are interested in particular in the geometric method used by Newton in finding a geometric relation for such a curve, where he puts in place a finite figure derived from the geometry available, tangents, diameters, etc., and from this he constructs a similar figure composed of infinitesimal lengths both from linear and curvilinear increments, the vanishing ratio of which, for some chosen lengths, is equal to a fixed ratio in the macroscopic figure. Thus a differentiation has been performed, or fluxion found. I have added some labels in red to Newton's diagram, to make reading a little easier; however, if you wish, you can look at the unadulterated diagram in the Latin section.]

PROPOSITION XLVIII. THEOREM XVI.

If a wheel may stand at right angles on the outside of a sphere, and in the manner of rotation it may progress in a great circle ; the length of the curvilinear path, that some given point on the perimeter of the wheel made, from where it touched the sphere, (and which it is usual to call a cycloid or epicycloid) will be to double the versed sine of half the arc which it made in contact going between in this total time, as the sum of the diameters of the sphere and the wheel, to the radius of the sphere.

PROPOSITION XLIX. THEOREM XVII. If a wheel may stand at right angles on the concave inside of a sphere and by rotating it may progress on a great circle; the length of the curved path that some given point on the perimeter of the wheel made, from where it touched the sphere during this total time, will be to twice the versed sine of half the arc which it made in contact going between in the whole time, as the difference of the diameters of the sphere and the wheel to the radius of the sphere.



ABL shall be the sphere, C the centre of this, BPV the wheel resting on this, E the centre of the sphere, B the point of contact, and P the given point on the perimeter of the wheel. Consider that the wheel to go on a great circle ABL from A through B towards L, and thus to rotate going between so that the arcs AB and PB themselves in turn will always be equal, and that point P given on the perimeter of the wheel meanwhile describes the curvilinear path AP. Moreover AP shall have described the whole curvilinear path from where the wheel touched the sphere at A, and the length AP of this

path to twice the versed sine of the arc 12 PB , shall be as 2CE to CB. For the right line CE

(produced if there is a need) meets the wheel at V, and CP, BP, EP and VP may be joined, and the normal VF may be sent to CP produced. PH and VH may touch the circle at P and V meeting at H, and PH may cut VF in G, and the normals GI and HK may be sent to VP. Likewise from C and with some radius the circle onm may be drawn cutting the right line CP in n, the perimeter of the wheel BP in o, and the curved path AP in m; and with centre V and with the radius Vo a circle may be described cutting VP produced at q. Because the wheel, by always moving is rotating about the point of contact B, it is evident that the right line BP is perpendicular to that curved line AP that the point of



rotation P has described, and thus so that the right line VP may touch this curve at the point P. The radius of the [arc of the] circle nom , gradually increased or diminished is equal finally to the distance CP; and, because of the similarity of the vanishing figure Pnomq and the figure PFGVI, the final ratio of the vanishing line elements Pm, Pn, Po, Pq, that is, the ratio of the momentary changes of the curve AP, of the right line CP, of the circular arc BP, and of the right line VP, will be the same as of the lines PV, PF, PG, PI respectively. But since VF shall be perpendicular to CF and likewise VH to CV, and the angles HVG and VCF therefore equal ; and the angle VHG (on account of the right angles of the quadrilateral HVEP at V and P ) is equal to the angle CEP, the triangles VHG and CEP are similar; and thence it comes about that EP to CE thus as HG to HV or HP and thus as KI to KP, and on adding together or separately, as CB to CE thus PI to PK, and on doubling in the following as CB to 2CE thus PI to PV, and thus Pq to Pm.

[i.e. HG HGEP KIEC HV HP KP= = = (HV and HP are the common tangents from H) then

1 1 or EP[ EB ] CBKI PIEC KP EC PK,= + = + = ; hence 2

PqCB PIEC PV Pm= = . ]

Therefore the decrement of the line VP, that is, the increment of the line BV–VP to the increment of the curve AP is in the given ratio CB to 2CE, and therefore (by the Corol. of Lem. IV.) the lengths BV–VP and AP, arising from these increments, are in the same ratio. But, with the interval BV present, PV is the cosine of the angle BVP or 1

2 BEP , and thus BV–VP is the versed sine of the same angle ; and therefore in this wheel, the radius of which is 1

2 BV , BV–VP will be twice the versed sine of the arc 12 BP . Therefore AP is

to twice the versed sine of the arc 12 BP as 2CE to CB.

[ We may write this proportionality in the form : ( )( )

( )( ) 2 2; giving d PV d BV VP CB CB

CE CEd AP d AP BV VP AP− −= = − = × ; hence the arc length of the

rectifiable curve ( ) ( ) ( )2 2 1 22 21 2 1CE CE R

CB CB RAP BV VP BV cos BEP sinψρ ρ+= × − = × − = × − ,

starting from A, as required (see Whiteside, note 275 Vol. VI, for the outline of a comparable, but far more complicated, analytical derivation).] But we will call the line AP the cycloid outside the sphere in the first proposition and the cycloid inside the sphere in the following for the sake of distinction. Corol. I. Hence if the whole cycloid ASL is described and that may be bisected at S, the length of the part PS to the length VP (which is twice the sine of the angle VBP, with the radius EB present) is as 2CE to CB, and thus in the given ratio. Corol. 2. And the length of the semi-perimeter of the cycloid AS will be equal to the right line which is to the diameter of the wheel BV as 2CE to CB.



PROPOSITION L. PROBLEM XXXIII.

To arrange it so that the body of the pendulum may swing in a given cycloid. Within a sphere QVS, described from the C, the cycloid QRS may be given bisected in R and with its end points Q and S hence meeting the spherical surface there. CR may be drawn bisecting the arc QS in O, and that may be produced to A, so that CA shall be to CO as CO to CR. [Thus, if we let R CO= , and the radii of the two generating circles be given by 2 2r AO & ORρ= = the above terms introduced for the radii, this becomes

22

R r RR R ρ+

−= , in turn giving 222

rR R

ρρ−= ,

22; =R r R Rr

r R rρ ρ++= and 2 2

2r R r

r R rρ+ +

+= .]

With centre C and radius CA, the external sphere DAF may be described, and within this sphere by the rotation of a wheel, the diameter of which shall be AO, two semi-cycloids AQ and AS may be described, which touch the interior sphere at Q and S and meet the external sphere at A. From that point A, by a thread AR equaling the length APT, the body Q may be suspended and thus may swing between the semi-cycloids AQ and AS, so that as often as the pendulum is moving away from the perpendicular AR, with the upper part AP applied to that semi-cycloid APS towards which the motion is directed, and around that it may be wrapped as an obstacle, and with the remaining part PT to which the semi-cycloid has not yet got in the way, it may stretch out in a straight line; and the weight T is swinging on the given cycloid QRS. Q.E.F. [The cycloid and the lower evolute cycloid obey the normal/tangent to normal relation at any points T and P on this pair of curves. We have included the generating circles in red, not present in the original figure.] For the thread PT first may meet the cycloid QRS at T, and then the circle QOS at V, and CV may be drawn; and the perpendiculars BP and TW may be erected to the right line part of the thread PT from the end points P and T, crossing the right line CV in B and W. It is apparent, from the construction, and from the similar figures AS and SR arising, that these perpendiculars PB and TW cut off from CV the lengths VB and VW of the wheels with diameters equal to OA and OR. Therefore TP is to VP (twice the sine of the angle VBP multiplied by the radius 1

2 BV present [Both Cohen and Whiteside have misunderstood this point in their translations: you cannot equate a length to the sine of an angle.]) as BW to BV, or AO OR+ to AO, that is (since CA to CO, CO to CR, and AO to OR separately shall be proportionals) as CA CO+ to CA, or, if BV may be bisected in E, as 2CE to CB.



[ 2 2 22 22 2; or, rBW BW AO OR CA CO CETP R r

VP BV BV AO r CA R r CBρ++ + +

+= = = = = = , as above.] Hence (by Corol I. Prop. XLIX.) the length of the part of the right line of the thread PT is equal always to the arc PS of the cycloid, and the whole length [of the thread] APT is always equal to the arc APS of half the cycloid, that is (by Corol. 2. Prop. XLIX.), to the length AR. And therefore in turn if the thread always remains equal to the length AR, the point T will always be moving on the given cycloid QRS. Q.E.D. Corol. The thread AR is equal to the semi-cycloid arc AS, and thus has the same ratio to the radius of the external sphere AC as that similar semi-cycloid SR has to the radius of the internal radius CO. [ SRAR

AC CO= ] .

PROPOSITION LI. THEOREM XVIII. If a centripetal force acting in any direction towards the centre C of the sphere shall be as the distance of this place from the centre, and the body T may be oscillating by this force acting alone (in the manner described just now) on the perimeter of the cycloid QRS: I say that any whatever of the unequal oscillations are completed in equal intervals of time. For the perpendicular CX may fall on the tangent TW of the cycloid produced indefinitely and CT may be joined. Because the centripetal force by which the body T is impelled towards C is as the distance CT, and this (by Corol. 2 of the laws) is resolved into the parts CX and TX, of which CX by impelling the body directly from P stretches the thread PT and by the resistance of this it may cease to act completely, producing no other effect; but the other part TX, by acting on the body transversely or towards X, directly accelerates the motion of this on the cycloid; clearly because the acceleration of the body, proportional to this accelerating force, shall be as the length TX at individual instants, that is, on account of CV and WV given and TX and TW proportional to these [for we have the similar triangles VTW and CXW and CVTX

TW VW= ], as the length TW, that is (by Corol. I, Prop. XLIX.) as the length of the arc of the cycloid TR. Therefore with the two pendulums APT and Apt unequally drawn from the perpendicular AR and sent off at the same time, the accelerations of these always will be as the arcs to be described TR and tR. But the parts described from the start are as the accelerations, that is, as the whole [arcs] to be described at the start, and thereupon the parts which remain to be described and the subsequent accelerations, from these proportional parts, also are as the total [parts to be described subsequently] ; and thus henceforth. Therefore both the accelerations and the velocities arising from the parts and the parts requiring to be described, are as the whole



[arc remaining]; and thus the parts requiring to be described obey the given ratio, and to likewise vanish in turn, that is, the two oscillating bodies arrive at the perpendicular AR at the same time [i.e. both bodies arrive at R with the same speed and in the same time; a hall-mark of in-phase simple harmonic motion, where the period is independent of the amplitude]. Whenever in turn the ascents of the pendulums from the lowest place R, through the same cycloidal arcs made in a backwards motion, may be retarded separately by the same forces by which they were accelerated in the descent, it is apparent that the velocities of ascent and descent made through the same arcs are equal and thus for the times to become equal ; and therefore, since both parts of the cycloid RS and RQ lying on either side of the perpendicular shall be equal and similar, the two pendulums always complete their oscillations at the same times for a whole as well as for a half [oscillation]. Q.E.D. Corol. The force by which the body T is accelerated or retarded at some place T on the cycloid, is to the whole weight of the same body in the place with the greatest altitude S or Q, as the arc TR of the cycloid to the arc of the same SR or QR. [

Digression. Note from earlier, we have shown that an arc length ( )4

21RRAP sinψρ

ρ+= × − , where the

angle ψ is subtended by the contact chord at the centre of the generating circle, this is also the angle between the tangent and the chord. We indicate here the common cycloid inverted with the angle ψ now as customarily shown – twice the above, rolling on the upper

horizontal line, for which, with a generating circle of radius a, the coordinates are ( ) ( )2 2 ; 1 2x a sin y a cosψ ψ ψ= + = − . The gradient at some point on the curve is tanψ ,

and it is readily shown that the intrinsic equation of this curve is 4s a sinψ= , taking 0s = when 0ψ = . It is seen that the added complication of rotating the generating circle

on or in another circle of greater radius R to produce an epicycloid changes the constant 4a in the arc length formula to become 4aR

R as sinψ±= × in our definition of the angle; and the formula depends on where the origin has been chosen. Thus the length of a whole section of a simple cycloid from vertex to trough is 4a, with a similar formula for the epicycloid. We may consider such formulas in general to be of the form s k sinψ= . Note that a body P on a string acting as a pendulum drawn towards the point C, between the cusps of such a cycloid, of total length k, may have part of the string of length ( )1s k sinψ= − wrapped round the curve, while the remainder of length s k sinψ= is free, with corresponding results for Newton's epicycloids. We are interested



in the velocities of points along and perpendicular to both curves : it is clear that the components of the velocity and acceleration of the point P along the tangent of the Newton's upper cycloid are equal and opposite to the components of the velocity and acceleration of the point T along the normal of the lower curve.]

PROPOSITION LII. PROBLEM XXXIV.

To define both the velocities of pendulums at individual places and the times in which both whole oscillations as well as individual parts of oscillations may be completed. With some centre G, and with the radius GH equal to the arc of the cycloid RS, describe the semicircle HKM bisected by the radius GK. [At this point we have to imagine the sphere HKM, of which we see a portion, to be endowed with an abstract absolute force field of a special kind, so that a body L has the same force acting on it as the body T, the abstract force acting along the radius GH (so performing pure S.H.M. with the simplest possible geometry), while the other acts along

the tangent at T on the cycloid; these centripetal forces are also equal on the periphery MKH of the circle and on the sphere SOQ. The idea being that both bodies will execute S.H.M. ; in addition, the lengths GK, LI, and YZ represent the velocities of a body released from H towards G. Such a situation might arise for a particle that could pass through a hypothetical uniform earth without hindrance, such as a mass dropped through a hole passing all the way through a diameter of the earth, affected only by gravity, which in this case varies directly as the distance from centre.] And if a centripetal force proportional to the distances of the places from the centre, may tend towards the centre G, and let that force on the perimeter HIK be equal to the centripetal force on the perimeter of the sphere QOS tending towards the centre of this; and in the same time in which the pendulum T may be sent off from the highest place S, some body L may fall from H to G, because the forces by which the bodies may be acted on are equal from the beginning and with the intervals described TR and LG always proportional, and thus, if TR and LG may be equal at the places T and L; it is apparent



that the bodies describe the equal intervals ST and HL from the beginning, and thus from that at once to be urged on to progress equally , and to describe equal intervals. Whereby (by Prop. XXXVIII.) the time in which the body will describe the arc ST is to the time of one oscillation, as the arc HI, the time in which the body H may arrive at L, is to the semi-perimeter HKM, the time in which the body H may arrive at M.

[Thus : arcST arcHI

SQ HKM

T TT T= ; ( )

( )( )( )

T L

R G

d HL / dt d ST / dtv vv v d HG / dt d SR / dt= = = .]

And the velocity of the body of the pendulum at the place T it to the velocity of this at the lowest place R, (that is, the velocity of the body H at the place L to the velocity of this at the place G, or the instantaneous increment of the line HL to the instantaneous increment of the line HG, with the arcs HI and HK increasing by equal fluxes) as the applied

ordinate LI to the radius GK, or as 2 2SR TR− to SR. [ We have already considered the accelerations along the tangent PT . The free straight length PT is equal to the arc PS at any instant, and the arc described by the body P , or RT can be given by s k sinψ= , choosing the angle ψ as above, and we may take the velocity along the curve to be s k cosψ= , while the acceleration along the orbit is s k sin ksψ= − = − . This can be written in terms of the tangential velocity v as :

( )2 212 0

d v ksdvds dsv ks

++ = = , and hence ( )2 2 21 1

2 2constantv ks kS+ = = , since the velocity of

P is zero when the arc s S= ; we now regard first integration as the conservation of

energy equation. Hence, ( )2 2dsdtv k S s LI . k= = − = ; thus, the velocity at the point T

has been found. The time to travel from S to s, is given by the indefinite integral:

( ) ( ) ( )2 2 21 1 1

1ds du s

Sk k kS s ut arccos

− −= = =∫ ∫ , where limits can be applied as needed,

and sSu = ; (see Whiteside's note 286). Note that in the abstract force diagram, the radius

GH S= and we can write s S cos kt= , in which case the angle ktθ = , and we can identify k as the angular frequency, from which the period of the oscillation is given by 2

kT π= , where we recall that 4rR

R rk += ; note especially that the period is independent of

the amplitude. Thus we have Newton's results; note also that the part of the string wrapped round the cycloid arc behaves as a store of gravitational potential energy, for as the weight falls, more kinetic energy is fed into the system by the weight being allowed to fall further, and vice versa when it rises, than by the weight moving in a circular arc; and again, the centre of forces is at a finite distance C, and so we do not have a uniform gravitational field.] From which, since in the unequal oscillations, the arcs of the whole oscillations may be described in equal times proportional to the whole arcs of the oscillations; from the given



times, both the velocities and the arcs may be had, to be described in all the oscillations. Which were to be found first. Now bodies hanging from strings may be swinging in diverse cycloids described

between different spheres, of which the absolute forces are different also [i.e. those abstract forces giving rise to an S.H.M. above that do not specify the mechanism of the force]: and, if the absolute force of some sphere QOS may be called V, the accelerating force by which the pendulum is urged on the circumference of this sphere, where it begins to be moving directly towards the centre of this sphere, will be as the distance of that hanging body from that centre and the absolute force jointly on the sphere, that is, as CO V× [i.e. the original absolute force corresponding to CO is magnified by some factor V; now we have ( )dv k. ds HY dt CO Vdt= − = = − × ]. And thus the incremental line HY; which shall be as this accelerating force CO V× , described in the given time ; and if the normal YZ is erected to the circumference crossing at Z, the nascent arc HZ will denote that given time . But this nascent arc HZ is as GH HY× , [from similar triangles involving increments, HY:ZH :: ZH:MH , or 2 2 2ZH Sds GH .YH= = as the arc tends to zero] and thus the arc varies as GH CO V× × . [As dt d GH CO Vα θ α × × ]. From which the time of a whole oscillation in the cycloid QRS (since it shall be as the semi-periphery HKM, which [angle] may denote the time for that whole oscillation directly ; and as the arc HZ directly, which similarly may denote the given time inversely) shall be as GH directly and as GH CO V× × inversely, that is, on account of the equal

quantities GH and SR, as SRCO V× or (by the Coral. Prop. L.) as AR

AC V× .

[Thus, GH GH SR ARQRS CO V CO V AC VGH CO V

T α × × ×× ×= = = , since GH SR= ,

and SRARAC CO= ]

And thus the oscillations on the spheres and with all the cycloids, made with whatever absolute forces, are in a ratio composed directly from the square root ratio of the lengths of the string, and inversely in the square root ratio of the distances between the point of



suspension and centre of the sphere, and also inversely in the square root ratio of the force of the sphere. Q.E.D. Corol. 1. Hence also the times of the oscillations, of the falling and of the revolutions of the bodies can be compared among themselves. For if for the wheel, by which the cycloid will be described between the spheres, diameter may be put in place equal to the radius of the sphere, the cycloid becomes a right line passing through the centre of the sphere, and the oscillation now will be a descent and accent on this right line. From which both the descent time from some place to the centre, as well as the time for this equally by which a body may describe the quadrant of an arc by revolving uniformly about the centre of the sphere at some distance. For this time (by the second case) is to the time of a semi-oscillation on some cycloid QRS as I to AR

AC . Corol. 2. Hence also these propositions lead to what Wren and Huygens had found concerning the common cycloid. For if the diameter of the sphere may be increased indefinitely: the surface of this will be changed into a plane, and the centripetal force will act uniformly along lines perpendicular to this plane, and our cycloid will change into a cycloid of the common kind. But in this case the length of the arc of the cycloid, between that plane and the describing point, will emerge equal to four times the versed sine of half the arc between the plane and the point of the wheel describing the same ; as Wren found: And the pendulum between two cycloids of this kind will be oscillating in a similar and equal cycloid in equal times, as Huygens demonstrated. And also the time of descent of the weight, in the time of one oscillation, this will be as Huygens indicated. But the propositions from our demonstrations are adapted to the true constitution of the earth , just as wheels describe cycloids outside the sphere by going in great circles by the motion of nails fixed in the perimeters, and pendulums suspended lower in mines and caverns, must oscillate in cycloids within spheres, so that all the oscillations become isochronous. For gravity (as we will be teaching in the third book) decreases in progressing from the surface of the earth, upwards indeed in the square ratio of the distances from the centre of the earth, downwards truly in a simple ratio.



PROPOSITION LIII. PROBLEM XXXV.

With the quadratures of the curvilinear figures granted, to find the forces by which bodies on the given curves may perform isochronous oscillations. The body T may be oscillating on some curved line STRQ, the axis of which shall be AR passing through the centre of forces C. The line TX may be drawn which may touch the curve at any place of the body T, and on this tangent TX there may be taken TY equal to the arc TR. For the length of that arc will be known from the quadrature of the figure by common methods. [Thus, Newton's criterion for isochronous motion is that the acceleration at T is in proportion to the length of the arc TR]. From the point Y there may be drawn the right line YZ perpendicular to the tangent. CT may be crossing that perpendicular in Z, and the centripetal force [parallel to AT] shall be proportional to the right line TZ. Q E.I. For if the force, by which the body is drawn from T towards C, may be represented by the right line TZ taken proportional to this, this may be resolved into the forces TY, YZ; of which YZ by drawing the body along the length of the thread PT, no motion of this changes, but the other force TY directly either accelerates or decelerates the motion of this body on the curve STRQ. Hence, since this path TR requiring to be described shall always be as the accelerations or retardations of the body to be described in the proportional parts of two oscillations, which shall always be as these parts (of the greater and lesser), and therefore [these accelerations and decelerations] may be made as these parts likewise may be described, [as in the modern equivalent S.H.M. view, the

acceleration is proportional to the negative displacement; note that Newton considers an oscillation or swing to be a single motion clockwise or anticlockwise.] But bodies which at the same time always describe the proportional parts of the whole, likewise describe the whole. Q E.D. Corol. 1. Hence if the body T, hanging by the rectilinear thread AT from the centre A, may describe the circular arc STRQ, [we must assume the angle TAN is small, so that isochronous motion occurs for this simple pendulum] and meanwhile it may be urged downwards along parallel lines in turn by a certain force, which shall be to the uniform force of gravity, as the arc TR to the sine of this TN: the times of the

individual oscillations shall be equal. And indeed on account of the parallel lines TZ, AR,



the triangles ATN and ZTY are similar ; and therefore TZ will be to AT as TY to TN; that is, if the uniform force of gravity may be proposed by the given length AT; the force TZ, by which the isochronous oscillations may be produced, will be to the force of gravity AT, as the arc TR ,itself equal to TY, to the sine TN of that arc. [Note that the isochronous accelerating force is proportional to the arc TR, which is almost equal to the semi-chord TN for small oscillations, while AT and ZY are almost vertically downwards, with the tension AT as the weight of the pendulum bob; hence

= weight of bobTZ ATTY TN restoring forceα ]

Corol. 2. And therefore in clocks, if the forces impressed on the pendulum by the machinery to preserve the motion thus since with the force of gravity may be compared, so that the total force downwards always shall be as the line arising by dividing the multiple of the arc TR [ TY= ] and the radius AR[ AT= ] by the sine TN, all the oscillations will be isochronous.

PROPOSITION LIV. PROBLEM XXXVI. With the quadrature of the curvilinear figures granted, to find the times, by which bodies acted on by some centripetal force on some curved lines, described in a plane passing through the centre of forces, may descend or ascend. The body may descend from some place S, by a certain curve STtR in the plane passing through the given centre of forces C. Now CS may be joined and that divided into an innumerable number of equal parts, and Dd shall be some of these parts. With centre C and with the radii CD, Cd circles may be described, DT, dt, crossing the curved line STtR in T and t. And then from the given law of the centripetal force, and from the given height CS by which the body has fallen, the velocity of the body will be given at some other height CT (by Prop. XXXIX.). [We are to consider a body to slide along the given curve without friction from rest at S, under the action of a radial force f(r) acting along CT, at the point T, so that the velocity at the distance r is given by along the curve in the

line element Tt shall be

12

2 ( )r

R

v f r dr⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦∫ ,

whatever the shape of the curve. Hence, this [energy] integral has to be evaluated to obtain the velocity at T. Consequently, the time to arrive at T is given by a second



integration, and dsv cos tTCdt = , where the component of the radial velocity down the slope

is taken.] But the time, in which the body will describe the line element Tt, is as the length of that element, that is, directly as the secant of that angle tTC ; and inversely as the velocity. The applied ordinate DN perpendicular to the right line CS through the point D shall be proportional to this time, and on account of Dd given, the rectangle Dd DN× , that is the area DNnd, will be in the same proportion to the time. Therefore if PNn shall be that curved line that the point N always touches, and the asymptote of which shall be the right line SQ, standing perpendicularly on the right line CS: the area SQPND will be proportional to the time in which the body by falling has described the line; and therefore from that area found the time will given. Q.E.I.

PROPOSITION LV. THEOREM XIX.

If a body may be moving on some curved surface, the axis of which passes through the centre of the forces, and a perpendicular is sent from the body to the axis, and from some point a line parallel and equal to this line is drawn: I say that parallel line will describe an area proportional to the time. BKL shall be the curved surface, T the body revolving on that, STR the trajectory, that the body will describe on the same, S the start of the trajectory, OMK the axis of the curved surface, TN the right line perpendicular to the axis from the body, OP drawn equal and parallel to this from the point O, which is given on the axis ; AP the track of the trajectory described by the point P by the winding of OP in the plane AOP ; A corresponding to the start of the trace from the point S ; TC a right line drawn from the body to the centre ; TG the proportional part of the centripetal force TC, by which the body is urged to the centre C; TM a right line perpendicular to the curved surface; TI the proportional part of this pressing force, by which the body may be acted on in turn by the surface towards M; PTF a line passing through the body parallel to the axis, and GF and IH parallel right lines sent perpendicularly from the points G and I on that parallel line PRTF. Now I say, that the area AOP, described from the start of the motion by the radius OP, shall be proportional to the time. For if the force TG (by Corol. 2. of the laws) is resolved into the forces TF and FG; and the force TI into the forces TH and HI: But the forces TF and TH acting along the line PF perpendicular to the plane AOP in as much as they only change the motion of the body perpendicular to this plane. And thus in as much as the motion of this is body is made along the position of this plane; that is, the motion of the point P, by which the trace of the trajectory AP is described in this plane, is the same as if the forces TF and TR may be removed, and the body is being acted on by the forces FG and HI only ; and that is, likewise if the body in the plane AOP may describe the curve



AP, by a centripetal force tending towards the centre O and equal to the sum of the forces FG and HI,. But such a force will describe the area AOP (by Prop. I.) proportional to the time. Q E.D. Corol. By the same argument if the body, acted on forces tending towards two or more centres on the same given right line CO, may describe in free space some curved line ST; the area AOP always becomes proportional to the time.

PROPOSITION LVI. PROBLEM XXXVII.

With the quadrature of the curvilinear figure given, and with the law of the centripetal force tending towards the centre given, as well as the curved surface whose axis passes through that centre ;the trajectory is required to be found that the body describes on that same surface, advancing from a given place with a given velocity in a given direction on the surface. With everything in place which have been constructed in the above proposition, the body T may emerge from a given place S following a given right line in place in a trajectory required to be found STR, the trace of which in the plane BDO [D is called L in the previous diagram, and is used here in the original text] shall be AP. And from the given velocity of the body at the height SC, the velocity of this will be given at some other height TC. Since with that velocity, in the shortest time given, the body may describe a small part of its trajectory Tt, and let Pp be the trace of this described in the plane AOP. Op may be joined, and with the centre T of a small circle with the radius Tt of the trace described on the curved surface, in the plane AOP the ellipse pQ shall be described. And on account of the given circle with magnitude Tt, and the given distance TN or PO of this from the axis CO, that ellipse pQ will be given in kind and magnitude, and so in place according to the right line PO. And since the area POp shall be proportional to the time, and thus the angle POp may be given from the given time. And thence the common intersection p of the ellipse and of the right line OP will be given, together with the angle OPp in which the trace of the trajectory APp cuts the line OP. Hence truly (on bringing together Prop. XLI. with its Corol. 2.) an account of determining the curve APp may be readily apparent. Moreover from the individual points P of the trace, by raising perpendiculars PT to the plane AOP of the surface of the curve meeting in Q, the individual points T of the trajectory will be given. Q.E.I.


Book I Section XI. Translated and Annotated by Ian Bruce. Page 295

SECTION XI. Concerning the motion of bodies with centripetal forces mutually attracting each other.

Up to the present, I have explained the motion of bodies attracted to a fixed centre of force, yet scarcely such a force is extant in the nature of things. For the attractions are accustomed to be for bodies ; and the actions of pulling and attracting are always mutual and equal, by the third law: thus, if there shall be two bodies, so that neither shall it be possible to be attracting or to be attracted and to be at rest, but both shall be rotating around the common centre of gravity (by the fourth corollary of the laws), as if by mutual attraction : and if there shall be several bodies, which either may be attracted by a single body, and which likewise they may attract, or all may mutually attract each other; thus these must be moving among themselves, so that the common centre of gravity may be at rest, or may be moving uniformly in a direction. From which reason I now go on to explain the motion of bodies mutually attracting each other, by considering the centripetal forces as attractions, although perhaps more truly they may be called impulses, if we may speak physically. For now we may turn to mathematics, and therefore, with the physical arguments dismissed, we use familiar speech, by which we shall be able to be understood more clearly by mathematical readers. [The stand adopted by Newton has been explained in detail by Cohen in the introduction to his translation; essentially the physical world is to be understood from mathematical laws and considerations, and forces such as gravity, acting at a distance with no visible means of communicating forces, are to be understood by the mathematical relations they satisfy, rather than from some physical model involving unseen fluids and vortices, as in Descartes' model.]

PROPOSITION LVII. THEOREM XX. Two bodies attracting each other in turn describe similar figures, both around the

common centre of gravity and mutually around each other. For the distances of the bodies from the common centre of gravity are inversely proportional to the bodies [Newton means masses of bodies when he refers to bodies]; and thus in a given ratio one to the other, and on being put together in a given ratio to the total distance between the bodies. But these distances are carried around their common end [i.e. the centre of mass or gravity] by an equal angular motion, so that they do not therefore change their mutual inclination, always lying on a line. But right lines, which are reciprocally in a given ratio, and which are carried around their ends by an equal angular motion, describe completely similar figures about these same ends in planes, which together with these ends either are at rest, or may be moving in some non angular motion [i.e. the linear motion of the centre of mass.] Hence these figures which are described from the distances being turned through are similar. Q.E.D. [The analytical solutions of the two – body problem now presented in this section are solved to some extent in modern texts on dynamics, and Chandrasekhar gives proofs in



his work on Newton, from p. 207, onwards. Chandrasekhar thus solves the dynamical problem in the reference frame of a body P, and finds that the original force is augmented, for what Newton states depends on the bodies being in an inertial reference frame in which the forces and accelerations remain the same (thus, a kinematic change of viewpoint can be adopted) ; what we find in non-inertial frames, of course, is not surprising, as the rotating body P below is itself accelerating, and so 'fictitious' forces must be added to P to accommodate the correct motion, if we are to consider S as an inertial frame : for S's frame likewise is an accelerating reference; Chandrasekhar goes on to state that, ' Newton's proof, again couched in words, it is essentially the same....' . Thus, it seems appropriate to add some variables in red to Newton's diagrams that follow, and to indicate briefly the mathematical origins of the statements made and not demonstrated fully by Newton. We may note in modern terms that the left-hand diagram (a) refers either to an inertial frame in which the centre of mass of the system is at rest or moving uniformly in a straight line (note that Newton refrains from talking about straight lines, and discusses only directions, because otherwise it begs the question of the first law of motion, as discussed in the definitions and axioms of this work), while the right-hand diagram (b) considers s as a reference frame at rest. So we begin....]

PROPOSITION LVIII. THEOREM XXI. If two bodies attract each other by forces of some kind, and meanwhile they rotate about

their common centre of gravity: I say of the figure, which the bodies describe around each other mutually by moving thus, that it shall be similar and equal to the figure,

around either body at rest, to be described by the same forces. The bodies S and P are revolving around the common centre of gravity C, by going from S to T; and from P to Q . From a given point s, sp and sq may always be drawn equal and parallel to SP and TQ themselves ; and the curve pqv, that the point p will describe by revolving around the fixed point s, will be similar and equal to the curves, which the bodies S and P describe around each other mutually: and hence (by Theorem XX) similar to the curves ST and PQV,

which the same bodies will describe around the common centre of gravity C: and that because the proportions of the lines SC, CP, and SP or sp may be given in turn.



[Note from (a) that the orbits of the conic sections S and P generally are of differing sizes (here we will concentrate on ellipses), and the mass m1 in the case shown is greater than the mass m2 : if the force is proportional to the distance from the centre, then the two orbits will be concentric ellipses with the centre at the centre of gravity. In the inverse square case, the centre of mass is a common focus, the ellipses may intersect, but the bodies themselves must always be at opposite ends of a line such as TQ, intersecting both orbits. In both cases, the areas swept out by a radius are proportional to the times. In (b), according to C, the orbit of p is similar to that of P, but viewed from the stationary reference frame S, and a similar remark can be made about the orbit of s relative to p. If O is an arbitrary origin in (a), then the positions of S and P are given by

1 2 and R r R r+ + , where we consider 1 2 and OC R, CS r , CP r= = = as the vectors

shown; and the centre of gravity or mass is given by 1 1 2 2

1 2

m r m rm mR ++= . If the centre of

gravity is at rest, then 0R = and 1 1 2 20 m r m r= + , dispensing with vector notation, as the bodies lie on a straight line. We also have 1 1 2 20 m r m r= + , on differentiating again, so that the forces acting on S and P are equal and opposite, as required by the third law. If we choose C to be the origin, then we can write 1 1 2 2m r m r= , dispensing with signs : a useful relation. The force between the masses meanwhile may be represented by F, and this is a symmetric function of the masses, and depends on the distance between them. Hence we may write 1 1 2 2 and F m r F m r= = − , acting along the line joining SP in an attractive manner, positive going from left to right, so that S is urged forwards by a positive force, while P is urged backwards by an equal and opposite negative force at the instant shown. We also observe that though the forces are equal and opposite, the acceleration of the body varies inversely with its mass, and thus the larger mass accelerates less and produces the smaller orbit than the smaller mass. Now if we wish to consider the motion of P relative to S, then the relative displacement can be written as

2 1r r− , and once more dispensing with vectors, we have

the acceleration of P relative to S given by 2 1r r− , and this can be written as

1 2

2 1 1 22 1= = m mF F

SP sp m m m ma a r r F+− = − − = − , acting towards S. Hence, the force acting on P

relative to S in (a), or as p relative to s in (b), which is seen to be reduced to the previous immoveable cases treated, is given by ( ) 1 2

12 2 1= = m m

PS ps mF F m r r F+− = − towards s : that

is, the force F for the motion relative to C, has been augmented by the factor 1 2

2

m mm+ for

the motion of s relative to p. In a similar manner, we can write ( ) 1 2

21 1 2= m m

SP mF m r r F+− = , which is the augmented force on S due to P, now considered

at rest. Notice that these forces on P and p in diagrams (a) and (b) are no longer equal to each other, the one is given by 2 2 CPF m r F= − = and the other by 1 2

1= m m

ps mF F+− ; hence



the ratio of the accelerations on the same mass P is given by 1

1 2= mPC

m mps

aa + ; while the

nascent distances are given in the ratio ( )( )( )

21 1

21 2 2

= m t CPspm m t

RQrq

Δ

Δ+= , and the velocities are in

the ratio ( )( )( )

1 1

1 2 2= m tP

m m tp

vv

ΔΔ+

, where we consider the forces to act for different times.

Following Newton's arguments presented in case 1 below, due to the similar figures, we have : 2

1 2

rRQ CPrq sp r r+= = ; hence if we had the forces in this ratio [corresponding to

forces proportional to distance from the centre of the ellipses], then we would have : 2

1 2

CP

sp

F r CP CPF r r SP sp+= = = acting for the same small interval of time tΔ , giving rise to the

same centripetal accelerations, either 22

or pPCP sp

vv , from which = p

Pspv

v CP , giving rise to the

similar curves pqv and PQV, and the revolutions would be completed in the same time. But according to Newton, the forces are considered to be the same in each case, [we may presume that he means in the centre of mass inertial frame], so that the accelerations produced on P and p are equal, and the distances through which the body is drawn inwards is RQ in one case, and rq in the other, where rq RQ> , and hence the same acceleration must act for a longer interval to produce these nascent or infinitesimal displacements QR and qr ; thus we have, since ( ) ( )2 21 1

1 22 2and QR a t qr a tΔ Δ= = ,

then 11 2

2 1 2 1 2

mQR t rCPt r rqr sp m m

ΔΔ + +

= = = = . In this case, we have the similar curves pqv and

PQV produced, but not in the same times; as in this case, the time for one body to orbit the other t is not the same as the time T for both bodies to orbit around the centre of mass C. We can continue to apply the dynamic analysis to the situation with modified forces, and retaining Newton's insightful way of dealing with the nascent distances, to consider the ratio of the forces acting on P and p, which is the same as the ratio of the accelerations, augmented as discussed above : 1

2

CPP

p ps

a tvv a t

ΔΔ= , where

( )( )

( )( )( )

2 21 1 11

2 21 2 2 1 2 2

and CPCP

ps ps

a t m ta m QR CPa m m qr ps a t m m t

Δ Δ

Δ Δ+ += = = = and thus giving ( )1 2 1

21;m m CP t

tm psΔΔ

+= and

( )( )

21 2 11 1 1 2

22 1 2 1 2 1 21 1 2

= = = ;CPP

p ps

m m QR ma tv m m rv a t m m m m r rm qr m m

ΔΔ

++ + ++

= = × which Newton deduces below,

and thus it seems that Newton was using this formulation of the problem after all. ]



Case 1. That common centre of gravity C, by the fourth corollary of the laws, either is at rest or moving uniformly in a direction. We may put that initially to be at rest, and the two bodies may be located at s and p, with the immobile body at s, the mobile one at p, with the bodies S and P similar and equal to the bodies s and p. Then the right lines PR and pr may touch the curves PQ and pq in P and p, and CQ and sq may be produced to R and r. And on account of the similitude of the figures CPRQ and sprq, RQ will be to rq as CP to sp, and thus in a given ratio [indeed, 2

1 2

rRQ CPrq sp r r+= = ]. Hence if the force, by which

the body P is attracted towards the body S, and thus towards the intermediate centre C, should be in that same given ratio to the force, by which the body p is attracted towards the centre s [i.e. the above ratio 2

1 2

CP

sp

F rF r r+= ] ; these forces in equal times always attract

the bodies from the tangents PR and Pr to the arcs PQ and pq by intervals proportional to RQ and rq themselves, and thus the latter force effects, that the body p may rotate in the curve pqv, which shall be similar to the curve PQV, in which the former force effects that the body P may be revolving; and the revolutions may be completed in the same time. But since these forces are not inversely in the ratio CP to sp, rather (on account of the similitude and equality of the bodies S and s, P and p, and the equality of the distances SP and sp) but mutually equal to each other [as discussed above]; the bodies will be drawn equally in equal times from the tangents: and therefore, so that the latter body p may be drawn by the greater interval rq, a greater time is required, and that in the square root ratio of the intervals; therefore (by the tenth lemma) because the distances have been described in the squares ratio of the times from the beginning of the motion itself. Therefore we may put the velocity of the body p to be as the velocity of the body P in the square root ratio of the distance sp to the distance CP, thus so that in the times, which shall be in the same square root ratio, the arcs pq and PQ may be described, which are in a whole [ordinary] ratio : And the bodies P and p always attracted by equal forces describe the similar figures PQV, pqv, around the centres C and s at rest, of which the latter pqv is similar and equal to the figure, that the body P will describe about the mobile centre S. Q.E.D. Case 2. Now we may consider that the common centre of gravity, together with the distance in which the bodies may be moving among themselves, is progressing uniformly along a direction; and (by the law of the sixth corollary) all the motions may advance in this space as before, and thus the bodies describe the same figures around each other as at first, and therefore to the similar and equal figure pqv. Q.E.D. Corol. I. Hence two bodies with the forces proportional to their distances attracting each other mutually, (by Prop. X.) describe concentric ellipses both around the common centre of gravity and around each other, and vice versa, if such figures are described, then the forces are proportional to the distances.



Corol. 2. And two bodies, with forces inversely proportional to the square of the distance, describe (by Prop. XI. XII. XIII.) both around the common centre of gravity and about each other, conic sections having the focus at the centre, around which the figures are described. And conversely, if such figures are described, the centripetal forces are reciprocally as the squares of the distance. Corol. 3. Any two bodies rotating about the common centre of gravity, with the radii drawn both to that centre and between themselves, describe areas proportional to the times.

PROPOSITION LIX. THEOREM XXII. The periodic time of two bodies S and P, rotating about their common centre of gravity C, is to the periodic time of rotation of either of the bodies P, rotating about the other stationary body S, and with the figures, which the bodies describe mutually around each other, similar and equal to the figure described, as the square root ratio of [the mass of] the other body S, to the sum of the [masses of the] bodies S P+ . Indeed, from the demonstration of the above proposition, the times, in which any similar arcs PQ and pq are described, are in the square root ratio of the distances CP and SP or sp, that is, in the square root ratio of the body S to the sum of the bodies S P+ . And on adding together, the sum of all the times in which all the similar arcs PQ and pq are describes, that is, the total time, in which the whole similar figures may be described, are in the same square root ratio. Q.E.D. [See notes above. ]

PROPOSITION LX. THEOREM XXIII. If two bodies S and P, with forces inversely proportional to the square of their distance,

mutually attract each other about the common centre of gravity: I say that ellipse, which one of the bodies P will describe about the other ellipse in this motion S, the principal

axis is to that principal axis, which the same body P will describe around the other body S at rest, in the same periodic time, shall be as the sum of the two bodies S P+ to the first

of the two mean proportionals between this sum and that other body S. For if the ellipses were to be described equal to each other, the periodic times (by the above theorem) would be in the square root ratio of the body S to the sum of the bodies S P+ . The periodic time of the latter ellipse may be diminished in this ratio, and the periodic times may become; [i.e. and 1S

P S

St tT P S T

,+

+ ×= = ] ; but the principal axis of the

ellipse (by Prop. XV.) will be diminished in a ratio, which is in the three on two ratio of this, that is in a ratio which is the triplicate of S to S P+ [ For 2 2 2 2 3 3 3 3: and : : whereby by Kepler III, : = : T t S P : S T t A X , A X S P S= + = + ; now if two mean proportional B and C are taken between S P+ and S, then S P CB

B C S+ = =



hence S P+ will be in the triplicate ratio to B, that is ( )3 3: :S P S S P B+ = + , and hence

( )33 3 3: :A X S P B= + , and thus ( ) ( ): : or : :A X S P B X A B S P= + = + ]; and thus the principal axis will be to the principal axis of the other ellipse, as the first of the two mean proportionals between S P+ and S to S P+ . And conversely, the principal axis of the ellipse described about the mobile body will be to the principal axis described about the stationary ellipse, as S P+ to the first of the two mean proportions between S P+ and S. Q.E.D.

PROPOSITION LXI. THEOREM XXIV.

If two bodies may be mutually attracted by any forces, and neither disturbed nor impeded by any others, in whatever manner they may be moving ; the motions of these thus may be had, and, as if they may not attract with each other mutually, but each may be attracted by the same forces by some third body established at the common centre of gravity. And the law of the attracting forces will be with respect to the distance of the bodies from that common centre, and with respect to the whole distance between the bodies. For these forces, by which the bodies mutually pull on each other, stretch towards the common intermediate centre of gravity ; and thus they are the same, as if they spring from an intermediate body. Q.E.D. And since the ratio of the distance of whichever body from that common centre to the distance between the bodies may be given, the ratio will be given of any power of one distance to the same power of the other distance; and so that the ratio of whatever quantity, which may be derived from one distance and with whatever quantities given, to another quantity, which from the other distance and from just as many given quantities, and that given ratio of the distances to the former had similarly may be derived. Hence if the force, by which one body is pulled by the other, shall be directly or inversely as the distance of the bodies in turn ; either as any power of this distance ; or finally so that some quantity may be derived in some manner from this given distance and given quantities: the force will be the same, by which the body likewise is drawn to the common centre of gravity, likewise directly or inversely as the distance of attraction from that common centre, either as the same power of this distance, or finally as a quantity similarly derived from this distance and with similar given quantities. That is, the force of attraction will be the same law with respect to each distance. Q.E.D.



PROPOSITION LXII. PROBLEM XXXVIII.

To determine the motion of two bodies, which mutually attract each other with forces inversely proportional to the square of their distances, and which are sent off from given

places. Bodies (by the latest theorem) may be moved in the same way, as if they may be attracted by a third body put in place at the common centre of gravity ; and that centre from its own initial motion by hypothesis remains at rest; and therefore (by Corol. 4 of the laws) will always remain at rest. Therefore the motions of the bodies are required to be determined (by Prob. XXV.) in the same manner as if they may be urged by attracting forces at the centre, and the motions of the bodies mutually attracting each other will be found. Q.E.D.

PROPOSITION LXIII. PROBLEM XXXIX. To determine the motion of two bodies which attract each other with forces inversely proportional to the square of their distance, and with the places given, and the bodies

leave along given straight lines with given velocities. At the beginning with the given motions of the bodies, the uniform motion of the common centre of gravity is given, and so that the motion of the space, which together with this centre is moved uniformly in a direction, without any motion of the bodies with respect to this space. But the subsequent motions (by the fifth corollary of the laws, and the latest theorem) happen in this space in the same way, as if the space itself together with that common centre of gravity were at rest, and the bodies are not attracting each other mutually, but were attracted by a third body situated at that centre. Therefore the motion of either body is to be determined (by Problems 9 and 26) in this moving space, from the place given, along a given right line, with the given departure speed, and acted on by the centripetal force tending towards that centre: and likewise the motion of the other body about the same centre will be known. Since to this motion it is required to add the uniform motion of the space, and in that space the progressive rotational motion of the bodies found above, and the absolute motions of the bodies will be known in immobile space. Q.E.D. [Recall that Newton believes in the existence of a universal rest frame, that of the fixed stars, and relative to which all motions are absolute. Newton now proceeds from the generally soluble two-body problem to the only-soluble n-body problem.]



PROPOSITION LXIV. PROBLEM XL.

The motions of several bodies among themselves are required, with the forces by which the bodies mutually attract each other increasing in a simple ratio from the centres.

At first two bodies T and L, having a common centre of gravity D, may be put in place. These describe ellipses (by the first corollary of Theorem XXI.) having centres at D, the magnitude of which becomes known from Problem X. Now a third body S may attract the first two T and L with the accelerating forces ST and SL, and it may be attracted by these in turn. The force ST (by Corol. 2. of the laws) is resolved into the forces SD and DT; and the force SL into the forces SD and DL. But the forces DT and DL, which are as their sum TL, and thus as the accelerating forces by which the bodies T and L are attracted mutually, add to these the forces of the bodies T and L, each to the other in turn, composing forces proportional to the distances DT and DL, as at first, but with the forces greater than with the former forces; and thus (by Prop. X, Corol. I. and Prop. IV, Corol's.1. and 8) have the effect that these bodies will describe ellipses as before, but with a faster motion. The remaining accelerating forces SD and SD [contributed from the forces ST and SL], from the motive actions and SD T SD L× × , [i.e. in an analytical approach, such forces between the bodies 1 and 2 may be given by a generalised Hooke's law type formula : motive force displacement∝ − , provided there are only two bodies present, where the masse M1 and M2 may be incorporated into the constant of proportionality; or by considering a motive 3force displacementM∝ − × for forces that involve either body 1 or 2 but always body 3, some such scheme Newton has adopted, as this constant of proportionality must change if a third mass M3 is present, unless the masses are equal, etc.] which are as the bodies, by attracting these bodies equally and along the lines TI, LK, themselves parallel to DS, and in turn do not change the situation of these, but act so that they accelerate equally to the line IK; that taken drawn through the middle of the body S, and perpendicular to the line DS. But that access to the line IK may be impeding by arranging so that the system of bodies T and L from one side, and the body S from the other, with the correct velocities, may be rotating around the common centre of gravity C. From such a motion the body S, because with that the sum of the motive forces

and SD T SD L× × , of the proportional distance CS, tends towards the centre C, will describe an ellipse around the same C; and the point D, on account of the proportionals CS, CD, describes a similar ellipse out of the region. But the bodies T and L attracted by the motive forces and SD T SD L× × , the first body by the first force, the second by the second, equally and along the parallel lines TI and LK, as it has been said, proceed to



describe their own ellipses around the moving centre D (by the 5th and 6th corollaries of the laws), as at first. Q.E.D. Now a fourth body V may be added, and by a similar argument it may be concluded that this body and the point C describe ellipses about the common centre of gravity of all the bodies B ; with the former motions of the bodies T, L and S about the centres D and C remaining, but with an acceleration. And by the same method more bodies are allowed to be added. Q.E.D. Thus these may be themselves considered, as if the bodies T and L attract each other with greater or less accelerations than by which the remaining bodies for a [given] ratio of the distances. Let all the mutual accelerative attractions be in turn as the distances by the [masses of the] bodies attracted, and from the proceeding it may be readily deduced that all the bodies describe different ellipses in equal periodic times, around the common centre of gravity B, in a fixed plane. Q.E.D. [On page 217, Chandrasekhar produces the analytical solution that Newton must have worked through, by reducing the motion of each mass to an attraction about the common centre of mass, without any other accelerations needed, and these motions for each body are S.H.M.'s of the form :

acceleration of body total mass of all bodies displacement of body∝− × . Thus the basis is set for treating a three or more body problem where the forces follow an inverse square law, having established the reliability of a method of adding the forces.]

PROPOSITION LXV. THEOREM XXV. Several bodies, the forces of which decrease in the inverse square ratio of the distance

from their common centre of gravity, can move in ellipses amongst themselves ; and with the radii drawn to the focus describe areas almost proportional to the times.

In the above proposition the case has been shown where several bodies are moving forwards precisely in ellipses. From which the more the law of the forces departs from the law put in place there, from that the more the bodies mutually disturb the motions ; nor can it happen, following the law put in place here that bodies by mutually attracting each other, can be moving precisely in ellipses, unless by obeying in turn a certain proportion of the distances. But in the following cases it will not differ much from ellipses. Case: I. Put several smaller bodies to revolve around some large body at various distances from that body, and the bodies are attracted by the same body according to proportional individual absolute forces. And because the common centre of gravity of all is either at rest or may be moving in a direction uniformly (by the fourth Corollary of the laws), we may set the bodies in place to be rather small, so that the large body at no time departs sensibly from this centre : and the large body may be at rest, or moving uniformly



in a direction, without sensible error; but the smaller bodies may be revolving around this large body in ellipses, and with radii drawn to the same, they describe areas proportional to the times ; unless in so far as errors are induced, either by the large body receding from that common centre of gravity, or by the actions of the smaller bodies mutually between each other. But the smaller bodies can be diminished , until this error and the mutual interactions themselves, shall be less than in any given amount ; and thus the orbits can be made to agree with ellipses, and the areas may correspond to the times, with no tangible errors. Q.E.O. Case 2. Now we may put in place a system of smaller bodies revolving around some very large body in the manner now described, or some other system of two bodies revolving around each other progressing uniformly in direction, and meanwhile to be acted on laterally by the force of another by far greater body at a great distance. And because the equal accelerating forces do not act by changing the positions of the bodies in turn among themselves, with the motions of the parts maintained between themselves, but affect the system as a whole, by which the small bodies may be acted on along parallel lines, and may be moved together ; it is clear that, by the attractions from the great body, no change in the motion of the attraction of the bodies between themselves may arise, unless either from the inequality of the attracting accelerations or from the inclination of the lines in turn, along which the attractions happen. Therefore put all the attractive accelerations due to the great body to be inversely as the square of the distances between themselves ; and by increasing the distance of the great body, until the differences of the right lines drawn from that body to the rest of the bodies in respect of the length of these, and the inclinations in turn, may be made smaller than any given amount; then the motions of the parts of the system may persevere among themselves with minimal errors given, which may not be given any smaller. And because, on account of the smallness of the parts of these in turn with the great distance, the whole system is attracted in the manner of a single body ; and will be moved likewise by this attraction in the manner of a single body ; that is, it may describe with its centre of gravity some conic section about the great body (viz. a hyperbola or a parabola from a weaker attraction, an ellipse from a stronger attraction) and with a radius drawn to the great body, areas will be described proportional to the times, without any errors, except those arising from the distances of the parts, reasonably small, and to be minimized as it pleases. Q.E.O. It is permitted to proceed indefinitely to more composite cases. Corol. 1. In the second case, in which the greatest body of all approaches closer to the system of two or more bodies, the motions of the parts of the system among themselves are more disturbed by that ; because now the inclination of the lines drawn from the great body to these in turn is greater, and the greater the inequality of the proportion. Corol.2. Moreover the small bodies will be disturbed the most, on being put in place, in such a way that the attractive accelerations of the parts of the system towards the greatest body of all shall not be inversely in turn as the squares of the distances from that great body ; especially if the inequality of this proportion shall be greater than the



inequality of the proportions of the distances from the great body. For if the accelerative force [of the great body], by acting equally along parallel lines, disturbs nothing in the motion between themselves, it is necessary that a perturbation may arise from the inequality of the action, either it shall be a smaller disturbance for a greater body, or one of greater inequality for a lesser body. The excess of the impulses of the greater body, by acting on some bodies and by not acting on others, by necessity will change the position of these amongst themselves. And this perturbation added to the perturbation, which arises from the inclination and inequality of the lines, may return the whole major perturbation. Corol. 3. From which if the parts of this system may be moving in ellipses or in circles without significant disturbance ; it is evident, that the same bodies on attracting other bodies by accelerative forces, either may not to be acted on unless they are the lightest, or to be acted on equally, and approximately along parallel lines.

PROPOSITION LXVI. THEOREM XXVI. If three bodies, the forces of which decrease in the square ratio of the distances, mutually attract each other; and the accelerative attractions of any two on the third shall be reciprocally as the square of the distances; moreover with the smaller ones revolving around the greatest : I say that the inner of the two bodies revolving about the innermost and greatest body, by the radii drawn to the innermost itself, describes areas more proportional to the times, and a figure of an elliptic form, by having more of the radii meeting at the focus, if the greatest body may be disturbed by these attractions, than it would if that greatest body either was at rest and not attracted by the smaller bodies, or if it were attracted much more or much less, or disturbed much more or much less. It may almost be evident from the demonstration of the second corollary of the foregoing proposition ; but it may be established thus by a more widely compelling and distinct argument. Case 1 The smaller bodies P and S may be revolving in the same plane about the greatest body T, of which P may be described in the inner orbit PAB, and S in the outer orbit ESE. [Note that S may be presumed to be the sun, and is small only because of the great distance, while P is the moon revolving around the earth T in an almost circular orbit; also, point masses are assumed for these bodies.] Let SK be the mean distance of the bodies P and S; and the attractive acceleration of the body P towards S, at that average distance, may be expressed by the same line SK; [Thus, as in all of Newton's dynamics, some lines are geometrical lengths, others are forces or attractive accelerations, and some act as both, we will prefix some line sections by a word or short phrase not present in the original text, containing the word line or attractive acceleration or simply force if there is confusion ; in this case the line SK defines the unit of attractive acceleration, or force per unit mass.]



The ratio of the attractive accelerations SL to SK may be taken in the same square ratio SK to SP, and SL will be the attractive acceleration of the body P towards S at some distance SP.

[i.e. the attractive accelerations on the body P due to S are : 2

2

SL SKSPSK

= , so that when P is

at the average distance, the force SL is equal to the force SK ; if the distance SP < SK , the force SL is greater than the average and if P is more distant, i.e. if SP > SK, then the force SL is less than the average.] Join PT, and the line LM acts parallel to that line crossing ST in M, [which may need to be extended as here when SP < SK ]; and the attractive acceleration SL may be resolved (by Corol. 2 of the laws) into the attractions and SM LM . And thus the body P will be urged by three accelerative forces.

[i.e. LS LM MS= + gives two of the forces, and PT is the other.]

One force [ PT ] attracts P towards T, and arises from the mutual attraction of the bodies T and P. By this force alone the body P must describe equal areas in proportional times with the radius PT about the body T, either fixed, or disturbed by this attraction, and in an ellipse the focus for which is at the centre of the body T. This is apparent from Prop. XI, and the Corollaries 2 and 3 of Theorem XXI. The second force is by the attraction of the force LM, which because it attracts from P to T, may be added onto the first force and will coincide with that [in direction], and thus might bring about that the areas even still may be described in proportional times by Corol. 3 of Theorem XXI. But because it is not inversely proportional to the square of the distance PT, this force adding together with the first force differs from that proportion, and this with a greater variation; by which the proportion of this force is greater than the first force, with all else being equal.



[Note 495 adapted, L & J. : For from the construction, 2

2

SLSK SLSKSP SK

= = , and thus

3

3SK SL SK SL

SK SP SPSP.×

×= = But on account of the similar triangles MLS, TPS : SL LMSP PT= ; hence

3

3SKLM

PT SP= , and therefore the force LM is as

3

3SKSP

PT× , or with SK given, as 31

SPPT× ;

from which with the distance PT increased, so the force LM will increase] Hence since (by Prop. XI, and by Corol. 2 of Theorem XXI.) the force, by which an ellipse will be described about a focus T must attract towards that focus, and to be in the inverse square ratio of the distance PT; but that composite force, erring from that proportion, so makes the orbit PAB err from the elliptic form having a focus at T; and with that the more, so that from which the departure is greater from that proportion ; and thus also by how much greater is the proportion of the second force LM to the first force, with all else being equal. Now indeed the third force SM, by attracting the body P along a line parallel to the line ST itself, together with the previous forces comprises a force, which is no longer directed from P to T; and which may differ so much more from this determination, when the proportion of this third force to the former forces is greater, with all else being equal : and thus which will cause the areas no longer to be described by the body P in proportional times, with the radius TP ; and as the aberration from this proportionality may be so much greater, when the proportion of this third force to the other forces is greater. Truly the third force will increase the aberration of the orbit PAB from the previous elliptic form in a two fold manner, because that force is not directed from P to T, and also because it shall not be inversely proportional to the square of the distance PT. [Note 496, L & J. : For PT is to ST as the force LM is to the force SM, but from the previous note, the force LM is as

3

3SK PT

SP× , and hence the force SM is as

3

3SK ST

SP× . Whereby

the force SM, with SK and ST given, is as 31

SP.]

With which understood, it is evident, that the areas are made maximally in proportion to the times when the third force shall be a minimum, with the rest of the forces remaining constant; and so that the orbit PAB then can approach maximally to the previous elliptic form, where both the second force as well as the third, but particularly the third force, shall be a minimum, with the first force remaining. [Extra diagram and notes for this T based system : Black lines are geometric lines; green lines are forces exerted by S on T and on P; red lines are components of the excess or deficient force of S on P, resolved along PT and SN; PT and ST are overlapping lines.



Here we have detached the components of the PS force acting on P in our coloured diagram, showing the three forces considered acting on P ; we note that T(erra) the earth, is acted on by the S(un) directly at a great distance, by the force SN , moving slowly at a great distance, and so slowly pulling the two-body system slowly through a complete circle; and by the moon directly at P by the force variable force PT , while the force of the sun on the moon at P, (with the average force SK ), gives rise on resolution, during its orbit, to the indirect forces acting on the earth, and LM SN . These are the three forces per unit mass acting on P, and so really are accelerations, or gravitational field strengths. Also, the perturbing force on P at the position shown is LT LS TS= − , the difference of the moon-sun and the earth-sun accelerations ;and note that T is the centre of gravity of the body, while N is the position of the focus of the elliptical orbit of P.

It is interesting to observe the progression of Newton's figure, of which he must have been very proud, as it provides so much

information, and it expresses one of these happy moments in physics where a lot of ideas suddenly tumble out into the light from some obscure place, and at least a qualitative understanding of a situation can be grasped. Above is the diagram from the first edition

of the Principia, where SL is the perturbing force. And here on the left is the diagram from Brougham and Routh's book, where the original quadrilateral PLSM has become the parallelogram MLNE. However, they have not shown the resultant perturbing force.



And here is a later diagram from a 19th century book, that I have borrowed from the useful article on Lunar Theory, from that wonderful resource Wikipedia, for which we must thank the anonymous provider. Note that two opposite positions have now been put in place, one with the gravitational acceleration of the sun on the moon greater, and other less, than the

gravitational acceleration of the earth. Finally, here is a modern diagram from the same source, showing essentially the tidal forces on a particle in orbit or on the earth's surface at various positions.

End of translator's note. Back to Newton] The attractive acceleration of the body T towards S may be represented by the line SN; and if the accelerative attractions SM and SN were equal to each other; these, by attracting the bodies T and P equally along parallel lines, would make no change in the positions of these relative to each other. Now the motions of these bodies would be the same between each other (by Corol. VI of the laws) as if these attractions were removed. And by similar reasoning, if the attraction SN were less than with the attraction SM, that part of the attractive force SN of SM itself may be removed, and only the part MN may remain, by which the proportions of the times and of the areas, and by which the elliptic form may be disturbed. And similarly if the attraction SN should be greater than the attraction SM, there may arise from the difference only the perturbation force MN of the proportionality of the orbit. Thus, by the attraction SN, the attraction of the third body always reduces the third attraction SM above to the attraction MN, with the first and second attractions completely unchanged [thus, the common part of the attraction can be ignored, leaving only the part corresponding to the force MN] : and therefore the proportionality for the areas and the times also remain unchanged, and the orbit PAB then approaches as close as possible to the previous elliptic form, when the attraction MN either is zero, or that shall be made as small as possible ; or, when the accelerative attractions of the bodies P and T, made towards the body S approach as close as possible to being equal ; i.e., when the attraction SN is not zero, nor smaller than the minimum attraction of all SM, but as if it



were a mean between all of the maximum and minimum attractions of SM, not much greater nor much less than the attraction SK. Q.E.D. Case 2. Now the smaller bodies P and S may rotate about the largest body T in different planes [or, the new plane of the ellipse PAB is inclined to the above plane STP, which we can imagine as fixed]; and the force LM, by acting along the line PT situated in the plane of the orbit PAB, will have the same effect as before, nor will the body P be disturbed from its orbital plane [by this force]. [That is, the force SP is now resolved into components in the new plane SPT .] But the other force NM, by acting along a line which shall be parallel to ST itself, is (and therefore when the body S may be situated beyond the line of the nodes, to be inclined to the orbital plane PAB) besides the perturbation of the motion in longitude that now has explained before, may lead to a perturbation of the motion in latitude, by drawing the body P from its own orbital plane. [Thus, the forces and ST MN no longer lie in the same plane, and an extra perturbation arises changing the angle of this plane to the fixed plane]. And this perturbation, in some given situation of the bodies P and T to one another, will be as that force generating MN, and thus may emerge the minimum when MN is minimal, that is (as I have just explained) where the attraction SN is not much greater, nor much less than the attraction SK. Q.E.D. [The first ten corollaries that follow give qualitative arguments about the kinds of motions that can arise between the three bodies. People normally agree that the explanations supplied by Newton are quite inadequate for a proper understanding of the subject, and that he retained the accompanying analytical derivations that he had worked out; see Chandrasekhar on this point, at least until Book III. We offer here as part of this qualitative understanding, some of the notes supplied by Leseur & Janquier in their edition of the Principia. We will indicate briefly what each corollary sets out to establish if this is not apparent at once ; the whole theory can be applied to the moon as P, S as the sun, and T as the earth, but is presented in a more general way. Most of the technical terms used in the following can be found on line in the 1911 eleventh edition of the Encyclopaedia Britannica (Cambridge University Press), and from various astronomy websites. ] Corol. 1. [This corollary indicates that the inner body is affected less by the disturbing force; Newton show in Prop. XXVI of Book III that the components of this force acting along and perpendicular to the radius at a given point, for each body in orbit around T, are in fact proportional to this radius.] From these it is easily deduced, that if several smaller bodies P, S, R, &c. may be revolving around the greatest body T, the motions of the innermost body P will be disturbed minimally by the attractions of the exterior bodies, where the greatest body T



with all other things being equal, for a given ratio of the accelerative forces, is attracted and disturbed by the others, and as are the lesser bodies mutually between themselves. Corol.2. [This corollary indicates that the 'constant' areas are in fact slightly more at the syzygies and slightly less by the same amount at the quadratures.] For in a system of three bodies T, P, S, if the accelerative attractions of any two on the third shall be to each other inversely as the square of the distances ; the body P, with the radius PT, describes an area about the body T faster on account of the nearby conjunction A and the opposition B, than near the quadratures [i.e. the perpendicular positions on the diagram above] C, D. For all the force by which the body P may be acted on, (and the body T is not acted on, and which does not act along the line PT), either accelerates or retards the description of the areas, thus as the bodies come together or are moving apart. Such is the force NM. This, in the passage of the body P from C to A, attracts as a forward motion, and the motion accelerates; then as far as to D in moving apart, the motion is retarded; then acting together as far as to B, and finally by acting in opposition in passing from B to C. [Note 498 from L & J : Such is the force NM.... If we may suppose the orbit CADB to be almost a circle, and the distance SD a maximum with respect to the radius PT, there will be almost SC SK ST SN= = = , and hence NM TM= . Again with the body P at the quadratures C and D, there is SC SP SK= = ; whereby since there shall be

2

2SL SK

SPSK= ,

by the construction of Prop.66, , there will be SL SK SC= = at the quadratures and LM coincides with CT or PT, and thus TM or NM vanishes. Therefore there will be no difference in the forces SM and SN at the quadratures, and thus the body P is disturbed by the remaining forces, and is attracted towards the centre T, describes there by the radius drawn, areas proportional to the times : [we can think of both T and P accelerating equally towards S at these points, and the non-central force corresponding to LM is taken as negligible.] Moreover, when the body P is in the hemisphere CAD beyond the quadratures, the force SM is greater than the force SN and the body P is drawn by the difference of the forces along a direction parallel to TS itself.



Let Pm be equal and parallel to NM itself, and by sending from m a perpendicular to the radius TP produced, the force Pm, or NM, is resolved into the two forces Pn and nm, the former of which Pn by drawing along the direction of the radius TP, the motion of the body P does not change in longitude – i.e. the rate at which the angle P rotates through in its orbit, nor will it disturb the equality of the area described; truly the latter force nm, by attracting along the direction of the line nm, perpendicular to the radius TP, that is, along the direction of the tangent at P, accelerates the motion in the longitude in the first quadrant CA and retards it in the second quadrant AD. [Assuming the northern hemisphere view as an anticlockwise rotation of P about T.] In the other hemisphere DBC, the force SM is less than the force SN, because the body P is at a greater distance from the body S than the body T, from which if the perturbing forces may be considering only on the body P, the difference of the forces SM and SN will be negative or taken away, or because it is the same force, acting in the opposite direction. For both the bodies T and P may be considered to be urged by the force SN equal and parallel to itself everywhere, and they can move among themselves as if all that force were absent, by Cor. 6. of the laws of motion; The body P may be drawn by the force NM along the opposite direction to the force SN, and by that action the motion of the bodies will be changed between themselves; but also by that action the force SN which was being considered to draw the body P, has been reduced to the force SM , which is the reverse force acting while the force SN acts on T. Therefore if the motion of the bodies T and P may be judged between themselves, so that the body P may be urged by the difference of the forces NM, acting in the opposite direction in the hemisphere, the true changes in the motions of the bodies T and P between themselves will be obtained. arising from the actions SN and SM. Finally, the body P may be considered in the

hemisphere DBC as if urged by the force NM along the direction Pm parallel to NM itself by acting from P towards m; and thus, if the force Pm may be resolved into two forces as has been done in the other hemisphere, it is evident the longitudinal motion will be accelerated in the quadrant DB, and retarded in the quadrant BC.] Corol.3. And by the same argument it is apparent that the body P, with all else being equal, will be moving faster in conjunction and in opposition than at quadrature. [But as we have seen above, in opposite directions.]



Corol. 4. For the orbit of the body P, with all else being equal, is more curved at quadrature than in conjunction and opposition. For swifter bodies are deflected less from a straight path. And in addition the force KL, or NM, in conjunction and in opposition is opposite to the force, by which the body T attracts the body P; and thus that force is decreased; but the body P is deflected less from a right path, where it is less urged towards the body T. [Note 499, L & J : And besides the force KL......With everything in place as above, the right lines SL and SM are almost parallel, and hence TM PL= and LM PT= approximately; whereby P coincides with A and K with T, there becomes LM AT PK= = , and NM or TM PL AT KL= + + , and NM LM KL− = , that is, the whole disturbing force by which the body P in conjunction with A is withdrawn from the body towards S, is as KL approximately ; for the force LM attracting P towards T and by the force NM is withdrawn from the body T towards S. Likewise it may be shown in the same manner, with the body P in opposition to the position B. ] Corol. 5. From which the body P, with all else being equal, departs further from the body T at the quadratures, than at conjunction and at opposition [i.e. the elliptical shape is made slightly prolate from perturbation]. These thus are considered with the exclusion of the eccentricity from the motion. For if the orbit of the body P shall be exocentric, the eccentricity of that (as will be shown in Corol. 9 of this work soon) emerges the maximum when the apsides are at the syzygies ; and thus it is possible to happen that the body P, calling at the greater apside, may be further from the body T at the syzygies than at the quadratures. [Note 499q, L & J : From which the body P, ...... For since the orbit of the body P shall be more curved at the quadratures C or D than at the syzygies A and B (by Corollary 4), it is necessary, with all else equal, that at the syzygies A and B shall be more squeezed than at the quadratures C and D to the image of the ellipse, the centre of which shall be T, CD the major axis, and AB the minor axis. Thus, these may be found if , with the exclusion of the perturbing forces, the orbit of the body P were a circle of which the centre were T.] Corol. 6. [In which changes in the Kepler Law III are accounted for in terms of perturbing factors. This is really a 'second order' effect : see Chandrasekhar p. 245 for a detailed account.] Because the centripetal force of the central body T, by which the body P may be held in its orbit, is increased at the quadratures by the addition of the force LM, and diminished at the syzygies by taking away the force KL, and on account of the magnitude of the force KL being greater than LM, it is more diminished more [at A and B ]than increased [at C and D]; but that centripetal force (by Corol. 1. Prop. IV.) is in a ratio compounded from the simple ratio of the radius TP directly and from the ratio of the inverse of the square of the time [i.e. 2

radiusperiod

PTC.F .∝ ]: it is apparent that the

compounded ratio be diminished by the action of the force KL; and thus the periodic time, if the radius of the orbit TP may remain, to be increased, and that in the square root



ratio, by which that centripetal force is diminished: and thus by diminishing or increasing this radius, the periodic time becomes greater, or to be diminished less than in the three on two power of this radius (by Corol. VI. Prop. IV.) If that force of the central body were to become gradually weaker, the body P attracted always less and less continually recedes further from the centre T; and on the other hand, if that force were increased it would draw closer. Therefore if the action of the distant body S, by which that force is diminished, were increased and diminished in turn: likewise the radius TP will be increased and diminished in turn; and the periodic time will be increased and diminished in a ratio composed from the three on two power of the radius, and from the square root ratio by which that centripetal force of the central body T; by increasing or decreasing the action of the distant body S, is diminished or increased. [Note 500, L & J : And because the magnitude of the force KL, ...... If the mean distance SK or ST were made large with respect to the radius of the orbit TP of the orbit PAB, in some position of the body P, the force LM will be approximately to the force NM as the whole sine to the triple sine of the angle of the distance of the body P from the quadrature. For on account of the increase of the distance of the body S (by hypothesis) the lines SL and SM are almost parallel and hence

or and LM PT , NM TM PL, SP SK= = = ; and since ST shall be perpendicular to the line of the quadratures CD, also SK will be normal to the same line, and with the radius PT present, PK will be the sine of the angle PTC, that is, the sine of the of the angle of the distance of the body P from the quadrature C approximately. Again (by Prop. 66)

2

2SL SKSK SP

= , and thus 2 2

2SL SK SK SP

SK SP− −= , that is,

2 22 2 2SP SPKL PK PK

SK SP SKSP SPPK SK PK= × + = × = = , on account of

and 2SK SP, SK SP SP.= + = Whereby there will be 2KL PK= , and PL or 3NM PK= , that is, the force LM or PT to the force NM or PL as the whole sine PT to 3PK the triple sine of the angle of the distance of the body P from the nearby quadrature. Extra L & J Corollary: The (extra) force KL at the conjunction A, is to the similar force K'L' at the opposition B –K' and L' are not shown, almost as AT to TB; i.e. KL AT

K' M TB= ,

[for the extra force FA at A is proportional to ( )2 2 2 2

1 1 AT ST SASA ST SA .ST

+− = , while the extra force

FB at B is proportional in the same way to ( )2 2 2 2

1 1 BT ST SBSB ST SB .ST

+− = in the opposite direction

; hence ( )

( )( )( )

2 2

2 2

11

AT ST SA STSASA .STA

BT ST SB STB SBSB .ST

F SBTA TAF TB SA TB

+

+

+

+= = × × ≈ , as SA, ST, and ST are almost equal. ] and if

the orbit PAB were circular or almost circular, the force KL at the syzygies will be almost twice as great as the force LM at the quadratures. For with the body P turning about the syzygies, there shall be PK AT PT LM= = = , and hence NM or PL becomes

3 and 2LM , KL LM .= = Yet with the same positions, the force NM is a maximum at the syzygies, because there PK becomes a maximum or it emerges AT= , and 3NM AT .=



From which on account of the magnitude of the force KL, the centripetal force of the central body T is more diminished than augmented, and thus is to be thought to be diminished by the action of the body S.] Corol. 7. [In which the detailed motion of the apsidal line is discussed.] Also from what has been presented it follows, that the axis of the ellipse described by the body P, or the line of the apsides, as far as the angular motion is concerned, in turn moves forwards and backwards, but yet it progresses more forwards than backwards, and by the excess of the progression in succession it is carried forwards. For the force which acts on the body T at the quadratures, where the force MN vanishes, is composed from the force LM and the centripetal force, by which the body T attracts the body P. The former force LM, if the distance PT may be increased, may be increased in almost the same ratio with this distance, and the latter force decreases in that square ratio, and thus the sum of the two forces decreases in less than in the square ratio of the distance PT, and therefore (by Corol. I. Prop. XLV.) effects that the upper or greater apside, may be regressing. Truly in conjunction and in opposition the force, by which the body P is urged towards the body T, is the difference between the forces, by which the body T attracts the body P, and the force KL ; and that difference, because that force KL may be increased approximately in the ratio of the distance PT, decreases in more than the inverse square of the distance PT, and thus (by Corol. I Prop. XLV.) has the effect that the greater apside progresses. In the places between the syzygies [or conjunctions] and the quadratures the motion may depend on increases from both causes taken together, and thus so that either from the excess of one or the other it may progress or regress. From which since the force KL at the syzygies shall be as if twice as large as the force LM at the quadratures, the excess will belong to the force KL, and an increase will be carried forwards in succession. But the truth of this and of the preceding corollary may be understood more easily by considering the system of the two bodies T and P with several bodies S, S, S, &c. in place in the orbit ESE, surrounded on every side. In as much as the action of T with the actions of these may reduce the action of T on both sides, and it may decrease in a ratio more than the square of the distance. Corol. 8. [In which the motion of the apsidal line is discussed at the syzygies and at the quadratures]. But since the progression or regression of the apsides depends on the decrease of the centripetal force, that is, being made in either a greater or lesser ratio than in the square ratio of the distance TP, in the transition of the body from the lower apside [i.e. closer to the focus of the ellipse] to the higher apside; and thus it shall be a maximum when the proportion of the force at the higher apside to the force at the lower apside departs in the inverse square of the distances ; it is evident that the apsides in the conjunctions of this, by removing the force KL or NM LM− , to be progressing faster, and at their quadratures



to recede slower, by the addition of the force LM. Truly on account of the long period of time, in which the velocity of the progression or the slow regression may be continued, this inequality becomes maximally long. Corol. 9. [In which the variation of the eccentricity is discussed]. If some body, by some force inversely proportional to the square of its distance from the centre, may be revolving about this centre in an ellipse ; and soon, in descending from the upper apside or by increasing from the lower apside, that force may be increasing to a new force by always approaching in a ratio diminishing more than the square of the distance: it is evident that the body, by always approaching towards the centre by the impulse of that new force, may be inclined more towards the centre, than if it were acted on only by the decrease in the square ratio of the distance ; and thus it may describe a more inner elliptic orbit, and at the inner apside it may accelerate closer to the centre than before. Therefore this orbit by the influence of this new force, may become more exocentric. If now the force, in the recession of the body from the lower apside to the upper apside, may decrease in the same steps by which before it had increased, may return the body to the former distance, and thus if the force may decrease in a greater ratio, now the body attracted less may ascend to a greater distance and thus the eccentricity of the orbit will be increased even more at this stage. Whereby if the ratio of the increase and decrease of the individual centripetal force may be increased in the rotations, the eccentricity will always be increased; and on the other hand, that will be decreased the same, if that ratio may decrease. Now truly in a system of bodies T, P, S, when the apsides of the orbit PAB are at the quadratures, that ratio of the increase and decrease is a minimum, and it shall be a maximum when the apsides are in conjunction [syzygies]. If the apsides may be set up at the quadratures, the ratio near the apsides is smaller, and near the syzygies greater than the square of the distances, and from that increased ratio a direct motion or the line of the apsides arises, as had been said just now. But if the ratio of all the increase or decrease may be considered in the motion between the apsides, this is less than the square of the distances. The force at the lower apside is to the force at the upper apside in a ratio less than the square of the distance of the upper apside from the focus of the ellipse to the distance of the lower apside from the same focus: and conversely, when the apsides may be put in place at the syzygies, the force at the lower apside is to the force at the upper in a ratio greater than the square of the distances. For the forces LM added at the quadratures to the forces of the body T compound forces in a smaller ratio, and the forces KL at the syzygies taken from the forces of the body T leave forces in a greater ratio. Therefore the ratio of the whole decrease and increase, in passing between the apsides, is a minimum at the quadratures and a maximum at the syzygies: and therefore it will always be increased in the passing of the apsides from the quadratures to the syzygies, and the ellipse increases in



eccentricity; and in the transition from the syzygies to the quadratures the ratio will always be diminished, and the eccentricity diminished. Corol.10. [In which the variation of the inclination is discussed]. In order that we may enter into the error [i.e. the nature of the disturbing forces] in the latitude, we may imagine the plane of the orbit EST to remain at rest; and the cause of the errors is evident from the exposition, because from the forces NM and ML, which are that whole cause, the force ML by always acting along the plane of the orbit PAB, at no time disturbs the motion in the latitude ; each force NM, when the nodes are at the syzygies, also may be acting along the same plane of the orbit, and so does not disturb this motion either ; truly when the bodies are at the quadratures, this force disturbs these motions greatly, and the body P is always attracted from the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and it may augment the same in turn in the passage from the syzygies to the quadratures. From which it may be that with the body present at the syzygies the smallest inclination of all may emerge, and it may be returned to the former magnitude almost, when the body approaches close to a node. But if the nodes may be put in place at the octants after the quadrants, that is between C and A, V and H, it may be understood from the manner shown, that in the transition of the body P from either body thus to 900, the inclination of the plane is continually diminished; then in a transition through approximately 450, as far as to the nearest quadrant, the inclination will be increased, and again it is diminished after passing through another transition of 450 , as far as to the nearest node. And thus the inclination is diminished more than it is increased, and therefore it is always less in the subsequent node than at the preceding one. An by similar reasoning, the inclination is increased more than it is diminished, when the nodes are in the alternate octants between A and D, B and C. Therefore the greatest inclination of all is when the nodes are at the syzygies. In the passage of these from the syzygies to the quadratures, the inclination of the body is diminished by the individual influences to the nodes ; and it shall be the least all when the nodes are at the quadrants, and the body at the syzygies: then the inclination increases by the same steps by which previously it decreased; and under the influences [of the other bodies acting] it reverts to the original magnitude with the nodes at the nearby syzygies.



[Note 507 L & J : If the nodes of the body P are situated at the quadratures C and D, the angle of inclination of the orbit to the fixed plane EST is always diminished in the

passage of the body from the quadratures to the syzygies, truly to be enlarged in the passage of the body from the syzygies to the quadratures, and in each passage the nodes are regressed. For let CAD be the part of the orbit PAB raised higher above the fixed plane of the orbit EST, and truly CBD may be understood to be the part depressed below ; through the place of the body P the right line Pm acts parallel to the line TS, showing the direction of the force NM, and the body P may be carried first from a node or quadrature to the conjunction A, and because the body P is urged by the force of rotation through the arc Pp, in an element of time, by the force put in place, it will describe the line element Pπ which is not in the plane CPT, but curved away from that towards Pm, and thus the body is moving in the plane TPπ that produced with the plane EST does not cross at C but beyond C towards the opposition B. With centre C and interval TP the circle CaDb is described in the plane EST, in the plane CPD the arc of the circle PC, and in the plane TPπ the arc Pc crossing the circle CaDb at c. And because the force NM is a minimum with respect to the force of rotation of the body P, the angle CPc, of the inclination of the planes CPT and cPT is the smallest possible or infinitesimal, and the arc Pc only differs from the arc PC by an infinitesimal quantity; whereby since, by hypothesis, the arc PC differs from the quadrant CA by a finite amount PA, the sum of the arcs PC and Pc is less than a semicircle, and hence in the spherical triangle CPC, the exterior angle PCa (by Prop.13 of the Sphericorum of Menelaus, or by the Sphericorum of Wolf) is greater than the internal angle PcC, that is, the inclination of the plane cPT to the plane EST is less than the inclination of the plane CPT to the same plane EST. Therefore in the passage of the body P from the quadrature C to the conjunction A the inclination of the orbit is always diminished, and because the node C is transferred to c, there becomes because of the way in which the rotation of the body, the nodes are returned. In the same way the diminution of the inclination and the nodes to be returned in the passage of the body from the quadrature D to the opposite B is demonstrated. Now the body may be carried from the conjunction A to the vicinity of the quadrature D, and at some place P, by the squared



force, certainly it is urged by the force of rotation through the arc Pp and by the force NM along the right line Pm, and thus it will describe the line element Pπ , which leans towards Pm from the arc Pp. Whereby if from the centre T and with the radius TP, three arcs PD, aD, and Pd are described, in the same way it will be shown that the node D to be transferred to the previous at d, and the angle Pda to be greater than the angle the internal angle PDd, that is, the inclination of the orbit is to be augmented in the transition of the body P, from the conjunction to the nearby quadrature, and in the same way it is shown to happen in the transition from the opposition B to the quadrature C. Q.e.d.] Corol. 11. [In which the variation of the ascending node is discussed]. Because the body P, when the nodes are in the quadratures, are perpetually attracted from the plane of its orbit, and that in part towards S in its transition from the node C through the conjunction A to the node D ; and in the opposite part in the transition from the node D through the opposition B to the node C : it is evident, that in its motion from the node C the body constantly recedes from the orbit of its first plane CD, until then it has arrived at the nearest node; and thus at this node, with the greatest distance from that first plane CD, it passes through the plane of the orbit EST not in its plane with the other node D, but at some point thence it turns in the direction of the body S, each hence is turning towards the new position of a node. And by a similar argument the nodes go on to recede in the transition of the body from this node to the neighbouring node. Therefore the nodes continually recede from the quadratures put in place; the nodes are at rest at the syzygies, where nothing in the latitude disturbs the motion; at the intermediate positions, participants of each condition, the nodes recede more slowly: and thus, always either by receding, or by stationary individual revolutions, they are carried to the preceding nodes. Corol.12. All these errors described in these corollaries are a little greater at the conjunction of the bodies P and S, than at the opposition of these; and that on account of the greater generating forces NM and ML. Corol. 13. [In which the motion of the body S, so far ignored, is discussed]. Whenever ratios of these corollaries do not depend on the magnitude of the body S, all the preceding will be obtained, when only the magnitude of the body S is put in place, so that the system of the two bodies T and P may be rotating about the centre of this. And from the increase in the body S, and thus with the increase of the centripetal force, from which the errors of the body P arise, all these errors emerge, with the distances greater in this case than in the other, where the body S is revolving around the system of the bodies P and T. Corol.14. [In which the formula for S's perturbation is introduced].



But since the forces NM and ML, when the body S is remote, shall be approximately as the force SK and the ratio PT to ST conjointly, that is, if there may be given both the distance PT, as well as the absolute force of the body S, inversely as ST cubed ; but these forces NM and ML the cause of all the errors and effects, which have been acted on in the preceding corollaries: it is evident, that these have affected everything, with the system of the bodies T and P in place, and only with the distance ST chanced and with the absolute force of the body S, shall be composed approximately in the ratio from the direct ratio of the absolute force of the body S, and inversely in the triplicate ratio of the inverse distance ST. From which if the system of bodies T and P may be revolving about the distant body S; these forces NM and ML, and the effects of these will be will be (by Corol. 2. and 6, Prop. IV.) reciprocally in the square ratio of the periodic times. And thence also, if the magnitude of the body S shall be the proportional of the absolute force itself, these forces NM and ML and the effects of these, shall be directly as the cube of the apparent diameter of the distant body S observed from the body T, and conversely. For these ratios are the same, and composed from the above ratio. [Note 512, L & J : On account of the great distance of the body S, LS will be almost parallel to MS, and SN ST SK= = , and ML PT= ; and because NM at the syzygies is as ML at the quadratures. If increased or diminished by the action of the body S, the orbit CADB hence together with the depending lines PT, NM, ML may be increased or diminished (by Cor.6 of this Prop. 66) , these three lines will be increased or diminished in almost the same ratio between themselves (with all else being equal). But the force ML to the force SK is as the right line ML to the right line SK, or approximately as PT to ST ; whereby the force ML (and thus also the force NM) is nearly as the force SK and the ratio PT to AD, jointly that is, if the accelerative force SK may be called A, as A PT

ST× . Again

with the given absolute force of the body S, the accelerative force A at the distance SK or ST is as 2

1ST

( by hypothesis). Whereby the forces NM, ML, with the given absolute force

of the body S, are as 3PTST

; that is (if the distance PT may be given), inversely as 3ST .

Indeed if the variable shall be the absolute force V of the body S, the accelerative force A will be as directly as the absolute force and inversely as the square of the distance ST, (for with the absolute force of the body S, the accelerative force is inversely as 2ST , and with the distance ST remaining, the accelerative force is as the absolute force directly, and therefore likewise with the variations of the absolute force and the distance, the accelerative force is as the absolute force directly and the square of the distance inversely). Whereby if in place of the accelerative force A that ratio composed from the factor A PT

ST× may be put in place, the forces NM, ML will be approximately as 3

V PTST× , or

with PT given, as 3V

ST , that is in a ratio composed from the direct ratio of the absolute

force of the body S, and from the inverse cube of the distance ST. But the absolute force of the body S, is as shown in the ratio composed of the accelerative force A and the square of the distance ST, and the accelerative force A at the distance ST is (by Corol.2 Prop. 4) in a ratio composed from the direct ratio of the distance ST and in the inverse square ratio of the periodic time of the body T around S to the distance ST of the circle



described, and thus the absolute force of the body S is as the cube of the distance ST directly, and the square of the periodic time of the body T inversely. Whereby the forces NM, ML (and the effects of these) which are directly as the absolute force, and inversely as the cube of the distances, are inversely in the square ratio of the periodic times of the body T. End of note.] Corol. 15. [In which Newton infers the proportionality of perturbing force to the radius of the orbit of P about T.] And because if, with the forms of the orbits ESE and PAB, with the proportions and inclination to each other remaining constant, the magnitude of these may be changed, and if the forces on the bodies S and T may either remain or be changed in some given ratio; these forces (that is, the force of the body T, by which the body may be deflected from a straight path into the orbit PAB, and the force of the body S, from which the same body P is forced to deviate from that) always act in the same way, and in the same proportion: it is necessary that all effects shall be similar and the times of the effects to be proportional; that is, so that all linear errors shall be as the diameters of the orbits, truly the angles shall be as at first, and the times of similar linear errors, or equal angular errors, shall be as the periodic times of the orbits. [Note from L & J : That is, if the absolute forces of the bodies S and T either may remain or be changed in some given ratio, and the magnitude thus may change of the orbits ESE and PAB, so that it may always remain similar to the orbit ESE itself, thus also so that of the orbit PAB itself, and the inclination of these orbits may not change, nor the proportion of the ratio of the axis of one orbit to the axes of another or of any lines whatever in one orbit to the homologous lines in another orbit. End of L & J note. Translator's note : these are not errors as we now understand the word, but disturbances by forces that alter the orbits in some way.] Corol.16. [In which the mean motion of the increase of the line of apses and the mean regression of the line of ascending nodes are to be the same, apart from sign]. From which, if the forms of the orbits and the inclination to each other may be given, and the magnitudes, forces and distances of the bodies may be changed in some manner ; from the given errors and from the errors in the times in one case, the errors and the error in the time can be deduced closely : but this is shorter by this method. The forces NM and ML, with the rest remaining, are as the radius TP, and the periodic effects of these (by Corol. 2, Lemma X.) as the forces and the squares of the periodic times of the body P taken together. These are the linear errors [disturbing the orbit] of the body P; and hence the angular errors have been viewed from the centre T (that is, both the motion of the apsides and of the nodes, as well as all the errors appearing from the longitude and latitude), in whatever revolution of the body P, as the square of the time of revolution approximately. These ratios may be taken together with the ratios of Corollary XlV and in some system of the bodies T, P, S, where P is revolving around the vicinity of T itself,



and T around the distant body S, the angular errors of the body P, appearing from the centre T, will be, in the individual revolutions of that body P, as the squares of the periodic times of the body P directly and the square of the periodic time of the body T inversely. And thence the mean motion of the apsides will be in a given ratio to the mean motion of the nodes ; and each motion will be as the periodic time of the body P directly and the square of the periodic time of the body T inversely. By increasing or lessening the eccentricity and the inclination of the orbit PAB, the motion of the apsides and of the nodes are not changed appreciably, unless where the same are exceedingly large. Corol. 17. [In which the gravitational forces of S and T may be compared with their distances and periods]. But since the line LM may now be greater, then less than the radius PT, the mean force LM may be expressed by that radius PT; and [from the similar triangles SPT and SLM] this will be to the mean force SK or SN (that is allowed to express by ST [recall that N is the focus of the elliptical orbit of P, T the centre of gravity of the fixed body T ]) as the length PT to the length ST.

[i.e. LM PTSK ST= ]

But the mean force SN or ST, by which the body T is held in its orbit around S, to the force, by which the body P is held in its orbit around in T, is in a ratio compounded from the ratio of the radius ST to the radius PT, and the square ratio of the periodic time of the body P around T [which confusingly I have called T], to the periodic time of the body T [t] around S.

[i.e. ( )2ST ST TPT tPT

∝ × .]

And from the equality, the mean force LM to the force, by which the body P in held in its orbit around T (or by which the same body P, in the same periodic time may be able to revolve around some fixed point T at a distance PT) is in that square ratio of the periodic times. Therefore with the given periodic times together with the distance PT, the mean force LM may be given; and with that given, the force MN is also given approximately by the analogy of the lines PT and MN. Corol. 18. [Some relevant outcomes of the theory are now discussed, relating to the tides and associated matters, in the final corollaries. These are of considerable interest]. From the same laws, by which the body P is revolving about the body T, we may imagine many fluid bodies to be moving around the same body T at equal distances from that; then from these made touching each other a ring of fluid may be formed, round and concentric to the body T; and the individual parts of the ring, on completing all their motion according to the law of the body P, approach closer to the body T, and faster at the conjunction and opposition of these and the body S, than at the quadratures. And the nodes of this annulus, or the intersections of this with the plane of the orbiting body S or T, are at rest at the syzygies ; truly beyond the syzygies they will be moving as before,



and most quickly indeed at the quadratures, more slowly at other places. Also the inclination of the annulus will vary in inclination, and the axis of this will oscillate in the individual revolutions, and with a complete revolution completed it will return to the former position, unless as far as it will be carried around by the precession of the nodes. Corol. 19. We may now consider the globe of the body T; agreed not to be fluid matter, to be made bigger and to be extended as far as to this annulus, and with the hollow evacuated around to contain water, and with the same motion to revolve with the same period around its axis uniformly. This liquid will be accelerated and retarded (as in the above corollary) in turn ; at the syzygies it will be faster, and at the quadratures slower than the surface of the globe, and thus it will flow to and fro in the manner of the sea. Water, by revolving around the centre of a globe at rest, if the attraction of the body S may be taken away, will acquire no motion of flowing backwards and forwards. The reasoning is the same for a globe progressing uniformly in a direction, and meanwhile turning about its centre (by Corol. 5 of the laws) and as of a globe drawn uniformly from its course (by Corol. 6. of the laws). But the body S may approach, and from the inequality of this attraction soon the water will be disturbed. And also the water to be attracted more nearer than the body, lesser with that more remote. Moreover the force LM pulls the water down at the quadratures, and will make that descend as far as the syzygies ; and the force KL draws the same water up at the syzygies, and stop the descent of this, and will make that ascend as far as the quadratures: unless in so far as the motion of flux and reflux of the water may be directed by the channel, and may be retarded a little by friction. Corol. 20. If the annulus now may be rigid, and the globe may be made smaller, the flow to and fro will stop ; but that inclination of the motion and of the precession of the nodes will still remain. The globe may have the same axis with the annulus, and the gyrations may be completed in the same times, and the annulus may touch the interior itself with its own surface, and may adhere to that; and by sharing the motion of this each structure will be oscillating, and the nodes will be regressing. For the globe, as soon will be said [Cor. XXII], may be indifferent to all forces being accepted. The maximum angle of the inclination of the annulus with the globe of the orbit is when the nodes are at the syzygies. Thence in the progression of the nodes to the quadratures the annulus tries to minimize its own inclination, and by that trial it impresses the motion on the whole globe. The globe retains the impressed motion, until the annulus then by attempting motion in the opposite sense may hence remove that motion, and a new motion is impressed in the opposite direction: And on this account the maximum motion of the decrease of the inclination shall be at the quadrature of the nodes, and the smallest angle of inclination in the octants after the quadratures; then the maximum motion of re-inclination at the syzygies, and the maximum angle in the next octant. And the same account is given of the globe with the ring removed, which is a little higher at the equatorial or higher regions than near the poles, or is made from a slightly denser material. For this excess of the material in the equatorial regions supplies that of the ring. And nevertheless, with any increase of the centripetal force of this globe, all the parts of this are supposed to be attracted downwards, according to the manner of the gravitating parts of the earth, yet



the phenomena of this and of the preceding corollary hence scarcely are changed, except that the places of the maxima and minima height of the water will be different. For now the water will be sustained and remain in its own orbit, not by its centrifugal force, but by the hollow in which is flows. And in addition the force LM pulls the water downwards maximally at the quadratures, and the force KL or NM LM− draws the same upwards maximally at the syzygies. And these forces taken together stop the water being drawn downwards and begin to draw the water upwards in the octants before the syzygies, and they stop drawing the water upwards and begin to draw the water downwards at the octant after the syzygies. And hence the maximum height of the water can come about in the octants after the syzygies, and the minima in the octants after the quadrutures approximately; unless as it were the motion of rising or falling impressed by these forces, for the water to persist with a little daily motion, either by a force in place or by the hindrance of some channel it might have stopped a little quicker. Corol. 21. By the same account, by which the excess matter of the globe lying around the equator caused the nodes move backwards [or regress], and thus by the increase of this matter this regression in turn may be increased, again truly with diminution it is diminished, and on being removed is completely removed; if more than the excess matter may be removed, that is, if the globe near the equator is either further lowered, or rarer than around the poles, the motion of the nodes will arises in succession. Corol. 22. And thus in turn, the constitution of the globe is understood from the motion of the nodes. Without doubt if the same poles of the globe are maintained constantly, and the motion shall be as previously [i.e. retrograde], there is an excess of matter near the equator ; if it is in succession, there is deficiency. Put a uniform and perfectly round globe first at rest in free space; then propelled from its place by some oblique impulse on its surface, and the motion thence considered partially to be circular and partially along a direction. Because this globe has all axis passing through its centre indifferently, there is no propensity for one axis or any other axis to be in place, it is evident that neither this particular axis, nor the inclination of the axis, at any time will be changed by its own force. Now the globe may be impelled obliquely, in the same part of the surface, as the first, by some new impulse; and since the effect of impulses coming sooner or later changes nothing, it is evident that these two successive impulses impressed in succession produce the same motion, as if they were impressed together, that is , the same [motion is produced], as if the globe had been struck by a simple force composed from each impulse (by Corol. 2 of the laws), and thus a simple force about the given inclination of the axis. And the reasoning is the same of a second impulse made at some other place on the equator of the first motion ; or of the first impulse made at some place on the equator of the motion generated by the second impulse without the first ; and of both impulses made at any places : these will generate the same circular motion as if they were impressed once simultaneously at the intersection of the equators of their motions, which they would generate by themselves. Therefore a homogeneous and perfect globe will not retain several distinct motions, but adds together all the impressed forces and reduces that to one, and just as within itself, it always rotates in a simple and uniform motion about a single axis, always with an invariable inclination given. Moreover neither will a



centripetal force be able to change the inclination of the axis, nor the velocity of the rotation. If a globe may be considered to be divided into two hemispheres by some plane through its centre, and in which a force is directed passing through its centre ; that force will always urge each of the hemispheres equally, and therefore the globe, by which the motion of rotation, will be inclined in no direction. Truly there may be added at some place between the pole and the equator new matter in the form of raised mountains, and these, by always trying to be receding from the centre of its motion, will disturb the motion of the globe, and it may happen that the poles of this globe may wander on the surface, and they may always describe circles about themselves and about the points opposite to themselves. Nor may the immensity of these wanderings be themselves corrected, except by locating that mountain either at one pole or the other, in which case (by Corol. XXI.) the nodes of the equator will be progressing ; or at the equator, by which account the nodes will be regressing (by Corol. XX.) ; or finally by adding new matter to the other part of the axis, by which the mountain will be balanced in its motion, and with this agreed upon the nodes either will be progressing, or receding, provided that the mountain and these new matters are either nearer to the pole or to the equator.

PROPOSITION LXVII. THEOREM XXVII. With the same laws of attraction in place, I say that a body beyond S, around the common centre of gravity O between P and T, by radii drawn to that centre, will describe areas more proportional to the times and the orbit approaching more to the form of an ellipse having the focus at the centre, than can be described around the innermost and greatest body T, with radii drawn to that itself. For the attractions of the body S towards T and P are composed of this absolute attraction, which is directed more to the common centre of gravity O of the bodies T and P , than to the greatest body T , with every square of the distance SO more inversely proportional, than to the square of the distance ST: as the matter on examination will be readily agreed upon.

PROPOSITION LXVIII. THEOREM XXVIII. With the same laws of attraction in places , I say that the external body S, with radii drawn to that centre, will describe areas more proportional to the times, about the common centre of gravity O of the interior bodies P and T, and an orbit more approaching to the form of an ellipse having the focus at the same centre, if the innermost body and greatest body and the others thus may be disturbed by these attractions, than if that either were at rest and not being attracted, or much less or much more attracted, and consequently either much more or much less disturbed. This may be shown in almost the same manner as Prop. LXVI, by a more lengthy argument, which therefore I shall pass by. It will suffice to consider the matter thus. From the demonstration of the latest proposition it is apparent that the centre, towards which



the body S is urged by the forces conjointly, is very close to the common centre of gravity of these two. If this centre should coincide with that common centre, and the common centre of gravity of the three bodies may remain at rest; accurate ellipses are described by the body from the one side, and the common centre of gravity of the other two from the other side, about the common centre of gravity of everything at rest. This may be apparent from the second corollary of Proposition LVIII, taken with the demonstrations in Prop. LXIV. & LXV. This elliptical motion may itself be disturbed a little by the distance of the centre of the two from the centre, to which the third body S is attracted. Besides a motion may be given to the common centre of the three, and it will be increased by the perturbation, Hence the perturbation is a minimum, when the common centre of the three is at rest ; that is, when the innermost and largest body T is attracted by the law of the others: and the perturbation shall always be the largest, when that common centre of the three, by the smallest decrease in the motion of the body T , begins to move, and it is disturbed more and more henceforth. Corol. And hence, if several small bodies were revolving around the largest, it is possible to deduce that the orbits described approach closer and closer to ellipses, and the descriptions of the areas become more equal, if all the bodies by accelerative forces, which are as the absolute forces of these directly and inversely as the square of the distances, and may mutually attract and perturb each other, and also the focus of each orbit may be deduced to be at the common centre of gravity of all the interior bodies (without doubt the focus of the first and innermost at the centre of gravity of the first and largest body; that of the second orbit, at the common centre of gravity of the two innermost bodies; that of the third, at the common centre of gravity of the three innermost , and thus henceforth), as if the innermost body were at rest and may be situated at the common focus of all the orbits.

PROPOSITION LXIX. THEOREM XXIX. In a system of several bodies A, B, C, D, &., if some body A shall attract all the others B, C, D, &c. by accelerative forces which are inversely as the squares of the distances from the attracting body; and another B also attracts the others A, C, D, &c. by forces which are inversely as the squares of the distances from the attracting body: the absolute forces of attraction of the bodies A, B to each other, as are the bodies themselves A, B, of which they are the forces. For the accelerative attractions of all the bodies B, C, D towards A, with equal distances, by hypothesis are equal to each other; and similarly the acclerative attractions of all the bodies towards B, with equal distances, in turn are equal to each other. Moreover the absolute force of attraction of the body A is to the absolute force of attraction of the body B, as the accelerative attraction of all the bodies towards A to the accelerative attraction of all the bodies towards B, with the distances equal; and thus is the accelerative attraction of the body B towards A, to the accelerative attraction of the body A towards B. But the accelerative attraction of the body B towards A is to the accelerative attraction of the body A towards B, as the mass of the body A to the mass of the body B ; therefore because the motive forces, which (by the second, seventh and



eighth definitions) are as the accelerative forces and the bodies attracted jointly, here they are (by the third law of motion) in turn equal to each other. Therefore the absolute attractive force of the body A is to the absolute attractive force of the body B, as the mass of the body A to the mass of the body B. Q. E. D. Corol. I. Hence if the individual bodies of the system A, B, C, D, &c. seen to attract all the rest separately by accelerative forces, which are inversely as the squares of the distances from the attracting body ; the absolute forces of all the bodies will be in turn as the bodies themselves. Corol. 2. By the same argument, if the individual bodies of the system A, B,C, D, &c. separately observed attract all the others with accelerative forces, which are either inversely, or directly in the ratio of any powers of the distances from the attracting body, or which are defined to be following some common law by attracting each other from the distances ; it is agreed that the absolute forces of these bodies are as the bodies. Corol.3. In a system of bodies, the forces of which decrease in the square ratio of the distances, if the lesser may be revolving around the largest one in ellipses, having the common focus of which at the centre of the largest of these, as it can happen that they are revolving most accurately ; and with the radii drawn to that largest body, areas may be described exactly proportional to the times : the absolute forces of these bodies will be in turn, either accurately or approximately, in the ratio of the bodies; and conversely. This is apparent from the Corollary of Prop. LXVIII, taken with the corollary I of this Prop.

Scholium. We are led by these propositions to an analogy between the centripetal forces, and the central bodies, to which these forces are accustomed to be directed. For it is agreeable to the reason, that the forces which are directed towards the bodies, depend on the nature and quantity of the same, as happens in experiments with magnets. And as often as cases of this kind arise, the attractions of the bodies are to be considered, by designating appropriate forces to the individual members of these, and by gathering together the sum of the forces. I may talk about this attraction generally by taking some known attempt of the bodies to approach each other: either that attempt comes about itself from the action of the bodies, or they reach towards each other mutually, or by some agent [spirit in the original] of agitation sent between themselves in turn; or this arises from the action of the ether, or of the air, or by some medium whatsoever, originating either from bodies or not from bodies, by impelling the floating bodies in some manner towards each other in turn. In the same general sense used I may talk not about the kind of forces and physical quantities. but the amounts and mathematical proportions of these set out in this tract; as I have explained in the definitions. In mathematics it is required to investigate the magnitudes of forces and the ratios of these , which follow from some conditions put in place: then, when we descend to the level of physics, we compare these ratios with the phenomena; so that it may become known which conditions may be agreed upon with the



kinds of the individual attractive forces of the bodies. And then finally concerning the kinds of bodies, it will be allowed to argue without risk about the causes and reasons from physics. Therefore we may see by which forces spherical bodies, now agreed upon in the manner established from the attraction of small bodies, must act mutually on each other ; and thus what kind of motions they may thence follow.


Book I Section XII. Translated and Annotated by Ian Bruce. Page 348

SECTION XII. Concerning the attractive forces of spherical bodies.

PROPOSITION LXX. THEOREM XXX.

If the individual points of the surface of a some sphere may be drawn to the centre by equal forces inversely proportional to the square of the distance from the points : I say that a corpuscle within the surface agreed upon is not attracted by these forces in any direction. Let HIKL be that spherical surface, and P a corpuscle placed within. Through P there may be drawn two lines to this surface HK and IL, the intercepting arcs HI and KL as minimum; and, on account of the similar triangles HPI. LPK (by Corol. 3, Lem. VII.), these arcs will be proportional to the distances HP and LP , and whatever the small parts of the spherical surface, proportional to HI and KL, with right lines drawn passing through the point P and terminating there on all sides, and they will be in that duplicate ratio. Therefore the forces of these particles acting on the body P are equal to each other. For they are directly as the particles, and inversely as the square of the distances. And these two ratios compound a ratio of equality. Therefore the attractions, acting equally on the opposite parts, mutually cancel each other out. And by a similar argument, all the attractions from the contrary parts for the whole spherical surface cancel each other by opposite attractions. Therefore the body P in no part is impelled by these attractions. Q.E,D. [Note 515 (o) L & J : For the angles HPI , LPK vertically opposite are equal; and the angles HIL, LKH resting on the same arc are equal, (by Prop. 27. Book 3, Eucl.) For the evanescent arcs IH, KL, can be taken for the chords themselves (by Cor. 3. Lem. 7) Whereby the arcs HI, KL with the distances HP, LP are proportional, and hence if an innumerable number of right lines may be understood to be drawn to the spherical surface through the point P to the smallest possible arcs as terminals HI, KL, these right lines will form similar solid figures – pyramids or cones, the bases of which will be similar on the surface of the sphere, and hence (by Lemma 5) they will be in the duplicate ratio of the sides HI, HL, or of the distances HP, LP. Hence the forces, etc..... ]



PROPOSITION LXXI. THEOREM XXXI.

With the same in place, I say that a corpuscle put in place outside the surface of the sphere is attracted to the centre of the sphere, by a force inversely proportional to the square of its distance from the same centre. Let AHKB, abhk be two spherical surfaces described, with centres S, s, diameters AB,

ab, & P, p corpuscles situated outside, on these diameters produced [in the case shown, PA pa> ]. The lines PHK, PIL, phk, pil may be drawn from the corpuscles, taking the equal arcs HK, hk and IL, il from the great circles AHB, ahb. And the perpendiculars SD, sd; SE, se; IR, ir; may be sent to these of which SD, sd may cut PL, pl in F and f: also the perpendiculars IQ, iq may be sent to the diameters. The angles DPE, dpe may be vanishing: and on account of the equality DS and ds, ES and es, the lines PE, PF and pe, pf and the smallest lines DF, df may be considered as equal; [We have taken the liberty of marking these equal chords and perpendiculars in different colours on both diagrams; unmarked diagrams are of course available in the Latin section of this file]; evidently the ultimate ratio of these is equality, with the angles DPE, dpe likewise evanescent. And thus with these put in place, PI to PF will be as RI to DF, & pf to pi as df or DF to ri

[i.e. ; pi riPI RIPF DF pf df= = ] ;

and from the equality, to PI pf PF pi× × is as RI to ri,

[i.e. ;PI pf RI df RI HIPF pi DF ri ri hi× ×× ×= = = ]

that is (by Corol. 3. Lem. VII.) as the arc IH to the arc ih [since ;RI HIri hi= from the similar

triangles RHI and rhi] . Again, PI is to PS as IQ to SE, and ps to pi as se or SE to iq; [i.e. or = ; and IQ ps se SEPI

PS SE pi iq= ],

and from the equality is to PI ps PS pi× × , as IQ to iq;

[i.e. = ;PI ps IQPS pi iq×× ],

And with the ratios taken together 2PI pf ps× × to 2pi PF PS× × , as to IH IQ ih iq× × ;



[i.e. 2

2 ;PI ps PI pf PI pf ps IH IQPS pi PF pi ih iqpi PF PS× × × × ×× × ×× ×

× = = ],

that is, as the circular surface, that the arc IH will describe by rotating about the diameter AB of the semicircle AKB, to the circular surface, that the arc ih will describe by rotating about the diameter ab of the semicircle akb. And the forces, by which these surfaces attract the corpuscles P and p along the lines tending towards themselves, are (by hypothesis) directly as the surfaces themselves, and inversely as the squares of the distances of the surfaces from the bodies, that is, as to pf ps PF PS× × .

[i.e. 2 2: ;IH IQ ih iq pf psPF PSPI pi

× × ××= ]

And these forces are to the oblique parts of these, which (by the resolution of the forces made according to Corol 2. of the laws by resolution of the forces) act along the lines PS, Ps to the centre, as PI to PQ, and pi to pq; that is (on account of the similar triangles PIQ and PSF, piq and psf) as PS to PF and ps to pf. Thus, from the equality, the attraction of the corpuscle P towards S shall be to the attraction of the corpuscle p towards s, as

to PF pf ps pf PF PSPS ps× × × × that is, as 2 2 to ps PS .

[i.e. 2

2horizontal attraction of left segment

horizontal attraction of right segmentPFPSpfps

pf pspf ps cos PFS psPF PS cos pfs PFPF PS

× ×× ×× × × ×

= = = .]

And by a similar argument the forces, by which the surfaces by the rotation of the described arcs KL, kl pull on the corpuscles, will be as 2 2 to ps PS . And it will be possible to distinguish the forces of all the circular surfaces in the same ratio in which each spherical surface will be, by always taking sd equal to SD and se equal to SE. And, by putting these together, the forces of all the spherical surfaces exercised on the bodies will be in the same ratio. Q.E.D. [When this proof is understood, it is seen to be very simple, and this was probably the only way in which the problem could be solved at the time, without resorting to a full integration ; the distances of the corpuscles from the shells is the only variable, and other chords and perpendiculars, etc. maintain the same lengths in each shell, allowing the summation to proceed, and the final ratio required to emerge.]



PROPOSITION LXXII. THEOREM XXXII.

If equal centripetal forces attract particular points of some sphere decreasing in the inverse ratio of the distance from these points; and both the density of the sphere may be given, as well as the ratio of the diameter of the sphere to the distance of [such]a corpuscle from the centre of this : I say that the force, by which the corpuscle is attracted, will be proportional to the radius of the sphere. For consider two separate corpuscles attracted by two spheres, one corpuscle by the one sphere, and the other corpuscle by the other sphere, and the distances of these from the centres of the spheres are proportional to the diameters of the spheres respectively, the spheres moreover are resolved into similar particles and similarly put in place with the corpuscles. And the attractions of the one corpuscle made towards the individual particles of the one sphere, will be to the attraction of the other corpuscle towards just as many particles of the other sphere, in a ratio compounded from the ratio of the particles directly, and in the ratio of the inverse squares of the distances. But the [number of the] particles are as the [volumes or masses of the] spheres, that is, in the triplicate [i.e. cubic] ratio of the diameters, and the distances are as the diameters ; and the first ratio directly together with the second ratio inversely twice is the ratio of the diameter to the diameter. Q.E.D. [Consider two spheres A and B with radii RA and RB; corpuscles are situated at distances CA and CB from their respective spheres, then by hypothesis, =A A

B B

C RC R ; let NA and NB be

the number of particles or corpuscles in the respective spheres, then the respective FA and

FB forces will be in the ratio 2 2 3 2

2 2 3 2=A A B A B A B A

B B BA A B A B

F N C M C R R RF N MC C R R R

× = × = × = .]

Corol. I. Hence if the corpuscles may be revolving in circles about the spheres constantly attracted equally by the matter ; and the distances from the centres shall be proportional to the diameters of the same: The periodic times shall be equal. [Recall from previously, such S.H.M. motion has the period independent of the amplitude or radius.] Corol. 2. And conversely, if the periodic times are equal; the distances shall be proportional to the diameters. These two are in agreement by Corol. 3. Prop. IV. Corol. 3. If equal centripetal forces may attract the individual points [corpuscles] of any two similar solids of equal density, decreasing in the squared ratio of the distances from the points ; the forces, by which the corpuscles will be attracted by the same, by these two similar solids in place, will be in turn as the diameters of the solids.



PROPOSITION LXXIII. THEOREM XXXIII.

If for some given sphere, equal centripetal forces attract particular points decreasing in the square ratio of the distances from the points : I say that a corpuscle within the sphere constituted is attracted by a force proportional to its distance from the centre. A corpuscle P will be located in the sphere ABCD described with centre S; and from the same centre S, with the interval SP, consider the spherical interior PEQF to be described. It is evident, (by Prop. LXX.) that the concentric spherical surfaces, from which the difference of the spheres AEBF is composed, with their attractions destroyed by opposing attractions, do not act on the body P. Only the attraction of the interior sphere PEQF remains. And (by Prop. LXXII.) this is as the distance PS. Q.E.D. [ In the diagram constructed here, the original sphere is replaced by two spheres of the same material having radii OB and O'B', while the corpuscles considered are at the positions P and P' with the radii OA and O'A' . From the first proposition in this section (no. 70), the corpuscles at P and P' experience no force from the outer particles beyond the radii OB and OB', considering these outer particles as lying on concentric shells; meanwhile, from the second proposition in this section (no. 71), the inner particles lying in their concentric shells can be replaced by equivalent masses at the origins O and O', exerting forces proportional to their respective masses, and inversely as the distances OP and O'P' squared in turn. The ratio of these forces in an obvious notation is given by :

3 3

2 2OP A A A B

O' P' A BA A

F R R' R RF' R' R'R R'

.ρ ρ= ÷ = = ]

Scholium.

The surfaces, from which the solids are composed, here are not purely mathematical, but thus tenuous shells [orbs], so that the thickness of these shall be equivalent to zero; without doubt the evanescent shells, from which the sphere finally will be composed when the number of these shells may be increased indefinitely and the thickness is minimised. Similarly the points, from which the lines, surfaces, and volumes are said to be composed, are understood to equal particles of insignificant magnitude.



PROPOSITION LXXIV. THEOREM XXXIV.

With the same in place, I say that a corpuscle constituted beyond the sphere is attracted by a force inversely proportional to the square of its distance from the centre of this sphere. For the sphere may be separated into innumerable concentric spherical surfaces, and the attractions of the corpuscle arising from the individual surfaces will be inversely as the square of the distance from the centre (by prop. LXXI.) And on compounding, the sum of the attractions, that is the attraction of the corpuscle by the whole sphere, happens in the same ratio. Q. E. D.

[Thus, the attractive force 21

distance∝ ]

Corol. I. Hence at equal distances from the centres of homogeneous spheres, the attractions are as the [masses of the] spheres. For (by Prop. LXXII.) if the distances are proportional to the diameters of the spheres, the forces will be as the diameters. The greater distance may be reduced into that ratio [of the radii]; and with the distances now made equal [to those of the respective spheres], the attraction will be increased in that ratio squared ; and thus it will be to the other attraction in that ratio cubed, that is, in the ratio of the spheres. [ See Chandrasekhar on this point : i.e. the forces are in the ratio of the cubes of the radii for equal distances from the spheres.] Corol. 2. In which any attractions are as the [masses of the] applicable spheres to the squares of the distances. Corol. 3. If a corpuscle, situated outside a homogenous sphere, is drawn by a force inversely proportional to the square of its distance from the centre itself, but the sphere may be constructed from attracting particles ; the force of each particle will decrease in the square ratio of the distance from the particle. [As Chandrasakher points out on p. 278, this amounts to a statement of the universal law of gravitation, but applied to this circumstance.]

PROPOSITION LXXV. THEOREM XXXV. If equal centripetal forces may extend to the individual particles of a given sphere, decreasing in the inverse square ratio of the distances from the points ; I say that any other similar sphere, by the same is attracted by a force inversely proportional to the square of the distance of the centres. For the attraction of each particle is inversely as the square of its distance from the centre of the attracting sphere, (by Prop. LXXIV.) and therefore [the attraction of the particles in a sphere between themselves] is the same, as if the total attracting force may spring from a single corpuscle situated at the centre of this sphere. But this attraction is



just as great, as would be the case as if in turn it were attracted by a force of the same size by all the other particles of the sphere. But the attraction of that sphere would be (by Prop. LXXIV.) inversely proportional to the square of its distance from the centre of the other sphere; and thus the attraction of the sphere is equal to this in the same ratio. Q.E.D. Corol. I. The attractions of spheres, towards other homogeneous spheres, are as the applied attracting spheres to the squares of the distances of their centres from the centres of these which they attract. Corol.2. The same becomes apparent, when the attracted sphere also attracts. In as much as the individual points of this will attract the individual points of the other sphere with the same force, from which by these themselves in turn they are attracted; and thus as in any attraction both the attracting point, as well as the attracted point, may be acted on – by law 3, there will be a mutual pair of attracting forces with the proportions conserved. Corol. 3. Everything prevails the same, which have been shown above concerning the motions of bodies about the focal points of conic sections, when the attracting sphere is located at the focus, and the bodies will be moving outside the sphere. Corol. 4. That truly, which may be shown concerning the motion of bodies about the centre of a conic section, will prevail when the motions are taking place within the sphere.

PROPOSITION LXXVI. THEOREM XXXVI. If spheres, in a progression from the centre to the circumference, dissimilar in some manner (as far as the density of the material and the attractive force are concerned), by progressing around, are all alike on all sides at a given distance from the centre; and he attractive force of each point decreases in the square ratio of the attracting body : I say that the total force, by which one sphere of this kind attracts another, shall be inversely proportional to the square of the distance from the centre.



Let AB, CD, E F, &c. be some similar concentric spheres the interiors of which added to the outsides put together a material denser towards the centre, or taken away leave behind a more tenuous medium; and these individual spheres (by Prop. LXXV.) will attract other individual similar concentric spheres GH, IK, LM, &c., with forces inversely proportional to the distance SP. And by adding together or by dividing apart, the sum of the forces of all these, or the excess of some above the others, that is the force by which the whole sphere, composed from any number of concentric spheres or differences AB, attracts the whole composed from some number of concentric spheres or differences GH, will be in the same ratio. Thus the number of concentric spheres may be increased indefinitely, to that the density of the material together with the attractive force, in progressing from the circumference to the centre, may increase of decrease following some law; and non attracting material may be added whenever the density is deficient, so that the spheres may acquire some optimum form; and the force, by which one of these may attract the other, even now will be by the above argument, in that inverse ratio of the square of the distance. Q.E.D. Corol. 1. Hence if many spheres of this kind, themselves in turn similar through all, mutually may attract each other ; the accelerative attractions of individuals on individuals will be, at whatever equal distances of the attractions, as the attracting spheres. Corol. 2. And with any unequal distances, as the applicable attracting spheres to the squares of the distances between the centres. Corol. 3. Truly the motive attractions [i.e. the attractive forces], or the weights of the spheres on the spheres will be, at equal distances of the centres, as the attracting and attracted spheres conjointly, that is as the content within the spheres produced by multiplication. Corol. 4. And in accordance with unequal distances, directly as that product and inversely as the square of the distances between the centres.



Corol.5. The same will emerge when the attraction arises on each sphere by virtue of the mutual attraction exercised by one sphere on the other. For with both forces the attraction forms a pair, with the proportion maintained. Corol. 6. If some spheres of this kind may be revolving around others at rest, individual spheres around individual spheres; and the distances between the centres of the revolving and the resting spheres proportional to the resting diameters ; the periodic times will be equal. Corol.7. And conversely, if the periodic times are equal ; the distances will be proportional to the diameters. Corol. 8. Everything will be obtained the same, which have been shown above concerning the motion of bodies around the foci of conic sections ; where the attracting sphere, of the form and of each condition now described, may be located at the focus. Corol. 9. And as also where the attracting spheres are rotating, with any of the conditions now described.

PROPOSITION LXXVII. THEOREM XXXVII. If centripetal forces attract the individual points of the spheres proportional to the distances between the points from the attracting bodies : I say that the composite force, by which the two spheres will attract each other mutually, is as the distance between the spheres. Case. 1. Let AEBF be the sphere; S the centre of this; P an attracted corpuscle, PASB the axis of the sphere passing through the centre of the corpuscle ; EF, ef two planes, by which the sphere may be cut, perpendicular to this axis, and hence thereupon equally distant from the centre of the sphere ; G, g the intersections of the planes and the axis ; and H some point in the plane EF. The centripetal force at the corpuscle P to the point H, acts along the line PH , is as the distance PH; and (by Corol. 2. of the laws) along the line PG, or towards the centre S, as the length PG. Therefore the force of all the points in the plane EF, that is of the whole plane, by which the corpuscle P is attracted towards S, is as the distance PG multiplied by the number of points, that is, as that solid contained under the plane EF itself and that distance PG. And similarly the force of the plane ef, by which the corpuscle P is attracted towards the centre S, is as that plane taken by its distance Pg, or equally as the plane EF taken by that distance Pg; and the sum of forces of each plane as the plane EF taken by the sum of the distances PG Pg+ , that is, as that plane multiplied into twice the distance PS between the centre and the corpuscle, that is, as twice the plane EF multiplied into



the difference PS, or as the sum of the equal planes EF ef+ multiplied by the same distance. And by a similar argument, all the forces of the planes in the whole sphere, hence in this manner at equal distances from the centre of the sphere, as the sums of the planes multiplied by the distance PS, that is, as the whole sphere and as the distance PS conjointly. Q.E.D. Case 2. Now the corpuscle P may attract the sphere AEBF. And by the same argument it will be approved that the force, by which that sphere is attracted, will be as the distance PS. Q.E.D. Case 3. Now the other sphere may be put together from innumerable particles P; and because the force, by which any corpuscle whatever is attracted, is as the distance of the corpuscle from the centre of the first sphere and as the same sphere taken conjointly, and thus is the same, as if the whole may be produced from a single corpuscle at the centre of the sphere ; the whole force, by which all the corpuscles are attracted to the second sphere, will be the same as if that sphere may be attracted by the force produced from a single corpuscle at the centre of the first sphere, and therefore is proportional to the distance between the centres of the spheres. Q.E.D. Case 4. The spheres may attract each other mutually, and the double force will maintain the previous proportion. Q.E.D. Case 5. Now the corpuscle p may be located within the sphere AEBF; and because the force of the plane ef at the corpuscle is as the solid contained under that plane and the distance pG; and the force of the opposite plane EF is as the solid contained under that plane and the distance pG; the force composed from each will be as the difference of the solids, that is, as the sum of the equal planes multiplied by half the difference of the distances, that is, as half that multiplied into the distance pS of the corpuscle from the centre of the sphere. And by a similar argument, the attraction of all the planes EF, ef in the whole sphere, that is, the attraction of the whole sphere, is jointly as the sum of all the planes, or the whole sphere, and as the distance pS of the corpuscle from the centre of the sphere. Q.E.D. [ Note 517(s) Leseur & Janquier : the force composed from each will be as the difference of the solids, that is, as ef pg EF pG.× − × But Sg SG= , and thus

2pg pG pS SG pG pS;− = + − = whereby since also there shall be EF ef= , there will be 2ef pg EF pG ef pg pG ef pS ef EF pS.× − × = × − = × = + × If the point G is placed

between p and S, the total force will be as ef pg EF pG× + × , and because there is



always Sg SG= , and in this case 2pg pG pS SG pG pS ,+ = + + = similarly the total force will be found as ef EF pS+ × .] Case 6. And if the new sphere may be composed from innumerable corpuscles p, situated within the first sphere AEBF; it will be approved as before that the attraction, either a simple one of one sphere towards the other, or mutually each in turn towards the other, will be as the distance of the centres pS. Q.E.D.

PROPOSITION LXXVIII. THEOREM XXXVIII.

If the spheres in the progression from the centre shall be dissimilar in some manner and unequal, but truly in progressing around, all at a given distance from the centre, they shall be the same on all sides; and the attractive force of each point shall be as the distance of the attracted body : I say that the total force by which two spheres of this kind mutually attract each other shall be proportional to the distance between the centres of the spheres. This may be demonstrated from the previous proposition in the same manner, by which Proposition LXXVI was shown from Proposition LXXV. Corol. Those matters have been shown above in Propositions X. and LXIV. concerning the motion of bodies about the centre of conic sections, where all the attractions emerge made from the force of spherical bodies under the conditions now described, and the attracted bodies are spheres of the same description.

Scholium. I have now explained the two most significant cases of the attractions ; without doubt where the centripetal forces decrease in the square ratio of the distances, or increase in the simple ratio of the distances ; in each case putting into effect that bodies may rotate in conic sections, and the component centripetal forces of spherical bodies by the same law, in receding from the centre, decreasing or increasing with these themselves : which is noteworthy. It would be tedious to run through the remaining cases one by one, which show less elegant conclusions. I am inclined to understand and to determine everything by a general method, as follows.



LEMMA XXIX.

If some circle AEB be described with centre S, and with centre P the two circular arcs EF, ef are described, cutting the first in E and e, and the line PS in F, f; and the perpendiculars ED and ed may be sent to PS: I say that, if the separation of the arcs EF and ef may be understood to be infinitely small, the final ratio of the vanishing line Dd to the vanishing line Ff shall be that, which the line PE has to the line PS. For if the line Pe may cut the arc EF in q; and the right line Ee, which will coincide

with the vanishing arc Ee, produced meets the right line PS in T; and from S the normal SG may be sent to PE : on account of the similar triangles DTE, dTe, DES; Dd will be to Ee, as DT to TE, or DE to ES; and similar on account of the triangles Eeq, ESG (by Lem. VIII. & Corol 3. Lem. VII.), Ee will be to eq or Ff as ES to SG; and from the equality, Dd to Ff as DE to SG; that is (on account of the similar triangles PDE, PGS) as PE to PS. Q.E.D. [ Note initially, as was common practice for Newton, the increment in a length from a

point was given a small letter, such as Ee, etc. From similar triangles, Dd DT DEEe ET SE= = ;

Ee Ee ESeq Ff SG= = . Thus, Dd DE PE

Ff SG PS= = . It is thus apparent that a ratio such as DdFf is one of

vanishing quantities, where the sides of generating triangle tend towards zero and would be termed by us a differential ratio; it is also usual that such a vanishing ratio is set equal to some finite ratio found geometrically. We may also mention here again, that such quantities are considered by Newton as vanishing : not about to vanish, in which case they would be represented by finite difference, and not vanished, in which case they would all be the meaningless zero on zero, but in the act of vanishing together, in which



circumstance they adopt the final ratio; which we might term now the limiting value of the ratio, which means the same thing in different words. In the annotated diagram, were g and a are constants, this becomes in modern terms : fdx a sin

df g sin gθϕ= = .

To show this, we may revert to modern analysis, as the original calculation has not been supplied in this manner. Thus initially, we may consider the circle AEB, with P the origin of coordinates and the radius a; the coordinates of any point E on the circle are and x g acos y a sinθ θ= + = ; and the implicit equation of the circle AEB is :

( )( )2 2 2x g y a− + = . However, from triangle EPS, we have ( )f ga

sin sin sinθ φ θ φ−= = , and

hence ( ) ( )f sin a sin a sing g sinsin a sin

θ θ θφθ φ θ φ− −

= = = . Also,

( )( )22 2 2 2 22f g acos a sin g ag cos aθ θ θ= + + = + + ; hence

( )fdf ag sin d gd a cos gdxθ θ θ= − = = , giving fdx a sindf g g sin

θφ= = , as required and ( )f ,φ

may be taken as the independent variables for future use in integrations over the spherical volume. To turn the circle integrations into spherical integrations, it is necessary to multiply only to multiply by 2π . We will not do so, as this factor disappears in ratios. This derivation above is no more and no less than the original result for the Lemma obtained above by Newton, but written in more familiar terms.]

PROPOSITION LXXIX. THEOREM XXXIX. If the surface EFfe is already vanishing on account of an infinite diminution of width, with the rotation of this around the axis PS, it may describe a concave-convex spherical solid, and equal centripetal forces are extended to the individual particles of this: I say that the force, by which that solid attracts a corpuscle situated on the axis at P, is in a ratio composed from the ratio of the solid 2DE Ff× , and in the ratio of that force by which the given particle at the place Ff may attract the same corpuscle. For if in the first place, we may consider the force of the spherical surface FE, which may be generated by the rotation of the arc FE, and which may be cut somewhere by the line de in r; the part of the surface of the ring, generated by the rotation of the arc rE , will be as the line increment Dd, with the radius of the sphere PE remaining constant (as Archimedes demonstrated in his book, On the Sphere & Cylinder.) [Imagine rE coming out of the plane of the page as it rotates about the axis PS forming a ring of radius DE and width Dd, with P the apex of the twin cone; as we move towards F, concentric rings of width Dd are inscribed in the inner surface of the sphere centre P, radius PE, until an incremental circle is found at F; the diagram is a vertical section through this sphere and the spherical body AEB ]



[Note 518(x) Leseur & Janquier : The demonstration is easy. For because the angle PEr is right (from the nature of the circle) the angle DEr is equal to the angle DPE, on account of the sum of the angles being equal to the right angle DPE PED PEr+ . From which, if from the point r it may be considered to send a perpendicular (shown here in red) to the line DE , that is equal to the line increment Dd, there may be put in place a vanishing triangle similar to triangle EPD, and thus there will be and Dd PE DdDE

PE Er DEEr ×= = , but the circular region generated by the rotation of the arc rE is as the rectangle Er DE× ; whereby if in this rectangle in place of Er there may be substituted the value found in this manner, the region will be as PE Dd× , that is, on account of the given radius PE, as Dd. Q.E.D.] And the force of this element of area, exercised on the conical surface in place along the lines PE or Pr, shall be as this part [i.e. incremental area] of the annular surface itself; that is, as the line increment Dd, or, because it is likewise, as the rectangle under the sphere with the given radius PE and that line increment Dd : but along the line PS attracting towards the centre S in the smaller ratio PD to PE, and thus as PD Dd× . [Note 518(z) Leseur & Janquier : For if the force acting along the direction PE may be set out by the line PE, the part of this force which acts along the direction PS, may be set out by the line PD; PE DdPE

PD PD Dd ,××= that hence will show the force along that direction PD, but

the oblique forces ED from each part of the axis PB mutually destroy each other; end of L & J note. Thus, the force normal to the element PE Dd× is proportional to

PDPEPE Dd PD Dd× × = × .]

Now the line DF may be considered to be divided into innumerable equal small parts, which one to the other may be called Dd; and the surface FE may be divided into an equal number of rings, the forces of which are as the sums of all the terms PD Dd× , that is as 2 21 1

2 2PF PD− , and thus as 2DE . [The force exerted on P due to this set of rings of successive width dx on the PS axis is the sum of forces parallel to the axis , any of which has the form PD Dd× .] [Note 519(b) Leseur & Janquier : Clearly all PD, while they are being changed from PD into PF by increasing uniformly make an arithmetic progression, because all the small parts Dd by which the line PD may be increased are equal : therefore the sum of all PD will be in that ratio found from which the sums of the arithmetical progressions may be obtained, certainly by being multiplied jointly by the number of terms of the progression, and by taking half of the product. Truly the first term of this progression is PD, the final PF and DF the number of terms, if indeed DF is the sum of the equal vanishing increments of the line PD, therefore the sum of all the PD is 2

PF PD DF+ × or (because DF

is the difference of the lines PF and PD) the sum of all the PD is 2PF PD PF PD+ × −= ; but

(by 6.2. Eucl. Elements) the sum and the difference of two lines is equal to the difference



of the squares of these, hence 2 21 12 2 2

PF PD PF PD PF PD+ × − = − , and the sum of all 2 21 1

2 2PF PD Dd= − × , but Dd is the increment which is assumed as constant in all these cases, therefore the forces of the whole surface FE which are as the sum of all the terms PD Dd× ,are as 2 21 1

2 2PF PD− or as 2 2PF PD− ; but 2 2PF PE= by construction,

and 2 2 2PE PD DE− = (by 47.1 Eucl. Elements) hence the forces of the surface FE, are as DE2. The same otherwise: Let the given radius PE f= (not to be confused with the increment Ff), the variable FD x= , the fluxion and Dd dx, PD f x= = − , and thus (the limits of integration, to the left, are 0x = and x DF= ) : PD Dd fdx xdx× = − , and with the

fluents (integrals) taken on both sides : 221

2 2 2 fx xx DESum PD Dd fx xx −× = − = = , (Prop.

13. Book 6. Eucl. Elements). (For ( )22 2 22f f x fx x ED− − = − = ). Whereby the force of the surface arising by rotating the arc FE will be as DE2 . Extra note : this integral with the limits 0 and x is understood to be prefixed by a force, which is constant under these circumstances, so that the original double integral can be expressed as the product of two integrals.] Now the surface FE may be multiplied by the [incremental] altitude Ff; and the force of the solid EFfe exercised on the corpuscle P becomes as 2DE Ff× : for example if Ff may be given as the force some small particle exerts on the corpuscle P at the distance PF . But if that force may not be given, the force of the solid EFfe becomes as the solid

2DE Ff× , and that force is not given conjointly. Q.E.D. [i.e. we have not yet been given the force, just a very cunning way of evaluating its volume integral from a particular form of an increment of the sphere. Incidentally, Chandrasekhar seems to have missed the point here, as he sets out on a conventional double integration, which Newton has avoided by integrating over constant force shells in an almost trivial manner. It is a short step for us to consider these as equipotential surfaces; however, it cannot be assumed more than coincidence that such constant energy surfaces have appeared in Newton's calculations ; the idea of potential energy was just not present at the time, and when such energy related ideas arise. they must be considered merely as useful devices that ease calculations, rather than some fundamental insight. Chandrasekhar's book is full of such anachronisms, which does little to enhance one's understanding of the evolution of ideas that occurred at this time. One way to explain in ordinary language what Newton has done, is to consider a hypothetical onion, with uniform layers or shells; P lies at the centre of this onion; now imagine a device that can cut out a spherical shape from the body of the onion; retaining P at its original position and consider this dissected sphere, also at its original position: this now consists of layers or parts of shells each of which is equidistant from P, and an integration over the sphere can now be performed, so finding the total force using this unusual by highly effective way of dissecting the sphere into elements equidistant from P. Newton's originality



continues to amaze. Later in this section, he introduces the idea of inversion, so that the forces due to all the points of the sphere can be found on some corpuscle located within the sphere. ]

PROPOSITION LXXX. THEOREM XL.

If equal centripetal forces may be extended to the individual equal parts of a sphere ABE described with centre S, and to the axis AB of the sphere, on which some corpuscle may be placed at P, from individual points [such as] D perpendiculars DE may be erected crossing the sphere in E, and in these lengths [such as] DN may be taken, which shall be as the magnitude

2DE PSPE× and the force, that the particles of the sphere situated on the

axis at a distance PE may exercise on the corpuscle P, conjointly: I say that the total force, by which the corpuscle P is drawn towards the sphere, is as the area ANB taken under the axis of the sphere AB, and the curved line ANB, that always is a tangent at the point N. For indeed with everything in place which was constructed in the lemma and in the most recent theorem, consider the axis of the sphere AB to be divided into innumerable equal parts Dd, and the whole sphere to be divided into just as many concave – convex sheets EFfe; and the perpendicular dn may be erected.

By the above theorem the force, by which the shell EFfe attracts the corpuscle P, is as

2DE Ff × and as the force of the single particle Dd at the distance PE or PF exercised conjointly [with this mass or weight factor.] But (by the most recent lemma : Dd PE

Ff PS= )

Dd is to Pf as PE to PS, and thence Ff is equal to PS DdPE× ; and 2DE Ff× equals

2DE PSPEDd ×× , and therefore the force of the shell EFfe is as

2DE PSPEDd ×× , and the force

of the particle at the distance PF exercised conjointly, that is (by hypothesis) as DN Dd× ,[ i.e.

2DE PSPEDd DN Dd ×× = × ], or the vanishing area DNnd. Therefore the

forces of all the shells, exercised on the body P, are as all the areas DNnd, that is, the total force of the sphere is as the total area ANB. Q.E.D.



Corol. I. Hence if the centripetal force, attracting the individual particles, always remains the same at all distances, and DN becomes as

2DE PSPE× ; the total force, by which the

corpuscle is attracted to the sphere, shall be as the area ANB. Corol. 2. If the centripetal force of the particles shall be inversely as the distances of the corpuscles attracting each other, and DN becomes as

2

2DE PS

PE× ; the force, by which the

corpuscle P is attracted to the whole sphere, shall be as the area ANB. Corol. 3. If the centripetal force of the particles shall be inversely as the cube of the distances by which they attract each other, DN shall be as

2

4DE PS

PE× ; and the force, by

which the corpuscle is attracted by the whole sphere, shall be as the area ANB. Corol. 4. And generally if the centripetal force drawing the individual particles of the sphere may be put to be inversely as the magnitude V, moreover DN becomes as

2DE PSPE V

×× ;

and the force, by which the corpuscle is attracted to the whole sphere, is as the area ANB.

PROPOSITION LXXXI. PROBLEM XLI. With everything remaining in place, it is required to measure the area ANB.

From the point P the right line PH may be drawn tangent at H, and the normal HI may be sent to the axes PAB, and PI may be bisected in L; and (by Prop. XII. Book 2. Eucl. Elem.) 2PE will be equal to 2 2 2PS SE PS.SD+ + . But 2SE or 2SH (on account of the similar triangles SPH and SHI) is equal to the rectangle PS.SI [ i.e.

; or IS HS IS aHS PS a g= = , giving

2agIS = , which

is, of course, constant.] . Therefore 2PE is equal to the rectangle contained by PS and PS +SI+2SD, that is, by PS and 2LS+2SD, that is, by PS and 2LD. Again 2DE is equal to 2 2SE SD− , or

2 2 22SE LS SL.LD LD− + − , that is, 22SL.LD LD AL.LB− − . For

2 2LS SE− or 2 2LS SA− , (by Prop. VI. Book 2. Eucl. Elem.) is equal to the rectangle AL.LB. And thus

22SL.LD LD – AL.LB− may be written for 2DE ; and the magnitude

2DE PSPE V

×× , which following the fourth

corollary of the preceding proposition, which is as the length of the applied ordinate DN, may be resolved into three parts :



22SL LD PS LD PS AL LB PSPE V PE V PE V× × × × ×× × ×− − ,

where if for V there may be written the centripetal force in the inverse ratio; and for PE the mean proportional between PS and 2LD; [for from above : 2 2PE PS.LD= ] these three parts produce the applied ordinates of just as many curved lines, the area of which become known by common methods. Q.E.F. [In more detail, for the first part : ( )22 2 2 2 2PE PS SD DE PS SE PS.SD= + + = + + ; but

2SE PS.SI= and hence ( )22 2 2 2PE PS SD DE PS PS.SI PS.SD= + + = + + ,

or ( )2 2PE PS PS SI SD= + + ; moreover, 2 2 2 2PS SI IS PI IS LI LS+ = + = + = ; hence, 2 2PE PS.LD= .

For the second part : ( )22 2 2 2DE SE SD SE LD LS= − = − − , giving 2 2 2 2SE LD LS LD.LS− − + . But

( )( )2 2 2 2LS SE LS SB LS SA LS SB LA.LB− = − = − + = ; hence, we may write

2 22DE LD.LS LD LA.LB= − − ; which is very pretty, as all the line sections lie on the axis.

Algebraically, ( ) ( )2 2 2

2 221 1ga a ag g g

PS g;PI g g ;PL LI= = − = − = = − ; and ( )2

22 1g ag

LS = + ;

( )2

22 1g ag

LD LS SD x= + = + + . While ( ) ( )2 2

2 22 21 1g ga ag g

LB a;LA a= + + = + − . Hence,

( )2

22 2 2 1 a

gLB LA ag− = + ; while ( )2 2

2

22

4 1g ag

LA.LB a= + − . Also, we have

( )2

22 1g ag

LS = + .]

Example 1. If the centripetal force attracting the individual particles of the sphere shall be inversely as the distance; write the distance PE for V; then 2PS LD× for 2PE , and DN becomes as

12 2

AL.LBLDSL LD− − .

Put DN equal to the double of this, or equal to 2 ALBLDSL LD− − : and the given [constant]

part of the ordinate 2SL multiplied by the length AB [i.e. the sum of the increments Dd] describes the area of the rectangle 2SL AB× ; and the variable part LD multiplied normally into the same length [Dd] by a continued motion, as by that law requiring the motion to be either increasing or decreasing between [the limits] , may always be equal to the LD, and it will describe the area

2 2

2LB LA− , that is, the area SL AB× ; which subtracted



from the first area 2SL AB× leaves the area SL AB× . [Here we

integrate ( ) ( )2 2

2 22 1 1g a ag a a

g gg a adx xdx ag

+

− −+ + = +∫ ∫ .]

But the third part AL.LB

LD , likewise with the same local motion multiplied normally into the same width, will describe the area under a hyperbola; which taken from the area SL AB× leaves the area sought AN.NB. From which will emerge the construction of such problems. Erect the perpendiculars Ll, Aa, Bb, at the points L, A, B of which Aa itself is equal to LB, and Bb to LA. [Thus, the area is required under the rectangular hyperbola 1

1 xy += between the points

(A,LB) and (B, LA) or (g–a, ( )2

22 1g ag

a+ + ) and (g+a, ( )2

22 1g ag

a+ − ) is required.]

With the asymptotes Ll, LB, the hyperbola ab is described through the points a and b. And with the chord ba drawn the area ab.ba enclosed [finally] is equal to the area sought AN.NB. [Here the integration is performed over 2

AL.LBLD Dd. This becomes:

( ) ( ) ( ) ( )22 2 22 2 2

4 2 2 2

222

12

2 4 4 11 21

g a g aag

x g aa A ag

a g a g a g aLA.LB a dx dx'A ALD A x' g agg ag x

Dd dx a dx' ln+

−

⎞⎛ + −⎜ ⎟ + − +⎝ ⎠+ −+⎞⎛ −+ +⎜ ⎟

⎝ ⎠

→ → − → − =∫ ∫

, where ( )22 22222 22 222 21 and 14 4 4

a ag g

g agAA g a ag

−⎞ ⎞⎛ ⎛= + − = + − =⎜ ⎟ ⎜ ⎟⎝ ⎝⎠ ⎠

. The total force is then the sum of

these component forces. Hence we have the total force F given by :

( ) ( )2

22 2 2 24

2 ( +g ) g ag ag

F ag a g a lnπ +−

⎞⎛= − −⎜ ⎟⎝ ⎠

.]

Example 2. If the centripetal force attracting the corpuscle towards the individual particles shall be inversely as the cube of the distance, or (that is likewise ) as that cube applied so some given plane ; write

3

22PEAS

for V, then as above write 2PS LD× for 2PE ;

and DN becomes as 2 2 2

22 2SL AS AS AL.LB ASPS LD PS PS LD× ×× ×

− − , that is (on account of the continued

proportionals PS, AS, SI) as 212 2

LS.SI AL.LB SILD LD

SI ×− − .

[For recall that 2 2SE SA PS.SI= = and hence i.e.

2

2 212 22 2

AS SL AL.LB SL SI SI AL.LB SIPS LD LDLD LD

DN × ×⎡ ⎤ ⎡ ⎤= − − = − −⎣ ⎦ ⎣ ⎦



If the three parts of this may be multiplied by AB, the first LS .SILD ,

[for ( )2 2

2 22 2

12 1

1g a

g

gLS .SI a aLD g g x⎞⎛ + +⎜ ⎟

⎝ ⎠

= × + × , etc.]

will generate a hyperbolic area; the second 12 SI the area

12 AB SI× ; the third 22

AL.LB SILD

× will generate the area

2 2AL.LB SI AL.LB SI

LA LB× ×− , that is 1

2 AB SI× . From the first there may be taken the sum of the second and the third, and the area sought ANB will remain. From such the construction of the problem emerges. Erect the perpendiculars Ll, Aa, Ss, Bb to the points L, A, S, B, of which Ss may be equal to SI, and through the point s to the asymptotes Ll, LB the hyperbola asb may be described crossing the perpendiculars Aa, Bb at a and b; and the rectangle 2AS.SI taken from the hyperbolic area AasbB leaves the area sought ANB. [Thus, Newton fits a section of a hyperbola to the forces at the ends A and B of the diameter of the sphere, and uses this area as a known amount, without mentioning natural logarithms.] Example 3. If the centripetal force, attracting the corpuscle to the individual particles of the sphere, decreases in the quadruple ratio of the distance from the particles ; write

4

32PEAS

for V, then 2PS LD× for PE, and DN becomes as

2 2 2

3 61 1 1

2 2 2 2 2SI SL SI SI AL.LB

SI SI LD SILD LD× ×× − × − × ;

The three parts of which multiplied by the length AB, produces just as many areas, viz.

2 2 2

3 32 1 1 1 1

2 2 2 2 into ; in ; and into SI SL SI SI ALB

SI LA LB SI SI LA LBLB LA× ×− − −

And these after due reduction become 2 32 22 2

3,SI SL SILI LISI ,& SI× + . Truly these emerge,

with the latter taken from the former, 34

3SILI . Hence the total force, by which the corpuscle

P is drawn to the centre of the sphere, is as 3SI

PI , that is, reciprocally as 3PS PI× . Q.E.I. [Chandrasekhar provides equivalent modern integrations for Newton's results on pp.291-293 of his book. Leseur & Janquier provide very longwinded derivations of these integrations, which have not been included here.] By the same method it is possible to determine the attraction of the corpuscle in place outside the sphere, but it will be more expeditious by the following theorem.



PROPOSITION LXXXII. THEOREM XLI.

In a sphere with centre S described with the radius SA, if the continued proportions SI, SA, SP may be taken: I say that the attraction of a corpuscle within the sphere, at some place I, to the attraction outside the sphere at the place P, is compounded in the ratio from the square root of the distances from the centre IS, PS, and in the square root ratio of the centripetal forces, at these places P and I, tending to the centre of attraction. For, if the centripetal forces of the particles of the sphere shall be reciprocally as the

distances of the corpuscles attracting each other; the force, by which a corpuscle situated at I is attracted by the whole sphere, to that by which it is attracted at P, is compounded from the ratio of the square root of the distance SI to the distance SP, and from the square root ratio of the centripetal force at the position I, arising from some particle situated at the centre, to the centripetal force at the position P arising from the same particle at the centre, that is, in the square root ratio of the distances SI, SP reciprocally in turn.

[i.e. PSI

P IS

FF SPF FSI

= × in an obvious notation.]

Then these square root ratios make a ratio of equality, and therefore the attractions at I and P from the whole sphere are made equal. By a similar computation, if the forces of the particles of the sphere are inversely in the square ratio of the distances, it may be deduced that the attraction at I shall be to the attraction at P, as the distance SP to the radius of the sphere SA: If these forces are inversely as the triplicate ratio of the distances, the attractions at I and P are in turn as SP2 to SA2 : If in the quadruplicate, as SP3 ad SA3. From which since the attraction at P, in this final case, were found inversely as 3PS PI× , the attraction at 1 will be inversely as 3SA PI× , that is (on account of 3SA ) inversely as PI. And the progression to infinity is similar. Thus truly the Theorem is demonstrated. Now with everything in place as before, and with a corpuscle present at some place P, the applied ordinate DN may be found, i.e.

2DE PSPE VDN ×

×= . Therefore if IE may be drawn, that ordinate for some other corpuscle in the place I, with everything changed requiring to be changed, in order that

2DE ISIE V

×× will emerge. Put the centripetal forces

emanating from some point of the sphere E, to be in turn with the distances IE, PE, as



to n nPE IE (where the number n designates the index of the powers PE and IE) and these ordinates [of the forces] become as

2 2 and n n

DE PS DE ISPE PE IE IE

× ×× ×

, the ratio of which in turn is as

to n nPS IE IE IS PE PE× × × × . Because on account of the continued proportionals SI, SE, SP, the triangles SPE and SEI are similar, hence there becomes IE to PE as IS to SE or SA; for the ratio IE to PE write the ratio IS to SA; and the ratio of the ordinates emerges to n nPS IE SA PE× × . [Thus, the ratio of the forces on like particles at P and I due to an incremental shell at E of the sphere at E,

2

2

n n n nPE

n n n nIE

F DE PS IE IE PS IS PS PSIE IE IE IEF PE IS SE SA IS SAPE PE DE IS PE PE PE

× × ×=× × ×

= = × × = × × = × ;

but ; or SISI SIIE IEPE SA PESP.SI SP

;= = = and SPSP SPSA SP.SI SI

= = hence 1

2 2 22 1

2 2

n nn n

PEn n n n

IE

F SPDE PS IE IE PS SI SIIE IEF PE ISPE PE DE IS PE SI SP SP

−

−× × ×

× × ×= = × × = × = .]

But PS to SA is as the square root ratio of the distances PS to SI; and the square root

to n nIE PE (on account of the proportionals IE to PE as IS to SA) is the ratio of the forces at the distances PS, IS. Therefore the ordinates, and therefore the areas which the ordinates describe, and from these proportional attractions, are in a ratio composed from the square roots of these ratios. Q.E.D. [The points P and I are inverse points with respect to the sphere of radius a; hence we have ( )2a g g a SP SI= × − = × . From the first example, the force acting on P due to the

whole sphere is ( ) ( )2

22 2 2 21 4

2 ( +g ) g ag ag

F ag a g a lnπ +−

⎞⎛= − −⎜ ⎟⎝ ⎠

while the force F2 acting on

a similar corpuscle within the sphere at the inverse point is given by the same formula, and thus the forces at P and I are equal in this case. Chandrasekhar shows that such image

forces at the image points R1 and R2 are related in general by ( ) 11 1

12 2

2 n

nF RF R

+

+= , where

21 2R R a× = , and the forces are proportional to the nth power of the distance ; in this case

1n = − , and thus the forces are equal; and the other cases Newton enumerates for higher powers are as given by this result. Thus, as Chandrasekhar points out, Newton was already familiar with the image method of inverse points applied to the gravitational cases of spheres as we will now find out, and later was discovered and used for solving electrostatics problems by William Thompson.]


Book I Section XII. Translated and Annotated by Ian Bruce. Page 370 PROPOSITION LXXXIII. PROBLEM XLII.

To find the force by which a corpuscle located at the centre of a sphere is attracted to any segment of this. Let P be some body at the centre of the sphere, and RBSD a segment of this sphere contained by the plane RDS and by the spherical surface RBS. The spherical surface EFG described with centre P may cut DB in F, and the segment may be separated into the parts BREFGS and FEDG. But that surface shall not be a purely mathematical one, but a physical one, having a minimum depth. This thickness may be called O, and this will be the surface (demonstrated by Archimedes) as PF DF O× × . [we can think of O as the incremental thickness.] Besides we may put the attractive forces of the particles of the sphere to be inversely as that power of the distances of which the index is n ; and the force, by which the surface EFG pulls on the body P, will be (by Prop. LXXIX.) as

2

nDE O

PF× that is, as

2

12

n nDF O DF OPF PF−

× ×− . [For

( )2 2 2 2DE PF PD PF DF DF= − = − × .] The perpendicular FN multiplied by O shall be drawn proportional to this ; and the curved area BDI, that the applied ordinate FN [that is in modern terms, the y coordinate] will describe by the motion through the length DB, will be as the total force RBSD attracting the body P. Q.E.I.

PROPOSITION LXXXIV. PROBLEM XLIII.

To find the force, by which a corpuscle, beyond the centre of the sphere at some place on the axis of the segment, is attracted by the same segment. The body P located on the axis of this ADB may be attracted by the segment EBK. With centre P and with the radius PE the spherical surface EFK is drawn, by which the segment may be separated into two parts EBKSE and EBKDE. The force of the first part may be found by Prop. LXXXI. and the force of the second part by Prop. LXXXIII ; and the sum of the forces will be the force of the whole segment EBKDE. Q.E.I.

Scholium. With the attractions of spherical bodies explained, now it may be permitted to go on to the laws of attraction of certain bodies similarly constructed from attracting particles ; but to treat these a little less than it is considered customary. Certain more general proposition concerning the forces of bodies of this kind will suffice to be added, from which motions thence originated, on account of these being of some little use in philosophical matters.


Book I Section XIII. Translated and Annotated by Ian Bruce. Page 385

SECTION XIII. Concerning the attractive forces of non–spherical bodies.

PROPOSITION LXXXV. THEOREM XLII.

If the attraction of a body by another shall be far greater, when it is attracted in contact, than when the bodies may be separated by the smallest distance from each other: then the forces of the attracting particles in the distant parts of the attracting body, decrease in a ratio greater than the square of the distances from the particles. For if the forces between the particles decrease in the inverse square ratio of the distances; the attraction towards a spherical body, [placed at the point of contact and drawn within the attracting body with the same curvature as the surface at this point], because that attraction (by Prop. LXXIV.) shall be inversely as the square of the distance of the attracted body from the centre of the sphere, will not be sensibly increased on being in contact; and it will be increased even less by the contact, if the attraction in the more distant parts from [the point of contact] of the attracting body may decrease in a smaller ratio. Therefore the proposition is apparent for the attraction of spheres. And there is the same reasoning for concave spherical shells attracting bodies externally. And the matter is agreed upon even more with hollow shells attracting bodies within themselves, since the attractions may be removed everywhere through the cavity by opposing attractions (by Prop. LXX), and thus are even zero at the contact point itself. For if from these spherical and spherical shells any parts may be taken from places remote from the point of the point of contact, and new parts may be added elsewhere : the shapes of these attracting bodies may be changed at will, yet the parts added or removed, since they shall be remote from the point of contact, will not notably increase the excess of the attraction which arises at the point of contact. Therefore the proposition may be agreed upon for all shapes of bodies. Q. E. D. [The reader may wish to reflect on the electrostatic analogy of this theorem.]

PROPOSITION LXXXVI. THEOREM XLIII. If the forces of the particles, from which the attracting body is composed, decrease in the remote parts of the attracting body in the triplicate or greater ratio of the distances from the particles : the attraction will be much stronger in contact, than when the attracted and attracting bodies may be separated in turn even at a minimal distance. For it is agreed in the approach of an attracted corpuscle to an attracting sphere of this kind that the attraction increases indefinitely, by the solution of Problem XLI in the second and third example shown. Likewise, it is easily deduced from that example and Theorem XLI taken together, concerning the attractions of bodies towards concave-convex shells, either with the attracted bodies gathered together outside the shells, or within the cavities of these. But also by adding or taking away any attracting matter from these spheres and shells somewhere beyond the point of contact, so that the attracting bodies may adopt there some assigned figure, the proposition will be agreed upon generally for all bodies. Q. E. D.



PROPOSITION LXXXVII. THEOREM XLIV.

If two bodies similar to each other, and consisting of equally attracting material, may each attract corpuscles to themselves, with these corpuscles proportional and similar to themselves : the accelerative attractions of the corpuscles for the whole bodies shall be as the accelerative attractions of the corpuscles for all the proportional particles of these, similarly put in place in the whole bodies. For if the bodies may be separated into particles which may be proportional to the whole, and similarly situated in the whole; there will be, as the attraction of some particle of one body to the attraction in the corresponding particle in the other body, thus the corresponding attractions in the individual particles of the first body to the individual particle attractions in the other ; and on adding these together, thus the attraction in the first whole body to the attraction in the second whole body. Q. E. D. [This proposition is an assertion that a body may be considered as consisting of particles, each of which, and collections of these, up to the whole body, behave in the same way under the influence of a mutual attraction with another body, similarly constituted: Newton's bodies are held together by gravitational attractions that extend to other bodies, wholly or in part. The same rules can be extended to other unspecified forces, obeying inverse power laws, that give rise to a scaling law, as set out by Chandrasekhar pp305-306 : Thus, if the force on a corpuscle at position R relative to an extended body, and summed over the constituent particles, each of which exerts a force, resolved along the axes, of which ( ) n

dvx R r

dF R density−

∝ × is a component, proportional to its mass or

volume, and varying inversely as the nth power of the distance, is given on replacing r r;R Rα α→ → , by 3( ) n

n dvx R r

dF R densityα α −−

∝ × × , which Newton now sets out for

the certain cases 2 3 4n , ,= .] Corol. 1. Therefore if the attractive forces of the particles, with the distances of the particles increased, may decrease in the ratio of some power of any of the distances; the accelerative attractions in the whole bodies shall be as the bodies directly, and these powers of the distances, inversely. So that if the forces of the particles may decrease in the ratio of the squares of the distances from the corpuscles attracted, the bodies moreover shall be as 3 3A & B . and thus both the cubic sides of bodies, as well as the distances of the attracting bodies from the bodies, as A & B: the accelerative attractions in the bodies will be as

3 3

2 2A BA B

& , that is, as the sides from these cubic bodies A & B. If the

forces of the particles may decrease in the triplicate ratio of the distances from the attracting bodies; the accelerative attractions in the whole bodies will be as

3 3

3 3A BA B

& , that

is, equal. If the forces may decrease in the quadruple ratio; the attractions between the



bodies will be as 3 3

4 4A BA B

& , that is , inversely as the cubic sides A & B. And thus for the

others. Corol.2. From which in turn, from the forces by which similar bodies attract corpuscles similarly put in place to themselves, the ratio of the decrease of the attractive forces of the particles can be deduced in the receding of the attracting corpuscles; but only if that decrease shall be directly or inversely in some ratio of the distances.

PROPOSITION LXXXVIlI. THEOREM XLV. If the attractive forces of the equal particles of some body shall be as the distances of the places from the particles : the force of the whole body will tend towards the centre of gravity of this ; and it will be the same as with the force of a globe agreed upon from the same material and equal in all respects, and having the centre of this at the centre of gravity. The particles A, B of the body RSTV may pull some corpuscle Z by forces, which, if the particles may be equal among themselves, shall be as the distances AZ, BZ; but if unequal particles may be put in place, the forces shall be as these particles and the distances of these AZ, BZ jointly, or (if I may say thus), as these particles respectively multiplied by their distances AZ, BZ . And these forces may be set out contained by these A AZ & B BZ× × . AB may be joined and with that line cut in G so that there shall be AG to BG as the particle B to the particle A; and G will be the common centre of gravity of the particles A & B . The force A AZ× (by Corol. 2. of the laws) is resolved into the forces and A GZ A AG× × and the force B BZ× into the forces and B GZ B BG× × . But the forces and A AG B BG× × are equal, on account of the proportionals A to B and BG to AG; therefore since they may be directed in opposite directions, the mutually cancel each other out. The forces remain and A GZ B Gz× × . These tend from Z towards the centre G, and they compound the force A B GZ+ × ; that is, the same force. and as if the attracting particles A & B may constitute the components of the common centre of gravity G of these, there the components of a globe. By the same argument, if a third particle C may be added and also it may be compounded with the force A B GZ+ × tending towards the centre G; hence the force arising tends towards the common centre of gravity of that globe at G and of the particle C; that is, to the common centre of gravity of the three particles A, B, C; and it will be the same as if the globe and the particle C may be present at that common centre, comprising a greater globe. And thus it may go on indefinitely. Therefore the total force of all the particles bodies of any kind is the same RSTV, as if that body, by maintaining the centre of gravity, may adopt the figure of a globe. Q. E. D.



Corol. Hence the motion of the attracted body Z will be the same, as if the attracting body RSTV were spherical : and therefore if that attracting body either were at rest, or progressing uniformly along a direction ; the attracted body will be moving in an ellipse having at its attracting centre the centre of gravity.

PROPOSITIO LXXXIX. THEOREMA XLVI. If several bodies shall consist of equal particles, of which the forces shall be as the distances of the places from the individual particles : the force from all the forces added together, by which a certain corpuscle is attracted, will tend towards the common centre of gravitational attraction ; and it will be the same, as if the particles might group together and be formed into a globe by that attraction, with the centre serving as a common centre of gravity . This may be demonstrated in the same manner as with the above proposition. Corol. Therefore the motion of an attracted body will be the same, as if the attracting bodies, with a common centre of gravity maintained, may coalesce and be formed into a globe. And thus if the common centre of gravity of the attracting bodies may either be at rest, or progressing uniformly along a right line; an attracted body will be moving in an ellipse, by having the centre in that common attracting centre of gravity.

PROPOSITION XC. PROBLEM XLIV. If equal centripetal forces may attract the individual points of any circle , increasing or decreasing in some ratio of the distances: to find the force, by which a corpuscle may be attracted at some position on a straight line, which remains perpendicular to the plane of the circle at its centre. With centre A and with some radius AD, a circle may be understood to be described in the plane, to which the right line AP is perpendicular ; and the force shall be required to be found, by which some corpuscle P on the same is attracted. A right line PE may be drawn from some point E of the circle to the attracted corpuscle P. On the right line PA, PF may be taken equal to PE itself, and the normal FK may be erected, which shall be as the force by which the point E attracts the corpuscle P. And let IKL be the curved line that always touches the point K. The same curved line may cross the plane of the circle at L. On PA there may be taken PH equal to PD, and a perpendicular to the aforementioned curve HI may be erected crossing at I; and the attraction of the corpuscle P to the circle will be as the area AHIL taken by the altitude AP. Q. E. I. And indeed on AE the line Ee may be taken as minimal [i.e. as the increment dx]. Pe may be joined, and on PE and PA there may be taken PC and Pf themselves equal to Pe.



And since the force, by which the annulus with centre A and radius AE described in the aforementioned plane, attracts P to some point E of the body itself, is put to be as FK, and hence the force, by which that point attracts the body P towards A, is as AP FK

PE× and

the force, by which the whole annulus attracts the body P towards A ,as the annulus and AP FK

PE× conjointly [for if ( )f r if the force due to the increment Ee along Pe, the

normal force due to the whole annulus is ( )2 area of annulis APPErf r cos FKπ θ ∝ × × ]; but

the annulus itself is as the rectangle under the radius AE and with the width Ee, and this rectangle (on account of the proportionals PE and AE, Ee and CE) is equal to the rectangle PE CE× or PE Ff× [for CE

sinAE Ee PE sin PE CE PF Ffθθ× = × = × = × ]; the force, by which the annulus itself attracts the body P towards A, will be as

and AP FKPFPE Ff ×× jointly, that is, as contained by Ff FK AP× × , or as the area FKkf

multiplied by AP. And therefore the sum of the forces, by which all the annuli in the circle, which is described with centre A and interval AD, attract the body P towards A, is as the total area AHIKL multiplied by AP. Q.E.D. [We note that ( )2Ff FK AP f r drπ× → × ∫ , where the area under the integral becomes

AHIL.] Corol. 1. Hence if the forces of the points decrease in the duplicate ratio of the distance, that is, if FK shall be as 2

1PF

,and thus the area AHIKL as 1 1PA PH− ; the attraction of the

corpuscle P into the circle will be as 1 PAPH− , that is, as AH

PH . Corol.2. And universally, if the forces of the points at the distances D shall be reciprocally as some power of some distances nD , that is, if FK shall be as 1

nD, and thus

AHIKL as 1 11 1

n nPA PH− −− ; the attraction of the corpuscle P towards the circle shall be as

2 11 1n nPA PH− −−

Corol. 3. And if the diameter of the circle may be increased to infinity, and the number n shall be greater than unity ; the attraction of the corpuscle P in the whole infinite plane shall be as 2nPA − , because therefore the other term PA vanishes.



PROPOSITION XCI. PROBLEM XLV.

To find the attraction of a body situated on the axis of a solid of rotation, to the individual points of which equal centripetal forces attract in some decreasing ratio of the distances. The corpuscle P is attracted towards the solid DECG, situated on the axis of this AB. This solid may be cut by some circle RFS perpendicular to its axis, and on the radius of this FS, by some other plane PALKB passing through the axis, the length FK may be taken for the force (by Prop. XC.), by which the corpuscle P is attracted proportionally in that circle. Moreover the point K may touch the line LKI, meeting the planes of the outermost circles AL and BI at L and I ; and the attraction of the corpuscle P towards the solid will be as the area LABI. Q.E.I. Corol. 1. From which if the solid shall be a cylinder described by revolving the parallelogram ADEB about the axis AB, and the centripetal forces tending towards the individual points of this shall be inversely as the square of the distances from the points : the attraction of the corpuscle P will be towards this cylinder as AB PE PD− + . For the applied ordinate FK [i.e. the increment of the force] (by Corol. I. Prop. XC.) will be as 1 PF

PR− [for the force due to the incremental circle is proportional to

( )1 1 1 PFPF PR PRPF × − = − ]. The part 1 which

multiplied by the length AB, will describe the area1 AB× : and the other part PF

PR multiplied by the length PB, will describe the area 1 by PE AD− , that which can be easily shown from the quadrature of the curve LKI ; and similarly the same part multiplied by the length PA will describe the area 1 by PD AD− , and multiplied by the difference of PB and PA, AB will describe the difference 1 by PE AD− of the areas. From the first content1 AB× there may be taken away the latter 1 by PE AD− , and there will remain the area LABI equal to 1 by AB PE PD− + . Therefore the force, proportional to this area, is as AB PE PD− + .



[for in a similar manner to the above : the incremental force dF due to the incremental cylinder circle with diameter RS and width dr from above will be equal to

( )2 21 1 12 2 2

PR PRdrz

r PF PRr rPF PF

rdr PF PFπ π π× × = = −∫ ∫ ; the sum of the increment forces due to

these incremental cylinders will be as

( ) ( ) ( )1 12 2

2 2

2 2 2 21 12 2PB

z a zPA

zdz PB PA a PB a PA AB PE PDπ π+

⎞⎛ − = − − + + + ∝ − +⎜ ⎟⎝ ⎠∫ ]

[There follows an extended note to this rather important following result.] Corol. 2. Hence also the force will become known, by which a spheroid AGBC attracts some body P, situated on its axis outside AB ; NKRM shall be the conic section of which

the applied ordinate is ER, perpendicular to PE itself, that may always be equal to the length PD, which is drawn to that point D, at which the applied line may cut the spheroid. From the vertices of the spheroid A and B to its axis AB, the perpendiculars AK and BM may be erected equal to AP and BP respectively, and therefore crossing the conic section at K & M; and KM may be joined, taking KMRK from the same segment. Moreover let the centre of the spheroid be S and the greatest radius SC: and the force, by which the spheroid attracts the body P, will be to the force, by which a sphere with the diameter AB described cuts the same body, as

2

2 2 2AS CS PS KMRK

PS CS AS× − ×

+ −to

3

23ASPS

. And from the

same fundamentals it is possible by computation to find the forces of the spheroidal segment. [The red lines and variables have been added to aid the following integration.] [We acknowledge the work done by Chandrasekhar in solving this problem in modern terms, which is introduced and paraphrased here somewhat. The problem has also been solved by a Leseur and Janquier, which we subsequently present fully in translation. First of all, the oblate spheroid is an ellipse rotated about the major axis GC, while the minor

axis is AB. Thus, we have the customary implicit equation of the ellipse : 22

2 2 1yxa b+ = ,

taking S as the origin, SG as the positive x-axis, and SA as the positive y axis. From



Newton's construction, we have given ; ; PA AK PE ER PB BM= = = ; also,

; ; ED x SE y PS Y= = = and EP Y y= − ; hence ( )22PD x Y y= + − . The incremental force due to an incremental slice of the oblate spheroid is given by Cor.1 above as

2 21 1 zdzPF PE

PR PD x zdz

+− → − = − in this case. The force due to the whole solid of revolution

is then, ignoring the 2π factor, is then given by 2 2

1Y b

zx z

Y b

F dz+

+−

⎞⎛= −⎜ ⎟⎝ ⎠∫ ; before this

integral can be evaluated, we must write x in terms of z : from 22

2 2 1yxa b+ = we have

( ) ( )2 222

2 2 22 21; b z a b zx

a b bx a− −+ = = − ;

hence( ) ( ) ( )

( )2 2 222

2 2 2 2 2 21

1 1

1 11 1 1

b z a b z b zza b b z e z

Y b Y b Y bz

aY b Y b Y b

F dz dz dz− − −

−

+ + +

− − −− − −

⎞⎛⎞ ⎞⎛ ⎛ ⎟⎜⎟ ⎟⎜ ⎜= − = − = − ⎟⎜⎟ ⎟⎜ ⎜ ⎟⎜⎝ ⎝⎠ ⎠ ⎝ ⎠

∫ ∫ ∫ ; recall that

the eccentricity is related to a and b by the equation ( )2 2 21b a e= − , hence

( )

( )2

2 21

1

11

b z

e z

Y b

Y b

F dz−

−

+

−−

⎞⎛⎟⎜

= − ⎟⎜⎟⎜

⎝ ⎠

∫ ; Chandrasekhar shows how an integral similar to this one is

equivalent to Newton's result, on page 316 of his book, Newton's Principia for the common reader. Rather than presenting that derivation here, we will delve into the other presentation by L & S, which is more relevant to Newton's approach, and although rather long, is presumably similar to the manner in which Newton derived the result. The fact that an elliptic integral corresponding to the area KRMK arises, which Newton could not resolve by conventional means, is of some interest.]

ADDITION : Leseur & Janquier derivation of Newton's result for the oblate spheroid: [Note 542 part (x)] NKRM shall be the conic section of which the applied ordinate is ER, perpendicular to PE itself, that may always be equal to the length PD, which is drawn to that point D, at which the applied line may cut the spheroid...... Let AP a= , and AS b= shall be the semi-axis of the given curve ACB, the rotation of which generates the spheroid, the other semi-diameter

SC c, AE x= = , there will be PE a x= + , and

from the equation of the ellipse [ ( )22

2 2 1x b EDb c− + = ],



2

22 22c

bED bx x= × − ;

from which the square of the ordinate ER [ = PD ] to the curve NKRM is given by 2 2

2 22 2 2 2 2 22 2c c

b bPD PE ED a ax x bx x= + = + + + × − ; therefore, since this equation for the

curve NKRM does not to rise beyond the second degree it is agreed that curve shall be one of the conic sections : moreover it will be an ellipse, if the quantity

2

22 2c

bx x− shall be

negative, which comes about when SC or c is greater than AS or b; Truly it will be a parabola if that quantity may vanish, and thus if c b= , which eventuates when the curve ACB is a circle; and then it will be a hyperbola if that quantity is positive, that is, if AS is the longer semi-axis. [Continuing, note 543 :] ACB shall be an ellipse the axis CS of which shall be greater than the axis AS, in which case the curve NKRM will be an ellipse [i.e. c > b above], and from this ratio, the axes and vertex of this curve NKRM will be determined. The semi-axis of this ellipse NKRM may be called ON s= , and the other semi-axis OT may be called t, the distance of the vertex N from the vertex A of the curve ACB may be called p, the abscissa NE will be

p x= + , and the square of the ordinate ER will be from the equation of the ellipse

2

22 22 2 2t

ssp sx p px x× + − − − ,

which from the hypothesis of the construction was found above

2 2

2 22 2 22 2c c

b ba ax x bx x= + + + × − .

The homogeneous terms of these values may be brought together, evidently constants with constants, these which include one variable with similar ones, etc., and three equations arise, with the variable omitted:

2 2 2 2 2

2 2 2 22 22 ; ; 1t c t c t

bs s b sa sp p a s p= × − + = × − − = − .

From that third equation, with each sign changed, with the first member reduced to a common denominator, and with the terms inverted, there becomes

2 2 2 2

2 2 2 2 22and s b b t

t c b c bs .

− −= = Then the second equation

2 2

2c tb s

a s p+ = × − , with the terms

multiplied by 2

2st

, with the reduction made of the first member to the same denominator,

and with the substitution made of the above value of 2

2st

found, will become



2 2b

c bs p ba cc

−− = × + . And then, with the terms of the first equation

2

22 22t

sa sp p= × −

multiplied 2

2st

, each with the sign changed and with s2 added, becomes finally 2

2 22 2 2 22b

c bs a s sp p

−− = − + , in which new equation the second term shall itself be the

square of the quantity s p− , with the value of this substituted in the first value found, and in place of s2 in the first term also with the value of this substituted, there becomes

( )2 2

2 2 2 22 2 2b b

c b c bt a ba c

− −× − = × + , and with each term divided by ( )

2

2 2b

c b− with a2

transposed, and by reducing the second term to a common denominator, and with the

common terms canceling each other, there shall be 2

2 22 2 22c

c bt a ab c

−= × + + , or because

PS a b= + , thus, there becomes 2

2 22 2 2 2c

c bt PS b c

−= × − + , certainly

2

2 22 2 2 2CS

CS ASOT PS AS CS

−= × − + , all of which terms are given, therefore with this

found the remaining terms pertaining to the ellipse can be conveniently found. With regard to the following note, from these we will determine the value of the quantity

2 2 2

2t s PO

t+ − that is equal to the quantity

2

2 2 2CS

PS AS CS− +, which thus may be put in

place with the above values found. From the third equation, there is 2 2

2 22 b t

c bs

−= , hence

there will be 2 2 2 2 2 2 2 2

2 2 2 22 2 b t c t b t c t

c b c bs t + −

− −+ = = , and thus

2 2 2 2

2 2 2 2 2s t c CS

t c b CS AS+

− −= = . Truly there

will be AO s p= − , and PO PA AO a s p= + = + − , and since there shall be

( ) ( )2 2 2s p c b ba c− = − × + , from the second equation, there is 2 22b

c bPO a ba c

−= + × + ,

with which value reduced to a common denominator, and with terms canceling out, there is

2

2 2c

c bPO a b

−= × + or

2

2 2CS

CS ASPS

−= × , and since there shall be

2

2 22 2 2 2CS

CS ASt PS AS CS

−= × − + , there is 2 2 2 2

PO PSt PS AS CS− +

= and

2

2 2 2 2 2 2PO CS PSt CS AS PS AS CS− − +

= × . From which there is finally

2 2 2 2 2 2

2 2 2 2 2 2 2 2s t PO CS CS PS

t CS AS CS AS PS AS CS+ −

− − − += − × , or ( )2 2

2 2 2 2 21CS PSCS AS PS AS CS− − +

= − , and on

being reduced to the same denominator, with terms canceling,

2 2 2 2

2 2 2 2 2 2 2 2CS AS CS CS

CS AS PS AS CS PS AS CS− +

− − + − += × = with the numerator and denominator divided by



2 2CS AS− . Therefore there is 2 2 2 2

2 2 2 2s t PO CS

t PS AS CS+ −

− += . Q.e.d.

[Continuing, note 544 :] Moreover, let the given curve ACB be a circle, thus so the spheroid arising from the rotation of this shall be a sphere, the curve NKRM then will be a parabola, for with everything that has been said remaining in place from the previous note, there will be as before, PE a x= + , and from the equation of the circle,

2 22EP bx x= − , from which there shall be PF squared = 2 2 2 2 2 22 2 2 2PE EF a ax x bx x a ax bx+ = + + + − = + + ; since therefore the ordinate ER to the curve NKRM may be taken equal to PF, the square of this ordinate will be equal to the abscissa itself multiplied by a constant quantity, but not increasing beyond the first order, which is a property of the parabola. Therefore the latus rectum of this parabola may be called 1, the distance of the vertex N from the vertex A of the curve ACB may be called p, the abscissa NE will be p x+ , and from the equation of the parabola, the square of the ordinate ER =1 1p x+ this value may be united with the value of the same ER2

found above, 2 2 2a ax bx+ + , the constant terms with constants, and those which include the variable with similar ones, become two equations 21 and 1 2 2 2p a , a b PS ,= = + =

and thus 2 2

2 2 2a PA

a b PSp += = ; and since from the equation of the parabola , there shall be 2 1ER p x= × + , there will be

2

2ER

PSp x NE+ = = ; and since the parabolic area between the abscissa, the ordinate, and the curve intercepted shall be equal to two thirds of the

rectangle of the abscissa by the ordinates, the area of the parabola 32 3

32 3ER ERPS PSNER = = ,

and because, from the construction, the ordinates erected at A and B shall equal PA and PB, the parabolic area will be

3

3PAPSNEK = , and the parabolic area

( )33 23 3

PA ASPBPS PSNBM += = , and the difference of these areas AKRMB corresponding to the

axis of the sphere AB, will be 2 2 36 12 8

3PA AS PA AS AS

PS× + × + , and then with the trapezium

AKMB removed, the segment of the remaining parabolic segment will be equal to 32

3PAPS ,

for the trapezium AKMB is equal to 12 AB AK BM× + or, since (because

12 and 2AB AS , AK PA, BM PB PA AS= = = = + ) it is equal to 22 2AS PA AS× + , and by

reduction to the denominator 3PS or 3 3PA AS+ it equals 2 2 36 12 6

3AS PA AS PA AS

PS× + × + , that

taken from the area AKRMB = 2 2 36 12 8

3PA AS PA AS AS

PS× + × + leaves

323ASPS . Q.e.d.

[Continuing, note 545 :] The force, by which the spheroid attracts the body P, will be to the force, by which a sphere with the diameter AB described cuts the same body, as

2

2 2 2AS CS PS KMRK

PS CS AS× − ×

+ −to

3

23ASPS

.

Just as the solution of this problem may be put in place, the second curve described AB, the ordinates of which for the single point E shall be equal to the force by which the



body P is attracted by a circle the radius of which is ED; that force is, by Cor. 1. Prop XC, as 1 PE

PD− , OE shall be the abscissa of this curve assumed from the point O (with the centre of the curve NKRM just noted 543 determined) and it may be called z, the fluxion [i.e. differential] of this will be dz, and thus the differential of the area of the curve which shows the force of the spheroid will be PE

PDdz dz− , and since there shall be and PE PO OE PO z PD ER= − = − = , the ordinates of the curve NKRM, from the

construction, and there shall be, as is easily deduced from note 543, 2 2tsER s z= − , the

differential of this area shall be 2 2 2 2t t

s s

POdz zdzs z s z

dz− −

− + . Of the positive terms 2 2t

s

zdzs z

dz−

+

the fluent [i.e. the indefinite integral] is 2 2stz s z+ − but as

2 2

2 22 2 2 2 and s s t s

t st tz OE s z s z ER= − = × − = , the corresponding differential of the

positive term is 2

2st

OE ER− , and the total area of the corresponding line OA is 2

2st

OA AK− , from which the corresponding second area of the part OB required to be

taken away, as the curve which the force of the spheroid is not lead to express, and which is

2

2st

OB BM− , and as by the construction and AK AP, BM PB BA AP= = = + , truly

there will be the integral 2 2 2 2

2 2 2s s s tt t t

OA OB AP BA AP AB AB AB +− − × − − = + = × .

The integral of the third term

2 2ts

POdzs z−

is found thus; the differential

of the elliptic sector of the ellipse TOK 12

2 2

stdz

s z−is multiplied by 2

2POt

and the

proposed term arises

2 2ts

POdzs z−

, from which the integral of

the term proposed is that elliptic sector TOK multiplied by 2

2POt

, but because

the area sought does not correspond to the whole area OA, but rather to the part AB of this, indeed the integral of the area sought found from the third term is the sector 2

2POt

TOK × with the

sector 22PO

tTOM × removed, or the

sector 22PO

tMOK × . But the sector

MOK is divided into the rectilinear figure MOK and the curved figure MRK; the triangle MOK becomes 1

2 PO AB× , for the right line MK may be produced and will extend to P,



because and PA AK PB BM= = , the whole triangle certainly 1 12 2OMP OP BM OP PB= × = × , and the triangle 1 1

2 2OKP OP AK OP AP= × = × , from which with triangle OKP subtracted from triangle OMP, there remains the triangle

( )1 12 2OMK OP PB AP OP AB= × − = × . From which the integral sought of this third term

is 2

2 2 2 22 2 21

2PO PO PO POt t t t

OP AB MRK AB MRK× × + × = × + × , which taken from the integral

of the positive terms 2 2

2s t

tAB +× becomes

2 2 2

2 22s t PO PO

t tAB MRK+ −× − × , since there shall

be therefore 2 2 2 2

2 2 2 2s t PO CS

t PS AS CS+ −

− += and 2 2 2 2

PO PSt PS AS CS− +

= by note 543, because

2AB AS= , the integral sought is 2

2 2 22 2AS CS PS MRK

PS AS CS× − ×

− +.

But if the curve ACB shall be a circle, truly the spheroid becomes a sphere, making CS AS= and the segment MRK becomes

3

223

ASPS

and thus that formula is changed into this 32 3 32 2 4 3

3 32 2 2 2 2

2 2 23

PS ASPSAS AS AS AS AS

PS AS AS PS PS

×× − −− +

= = , which expresses the force of the sphere; and thus

with the expression of the force of the spheroid and the force of the sphere divided by the common multiple 2; the force of the spheroid to the force of the sphere shall be as

2

2 2 2AS CS PS MRK

PS AS CS× − ×

− + to

3

23ASPS

. Q.e.d.

It is also possible to determine the force of the sphere, by this calculation, as first there shall be 2 the abscissa PA a, AB b, AE x, PF v,= = = = and there will be

2 2 22PE a ax x= + + , and 2 22EF bx x= − , from the equation of the circle, and thus 2 2 2 or 2 2PF v a ax bx= + + , from which there is found

( ) ( )2 2 2

2 2 and v a vdv vdva ba b a bx dx−+× + × +

= = = , and ( )2 222 and a ab v dx dv

PF a ba bPE + ++× +

= = . And thus, with

the derivative of the area expressed, the force of the sphere shall be by Cor. I Prop. XC, as PEdx

PFdx − , this differential [or fluxion] will be as ( )

2 2

22

2a ab v

a bdx dv+ +

× +− , of which the

integral [or fluent] is ( )

2 313

22

2

a v abv v

a bx Q+ +

× +− + , where Q is a constant, which must vanish on

putting 0x = and x a= , and thus ( ) ( )

3 2 3 3 21 43 32 2

2 2

2 20 and a a b a a a b

a b a bQ , Q+ + +

× + × +− + = = ; but the force

of the whole sphere may be obtained if there becomes or 2 and or 2x AB b v PB a b= = + , and thus there is

( )

3 2 3 2 3 2 2 384 13 3 3

22 4 2 4

22 a a b a a b a a b ab b

a bb + − − − − − −

× ++ which reduces to

( )

223

222 b

a bb

× +× , or on putting AS

for b and PS for a b+ , the total force of the sphere is 3

223

ASPS

. Q.e.i.



Corol. 3. But if the corpuscle may be placed within the spheroid on the axis, the attraction will be as its distance from the centre. This may be easily proven by this argument, whether the particle shall be on the axis, or on some other given diameter. Let AGOF be the attracting spheroid, S its centre, and P the attracted body. Through that body P there are drawn both a semi-diameter SPA, as well as any two right lines DE and FG, hence crossing the spheroid at D and E, F and G; PCM and HLN shall be the surfaces of two interior spheroids, similar and concentric with the exterior, the first of which may pass through the body P, and may cut the right lines DE and FG in B and C, the latter may cut the same right lines in H and I and in K and L. But all the spheroids have a common axis, and hence the parts of the right lines thus intersected are mutually equal to each other : DP and BE, FP and CG, DH and IE, FK and LG ; therefore as the right lines DE, PB and HI may be bisected at the same point, as well as the right lines FG, PC and K L. Now consider DPF and EPG to designate opposing cones, described with infinitely small vertical angles DPF and EPG, and the lines DH and EI to be infinitely small also; and the small parts cut of the spheroidal surfaces DHKF and GLIE, on account of the equality of the lines DH and EI, will be in turn as the squares of their distances from the corpuscle P, and therefore attract that corpuscle equally. And by the same reason, if the spaces DPF and EGCB may be divided into small parts by the surfaces of innumerable similar concentric spheroids having a common axis, all these each attract the body P equally on both sides in the opposite directions. Therefore the forces of the cone DPF and of the segment of the cone EGCB are equal, and by opposing each other cancel out. And the ratio of all the forces of the material beyond the inner spheroid PCBM. Therefore the body P is attracted only by the inner spheroid PCBM, and therefore (by Corol.3. Prop. LXXII.) the attraction of this is as the force, by which the body A is attracted by the whole spheroid AGOD, as the distance PS to the distance AS. Q.E.D.

PROPOSITION XCII. PROBLEM XLVI. For a given attracted body, to find the ratio of the decrease of the attraction of the centripetal forces at the individual points of this. From some given body either a sphere, a cylinder, or some other regular body is required to be formed, of which the law of attraction can be found, for any decrease of the ratio you please (by Prop. LXX, LXXXI, & XCI.). Then from the experiment performed the force of attraction at different distances can be found, and the law of attraction thence revealed for the whole body will give the ratio of the decrease of the forces of the individual parts, as was required to be found. [Such experiments were eventually performed by Henry Cavendish a century later.]



PROPOSITI XCIII. THEOREM XLVII.

If a solid may consist of a single plane surface [part], but with the remaining parts from an infinitude of planes, with equal particles attracting equally, the forces of which in the depths of the solid decrease in some ratio of the power greater than the square, and a corpuscle may be attracted by the force of the whole solid, by each of the plane parts put in place : I say that the attractive force of the solid, receding away from the plane surface of this, decreases in the ratio of some power, the root [i.e. base] of this power is the distance of the corpuscle from the plane, and the index by three less than the index of the power of the distances. Case 1. Let LGl be the plane in which the solid is terminated. But the solid may be put in place from plane parts of this towards I, and resolved into innumerable planes mHM, nIN, oKO, &c. parallel to GL itself. And in the first place the attracted body may be put in place at C outside the solid. But CGHI may be drawn from these innumerable perpendicular planes, and the attractive forces of the points of the solid may decrease in a ratio of the powers of the distances, the index of which shall be the number n not less than 3. Hence (by Corol. 3. Prop. XC.) the force, by which some plane mHM attracts the point C, is inversely as 3nCH − . In the plane mHM, the length HM may itself be taken reciprocally proportional of 2nCH − , and that force will be as HM. Similarly in the individual planes lGL, nIN, oKO, &c. the lengths may be taken GL, IN, KO, &c. reciprocally proportional to 2 2 2n n nCG ,CI ,CK ,− − − &c.; and the forces of the same planes will be as the lengths taken, and thus the sum of the forces as the sum of the lengths, that is, the whole force is as the area GLOK produced towards OK indefinitely. But that area (by the known methods of quadratures) is reciprocally as 3nCG − , and therefore the force of the whole solid is inversely as 3nCG − . Q. E. D. [Note from L & J : Let CH x,= then MH will be as 2

1nx − by hypothesis, and the element

of the area GLMH – corresponding to an element of the force- will be as 2ndx

x − , and thus

the area itself will be as ( ) 31

3 nn xQ −−− , for some constant Q, which vanishes when

x CG= . Whereby ( ) 31

3 nn CGQ −−= and the area GLMH, becomes as

( ) ( )3 31 1

3 3n nn CG n CH− −− −− . But since CH becomes infinite, the term ( ) 3

13 nn CH −−

vanishes and



the area GLOK becomes infinite as ( ) 31

3 nn CG −−, or on account of the given 3n − , inversely

as 3nCG − .] Case 2. Now the corpuscle C may be placed in that part of the plane lGL within the solid, and take the distance CK equal to the distance CG. And the part of the solid LGloKO, terminated by the parallel planes lGL, oKO, the corpuscle C situated in the middle will not be attracted by any part, with the counter actions of the opposing points mutually being removed from the equality. On that account the corpuscle C is attracted only by the force of the solid situated beyond the plane OK. But this force (by the first case ) is inversely as 3nCK − , that is (on account of the equality of CG and CK) inversely as 3nCG − . Q. E. D. Corol. I. Hence if the solid LGIN may be bounded by the two infinite parallel planes LG and IN on each side ; the attractive force of is known, by taking away from the attractive force of the whole infinite solid LGKO, the attractive force of the part beyond NIKO, produced from KO indefinitely. Corol. 2. If the ulterior part of this infinite solid, when the attraction of this taken with the attraction of the nearer part is hardly of any concern, it may be rejected : the attraction of that closer part may decrease in increasing the distance approximately in the ratio of the power 3nCG − . Corol. 3. And hence if some finite body, with a single plane part, may attract a corpuscle from the middle region of this plane, and the distances between the corpuscle collated with the dimensions of the attracting body shall be extremely small, it may be agreed moreover that the body be attracting with homogeneous particles, the attractive forces of which decrease in some ratio of the power greater than the square of the distances; the attractive force of the whole body will decrease approximately in the ratio of the powers, the base of which shall be that very small distance, and the index by three less than the index of the former power. Concerning a body constructed from particles, the attractive forces of which may decrease in the triplicate ratio of the distances, the assertion is not valid; because hence, in that case, the attraction of these further parts of the infinite body in the second corollary, is always infinitely greater than the attraction of the closer parts.



Scholium.

If some body may be attracted perpendicularly towards a given plane, and the motion of the body may be sought from some given law : the problem may be solved by seeking (by Prop. XXXIX) the right motion of the body descending to this plane, and (by Corol. 2 of the laws) compounding with that uniform motion, following lines made parallel to the same plane. And conversely, if the law of attraction of the body towards the plane along lines made perpendicular may be sought, so that the attracted body may be moving along some given curve from that condition, the problem may be solved by working according to the example of the third problem. Moreover the operations are accustomed to be drawn together by resolving the applied ordinates in converging series. Just as if to the base A [i.e. the x coordinate] the applied ordinate may be given a length B at some given angle, which shall be as some power of the base

mnA [i.e. or

m nn mB A A B= = ]; and the force by which the body shall be moving

along some curved line, following the position of the applied ordinate, either attracted to or repelled from the plane base, so that it will always touch the upper end of the applied ordinate [i.e. the y coordinate]: I may suppose the base to be increased from the minimum

O, and the applied ordinate ( )mnA O+ to be resolved into an infinite series

2

2m m n m nn n nm mm mn

n nnA OA OOA− −−+ + &c. and the term of this, in which O is of two

dimensions, I suppose to be proportional to the force, that is, the term 2

2m n

nmm mnnn OOA

−− .

Therefore the force sought is as 2m n

nmm mnnn A

−− , or what is moreover, as 2m n

mmm mnnn B

−− . So that if the applied ordinate may touch a parabola, with 2 and 1m , n= = arising: the force becomes as given 2B0, and thus may be given [as constant]. Therefore with a given force a body will be moving in a parabola, just as Galilio showed. But if the applied ordinate may touch a hyperbola, with 0 1 and 1m , n= − = arising ; the force becomes as 32A− or

32B : and thus the force, which shall be as the cube of the applied ordinate, will cause the body to move in a hyperbola. But with propositions of this kind dismissed, I go on to certain other kinds of motion, which I shall only touch on.


Book I Section XIV. Translated and Annotated by Ian Bruce. Page 411

SECTION XIV.

Concerning the motion of the smallest bodies, which may be set in motion by attracting centripetal forces towards the individual parts of some great body.

PROPOSITION XCIV. THEOREM XLVIII.

If two similar media may be distinguished in turn, with each space bounded by parallel planes, and a body in passing through this space is attracted or repelled perpendicularly towards either medium, and not set in motion or impeded by any other motion; moreover the attraction shall be the same at equal distances from each plane and taken in the same direction of each: I say that the sine of the incidence in each plane will be in a given ratio to the sine of the emergence from the other plane. Case I. Take two parallel planes Aa, Bb. A body is incident on the first plane Aa along the line GH, and in its whole passage through the space within the medium it may be attracted or repelled towards the medium of incidence, and from that action will describe the curved line HI, and may emerge along the line IK. At the plane of emergence Bb there may be erected the perpendicular IM, crossing both the line of incidence GH produced in M, as well as the plane of incidence Aa in R; and the line of emergence KI produced crosses HM in L. A circle may be described with centre L and radius LI, cutting HM at both P and Q, as well as MI produced in N; and initially if a uniform attraction or impulse may be put in place, the curve will be the parabola HI (from Galileo's demonstration) [i.e. a constant vertical force acts along a diameter of the parabola RI, while no force acts along the horizontal direction between the plate surfaces. In the original formulation of the parabola by Apollonius – see below, the latus rectum l, not drawn on this diagram, multiplied by the ordinate x, or distance along some diameter of the parabola from a point on the curve to the mid-point of a diameter 2y, are related in skew coordinates by 2l.x y= , similar to our 2 4y ax= .], a property of which is this : that the rectangle under the given latus rectum and the [ordinate]line 1M shall be equal to HM2; but also the line HM will be bisected in L. From which if a perpendicular LO may be sent to MI, MO and OR will be equal; and with the equal lines ON and OI added, the totals MN and IR become equal. Hence since IR may be given, MN is given also; and the rectangle NM.MI to the rectangle under the latus rectum by IM is in a given ratio to HM2. But the rectangle NM.MI is equal to the rectangle PM.MQ, that is, to the difference of the squares ML2 and PL2 or LI2; and HM2 has the given ratio

2

4ML : therefore the given

ratio 2 2 2 to ML – LI ML , and by converting the ratio LI2 to ML2, and with the square root taken, the ratio LI to ML is given. But in any triangle LMI, the sines of the angles are proportional to the opposite sides.



[i.e. 2 2 2

2 2 2 244 4PM .MQNM .MI ML –LI LI

HM HM ML / ML= = = − is in a given ratio, and thus LI

ML is given.]

Therefore the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given. Q. E. D. [Leseur & Janquire, note 551 (g) : With the diameter HT drawn through the point H, and with the right line HV the applied ordinate to the other diameter IR, and with IT the ordinate from the point I to the diameter HT, on account of the parallels MI, HT (by Theorem I Apol. de Parabola), and the parallels MH, IT (by Lem. 4, Apol. de Conic.),

and MI HT IT MH= = (by 34. Book I, Eu. Elem.) ; but (by Theorem I. de Parabola), the square of the ordinate TI is equal to the rectangle under the given latus rectum of the diameter HT and with the abscissa HT, therefore the rectangle under the given latus rectum and the line MI is equal to the square HM. And since HM is a tangent to the parabola at H and thus (by Cor. 1, Lem. 5, de Conic.) IM VI= and HV is parallel to LI, there will also be HL LM= . Q.e.d. Here the latus rectum is a fixed length l not drawn on the diagram, for which

2 2 etcl.TH TI ;l.LH XE ; .∝ ∝ , where TH, TI etc. are oblique ordinates, in the original formulation of Apollonius, and which also has a place when the chord becomes a tangent to the parabola.] Case 2. Now the body may pass successively through several spaces bounded by the planes AabB, BbcC, &c. and may be disturbed by a force which shall be uniform in every one apart, but which differ in different spaces; and now by the demonstration, the sine of incidence in the first plane Aa will be in a given ratio to the sine of emergence from the second plane Bb ; and this sine, which is the sine of incidence in the second plane Bb, will be in a given ratio to the line of emergence from the third plane Cc; and this sine will be in a given ratio to the sine of emergence from the fourth plane Dd, and thus indefinitely : and from the equation, the sine of incidence in the first plane to the sine of emergence from the final plane will be in a given ratio. Now the intervals between the planes may be minimised and the number may be increased indefinitely, so that from that attraction or from the action of the impulse, the following law assigned in some manner, will be



continually returned; and the ratio of the sine of incidence in the first plane to the line of emergence in the final plane, always proving to be given, also even now will be given. Q. E. D.

PROPOSITION XCV. THEOREM XLIX. With the same in place; I say that the velocity of the body before incidence is to the

velocity of the body after emergence, as the emergent sine to the incident sine. AH and Id may be taken equal, and the perpendiculars AG and dK may be erected meeting the lines of incidence and emergence GH and IK in G and K. On GH there may be taken TH equal to IK, and Tv may be sent normally to the plane Aa. And (by Corol. 2 of the laws) the motion of the body may be distinguished into two parts, the one perpendicular to the planes Aa, Bb, Cc, &c., the other parallel to the same. The force of attraction or of impulses, by acting along the perpendicular lines, changes no motion along the parallels, and therefore the body may complete equal intervals in equal times following the parallels, which are between the line AG and the point H, and between the point I and the line dK; that is, the lines GH and IK are described in equal times. Hence the velocity before incidence is to the velocity after incidence as GH to IK or TR, that is, as AR or Id to vH, that is (with respect to the radii TH or IK) as the sine of emergence to the sine of incidence.

PROPOSITIO XCVI. THEOREMA L. With the same in place, and because the motion before incidence shall be greater than after : I say that the body, on being inclined to the line of incidence, finally will be reflected, and the angle of reflection becomes equal to the angle of incidence.

For consider the body to describe parabolic arcs between the parallel planes Aa, Bb, Cc. &c. as above, and those shall be the arcs HP, PQ, QR, &c. And that line of incidence GH shall be oblique to the first plane Aa, so that the sine of incidence shall be to the radius of the circle, of which it is the sine, in that same ratio as the sine of incidence to the sine of emergence from the plane Dd, in the space DdeE : and on account of the sine of emergence now made equal to the radius, the angle of emergence will be a right angle



and thus the line of emergence will coincide with the plane Dd. The body may arrive at this plane at the point R; and because the line of emergence coincides with the same plane, it is evident that the body cannot progress further towards the plane Ee. But nor can it go on along the line of emergence Rd, as it is always attracted or repelled towards the medium of incidence. And thus it will be returned between the planes Cc, Dd, by describing the arc of the parabola QR2 of which the principal vertex is at R (just as Galileo has shown); and it will cut the plane Cc in the same angle at q, as before at Q; then by progressing in the parabolic arcs qp, ph, &c. by similar and equal arcs to the previous arcs QP, PH, it will cut the remaining planes in the same angles at p, h, &c. as before at P, H, &c. and finally it will emerge with the same obliquity at h, by which it began at H. Consider now the intervals between the planes Aa, Bb, Cc, Dd, Ee, &c. to be diminished indefinitely and the be increased in number indefinitely, so by that action of attraction or of impulse following some designated law it may be returned continually ; and the angle of emergence always arising equal to the angle of incidence, will remain even then equal to the same. Q. E. D.

Scholium. The reflection and refraction of light are not much dissimilar to these attractions, made following a given ratio of the secants, as Snell found, and by the ratio of the sines as consequence, as set out by Descartes. Just as light can be propagated from the sun both successively from the start and through space in a time of seven or eight minutes to arrive at the earth, now agreed upon through the phenomena of the moons of Jupiter, confirmed from the observations of different astronomers. But the rays present in air (as Grimaldi found some time ago, with the light admitted through a hole into a darkened chamber, and that itself I have tried too) in passing close to either the edges of opaque or transparent bodies (such as are circles and rectangular edges of gold, silver, and brass coins, or of knives, or the fractured edges of stones or glass) may be curved around bodies, as if attracted to the same ; and with these rays, which in passing approach closer are curved more, as if attracted more, as that itself I have carefully observed also. And those which pass at greater distances are curved less; and at greater distances they are curved a little towards the opposite direction and form bands of three colours. In the figure s may designate the shape edge of a knife, or of some kind of wedge AsB; and gowog, fnunf, emtme, dlsld are the rays, with the arcs curved towards the knife edge; and that more or less with the distance of these from the knife edge. But since there is a lack of curvature of the rays in air without the knife edge, also the rays which are incident on the knife edge must not be curved in the air before they reach the knife. And the account is the same of rays incident on glass. Therefore refraction happens, not at the



points of incidence, but by a little continuation of the rays, made partially in the air before they touch the glass, partially (lest I am mistaken) in the glass, after they have entered that: as with the incident rays ckzc, biyb,ahxa at r, q, p, and with the curvature traced out between k and z, i and y, h and x. Therefore on account of the analogy which there is between the propagation of rays of light and the progress of bodies, it is seen that the follow propositions be adjoined for optical uses; meanwhile concerning the nature of rays, (whether they may be bodies or not) nothing generally is disputed, but only that the trajectories of bodies are very alike to determining the trajectories of rays.

PROPOSITION XCVII. PROBLEM XLVII. Because that sine of incidence placed in some surface shall be in a given ratio to the sine of emergence; and because of the in-curving nature of the path of bodies made in the shortest space near that surface, that may be possible to consider as a point: to determine the surface, which all the corpuscles successively arising from a given place may be able to converge to, at another given place. Let A be the place from which the corpuscles diverge ; B the place at which they must converge [thus, a theory for a lens is presented here]; CDE the curved line which may describe the surface sought by rotating about the axis AB ; D and E any two points of this curve ; EF and EG perpendiculars sent to the paths AD and DB of the bodies . The point D may approach to the point E; and the final ratio of the line DF, by which AD may be increased, to the line DG, by which DB is being diminished, will be the same as the ratio which the sine of incidence has to the sine of emergence. [For from the added normal line in the diagram, we have

and :DG siniDF DFDE DE sin r DGsini sin r= = = as required.]

Therefore the ratio may be given of the increment of the line AD to the decrement of the line DB ; and therefore, if some point C may be taken on the axis AB, through which the curve CDE must pass, and the increment CM of AC itself may be taken, to the decrement of BC itself, CN in that given ratio, and with the centres A and B, and with the intervals AM and BN two circles may be described cutting each other mutually at D; that point D touches the curve sought CDE, and the same curve required to be touching everywhere will be determined. Q. E. I. Corol. 1. But on requiring now that the point A or B may go off to infinity, or to be transported to other parts of the point C, all these figures will be had, that Descartes set out in his optics and geometry relating to refractions. The invention of which Descartes has concealed, may be seen to be explained in this proposition. [Note (b) L & J relating to this corollary : Which lines indeed Descartes calls A5, A6, or A7, A8 in his Geometry. page 50 et seq., and these are called here by Newton CM, CN, and with the others the construction is as by that first author. (The interested reader may



wish to examine the Dover edition in translation of : The Geometry of Rene Descartes, circa p.110) From which it is evident, if the point C, between the points A and B, and the point N between C and M, the first Cartesian shall be situated to be described by the Newtonian construction; if for the remaining points A, C, B, M, the point N may be located between C and A, the second Cartesian oval will be obtained; truly if the point B may be moved to other regions of the point C beyond A, and the point C shall be between A and N, and M, the third Cartesian oval will be obtained, and with the same positions, if the point N shall be between C and A, the fourth Cartesian oval will be set out. Again, if the point A or B may go off to infinity so that the incident or refracted ray are parallel, then through the point M or N a perpendicular will be erected, that will cut a circle to be described with centre B or A, and with radius BN or AM , at some point D, of the curve CDE, which shall be either an ellipse of hyperbola, as may be apparent from an easy calculation, and these are the figures for which Descartes has described the use in optics in Chapter 8. (We may note here that the usual modern approach to establishing such curves, ideal of course for designing lenses free from spherical aberration, is to use Fermat's Principle of least time, where all the path lengths of the rays are equal, so that above we would have A Bn AD n DB× = × ; from which of course we have at once A Bn DF n DG× = × ); the different ovals which arise both for reflection and refraction are conic sections.)] Corol. 2. If a body incident on some surface CD, along a right line AD, acted on by some law, may emerge along some other right line DK, and the curved lines CP and CQ may be understood to be drawn from the point C always perpendicular to AD and DK themselves: the increments of the lines PD and QD, and thus these lines themselves arising from these increments PD and QD, will be as the sine of incidence and of emergence inversely: and conversely. [In this case the path length increases are again equal, or, the increase in one is reduced by the decrease in the other to equality. The added lines and letters R and S show how this can be proven, using the cyclic points RSDP and similar triangles. The Schaum Outline book Optics by Eugene Hecht is particularly illuminating on Descartes Ovoids at an elementary level. One wonders why Newton did not make use of Fermat's Principle, which would apply to small bodies as well as waves.]



PROPOSITION XCVIII. PROBLEM XLVIII.

With the same in place; and some attractive surface CD may be described around the axis AB, regular or irregular, through which bodies leaving from the given place A are able to pass through: to find another attractive surface EF, by which that body may be made to converge to the given place B. With AB joined it may cut the first surface in C and the second in E, at some assumed point. [The idea being that the refracting surface CD is present already, and a new element is to be added to the surface EF, to focus the particles travelling along AP at B : thus, the position of F is required to be found.] And on putting the sine of incidence on the first surface to the sine of emergence from the same, and with the sine of emergence from the second surface to the sine of incidence in the same, as some given quantity M to another given N: then produce AB to G, so that there shall be BG to CE as M N− to N; as well as produce AD to H, so that AH shall be equal to AG, as well also DF to K, so that there shall be DK to DH as N to M. Join KB, and with centre D and with radius DH describe the circle meeting KB produced in L, and draw BF itself parallel to D L : and the point F may touch the line EF [the emergent ray, but not a tangent at F], which rotated about the axis AB will describe the surface sought. Q. E. F. [So far, we have been introduced to the ratio M

N equal to the ratio of the sines of

incidence and emergence, so that we may write 1 1i eM Nsin sinθ θ= , then corresponding to

the Snell's law, we have 1 1 and i e

i en nM v N v= ∝ = ∝ , where ni, ne and vi, ve are the

refractive indices and velocities in the mediums i and e. The time to traverse the distance CE with the lower speed N is proportional to CE

N , which otherwise without the lens

would be traversed in a time proportional to CEM ; the extra time is given by

( )CE CE CEN M NM M N− = × − , and is equivalent to an extra distance BG and an extra time

BGM in the first medium, so that ( ) ; or CE M N BG CE BG

MN M N M N−

−= = . Thus, AH and AG correspond to the equal lengths that would be traversed in the actual equal times taken to traverse the system by any path, if no change in the medium occurs. Optically, it is the equivalent vacuum path distance. In the same way, a similar ratio results for the ray DF: or NDK DK DH

DH M N M= = , or, the time to traverse DK at the reduced speed N is the same as the time to traverse the free



space with the speed M. In turn, this means that the ratio of the sides of the respective triangle are as the ratio of the speeds, and so of angles of incidence and emergence. Finally, from the construction, DL is the equivalent free space distance from D to H. Hence, if we move the line DL up parallel to itself, a path DF in the slower medium and a path FB in the faster medium replaces the original path DL; when the ratio FK to FR has the required value, the appropriate point F has been reached. We now give Newton's explanation, in terms of ratios only; note that Newton often uses added or separated ratios to achieve his ends: that is , if then a c a a c

b d b b d, ±±= = ; thus, if for some k other than 1,

and then a c a ka ab d b kb bc ka d kb, ± ±± ±= = = = , or the original result follows simply by cross-

multiplying.] For consider the lines CP and CQ with respect to AD and DF themselves, and the lines ER, ES with FB, FD themselves to be everywhere perpendicular, and thus QS, is always equal to CE itself; [i.e. the times to traverse the sections CE and the composite section QS are always equal; for along ACEB, the angles of incidence and refraction are all 900, and the argument in the above note can be used to determine the length BG, from which the 'time of flight' can be found. The lengths in the following ratios have physical significance : thus, DL is the equivalent length in the first medium to travel from D to L, etc. These hypothetical lengths are shown dotted. Thus M DL FB

N FQ QD−−= arises from the same time to traverse the

numerator distance with speed M as does the denominator with speed N, etc.] and there will be (by Corol. 2. Prop. XCVII.), PD to QD as M to N, and thus as DL to DK or FB to FK;

[i.e. = = =PD M DL FBQD N DK FK ; the last step by similar triangles, ]

and on separating, as DL FB− or PH PD FB− − to FD or FQ QD− ;

[i.e. = = or = ;DL FB DL FB PH PD FB M DL FB PH PD FBDK FK FD FD N FQ QD FQ QD

− − − − − − −− − −= ]

and on adding, as PH FB− to FQ, that is (on account of PH and CG, QS and CE being equal) to CE BG FR CE FS+ − − .

[i.e. CE BG FRM PH FBN FQ CE FS

+ −−−= = ]

Truly (on account of the proportionals BG to CE and M N− to N) also there is CE BG+ to CE as M to N; and thus separated FR to FS as M to N; and therefore (by Corol. 2. Prop. XCVII.) the surface EF collects the body, incident on that second line itself DF , to go along in the line FR to the place B. Q. E. D. [Essentially, all the paths have the same time of traversal, or, they obey Fermat's Principle of least time; clearly the straight–through path is such a minimum, on which others can be gauged.]



Scholium.

It is permissible to go on by the same method to three or more surfaces. But spherical figures are the most convenient for use in optics. If spyglasses with objective and eyepiece may be constructed from two spherical glass figures, and water enclosed between them; it can come about that the refraction errors, which are present in the extreme surfaces of the glasses, may be corrected well enough by the refraction of the water. But such glass objectives are to be preferred than ellipses or hyperbolas, not only because they are they are easier and more accurate to be made, but also the pencils of rays beyond the axis of the glass in place may refract more accurately. But this is impeded from perfection by the diverse refractions of the different kinds of rays, by which the optics either by spherical or other figures is less than perfect. Unless the errors hence arising shall be corrected, all the labour involved in correcting the other errors will be to no avail.


Book II Section I. Translated and Annotated by Ian Bruce. Page 425

CONCERNING THE MOTION OF BODIES

BOOK TWO.

SECTION I.

Concerning the motion of bodies being resisted in the ratio of the velocities.


The motion [i.e. velocity] of a body removed by resistance, to which it is resisted in the

ratio of the velocity, is as the distance completed in moving. For since the loss of motion in equal small intervals of time shall be as the velocity, that is, as the small parts of the journey completed: the motion lost in the whole time shall be as the whole journey. Q. E. D. [Thus, in each increment of time tΔ for a given velocity v, we may write

; hence v xt tkv k v k xΔ Δ

Δ Δ Δ Δ= − = − = − thus on adding the losses over the whole journey, the whole loss in velocity is proportional to the whole distance gone.] Corol. Whereby if a body, freed from all weight, may be moving in free spaces by the action of its innate force only; and while the whole motion may be given at the start, as well as the remaining motion also, after the completion of some distances: the whole of the distance is given that the body is able to describe in an infinite time. For that distance will be to the distance now described, as the whole motion from the start, to that part of the motion lost. [Expressing this idea in modern terms, if the rate of decrease of the velocity in the time dt is proportional to the velocity, then we have for a body of unit mass,

( ) ( )dv t dxdt dta t kv k= = − = − , giving ( ) ( ) ( )0

0 ; 1vkt ktkv t v e x t e− −= = − . Thus, the whole

journey is given by ( ) 0vkx ∞ = , while ( )

( )1

1 ktxx t e−∞

−= ; again, ( )

0

0 1 ktv

v e−−is the whole motion at

the start to the motion lost, which is in the same ratio, as asserted. Newton makes use of



this simple idea in the following propositions. The luxury of the exponential function however, was not available to Newton; results had to be found without this convenience; use was made of the area under the rectangular hyperbola to measure the difference in times, as we shall see. Thus, we may write

( ) ( ) ( ) ( ) ( )0 0 0 0; ; ; o

t

vkt kt kt duv ua t kv e a e v t v e kx t v v t ln kt k− − −= − = − = = − = = ∫ ]

LEMMA I. Proportional quantities formed from their differences are continued proportionals.

A shall be to A B− as B to B C− and as C to C D− , &c., and by being converted there becomes A to B as B to C and C to D, &c. Q. E. D. [For if ,etc.CA B

A B B C C D− − −= = = ⋅⋅⋅ ; then 1 1 1 ,etc.CA BB A C B D C− − −+ = + = + = ⋅⋅⋅ ,

giving ,etc.CB DB A C B D C− − −= = = ⋅⋅⋅ then .....B C C C DA A B B

B B C C C D D D E− −−

− − −= = = = = .; from which the result follows.]

PROPOSITION II. THEOREM II. If, for a body resisted in the ratio of the velocity, and moving through a uniform medium by its innate force alone, and moreover equal time intervals may be taken, then the velocities at the beginnings of the individual time intervals are in a geometric progression, and the distances described in the individual time intervals are as the velocities. Case. I. The time may be divided up into small equal parts ; and if the force of resistance may act at the beginning of the increments of time by single impulses, which shall be as the velocity: the decrease of the velocity in the individual increments of time will be as the same velocities. Therefore the velocities from these differences are proportionals, and on this account continued proportionals, (by Lem. I. Book. II.). [For in ,etc.CA B

A B B C C D− − −= = = ⋅⋅⋅ , if etc. A B Cv , v , v , , are the velocities at the start of successive constant increments in time, then etc.A B A B C Bv v kv , v v kv ,− = − = ; and

1 11.....B C C C DA A B B

B B C C C D D D E

v v v v vv v v vv v v v v v v v v k r

− −−− − − −= = = = = = = , as the ratio of successive velocities is

a constant r or 11 k− , by hypothesis . Thus we may write

2 3 , etc.B A C B A D Av rv , v rv r v , v r v= = = = in an inductive manner, as we have shown above, forming a geometric progression. In the limit, these ratios form a continuous logarithmic or exponential curve.] Hence, if from a number of equal small time intervals, equal [larger] time intervals may be put in place, the velocities from the starts of these times will be as the terms in a



continued [geometric] progression, which are taken in jumps, with an equal number of intermediate terms omitted everywhere. But the ratios have been composed from these terms, by ratios with the same repeated equality of the intermediate terms between themselves, and therefore on that account these too are ratios composed between themselves, and the velocities proportional to these terms are in a geometric progression also; [i.e. the larger steps in time are in arithmetic proportion, and the corresponding steps in the velocity are still in geometric proportion]. Now these equal small time intervals may be diminished, and the number of these increased indefinitely, so that from that the impulse of the resistance may be rendered continuous, and the velocities from the beginnings of the equal times, always continued proportionals, in this case will be continued proportionals. Q. E. D. [Shades of Napier's algorithm for generating logarithms?] Case 2. And the differences of the velocities, that is, the parts of these removed by the individual time intervals, are in proportion to the whole : but the distance described in the individual times, are as the parts of the velocity removed (by Prop. I, Book. II.), and on that account also are as the total distance. Q.V.D. [As we have seen, in Eulerian or modern terms, ( ) ( ) ( )0

0 ; 1vkt ktkv t v e x t e− −= = − ; thus,

when the time is advanced by dt , the (negative) change in the velocity is proportional to the velocity

( ) ( ) ( )

( ) ( ) ( ) ( )

0

0

;and the increment of distance

;

k t

k t

dv t kv e dt v t

dx t v e dt dv t v t

−

−

= − ∝

= ∝ ∝

i.e. proportional to the velocity lost or distance gained. These ideas have been expressed in terms of ratios of succeeding increments by Newton, either singly or in groups with the same number in each. In addition, we may note that as the resistance is proportional to the velocity, it also is exponential in nature, and we might denote it by ( ) 0

ktR t R e−= . It is important to note here that the same exponential form of the decay occurs in the forced motion when gravity is present and the body is rising, although the multiplying constant is different.] Corol. Hence a hyperbola BG may be described with the rectangular asymptotes AC and CH, and AB, DG shall be perpendiculars to the asymptote AC, and then the velocity of the body for a resisting medium, with some initial motion [velocity], may be shown by a given [i.e. constant] line AC, and with some elapse in time the velocity is now given by the indefinite [i.e. variable] line DC: it is possible to express the time by that area ABGD, and the distance described by the [velocity] line AD in that time. For if that area may be increased by the motion of the point D uniformly in the manner of the time, then the right line DC will decrease in a geometric ratio in the manner of the velocity, and the parts of the right line described AC will decrease in the same ratio in equal times.



[ The hyperbola is a device used in the calculation in conjunction with the logarithmic scale on the ordinate axis to ascertain the like times involved when the velocity and distance traversed change from one value to another via a series of continued proportionals, arising from the nature of the exponential or logarithmic curve. The correspondence between the natural logarithm and the area of a section of the rectangular hyperbola has been demonstrated above, in the final theorem of Leseur & Janquier, originating from the work of Gregorius, as presented by de Sarasa : (see the paper by R.P. Burn in Historia Mathematica Vol.28, (2001), pp.1-17). Thus, the difference of the starting and finishing times are associated with the logarithm of the ratio of the corresponding velocities (or of the abscissas of the logarithmic curve) at these times, and this logarithm is in turn equal to the area under the hyperbola; for if the speed of the point at A is 0v , and at some later time t along the line CD the speed is ( ) 0

ktv t v e−= ; the change in distance gone and acceleration or force are given by similar formulas involving logarithm ratios, then e.g. the area under the corresponding hyperbolic section BADG

( )0

0

time

ktu v eduu

u v

.

−=

=

= ∝∫ See letter no. 297, Halley to Wallis, and subsequent notes in The

Correspondence of Isaac Newton, 1676-1687, edited by H.W.Turnbull; Vol. II, p.456-p.462, CUP 1960; see also :The Correspondence and Papers of Edmond Halley, E.F.MacPike, Taylor & Francis, 1930. ]

PROPOSITION III. PROBLEM I. To define the motion of a body, for which, while it may ascend or descend in right lines through a similar medium, it is resisted in the ratio of the velocity, and that body is acted on by uniform gravity also. With the body ascending, gravity may be put in place by some rectangle BACH, and the resistance of the medium at the beginning of the ascent by the rectangle BADE, taken on the opposite sides of the line AB. With rectilinear asymptotes AC and CH, a hyperbola may be described through the point B, cutting the perpendiculars DE and de in G and g; and by ascending in the time DGgd , the body will describe the distance EGge; and in the whole time of the ascent DGBA, the whole distance of the ascent EGB ; [Essentially, the whole motion of the ascent with an initial velocity 0v , and continued descent afterwards, is evaluated by making use of this diagram, after divinding the line under the hyperbola geometrically, the areas corrspond to equal times, and the forces, velocity changes, and distances gone are related by successively multipling these equal areas; for example, the force of gravity at the start of the latter downwards motion is represented by the line AC, which force subsequently diminishes geometrically or



exponentially due to the resistance increasing ; the total force at the start of the motion upwards is shown by the line DA, which increases with the resistance decreasing with the velocity to the maximum value at A; the actual trajectory is not shown on this diagram (the problems discussed here by Newton are set out below in the L & J treatment, the trajectory of a projectile under this law of resistance is shown below in Note 56, and thereafter). During the ascent, the velocities are set out to decrease in a geometrical progression along the line DA, giving equal areas under the hyperbola in arithmetic progression, which we have seen correspond to equal time intervals; during each equal increment of time, the velocity lost, in a constant ratio to the previous increment, and thus as a fraction of the starting value AD, is transformed into an equal increment in distance gone, so that the section BEG above the curve represents the whole distance gone upwards and the remaining area ADGB represents the loss in velocity. Newton's picture is hence a moving one, and we are given here only part of a graphical account of the solution of the associated differential equation. If we consider the hyperbola to be simply xy g= , with origin C, and the positive ordinate axis CD acting to the right in a diagram (not shown) reflected in the vertical axis CH, so that we are dealing with a more conventional situation, where g is the acceleration of gravity for unit mass, then B is the point ( )1g , . Furthermore conventionally, the

acceleration at some point D is given by 1 or dv dv kdvdt g kv k g kva g kv dt − −

+ += = − − = = , where

00

11 kv

g

gg kvGD + +

= = , and where at once we have the time to reach d,

( )00 0

11 1 11 and 1

kvgkvg

g kv kvDd DAk g kv k k gt ln ln t ln

+++ +

= = = + . In additon, the time to travel the

distance AI is given by 1 gAI k g kvt ln −= , and the terminal velocity is given by g

kv∞ = .

Now, the time is specified to be increasing in constant increments tΔ , and we need to find the velocities at these times; clearly the decrements g tΔ give the successive decreases from the initial velocity 0v due to the retardation of gravity 0v gt− , in addition, there is

the retarding resistive force initially, 0kv . Now, 0

gg kvGD += , ( )1

0Dd gkt

d k kv g kv e−= + − ,

the change in distance in the next small time increment is ( )0

Dd g tkttd k kx v t g kv e ΔΔΔ Δ −= = + − ; with the signs reversed, we see that the second term

in this formula corresponds to a constant change is distance, while the first term corresponds to a constant multiple of the speed decay resulting from the resistive force by tΔ , and thus the areas are as indicated, and as set out below in a laborious manner below by L & J. The argument of Newton below applies to the other branch of the problem. One notes of course that this is a graphical method, and the actual curve is not plotted.]



In the time of the descent ABKI the body falls the distance BFK, and in the time IKki the distance fallen is KFfk; and the velocities of the body (proportional to the resistance of the medium) in the periods of time of these will be ABED, ABed, nothing, ABFI, ABfi respectively; and the maximum velocity, that the body can acquire on falling will be BACH. The rectangle BACH may be resolved into innumerable [equal] rectangles Ak, Kl, Lm, Mn, etc. which shall be as equal increments in the velocity made in the same equal increments of time; and so that zero, Ak, Al, Am, An, etc. shall be the whole speeds [accomplished by multiplying these constant time increments by the constant acceleration of gravity], and thus (by hypothesis [on multiplying these speeds by k, ]) as the resistances of the medium from the beginning of the individual equal time intervals. Thus, AC becomes to AK or ABHC to ABkK as the force of gravity to the resistance at the beginning of the second time, and thence the resistances are taken away from the force of gravity, and ABHC, KkHC, LlHC, MmHC, etc. will remain as the absolute forces by which the body is acted on at the beginning of the individual time intervals, and thus (by the second law of motion) as the increments of the velocities, that is, as the rectangles Ak, Kl, Lm, Mn, &c., and on that account (by Lem. I. Book II.) in a geometric progression. Whereby if the right lines Kk, Ll, Mm, Nn, etc. produced meet the hyperbola in q,r,s,t, etc. the areas ABqK, KqrL, LrsM, MstN, etc. will be equal, and thus then always in the same ratio both with the times as well as the force of gravity. But the area ABqK (by Corol. 3. Lem. VII. & Lem. VIII. Book1.) is to the area Bkq as Kq to 1

2 kq or AC to 12 AK

that is, in proportion to the force of gravity to the resistance in the middle of the first time increment. And by a similar argument the areas qKLr, rLMs, sMNt, etc. are to the areas qklr, rlms; smnt, etc. as the forces of gravity to the resistances in the middle of the second, third, fourth, time increments, etc. Hence since the equal areas BAKq, qKLr, rLMs, sMNt, &c. shall be in the same ratio to the forces of gravity, the areas Bkq, qktr, rlms, smnt, etc. shall be to the resistances in the middle of the individual times, that is (per hypothesis) with the velocities, and thus analogous to the distances described. The sum of the analogous quantities may be taken, and the areas will be in the same ratio as the total distances described Bkq, Blr, Bms, Bnt, etc.; and also the areas ABqK, ABrL, ABsM, ABtN, etc. to the times. Therefore the body, while descending, in some time ABrL, will describe the distance Blr, and in the time LrtN the distance rlnt. Q. E. D. And the demonstration of the motion in the ascent is set out in a similar manner. Q. E. D. Corol. 1. Therefore the maximum velocity, that the falling body is able to acquire, is to the given velocity acquired by falling in some time, as the given force of gravity, by



which that body is continually urged on, to the force of resistance, from which it may be impeded up to the end of that time. [From ( ) 1g kt

kv e−= − , the maximum velocity becomes gmax kv = , and hence

( ) ( ) ( )1

1 max

ktv gv t kv te−−

= = .]

Corol. 2. But with the time increasing in an arithmetic progression, the sum of that maximum velocity and of the velocity in ascending, and also of the difference of these in the descent, is described by a geometric progression.

( )( ) ( )0 0[For 1 1 1 etc ]kv kvg g gkt ktk g k k gv e e .− −= + − + = +

Corol. 3. But also the difference of the distances, which are described in equal difference of the time, decrease in the same geometric progression. Corol. 4. Truly the distance described by the body is the difference of two distances, of which the one is as the time taken from the beginning of the descent, and the other as the velocity, which distances also at the beginning of the descent itself are equal to each other. [For ( ) ( ) ( )1ktg g gt ve v

k k k k g kx t t t .− −= + − = − = ]

[Notes by the translator: This is as in the previous corollary with the motion starting with some value 0v , except that a constant downwards force is added, in addition to the resistance decelerating the body to rest at AB, so that the velocity increments in the region EDAB become narrower geometrically, for constant time increments, and then afterwards accelerating the body, while the resistive force always opposes the motion in the downwards motion. A reference line AB is taken on the hyperbola representing the point where the body is momentarily at rest, the areas between the vertical lines mark out equal time intervals in the motion, while the starting point is DE on the axis DC; we may consider the asymptotes CD and CH as representing the axes of the dimensionless variables x and 1

xy = . As previously, the ascent time is found by multiplying the area under the hyperbola by a scaling constant; Thus, the time to rise to the maximum height, given by the area under the hyperbola BADE, is less than the time to fall, given by the area kiAB; both times are different from the time of the journey without the constant force. Again we may express this proposition in modern terms, if the rate of decrease of the velocity in the time dt is proportional to the velocity, then we have for a body of unit mass ascending, say under constant acceleration of gravity g acting downwards, with an

initial velocity upwards 0v : ( )dv t dxdt dtg kv g k= − − = − − , thus, the first equation gives

( ) ( )0 0

1 ; ordv t g kv g kv ktg kv k g kv g kvdt , ln t e .+ + −+ + += − → = − = In this case, the dimensionless variable



0

g kvg kvz ++= is used to find the time by integrating under the hyperbola. It follows from

( )dv t dxdt dtg k= − − that ( ) 0 giving dv t gdt kdx, v v gt kx= − − = − − , or ( )0v v g

k kx t−= − . In the first part of the motion, the speed decreases as the body rises, where the time is

given by 1 1

1

u

dzk k zlnu t− = − =∫ , and the speed is zero when ( )1

01 kAk gln v t+ = , where the

dimensionless variable z or its upper limit u is less than one. The distance ascended in the

time At is ( ) ( ) ( )0 0 0 02

1 10

g kk gv ln v v g kvg

Ak k g kkx ln v gt BADE BGDA BEG

− + += = − = − = − = .

The first term BADE represents the distance gone with no forces acting, while the second term is the distance removed by 'falling' under gravity in the same time. Similar reasoning can be applied for the downwards motion. For the second downward part of the motion, note that we have established above, for the rising motion, that

( ) ( )

( )

01 ; while subsequently, or

giving

d g kvg kv dvk g kv dt kdtln t g kv, g kv

ln g kv kt ,

−++ = = − − = −

− = −

for the falling motion ; hence ( ) or 1gkt ktkg kv ge v e− −− = = − . The distance gone in

time t is hence ( ) ( ) ( )1 ktg g gt ve vk k k k g kx t t t .

− −= − + = − = Newton's ratio, from

givesdvdt g kv v g t kv tΔ Δ Δ= − = − then becomes to : or g

kvg t k xΔ Δ− , or the ratio of the gravitational force to the resistive force, and the motions lost become equal to the distances gone.]



PROPOSITION IV. PROBLEM II.

Because a uniform force of gravity may be put in some similar medium, and it may act perpendicularly to the horizontal ; to define the motion of a projectile in the same, with the resistance experienced proportional to the velocity. The projectile may be sent off from some place D following some right line DP, and the velocity of the same may be set out by the length DP at the start of the motion. The perpendicular PC may be sent from the point P to the horizontal line DC, and DC may be cut at A, [for the greatest height reached in the trajectory] so that DA shall be to AC as the resistance of the medium, arising from the initial vertical motion, to the force of gravity; or (likewise) as the rectangle under DA and DP shall be to the rectangle under AC and CP as the whole resistance from the start of the motion to the force of gravity.

[i.e. ( )0kv upDAAC g= or as 0kvDA DP

AC CP g×× = , since

( )0 0PCDPkv up v= × ; this is a small modification of Cor. I,

Prop. III.] Some hyperbola GTBS is described with asymptotes DC and CP, cutting the perpendiculars DG and AB in G and B; and the parallelogram DGKC may be completed, the side of which GK may cut AB in Q; The line of length N may be taken in the ratio to QB in which DC shall be to CP; and at some point R of the line DC erect the perpendicular RT, that may meet the hyperbola in T, and the right lines EH, GK, DP in I, t and V , and on that line take Vr equal to tGT

N , or what is the same thing, take Rr equal to GTIEN ;

[i.e. since we have an obliquity factor N DC cosQB CP sin

αα= = , where α is the initial angle of

projection to the horizontal, and DC DRCP RV= , on account of the similar triangles DRV and

DCP; there will be N DRQB RV= , and thus DR QB

NRV ×= . But the rectangle

GEIt Gt GE DR QB GTIE tGT ,= × = × = + and thus DR QB tGTGTIE tGTN N NRV .× −= = −

Whereby if there is taken tGTNVr = , there will be GTIE

N RV Vr Rr.= − = ], and for the projectile to arrive at the point r in the time DRTG, describing the curved line DraF, that it may always be a tangent at the point r, moreover arriving at the maximum height a on the perpendicular AB, and afterwards always approaching towards the asymptote PC. And the velocity of this projectile at some point r is as the tangent of the curve rL. Q. E. I. [See the L. & J. notes in the other file for a complete explanation.]

For indeed N is to QB as DC to CP or DR to RV, and thus RV equals DR QBN× ,



[i.e. : DR QBN DC DRQB CP RV NRV ×= = = ; ] then Rr ( that is RV Vr− or DR QB tGT

N× − ) equals

DR AB RDGTN

× − . Now the time may be put in place by the area RDGT, and (by Corol. 2. of the laws) the motion may be separated into two parts, the one vertical, and the other horizontal. And since the resistance shall be as the motion, this also may be separated into two parts with parts proportional to and opposing the motion: and thus the length described by the motion horizontally, will be (by Prop. II. of this) as the line DR, truly with the height (by Prop. III. of this Book ) as the area DR AB RDGT× − , that is, as the line Rr. But the motion in the very beginning itself , the area RDGT, is equal to the rectangle DR AQ× and thus that line Rr

(or DR AB DR AQN

× − × then will be to DR as AB AQ− or QB to N, that is, as CP to DC ; and thus as the vertical motion to the longitudinal motion at the beginning. Therefore since Rr shall always be as the height, and DR always as the length, and Rr to DR at the start as the height to the length horizontally: it is necessary that Rr shall be always to DR as the height to the horizontal distance, and therefore so that the body may be moving on the line DraF, as it may always be a tangent at the point r. Q. E. D.

[Translator's note : See letter no. 297, Halley to Wallis, and subsequent notes in The Correspondence of Isaac Newton, 1676-1687, edited by H.W.Turnbull; Vol. II, p.456-p.462, CUP 1960. We give now the standard analytical solution. The initial velocity may be separated into its horizontal and vertical components 0 0 and v cos v sinα α , where 0v is the initial velocity, and α is the angle to the horizontal. The motions are described by the two equations from the second law for the horizontal and vertical motions :

and yx dvdvx ydt dtkv kv g= − = − − .

The solutions to both these have been found already ; for the horizontal motion throughout we have the equation for the speed :

00 and for the vertical up: yg kvkt kt

x g kv sinv v cos e eαα +− −+= = ; while for the vertical down we

have ( ) 1g kty kv e−= − . The maximum height is reached when ( )01 1 kv

k gt ln sinα= + , and

the height risen is found from ( )0dy g gktdt k kv sin eα −= + − , giving

( )( )10 1g gtkt

k k ky v sin eα −= + − − ; in which case the maximum height is

( )02

10 1 kvg

k gky v sin ln sinα α= − + , and the corresponding horizontal distances are given



by ( )10 1 kt

kx v cos eα −= − . Thus the time to reach a certain horizontal distance x is given

by ( )0

1 1k v cos kxt ln α−= The time of flight occurs when the downwards distance

( )( )10 1 0g gtkt

k k ky v sin eα −= + − − = ; this can only be solved approximately, as likewise

the range.] Corol. 1. Therefore Rr equals RDGTDR AB

N N× − : and thus if RT may be produced to X so

that RX shall be equal to DR to DR ABN× ; that is, if the parallelogram ACPY may be

completed and DY joined cutting CP at Z, and RT may be produced then it may meet DT in X; Xr will be equal to RDGT

N and therefore proportional to the time. [L& J: Note 57. For with DR AB

NRX ×= , RX will be to DR as AB given to N given, and thus the positions of the points X for the right line which passes through the point D, with DR vanishing everywhere, RX also vanishes. With the point R coinciding with A, there becomes RX or AY AB

DA N= , and by the properties of the

hyperbola as previously, or DC ABAC GD AQ= ; and

separating, DC ABDA BQ= , truly by construction,

BQCPDC N= , and thus by equality,

and hence .CP AB AYDA N DA AY CP= = = Thence if the parallelogram ACPY may be completed, and DY may be joined cutting CP in Z, DZ will be a right line that the point X always touches. Therefore since DR AB

NRX ×= , and DR AB RDGTDR AB

N NXr RX Rr × +×= − = − ; RDGTNXr = , and therefore, on account of N given,

Xr is as the area RDGT, and thus as the time in which the body arrives at the place r from the place D.] Corol. 2. From which if some innumerable lines CR may be taken, or, which is likewise, innumerable lines ZX may be taken in a geometric progression ; there will be just as many Xr in an arithmetic progression. And hence the curve DraF may be easily delineated by a table of logarithms.



Corol. 3. If from the vertex D, with the diameter DG produced downwards, and with the latus rectum which shall be to 2DP as the whole resistance from the start of the motion, to the force of gravity, a parabola may be constructed : the velocity with which the body may depart from the position D along the right line DP, as in a medium with uniform resistance may describe the curve DraF, that itself will be with some body to depart from the same place D, along the same right line DP, as in a space with no resistance may describe a parabola. For the latus rectum of this parabola, with the same initial motion, is

2DVVr ; and Vr is 2 or tGT DR Tt

N N× . But the right line which, if it be drawn, may touch the

hyperbola GTS at G, is parallel to DK itself, and thus Tt is CK DRDC× , & N is QB DC

CP× . And

therefore Vr is 2DRq CK CP

DCq QR× ×

× , that is (on account of the proportionals DR and DC, DV and

DP) 2DVq CK CP

DPq QB× ×

× and the latus rectum DVquad .Vr produced 2DPq QB

CK CP×× , that is (on account of

the proportionals QB and CK, DA and AC), 2DPq DAAC CP

×× , and thus to 2DP, as DP DA× to

CP AC× ; that is, as the resistance to gravity. Q. E. D. Corol. 4. From which if the body may be projected from some place D, with a given velocity, along some given right line DP in place , and the resistance of the medium from the start of the motion may be given: it is possible to find the curve DraF, that the same body describes. For from the given velocity the latus rectum of the parabola is given, as has been noted. And on taking 2DP to that latus rectum, as the force of gravity is to the force of resistance, DP may be given. Then by cutting DC in A, so that shall be to CP AC DP VA× × in that same ratio of gravity to resistance, the point A will be given. And thence the curve DraF may be given. Corol. 5 . And conversely, if the curve DraF may be given, both the velocity of the body and the resistance of the medium will be given at the individual points r. For from the given ratio to CP AC DP VA× × , then the resistance of the medium from the start of the motion, as well as the latus rectum of the parabola is given: and thence also the velocity from the beginning of the motion is given. Then from the length of the tangent rL, and the velocity proportional to this is given, and the resistance proportional to the velocity at some place r.



Corol. 6. But since the length 2DP shall be to the latus rectum of the parabola as gravity to the resistance at D; and from the increase in the velocity the resistance may be augmented in the same ratio, but the latus rectum if the parabola may be augmented in that square ratio: it is apparent the length to be increased in that simple ratio, and thus always to be proportional to the velocity, nor to be increased or decreased by a change in the angle CDP, unless also the velocity may be changed. Corol. 7. From which a method of determining the curve DraF approximately from phenomena is apparent, and thence by deducing the resistance and velocity by which a body is projected. Two similar and equal bodies may be projected with the same velocity from the same place D, following different angles CDP, CDp and the locations may be known F, f, where they are incident on the horizontal plane DC. Then, with some length assumed for DP or Dp, which resistance in some ratio to gravity may be put in place at D, and that ratio may be established by some length SM. Then by calculation, from that assumed length DP, the lengths DF, Df, from the ratio Ff

DF are found by calculation: the same ratio found by experiment may be taken, and the difference may be put in place by the perpendicular MN. Do the same over again and a third time, always by assuming a new ratio SM of the resistance to gravity, and by putting in place a new difference MN. But the differences may be drawn positive on one side of the line SM and negative on the other; and through the points N, N, N there may be drawn a curve of the form NNN cutting the right line SMMM in X, and SX will be the true ratio of the resistance to gravity, as it was required to find. From that ratio the length DF is required to be deduced by calculation ; and the length, which shall be to the assumed length DP, as the length DF known through experiment to the length DF found in this manner, will be the true length DP. With which found, both the curved line DraF will then be had that the body will describe, as well as the velocity and resistance of the body at individual places.



Scholium.

Besides, the resistance of bodies to be in the ratio of the velocities, is a hypothesis more mathematical than natural. In mediums, which are free from all stiffness, the resistance of the bodies are in the square ratio of the velocities. And indeed by the action of faster bodies there is communicated to the same quantity of the medium, in a smaller time, a greater motion on account of the greater velocities ; and thus in equal times, on account of the greater quantity of the medium disturbed, a greater motion is communicated in the square ratio; and the resistance (by laws of motion II & III ) is as the motion communicated. Therefore we may see how a motion arises from this law of resistance.


Book II Section II. Translated and Annotated by Ian Bruce. Page 446

SECTlON II.

Concerning the motion of bodies resisted in the square ratio of the velocities.

PROPOSITION V. THEOREM III. If there is resistance to a body in the square ratio of the velocity, and likewise it may be moving only by the inertial force through the same medium, truly the times in going from the smaller to the greater bounds may be taken in a geometric progression : I say that the velocities from the beginning of the individual times are inversely in the same geometric progression, and that the distances, which are described in the individual times, are equal. For because the resistance of the medium is proportional to the square of the velocity, and the decrease of the velocity is proportional to the resistance ; if the time may be divided into innumerable equal parts, the differences of the same velocities will be proportional to the squares of the velocities from the beginnings of the individual times. Let these elements of time AK, KL, LM, &c. be taken on the right line CD, and the perpendiculars may be erected AB, kK, Ll, Mm, &c., meeting the hyperbola BklmG described, (with centre C and rectilinear asymptotes CD, CH) at B, k, l, m, &c., [where these vertical lines are taken as the velocities at these times] and AB shall be to Kk as CK to CA, and on separating the ratio, AB Kk− shall be to Kk as AK to CA, & in turn AB Kk− shall be to AK as Kk to CA, and thus as to AB Kk AB CA× × . From which, since AK and AB CA× may be given, AB Kk− will be as AB Kk× , and finally, when AB and Kk merge, as 2AB . And by like arguments

etcKk Ll , Ll Mm, .− − will be as 2 2Kk ,Ll .etc. Therefore the squares of the lines AB, Kk, Ll, Mm are as the differences of the same and therefore since the squares of the velocities also were as the differences of these, the progression of both shall be similar. [Thus, in terms of Newton's hyperbola :

or ; or

and = ; hence ; leading to the incremental change

in the ordinate being proportional to the square of the ordina

CK CK CA AB kK AB kKAB AKCA kK CA kK CA kK

kK AB kK AB KkCA AK AB CA

CA AB CK kK

AB Kk AB Kk

− − −

− ××

× = × = = =

= − ∝ ×

te when .Kk AB→

The ordinates of the curve correspond to the velocities, and the abscissae correspond to the times : thus, the difference in the velocity between increments is proportional to the product of the velocities at the start of the increments. Newton has shown that the difference of neighbouring velocities is proportional to the product of these velocities, as required, and hence the gradient is proportional to the resistive force or acceleration. Now, in analytical terms, the general drift of the solution follows if we consider :



( )20

2 1 10 or giving dv dv

dt v vvkv kdt k t t= − = − − = − , for the neighbouring velocities

( ) ( ) ( ) ( ) 20 0 0 02 etc. ; then AB v , Kk v t , Ll v t , v t v v kv v t kvΔ Δ Δ Δ Δ= = = − = = − → − , as

above; and from ( )0

1 10v v k t t− = − , the velocity varies inversely with the time, on

choosing 0

10v kt= , with the starting point 0 0t t , v v= = ; where for equal small changes in

the time, 2v kv tΔ Δ= − . Again, to pass on to the next integration, we may write :

( )0

0

20

1

or giving and or e ;

giving ; thus, the velocity decays exponentially with distance.]

kxdv dv dx dv dv dv vdt dx dt dx dx v v

vk v

. v kv kv kdx, ln kx v x v

x ln

−= = = − = − = − = − =

=

With which demonstrated, it is a consequence also that the areas described by these lines shall be in a similar progression with the distances which are described by these velocities. Therefore if the initial velocity of the first time AK interval may be represented by the line AB, and if the initial velocity of the second KL by the line Kk, and with the distance from the first time described by the area AKkB; all the subsequent velocities may be represented by the following lines Ll, Mm, &c., and the distances described by the areas Kl, Lm, &c. And by adding these together, if the whole time may be expressed by the sum AM of all their parts, the whole length described may be represented by the sum of the parts AMmB. Now consider the time AM divided thus into the parts AK, KL, LM, &c. so that CA, CK, CL, CM, etc. shall be in a geometric progression ; and those parts will be in the same progression, and the velocities AB, Kk, Ll, Mm, &c. shall be [, from the nature of the hyperbola,] in the same progression inverted, and equal distances described Ak, Kl, Lm, &c. Q.E.D. [Thus AK, KL, LM, etc., which are the differences of the lines CA, CK, CM, etc., are in the same progression. For the differences of any geometric progression are in the same geometric progression. Thus if there shall be :

, etc ; on taking the previous term from each: , etcCA CK CL CA AK KLCK CL CM CK KL LM. .= = = = , then the differences in the times are also in a geometric progression ; ] Corol. 1. Therefore it may be apparent, if the time may be represented by some part of the asymptote AD, and the velocity at the beginning of the time by the applied ordinate AB; then the velocity will be represented at the end of the time by the ordinate DG, and the whole distance described by the adjacent hyperbolic area ABGD; and also the distance, that some body can describe in the same time AD. with the first velocity AB, in a non-resisting medium, is given by the rectangle AB AD× . [Thus the mathematics is far simpler at this point than in the first section; the hyperbola is the velocity vs time curve itself, the area within a sector is the distance gone, and the gradient is proportional to the acceleration.]



Corol. 2. From which the distance is given in a resisting medium, by taking that likewise to the distance that it may be able to describe in a non-resisting medium with a uniform velocity AB, as the hyperbolic area ABGD to the rectangle AB AD× . Corol. 3. Also the resistance of the medium is given, by putting in place that initial motion to be itself equal to a uniform centripetal force, which by a body falling in the time AC, may be able to generate the velocity AB, in a non resisting medium. For if BT may be drawn touching the hyperbola in B, and crossing the asymptote at T; the right line AT will be equal to AC [from a property of the hyperbola], and the time may be represented, in which the initial resistance uniformly continued may be able to remove the whole velocity AB. Corol. 4. And thence also the proportion of this resistance to the force of gravity is given, or some other centripetal force given. Corol 5. And in turn, if the proportion of the resistance may be given to some centripetal force, then the time AC is given, in which a centripetal force equal to the resistance may be able to generate some velocity AB : and thence the point B may be given through which the hyperbola must be drawn, with the asymptotes CH, CD ; and so that a body by beginning its motion with that velocity AB, can describe that distance ABGD in some time AD, in a medium with a similar resistance.

PROPOSITION VI. THEOREM IV. Equal homogeneous spherical bodies, impeded by resistances in the square ratio of the velocities, and moving by the inertial force alone, always describe equal distances in times which are inversely as the velocities at the start of the times, and lose proportional parts of their whole velocities. With the rectangular asymptotes CD, CH, some hyperbola BbEe is described cut by the perpendiculars AB, ab, DE, de, in B, b, E, e; the initial velocities are represented by the perpendiculars AB, DE, and times by the lines Aa, Dd. Therefore as Aa is to Dd thus (by hypothesis) DE is to AB, and thus (from the nature of the hyperbola) as CA to CD; and on adding together, thus as Ca to Cd. Therefore the areas ABba, DEed, that is, the distances described are equal between themselves, and the initial velocities AB, DE are proportional both to the final ones ab, de, and therefore with the parts separated, also to the parts of these lost AB ab, DE de− − . Q.E.D. [ Aa CA CA Aa CaDE

Dd AB CD CD Dd Cd++= = = = . Thus, CA

CD represents the ratio of the time intervals in which

the velocity ratio equally changes according to DEAB , and also the changes to the velocities

ratio in the incremental times Aa and Dd: Now ( ) ( )( )010e or vkx

k v xv x v x ln−= = gives



here in an obvious notation, ( ) ( )1 1 and A D

a d

v vAa Ddk v k vx ln x ln= = ; since the velocity ratios

are equal, then so are the distances gone.]

PROPOSITION VII. THEOREM V. Spherical bodies in which there is resistance in the square ratio of the velocities, in times, which are directly as the initial motions and inversely as the initial resistances, will lose proportional parts of the whole motion, and they will describe distances proportional jointly to these times and to these initial velocities. For the parts of the motions lost are as the resistances and the times conjointly. Therefore as these parts shall be proportional to the whole, the resistance and the time conjointly must be as the motion. Hence the time will be as the motion directly and the resistance inversely. Whereby with the small differences [i.e. differentials] of the times taken in that ratio, the bodies will always lose small parts proportional to the whole motion, and thus they will retain velocities always proportional to their initial velocities. And on account of the given ratio of the velocities, they always describe distances which are as the initial velocities and times conjointly. Q. E. D. [Here we follow the demonstration of Leseur & Janquier, in Note 89: The whole demonstration of this proposition is set out by analysis in this manner. Let the mass of some globe be m, the initial velocity of the motion c, at the end of the time t it shall be v, with the initial resistance of the motion r, and because the resistances of the body at different places shall be as the square of the velocity, by hypothesis, there will be

2

22 2 to as to the resistance at the end of the time , which will be rv

cc v r t .But the

resistance 2

2rvc

is as the decrement of the motion mdv− directly, and as the time dt

inversely, that is 2

2rv mdv

dtc= − , and hence

2

2mc dv

rvdt = − , and with the fluents taken [i.e.

integrals], 2mc

rvt Q .= + There is put 0 and mcrt , Q ,= = − with which value substituted

there becomes 2mc mcvrvt .−= The time may be taken directly as the first motion directly

and as the first resistance r inversely, that is, mcrt ∝ , and there will be

2 2; and thus mc mc mcvr rv mcv mc mcv−∝ ∝ − , and on dividing by c, mv mc mv∝ − ; that is,

the motion lost is as the initial motion; and hence on account of the given mass m, there will be also, as c c v− : that is the velocity lost as the initial velocity; thus also there will be as or c c c v, v,− + that is, the first velocity c is in a given ratio to the remaining velocity v. Now if the distance described is called s in the time t, there will be ds vdt= , and because v is as the given c, there will be ds as cdt, and with the fluents taken on account of c given, there becomes s as ct. Q.e.d. Note 90. Because the distance s ct∝ , and mc

rt ∝ , also there will be mccrs ∝ ; the mass m

of a globe of which the diameter shall be D, and with the density for the globe given as



the mass to the volume, that is, as the cube of diameter D3; whereby there becomes 3D ccrs .∝ If in addition with the velocity c given, the resistance r is as the diameter D, the

index of the power of which is n, that is nr D∝ , and hence with the velocity not given, the resistance r, as 2nD c , will be as

3 2

23 or as n

nD cD c

s , D .−∝ From which the Corollaries

following are apparent.] Corol. 1.Therefore if for bodies having the same velocity there is resistance in the square ratio of the diameters : homogeneous globes moving with some velocity or other, on describing distances proportional to their diameters, will lose parts of the motions proportional to the wholes. For the motion of any globe will be as the velocity and the mass conjointly, that is, as the velocity and the cube of the diameter, the resistance (by hypothesis) will be as the square of the diameter and the square of the velocity conjointly, and the time (by this proposition) is in the first ratio directly, and in the second inversely ; that is, as the diameter directly and the velocity inversely , and thus the distance, in proportion to the velocity and the time, is as the diameter. Corol. 2, If there is resistance for bodies moving with equal velocities in the three on two ratio of the diameters : homogeneous globes moving with some velocity or other, by describing distances in the three on two ratio of the diameters, lose parts of the motions proportional to the wholes. Carol. 3. And generally, if there is resistance for equally fast bodies in the ratio of some power of the diameters : the distances in which homogeneous globes, with some velocity of the motion, will lose parts of the motions proportional to the wholes, will be as the cubes of the diameters to that applicable power. Let the diameters be D and E; and if the resistances, when the velocities are put equal, shall be as and n nD E : the distances in which the globes, moving with some velocity or other, will lose parts of the motions proportional to the wholes, will be as 3 3 and n nD E− − . And therefore homogeneous globes by describing proportional distances 3 3 and n nD E− − themselves, will retain velocities in the same ratio in turn and from the beginning. Corol.4. For if the globes may not be homogeneous, the distance described by the denser globe must be increased in the ratio of the density. For the motion, with equal velocity, is greater in the ratio of the density, and the time (by this proposition) may be increased in the ratio of the motion directly, and the distance described in the ratio of the time. Corol. 5 And if globes may be moving in different mediums, the distance in the medium, that resists more than the other parts, will be diminished in the ratio of the greater resistance. For the time (by this proposition) may be diminished in the ratio of the increased resistance, and the distance in the ratio of the time.



LEMMA II.

The moment of a generating quantity [genitam] is equal to the moments arising from the generating quantities of the individual parts, with the indices of the powers of the same parts and coefficients continually multiplied out. [Genitam is Newton's word describing a function of several variables, or more roughly as befits the times, a formula with several variables present ; the Latin word means : that brought forth, born, etc.; by moment Newton means the small momentary quantity that may be associated with a given generating quantity which itself is allowed to increase by its own moments ; Newton considered time as the free variable against which all other changes are to be measured. Please note that it is an anachronism to use the word function in the Principia; it was especially the arrival of Euler on the scene that started the change in people's ways of thinking about such things. Whereas for example, we think of a table of sines as representing special values of the sine function for discrete angles, at the time it was commonplace to consider such as a compilation of the ratios opp./hypot. for the collection of right-angled triangles with the corresponding angles, and nothing more ; likewise logarithms, etc. Some mathematicians such as Newton clearly thought further than this, of interpolations, and so forth, but there was the enormous task of setting up a framework to accommodate such activities by devising suitable notations, and so on. Thus Newton stumbles forward trying to find an appropriate name for something important, that we now call a function. The interested observer may note that Newton's proofs of his propositions are little more than word descriptions of the analytical calculations he has performed : such proofs are very hard if not impossible at times to understand without the underlying calculus based methodology. So if you wonder : how on earth did he come upon this? The answer must be : he didn't do it this way initially, the proof presented is an afterthought, where he paints a pretty geometrical or graphical picture at the end of the analytical process. Hence, to understand Newton, read the enounciation of the proposition, work it out for yourself using calculus of the Leibniz kind, then read Newton's explanation, and you will, hopefully, understand everything. These thoughts are those of the translator: you may or may not agree with them; in any case, each person is entitled to their own opinion.] I call a quantity a generating quantity [i.e. gentitam] , which arises from several parts or terms in arithmetic by multiplication, division, and extraction of roots, without addition or subtraction, or in geometry both by the discovery of contained quantities and sides [in the sense side of a square or cube, or the square , cube root, etc.], or of mean and extreme proportions, again without addition or subtraction. Products, quotients, roots, rectangles, squares, cubes, sides squared, sides cubed, and the like. These quantities I consider here, as indeterminate and variable, and as if continually increasing or decreasing in a state of motion or flux, and the momentary increments or decrements of these I understand by the name of moments : thus so that the increments may be obtained from added or positive quantities, and the decrements for subtracted or negative quantities. Yet beware that you have to understand small finite parts. The small finite parts are not the moments, but quantities arising themselves from the moments. Now the origins of finite magnitudes are



required to be understood. For the magnitude of moments will not be seen in this Lemma, but the first proportion of these arising. It returns to the same thing if in place of moments there may be taken either the velocities of increments or decrements (which motion also is allowed to be called the changes and fluxions of quantities) or some finite quantity proportional to these velocities. But the coefficient is the quantity which arises from some part generated by applying the genitam to that part. Therefore the sense of the lemma is, that if the moments of some quantities A, B, C, &c. shall always be increasing or decreasing by a motion , or from these the proportional velocities of the changes may be called a, b, c, &c.; the moment or change of the generating quantity of the rectangle AB were Ba bA+ , and the moment of the contained generating quantity ABC were CaBC bA cAB+ + : and the moments of the powers of the generating quantities

3 1 21 13 32 2 22 3 4 1 2 A , A , A , A , A ,A ,A ,A , A , & A−− − , were

2 11 13 32 2

32

2 3 231 2 22 2 3 3

3 12

2 3 4 , ,

2 ,& respecive.

.aA, aA , aA , aA , aA aA aA , aA ,

A aA

− −− −

−−

−

− −

And generally, so that the moment of any powernmA were

n mmn

m aA−

. Likewise so that the

moment of the generating quantity 2A B were 22aAB bA+ ; and the moment of the generating quantity 3 4 2A B C 2 4 2 3 3 2 3 43 4 2aA B C bA B C cA B C+ + ; and the moment of the generating quantity

3

2AB

or 3 2A B− becomes 2 2 3 33 2aA B bA B− −− and thus with the rest.

Truly, the lemma may be demonstrated in this manner. Case 1. Some rectangle AB always increased by a motion, where half of the moments 1 12 2 and a b may be lacking from the sides A and B, which becomes 1 1

2 2 by A a B b− − , or 1 1 12 2 4AB aB bA ab− − + ; and so that the initial sides A and B increased by the other

moments added on, becomes 1 12 2 by A a B b+ + or 1 1 1

2 2 4AB aB bA ab+ + + .The first rectangle may be subtracted from this rectangle, and there will remain AaB b+ . Therefore the increment aB bA+ of the rectangle is generated from the whole increments of the sides a and b. Q.E.D. Case 2. AB may always be put equal to G, and the moment of the volume ABC or GC (by case 1, will be gC cG+ , that is (if for G and g there may be written and AB aB bA+ ) aBC bAC cAB+ + . And the account is the same for a generating quantity containing any number of sides [variables]. Q.E.D. Case 3. The sides A, B, C may always themselves be put mutually equal ; and of 2A , that is of the rectangle AB, the moment aB bA+ will be 2aA, but of 3A , that is the content of



ABC, the moment 2 will be 3aBC bAC cAB aA+ + . And by the same argument the moment of any power nA is 1nnaA . Q.E.D.−

Case 4. From which since 1A by A shall be 1, the moment of 1

A taken with A, together

with 1A multiplied by a will be the moment of 1, that is, zero.

Therefore the moment of 1A or of 1A− itself is 2

aA− . And generally since 1

nA by nA shall

be 1, the moment of 1nA itself taken by nA together with 1

nA with 1nnaA − will be zero.

And therefore the moment of 1nA or of nA− itself will be 1n

naA +− . Q.E.D.

Case 5. And since

12A by

12A shall be A, the moment of

12A itself multiplied by

122A

will be a, and thus by case 3: the moment of 12A will be 1

22

a

A, or

121

2 aA− . And generally if

mnA may be put equal to B, mA will be equal to nB , and thus 1mmaA − is equal to 1nnbB − ,

and 1maA− equals 1nbB− or mnnbA− , and thus

m nnm

n aA−

equals b, that is, equal to the

moment of mnA . Q.E.D.

Case 6. Therefore the moment of any generated quantity m nA B is the moment of mA taken with nB , together with the moment of nB taken with mA , that is

1 1m n n mmaA B nbB A− −+ ; and thus the power of the indices m and n shall be either whole or fractional numbers, and either positive or negative. And the account is the same starting from several contained powers. Q.E.D. Corol. I. Hence in continued proportions, if one term is given ; the moments of the remaining the terms will be as the same terms multiplied by the number of intervals between these and the given term. Let A, B, C, D, E, F be continued proportionals ; and if the term C is given, the moments of the remaining terms will be amongst themselves as 2 2 3A, B, D, E , F− − . [For, since A, B, C, D, E, F are continued proportionals

2 2 1: : CDD C C B C D−= = = and

similarly it is found that 3 2 3

2 23 2 etcC D D

CD CA C D , E , F , .−= = = = Whereby on account of C

given, the moment of this is zero, the moments of the remaining terms will be (by cases 2 & 3),

2

23 3 2 2 2 32 dD dD

C CdC D , dC D ,d , , ,− −− − and on multiplying the individual terms by D

d ,

the proportions of the terms will remain : 32

23 2 2 1 322 that is 2 2 3 DD

C CC D , C D ,D, , , A, B,D, E, F.− −− − − − But the number of terms

between the term A and the given term C is 2, and thus is the interval between E and C; 1 is the interval between B and C, and between C and D; and 3 is the number of intervals between C and F. Whereby the truth of the Corollaries is verified.]



Corol. 2. And if in four proportionalities the two middle moments may be given, the moments of the extremes will be as the same extremes. The same is understood concerning the sides of any given rectangle. [Let : : or A B C D B C A D= × = × , and B C× , the given rectangle, will be by Case 1.

0 and hence aD dA , aD dA+ = = − , and thus : :a d A D.− = ] Corol. 3. And if the sum or difference of two squares may be given, the moments of the sides will be inversely as the sides. [Let the sum of two squares equal the given square, or

2 2 2 then, by Case 3, 2 2 0 A B C aA bB ,+ = + = and thus and hence : :aA bB a b B A.= − − = In these two corollaries it is necessary that with one of

the variables increasing, the other is decreasing, and thus while the moment of one is positive, the other moment is negative.]

Scholium.

In a certain letter to our countryman D.J. Collins, dated10th December, 1672, since I may have been establishing a method of tangents that I suspected to be the same as that of Slusius, that had not yet been made public; I adjoined : ........This is one particular instance or rather corollary of a general method, which can be brought to bear on any troublesome calculation, not only to drawing tangents to some curves, either geometrical or mechanical, or respecting right lines or any other curves, truly also to the resolution of other more obstruse general problems concerning curvatures, areas, lengths, centres of gravity of curves, etc. nor (as Hudden's method of maximas and minimas ) is it restricted only to those equations in which irrational quantities are absent. I have carefully combined this method with that other so that I may reduce the equations arising to infinite series. The letter up to this point. And these final words are with regard to that treatise I had written about these things in the year 1671. Truly the fundamentals of this general method are contained in the preceding lemma. [Thus Newton puts his stamp on Calculus as being his invention, in the third edition of the Principia; earlier editions had contained references to Leibniz. Certainly, we must grant to Newton the wonderful ideas expressed in the above Lemma, which should be read by any serious student of mathematics : How many have or ever will ? However, we must also grant to Leibniz the invention of the notation which has stood the test of time, and which has been a cornerstone of the theory ever since. Both men made major contributions to the fledgling art, which had its beginnings in the work of others [see for example, Boyer's The History of the Calculus....Dover.]; in a world devoid of adverse human passions and frailties, one might hope that they could have shook hands, and said: We did this together. Such an event never happened; neither conceded honour to the other; Leibniz died a lonely old man in 1717, perhaps broken by the dispute: only his valet came to the funeral, while Newton reaped every award available to him, and now



lies finally at rest in Westminster Abbey, the tomb jealously guarded by clerics who will stop you taking a photo, if they can ; one wonders who is treated with the greater respect regarding this controversy today. A useful commentary on the dispute can be found by anyone, thanks to the generosity of JSTOR: The manuscripts of Leibniz on his discovery of the differential calculus. J.M. Child. The Monist, Vol. 26, No. 4 (Oct., 1916).]

PROPOSITION VIII. THEOREM VI. If a body in a uniform medium, with gravity acting uniformly, may rise or fall in a straight line, and the whole distance may be separated into equal parts, and in which at the beginning of the individual parts (with the resistance of the medium added to the force of gravity, when the body rises, or by subtracting the same when the body falls) the absolute forces may be found; I say that these absolute forces are in geometric progression. For the force of gravity may be shown by the given line AC [in the descent of the body from A at rest]; the resistance by the variable line AK; the absolute force in the descent of the body by the difference KC; the velocity of the body by the line AP, which shall be the mean proportional between AK and AC [thus, Newton selects 2AP AK AC= × to obtain a geometric progression, and AC is fixed, so that the resistance is proportional to the velocity squared, and AC is taken as the constant of proportionality], and thus in the square root ratio of the resistance; the increment of the resistance made in a small time interval by the line KL, and the increment of the velocity at the same time by the short line PQ; and some hyperbola BNS will be described with the centre C and with the rectangular asymptotes CA, CH, with the perpendiculars AB, KN, LO erected meeting at B, N, O. Because AK is as AP2, the moment of the one KL will be as the moment of the other 2AP.PQ. that is, as AP to KC; [Thus, 2AP PQ AC KL× = × , or 2vdv gdR= ]; for the increment of the velocity PQ (by Law II of motion) is proportional to the force arising KC. The ratio of KL may be compounded with the ratio of KN, and the rectangle KL KN× [i.e. the increment in the area] becomes as AP KC KN× × ; that is, on account of the given rectangle KC KN× , [from the nature of the hyperbola], KL KN× becomes as AP. And the vanishing ratio of the area of the hyperbola KNOL to the rectangle KL KN× is one of equality, when the points K and L coalesce. Therefore that vanishing hyperbolic area is as AP. Therefore the whole hyperbolic area ABOL, compounded from the individual areas KNOL, is always proportional to the velocity AP, and therefore proportional to the distance described with that velocity [in each incremental area]. Now that area may be divided in to the equal parts ABMI; IMNK; KNOL, &c. and the absolute forces AC, IC, KC, LC, &c. will be in a geometric progression. Q.E.D. And by a similar argument [see following note], in the ascent of the body, by taking, at the contrary part of the point A, equal areas ABmi, imnk, knol, &c. and it will be agreed



that the absolute forces AC, iC, kC, IC, &c. are continued proportionals. And thus if all the distances in the ascent and in the descent may be taken equal; all the absolute forces lC, kC, iC, AC, IC, KC, LC, &c. will be continued proportionals, Q.E.D. [In modern terms, we have the force equation : 2dv dv

dt dxv g kv= = ± − ; initially with gravity

absent as above, the equation can be integrated at once to give : 0

1 1v v kt− = . Again, from

2dvdxv kv= − we find : ( ) 0

kxv x v e−= , and from which equations we can find at once:

( ) ( )101kx t ln kv t= + .

With the body projected downwards , the equation becomes with gravity acting :

( ) ( )2

2 20

12 2 2 201 1

1 , or giving 2 ; or kg

k kg g

v kxdv dv k vdvdt dx g v v

v g v gdx, ln kx g kv g kv e− −

− −⎞⎛= = − = = − − = −⎜ ⎟

⎝ ⎠ this equation is also true if the body is projected upwards with the sign of g changed, and shows that the velocities are in a geometric progression.] [L & J note : And by a similar argument.... For the force of gravity may be set out by the given line AC, the resistance by the indefinite line Al, the absolute force in the ascent of the body by the sum Cl, the velocity of the body by the line Ap which shall be the mean proportional between Al and AC, and thus in the square root ratio of the resistance ; the decrement of the resistance in the given small part of the time by the small line pq; and the hyperbola SBo will be described as above; because Al is as Ap2 the moment of this kl will be as the moment of that 2Apq, that is, as Ap into lC; for the velocity of the decrement pq (by the Second Law of Motion) is proportional to the force generated lC, the ratio of this kl may be put together with the ratio of that lo, and the rectangle kl lo× as Ap lC lo× × , that is, on account of the rectangle lC lo× , as Ap. Therefore, with the points k, l joined together, the hyperbolic area knol kl lo= × is as Ap. Therefore with the whole hyperbolic area 2ABol taken together from the individual knol always proportional to the velocity Ap, and therefore is proportional to the distance described with that velocity. Now the area may be divided up into equal parts ABmi; imnk, knol, etc., and the absolute forces AC, iC, kC, lC, etc. are in a geometric progression. Q.e.d. ] Corol. I. Hence if the distance described may be displayed by the hyperbolic area ABNK; the force of gravity, the velocity of the body and the resistance of the medium are able to be shown by AC, AP, and AK respectively , & vice versa. [L & J note : And vice versa. In a similar manner if in the ascension of the body, the distance described until the motion is exhausted may be shown by the hyperbolic area ABnk ; the force of gravity, the velocity of the body, and the resistance of the medium can be put in place by the lines AC, Ap, and Ak.] Corol. 2. And the line AC shows the maximum velocity, that the body by descending indefinitely, can acquire at some time. [L & J note : For indeed AP AC= , and because, by construction, 2AP AK AC= × , there will also be AK AC= , and thus with the ordinate KN meeting the asymptote CH, the



hyperbolic area ABNK becomes infinite , and the distance described by descending in this proportion also will be infinite, truly the weight, the resistance, and the velocity of the body may be shown by the line AC, and then the resistance equals the weight, and therefore the velocity AC is a maximum.] Corol. 3. Therefore if the resistance of the medium may be known for some given velocity, the maximum velocity may be found, by taking that to that known given velocity in the square root ratio, that the force of gravity has to that known resistance of the medium. [Thus, when the velocity is a max., we have

2 2; and ; then max

given

v mgmax given Rv

kv mg kv R= = = .]

PROPOSITION IX. THEOREM VII. With all now demonstrated in place, I say that, if the tangents of the angles of a circular section and of a hyperbolic section may be taken, with proportional velocities, with a radius of the correct magnitude present: the time for any ascent to the highest point will be as the sector of the circle, and the time for any descent from the highest place will be as the sector of the hyperbola. To the right line AC, by which the force of gravity is shown, the perpendicular and equal right line AD may be drawn. With centre D and with semi diameter AD both the quadrant of the circle AtE may be drawn, as well as the rectangular hyperbola AVZ having the axis AX, the principle vertex A, and the asymptote DC. Now Dp and DP may be drawn and the sector of the circle AtD will be to the whole time ascending to the highest place; and the hyperbolic sector ATD will be as the whole time in descending from the highest place : But only if the tangents Ap, AP of the sectors shall be as the velocities. Case 1. For Dvq may be drawn separating the moments of the sector ADt and of the triangle ADp, or the increments that likewise describe the incremental parts tDv and qDp. Since these increments, on account of the common angle D, are in the square ratio of the

sides, the increment tDv will be as 2

2qDp tD

pD× (see first note below), that is, on account of

the given tD, as 2qDppD

. But 2pD is as 2 2AD Ap+ , that is,



[Recall from the last proposition that Ap represents the variable velocity of the body, and Ak the associated resistance, proportional to the velocity squared.] 2 or ; and AD AD Ak , AD Ck qDp+ × × is 1

2 AD pq× .

[For AC Ak× , or 2AD Ak Ap× = , by Prop. VIII, and

( )2 =AD AD Ak AD AC Ak AD Ck.+ × = × + × ]

Therefore the increment of the sector tDv is as pqCk ; that is, as the decrement of the

velocity pq directly, and that force Ck which may diminish the velocity ; and thus inversely as the element of time corresponding to the decrement of the velocity. And on adding these together, the sum of all the increments tDv in the sector ADt shall be as the sum of all the individual increments of time with the velocity decreasing Ap, with the corresponding elements pq removed, until that velocity may have diminished to zero ; that is, the whole sector ADt is as the total time of ascending to the highest place. Q. E. D. [In modern terms, ( ) ( )( )

21 1

1 gives k kg g

dv k dvdt g v v

g v gdt− +

= − = , and hence

( )( ) 21 11 1 and

k kk kg gg g

kgdv dv dvv vv v

gdt dt− +− +

= + = , giving on integration

0

0

1 121 1

k kg g

k kg g

v v kgv v

ln ln t− +

− +

⎞ ⎞⎛ ⎛+ =⎟ ⎟⎜ ⎜

⎝ ⎝⎠ ⎠, provided 0g > , or if 0g < :

( ) 22

11 giving k

g

dv k dvdt g v

g v gdt+

= + = − , and on integration,

1 10

k kg gtan v tan v kgt− −− = − . In the first case, the body is moving down, and in the

second case, it is moving up.] Case 2. DQV may be drawn cutting off the small parts TDV of the sector DAV, as well as the minimal part PDQ of the triangle DAQ ; and these small parts will be in turn as DT2 to DP2, that is (if TX and AP may be parallel) as DX2 to DA2 or TX2 to AP2, [on account of the similar triangles DTX, DPA], and on separating as 2 2 2 2 to DX TX DA AP− − . But from the nature of the hyperbola, [see second note below], 2 2 2 is DX TX AD− , and by hypothesis AP2 is AD AK× .Therefore the small parts are in turn as AD2 to

2AD AD AK− × ; that is, as AD to AD AK− or AC to CK: and thus the small part of the sector TDV is PDQ AC

CK× ; [on account of AC to AD. For indeed 1

2PDQ AD PQ= × , and

thus 12 AD PQ AC

CKTDV × ×= .] ; and thus on account of AC and AD given, as PQCD , that is,

directly as the increment of the velocity, and inversely as the force generating the increment ; and thus as the increment of the corresponding particle of time. And on adding together the sum of the elements of time, in which all the elements PQ of the



velocity AP may be generated, as the sum of the sectors ATD, that is, the total time as the whole sector. Q. E. D. [Here we follow the demonstration of Leseur & Janquier note one: In the square ratio of the sides ..... For if from the point q there is drawn to Dp the small line qr itself parallel to vt, the two vanishing triangles Dqr, Dvt are similar and [their areas] in the ratio of the squares of the sides Dq, Dv (from Euclid's Elements, Prop. XIX, Book VI) and triangle Dqp is equal to triangle Dqr with pr vanishing with respect to Dq ; therefore pD2 is to tD2, or

AD2, as triangle qDp to triangle tDv, and thus 2

2AD qDp

pDtDv ×= , from which on account of

the given radius of the circle AD, the element tDv is as 2qDppD

.

Note 2 : but from the nature of the hyperbola, .... Since (by Theorem II, Concerning the Hyperbola: Apollonius), the rectangle ( )2AD AX AX+ × , is to the square of the ordinate TX, as the transverse side is to the latus rectum, truly this hyperbola is equilateral, there will be (by Theorem II Concerning the Hyperbola: Apollonius)

( )2 2TX AD AX AX= + × . But ( ) 2 22AD AX AX DX DA+ × = − , (by Prop. VI, Book II, Elements),

2 2 2 2 2 2 and hence TX DX DA DX TX DA= − − = .] Corol. I. Hence if AB may be equal to the fourth part of AC, the distance that the body will describe in some time by falling, will be to the distance, that the body with the maximum velocity AC, can describe in the same time by progressing uniformly, as the area ABNK it can be shown to describe by falling which distance, to the area ATD, so that the time may be put in place. For since there shall be AC to AP as AP to AK, (per corol. I, Lem. II. of this section) LK will be to PQ as 2AK to AP, that is, as 2AP to AC, and thence LK to 1

2 PQ as AP to 14 AC or AB; and KN is to AC or AD, as AB to CK ; [property of the

hyperbola : Th. IV, de Hyperb.]; and thus from equation LKNO to DPQ as AP to CK. But there was DPQ to DTV as CK to AC. Again therefore from the equality LKNO is to DTV as AP to AC; that is, as the velocity of the falling body to the maximum velocity that the body by falling is able to acquire. Therefore since the moments of the areas ABNK and ATD , LKNO & DTV are as the velocities, all the parts of these areas taken together, likewise generated, as the distances likewise described, and thus the whole area generated from the beginning ABNK and ATD to the whole distance described from the beginning of the descent. Q. E. D.



Corol. 2. Likewise it follows also from the distance that is described in the ascent. Without doubt that whole distance shall be to the distance described in the same time with a uniform velocity AC, as the area ABnk is to the sector ADt. Corol. 3. The velocity of the body falling in the time ATD is to the velocity, that it may acquire in the same time in a distance without resistance, as the triangle APD to the hyperbolic sector ATD. For the velocity in the non-resisting medium may be as the time ATD, and in the resisting medium it is as AP, that is, as the triangle APD. And these velocities at the starts of the descents are equal to each other, thus as these areas ATD, APD. [ S & J note 91, and following note: The velocity Ap of the body ascending in the medium with resistance to the maximum height ABnk, is to the velocity AP of the body in the same medium descending from rest through an equal distance ABNK, as the secant of the angle ADp to the radius, or what amounts to the same, as the tangent Ap of the angle ADp, to the sine of the same. For because, by hypothesis, the area ABNK, is equal to ABnk, there will be

and on separating CK AC Ak AKAC CK AC CK= = , and on interchanging, Ak AC Ck AC Ak

AK CK AC AC+= = = ;

and thus 2

2Ak AC AC Ak ACAK AC AC

× + ×× = ; but by Prop. VIII, 2 2 and AC Ak Ap , AC AK AP× = × = .

Whereby 2 2 2 2

2 2 2Ap AC Ap DpAP AC AC

+= = , and hence Ap Dp DpAP AC AD .= = Q.e.d.

L & J Note : And these velocities at the starts of the descents are equal to each other on account of no resistance with respect to gravity, when a velocity arises. Therefore since the velocities in a non resisting medium shall always be among themselves as the areas ATD, and in a resisting medium they shall be as the triangle APD, the velocity acquired in a finite time in a medium with resistance ATD will be to the initial velocity in that medium with resistance will be as the finite triangle APD, to the triangle nascent triangle APD, and the initial velocity of descent in the non-resisting medium to the velocity acquired in the same medium in the finite time ATD will be as the nascent area ATD (equal to the nascent area APD) to the finite area ATD ; whereby, from the equality, the velocity of the body falling in the finite time ATD in the medium with resistance is to that velocity it may acquire falling in the same time in the non-resisting medium as the triangle APD to the sector of hyperbola ATD. ] Corol. 4. By the same argument the velocity in the ascent is to the velocity, by which the in the same time in a non-resisting distance may be able to lose all of its motion by ascending, as the triangle ApD to the circular sector AtD; or as the rectangle Ap to the arc At. [L & J: By the same argument.... For the velocity in the non-resisting medium becomes as the time AtD, and in the resisting medium it is as Ap, that is, as the triangle ApD on account of the given AD, and these velocities at the end of the ascent when the vanish are equal to each other, hence as the vanishing areas AtD, ApD; but the triangle

12ApD AD Ap= × , and the sector of the circle 1

2AtD AD At.= × Whereby ApD is to AtD as Ap to At.]



Corol. 5. Therefore the time, in which a body may acquire a velocity AP by falling in a resisting medium, is to the time, in which it may acquire a maximum velocity AC by falling in a non-resisting distance, as the sector ADT to the triangle ADC [see following note one]: and the time, in which the velocity Ap may be lost in ascending in a resisting medium, is to the time in which the same velocity may be lost in a non-resisting distance, as the arc At to the tangent of this Ap [see following note two].

[ Extended S. & J. Notes 92 and two notes, 93, 94 & 95 : Hence if the velocity of ascent Ap in the resisting medium were equal to the maximum velocity AC, the velocity will be Ap or AC, to the velocity in which the body in the same time in a non-resisting space may be able to lose all its upwards motion, as the triangle ACD, to the eighth part of the circle, or as the radius to the eighth part of the periphery, or what amounts to the same thing, as the square of the circumscribed to the circle to the area of the circle. For while there becomes Ap AC= , the triangle ApD is equal to the triangle ACD, and the sector AtD , the eighth part of the circle, and thus the arc At is the eighth part of the periphery, and the triangle ACD is to the sector AtD, as AC to the arc At, and therefore the triangle ACD, on account of AC AD= , is the eighth part of the circumscribed square to the circle. Note one : as the sector ADT to the triangle ADC ..... For since AP may show the velocity acquired in the time ATD in the resisting medium, AY is taken such as to show the velocity produced in the same time in the non-resisting medium, and there will be by Corol. 2 AP APD

AY ATD= , and since also AC may set out the maximum velocity, AYAC will be as

the time in which the first swiftness can be acquired in the non-resisting medium to the time in which the maximum velocity can also be acquired in the non-resisting medium; and since the time in which the swiftness AY is acquired, may be expressed by the area ATD, AY

AC as ATD to the area which may set out the time in which the maximum velocity

is acquired in the non-resisting medium, and thus since there shall be AP APDAY ATD= and

that areaAY ATDAC = , there will be from the equality that area

AP APDAC = ; but with the common altitude

DH taken AP APDAC ADC= , therefore the area which expresses the time in which the maximum

velocity is acquired in the medium without resistance, is the area ADC. From which it follows that the body in the resisting medium, cannot acquire the maximum velocity AC except by falling for an infinite time. For since there is made AP AC= , DT coincides with the asymptote DC of the hyperbola ATV, and the sector ADT becomes infinite. Note two : As the sector ADT to the triangle ADC.... For since AP may show the velocity acquired at the time ATD in the medium with resistance, AY may be taken such as to show the velocity produced at the same time in a medium without resistance, and there will be, by Corollary 2, AP APD

AY ATD= , and also since AC may show the maximum velocity, there will be AY to AC as the time in which the first speed AY may be acquired in the non-resisting medium can be acquired, to the time in which the maximum speed AC also in the non-resisting medium may be acquired, and since the time in which the speed AY is



acquired, may be expressed by the area ATD, AY to AC will be as ATD to the area which may show the time in which the maximum velocity in the non-resisting medium may be acquired, and thus since there shall be AP APD

AY ATD= , and that areaAY APDAC = , from the equality

that areaAP APDAC = , but with the common altitude DH taken, there is AP APD

AC ADC= ; therefore the area which will express the time in which the maximum velocity may be acquired, is the area ADC. From which it follows that the body in the resisting medium, to acquire the maximum velocity by falling is not possible unless in an infinite time. For since there is made AP AC= , DT will coincide with the asymptote DC of the hyperbola ATV, and the sector ADT shall be infinite. Note three : As the arc At, to the tangent of this Ap..... If indeed, by Corollary 4, the velocity Ap required to be made zero in the resisting medium in the time AtD, is to the velocity requiring to be made zero in the same time in the non-resisting space, as the triangle ApD is to the sector AtD; and also as the time in which the velocity Ap in the non-resisting medium may be reduced to zero to the time AtD in which the other velocity in the non-resisting space is reduced to zero, that likewise is with that in which the velocity Ap in the resisting space is reduced to zero. Whereby the time in which the velocity Ap, may vanish in the non-resisting medium is to the time AtD in the resisting medium in which the velocity may vanish, as triangle ApD to the sector AtD, or the tangent Ap to the arc At of this. Therefore the proposition is apparent. Extra Note 93. Hence the time in which the velocity of the body Ap can be lost by ascending in the resisting medium, is to the time in which the maximum velocity AC in the non-resisting medium may lose by ascending or acquired by descending as the sector of the circle AtD, to the triangle ADC, or as the arc At to the radius AD. For in the non-resisting medium the velocity Ap is to the velocity AC, as the time ApD, in which the velocity AC is generated or extinguished, that hence will be 1

2 or AC ApDAp AD AC× × , that

is, the triangle ADC. Therefore since the time in which the velocity Ap is extinguished in the resisting medium, may be shown by the sector AtD, the proposition is apparent. Extra Note 94. The time in which the body in the resisting medium may acquire the velocity AP by descending, of by ascending may lose the velocity Ap, is to the time in which it may acquire or lose the same velocity in the non-resisting medium, as the sector ADT, or ADt, to the triangle ADP, or ADp, respectively. And indeed, by Corollary 5 and note 93, the time in which the velocity AP may be generated in the non-resisting medium, or the velocity Ap removed, is to the time in which in which the maximum velocity AC may be generated or extinguished in the non-resisting medium, as ADT or ADt, to ADC; and the time in which the velocity AC may be generated or extinguished in the non-resisting space, is to the time in which the velocity AP or Ap may be generated or extinguished in the same non-resisting space, as AC to AP or Ap, and with the common altitude DA taken, as ADC to APD or ApD. Whereby, from the equality, the time in which the velocity AP may be generated in the medium with resistance, or the velocity Ap made



zero, is to the time in which the same velocity may be produced or removed in the same distance in the non-resisting medium, as ADT to ADP or ADp. Extra Note 95. If the speed Ap of the body ascending in the resisting medium were equal to the maximum AC, there will be ADp ADC= , and the sector ADt the eighth of the circle. Whereby the time in which the body ascending in the resisting medium can lose the maximum velocity AC is to the time in which it may lose the same in a non-resisting space, as the eighth of the circle to the triangle ADC, that is, as the area of the circle to the circumscribed square, or also as the 8th part of the periphery to the radius. We return to the Principia.] Corol. 6. Hence from the given time the description of the distances of ascent or descent is given. For the maximum velocity is given in an infinite descent (by Corol. 2. & 3, Theorem VI. Book. II.) and thus the time is given in which that velocity may be acquired by falling in a non-resisting medium. And on taking the sector ADT or ADt to the triangle ADC in the ratio of the time given to the time just found ; then there will be given the velocity AP or Ap, then the area ABNK or ABnk [note one below], which is to the sector ADT or ADt as the distance sought to the distance, that in the given time, with that maximum velocity now found before, it can describe uniformly. [L. & J. Note 96 : And thus the time is given ... For since the uniform accelerating forces, shall be as the velocities which they generate directly and the times in which these may be generated given inversely (13. Book I) to the uniform accelerating force by which the body may be acted on in some medium, or with the ratio of this force to some known other known force, for example, to the force of terrestrial gravity, and likewise with that given velocity that the accelerating force produced, will be giving the time in which that given velocity has arisen. For let the given accelerative force be to the known force of gravity, as a to b, the velocity c generated by that given accelerative force in the time x, and the velocity C that the force of gravity may generate in some given time t, there will be a c C

b x t/= . From which the time is found bctaCx = .

Note one : .....then the area ABNK or ABnk : For there is (from the dem. of Prop. VIII) and ApAC ACAP

AP AK Ap Ak,= = , and thus with AC and AP or Ap given, Ak or Ak are given, and

the corresponding areas ABNK, ABnk, which can be found by tables of logarithms. ] Corol. 7. And by going backwards, from the given ascent or descent distance ABnk or ABNK, the time ADt or ADT will be given. [Note 97 : And by going backwards....Without doubt the area ABnk or ABNK is required to be taken to the triangle ADC in the give ratio of the distance ascended or descended be to the square of the distance, that the body in the non-resisting medium will describe in falling to the maximum velocity it may acquire, and thus Ak or AK will be given. And hence Ap or AP or the velocity will be given; moreover from these the sector ADt or ADT may be given, or the time (by Corollary 5). For the distance that the body in the non-resisting medium will describe in descending so that the maximum velocity AC may be



acquired is called A, the time in which that distance is described T, the distance that it will describe in the resisting medium so that it may acquire the velocity AP, or the velocity lost Ap, is called a, the time t, and the distance that the body will describe in that time t and the maximum velocity AC by progressing uniformly shall be S, and because the body with the maximum velocity AC by progressing uniformly, it time T, will describe the distance 2A, there will be 2

S tA T= . But by Cor. 5 and note 93, or t ADT ADt

T ADC= , and thus or

2S ADT ADtA ADC= , and by Cor. 1 & 2, or s ABNK ABnk

S ADT ADt= . Whereby from the equality, or

2s ABNK ABnkA ADC= .]

PROPOSITION X. PROBLEM III. A uniform force of gravity may extend directly to the horizontal plane, and let the resistance [to motion] be conjointly as the density of the medium and as the square of the velocity: then both the density of the medium is required at individual places, which may be done so that the body may move along some given curved line, as well as the velocity of the body and the resistance of the medium at individual places. Let PQ lie in a perpendicular plane to the horizontal scheme ; PFHQ a curved line crossing this plane at the points P and Q; G, H, I, K four locations of the body on this curve going from F to Q ; and GB, HC, ID, KE four parallel ordinates sent to the horizontal plane from these points, and the lines meeting the horizontal line PQ standing at the points B, C, D, E ; and BC, CD, DE shall be equal distances between the ordinates. From the points G and H, the right lines GL, HN may be drawn touching the curve in G & H , & with the ordinates CH, DI projected up meeting at L & N & the parallelogram HCDM may be completed. And the times [see note one below], in which the body will describe the arcs GH, HI, will be in the square root ratio of the altitudes LH, NI, that the body may be able to describe in these times, by falling from the tangents; and the velocities will be directly as the lengths described GH, HI and inversely as the times. The times may be shown by T and t, and the velocities by and GH HI

T t ; and the decrement of

the velocity made in the time t may be put in place by GH HIT t− [see note two below].

This decrement arises from the resistance retarding the body, and with gravity accelerating the body. Gravity generates a velocity in the falling body and by describing a distance NI in falling, so that it may be able to describe twice that distance in the same time, as Galileo demonstrated, that is, a velocity 2NI

t : but with the body describing the arc HI [see note three below], that arc increases only by the length HI HN− or MI NI

HI× ; and thus it generates only the

velocity 2MI NIt HI

×× This velocity may be added to the

aforesaid decrement, and the decrement of the velocity will be had arising from the resistance only, clearly 2GH MI NIHI

T t t HI×

×− + . And hence since gravity in the same time generates the



velocity 2NIt from the body falling ; the resistance will be to gravity as 2GH MI NIHI

T t t HI×

×− +

to 2NIt or as 2 to 2t GH MI NI

T HIHI NI .× ×− + Now for the abscissas CB, CD, CE there may be written 2o, o, o− . For the ordinate CH there may be written P, and for MI there may be written some series

3 etcQo Roo So .+ + + And all the terms of the series after the first, clearly 3 etcRoo So .+ + shall be NI, and the ordinates DI, EK, & BG will be

3 3 3 c 2 4 8 0 etc etcP Qo Roo So & .P Qo Roo S .& P Qo Roo So .− − − − − − − + − + − , respectively. And by squaring the difference of the orders BG CH &CH DI− − , and to the square produced by adding the squares of BC, CD, the squares of the arcs GH, HI

3 32 etc 2 etcoo QQoo QRo . & oo QQoo QRo .+ − + + + + The roots of which

1 11 and 1QRoo QRoo

QQ QQo QQ , o QQ

+ ++ − + + shall be the arcs GH and HI. Besides if from

the ordinate CH there may be taken half the sum of the ordinates BG and DI, and from the ordinate DI there may be taken half the sum of the ordinates CH and EK, the sagitta of the arcs GI and HK will remain, 3 and 3Roo Roo So+ . And these shall be proportional to the small lines LH and NI, and thus in the square ratio of the infinitely small times T and t and thence the ratio

32R 3 2 is or ; and R Sot So t GH MI NI

T R R T HIHI++ × ×− + , by substituting the

values now found of these tT , GH, HI, MI & NI, there emerges 3

2 1SooR QQ+ . And since

2NI becomes 2Roo , the resistance now will be to gravity as 32 1Soo

R QQ+ to 2Roo , that

is, as 3 1 to 4S QQ RR+ . But this velocity is , whatever the body from any place H, going along the tangent HN, in a parabola by having a diameter HC and the latus rectum 1 or HNq QQ

NI R+ , then can be

moving in a vacuum. And the resistance is as the density of the medium and jointly with the square of the velocity, and therefore the density of the medium is as the resistance directly and

inversely as the square of the velocity, that is, as 3 14

S QQRR+ directly and 1 QQ

R+ inversely, that

is, as 1S

R QQ+. Q. E. I.

[L. & J. Note one : And the times..... For in the same moment of time that a body in place at G may describe the tangent GL by the in situ force of the motion, a body may fall through such a height LH under the uniform force of gravity in a non-resisting medium in that time itself; for the effect of the resistance diminishes that height itself by an infinitely small amount , which thus is not to be considered here, and thus the body is considered to describe the complete arc GH by a force composed from the in situ force of the motion and the force of gravity. And in a similar manner, in the same time that it will describe the arc HI, by the force of gravity it may fall through the height NI. Whereby (by Lemma X of Book I), the times in which the body will describe the arcs GH, HI, or in which it falls through the heights LH, NI, are in square root ratio of these heights.



L. & J. Note two : And the decrement of the velocity..... For if the velocity through the arc HI were the same as the velocity through the arc GH, the one may be shown by GH

T , the

other by HIt . Whereby if the velocity may decrease, the decrement of that made in the

time t may be expressed by GHHIt T− . But if indeed the velocity may increase, it may be

expressed by GH HIT t− ; this decrement or increment arises from the resistance of the body

by retardation along the direction of the tangent HN or directly opposite to the arc HI, and from the motion of the body under gravity, for the force of gravity of the body descending may be seen to be divided into two forces normal and tangent; the motion of the body along the curve may be accelerated by the tangential force, and which normal force may neither accelerate nor retard. Whereby if the resistance is greater than the force of gravity, the motion is retarded, if less it will be accelerated, and if equal, neither accelerated nor retarded. L. & J. Note three : but with the body....... For by the in situ force alone, the body may describe the tangent HN in the time t, and by the force of gravity alone the altitude NI, indeed with the forces taken together it will describe the arc HI. Whereby gravity may augment the distance along the direction HN or HI only by the amount HI HN− . But

MI NIHIHI HN ×− = . If indeed with centre H and radius HN, the arc of a circle NR may be

considered described, cutting HI in R, the two triangles IRN, IMH will be similar, on account of the MIH common to the triangle on each side, and the angle IRN, IMH right, and thus equal; from which there is : : or HI MI NI RI HI HN ;= − and therefore

MI NIHIHI HN ×− = . Therefore since RI shall be the distance described in the time t by the

tangential force of gravity, that velocity which that force may generate in the time t, may be expressed by 22 MI NIRI

t t HI×

×= . L. & J. extra notes : And there will be had the decrement of the velocity arising from the resistance alone, evidently 22 MI NIRI

t t HI×

×= , not only in that case in which the resistance may be greater than the tangential force of gravity, but also in that case in which that may be superior to the other. For let the decrement of the velocity arising from the resistance alone be V, since the increment of the velocity arising from the tangential gravitational force shall be 2MI NI

t HI×

× , in the first case there will be 2MI NI GH HIt HI T tV ××− = − , and thus

2GH MI NIHIT t t HIV ×

×= − + ; but in the second case there will be 2MI NI GHHI

t HI t TV×× − = − , and therefore 2MI NI GH HI

t HI T tV ××= + − ,

which is the same expression as the former. The resistance will be to gravity ...... For the accelerating and retarding forces are as the elements of the velocity which may be generated or removed in given moments of time.



There may be written –o, o, 2o.... For if the abscissas CD, CE may be taken positive, the abscissas in the opposite part CB, etc. must be assumed to be expressed negative. And for MI some series may be written.... For the difference of the fluxions MI of the ordinates CH, DN can be expressed by an infinite series 3 etcQo Roo So .+ + + , in which Q, R, S, etc. are finite quantities assumed here generally, and afterwards to be determined in the individual cases, and o is the nascent increment and a constant for the abscissa. And the ordinates..... For indeed 3 etcDI DM MI CH MI P Qo Roo So .= − = − = − − − − (by Hypothesis) ; and because 2CE o= , if in place of the value of the ordinate DI there may be written 2o , DI will become 32 4 8 etcEK P Qo Roo So .= − − − − ; and in a similar manner because CB o= − , if in the value of the ordinate DI in place of + o there may be written – o , there becomes

3 etcDI BG P Qo Roo So .= = + − + − ....the squares of the arcs GH, HI, etc. may be had..... For indeed, on account of the right angle HMI,

2 2 2HI HM MI= + , and 3 and etcHM CD o, MI CH DI Qo Roo So .= = = − = + + +

, and thus 2 2 2 2 3 2 42 etcHM oo,MI Q o QRo R o .= = + + + ; thence

2 2 2 2 32 etcHI o QQR o QRo .= + + + But the terms in

which there 4 5 etco , o , . that vanish before the others may be ignored and contribute nothing to that expression. Whereby with the extraction of the square root made

11 QRoo

QQHI o QQ

+= + + , with the remaining terms ignored; and in a

similar manner there is found 1

1 QRooQQ

GH o QQ +

= + − .

.....the sagitta of the arcs GI and HK will remain.... The chord GI may be joined cutting CH in V, and from the point I the perpendicular IS may be sent cutting CH in T. On account of the similar triangles ITV, ISG, there will be 1

2or as DC TVITIS DB GSi.e. , and thus

2 and 2 2 and 2 2GS VT , GB VT SB VT DI GB DI VT DI= = + = + + = + , whereby half the sum of the ordinates GB and DI is VT DI+ , or VC, which if taken from the ordinate CH, there will remain the sagitta VH of the arc GI. And by similar reasoning it is apparent that the sagittam IX of the arc HK is equal to the difference between the ordinates DI and half the sum of the ordinates CH and EK. ...And these are the small proportional lines LH and NI.... For with the points B, C, D, E and G, H, I, K joined, the figures NHIXH, LGHVG become similar, and the homologous



sides HV and IX, LH and NI are proportional; but the small lines LH, NI are as the squares of the times T , t (from the demonstration), by which the arcs GH, HI are described. ...And thence the ratio etct

T , . ..... For from the demonstration, 2 3

23 3t Roo So R SoIX

HV Roo RT+ += = =

and thus 33 3 RR SRot R So RR SRoT R RR R

++ += = = ; but 323 SRoRR SRo R+ = + , with

negligible terms ignored : whereby there will be 3

2 321

SRoRt SoT R R .+= = +

...... there emerges 32 1Soo

R QQ+ ..... For there is 32 1

11 Soo QQQRoot GH

T RQQo QQ +×

+= + − + ,

which with the term in which there is found o3 ignored, which vanishes before the rest.

From which there becomes 32 1 2

1Soo QQ QRoot GH

T R QQHI +×

+− = − ; but with

2 2 and MI Qo NI Roo= = ignored with the remainder of the terms of the series vanishing,

and hence 222

1QRoMI NI

HI QQ×

+= , with the term

1QRoo

QQ+ ignored in the value of the arc HI.

Whereby there will be 3 122

Soo QQt GH MI NIT HI RHI .+× ×− + =

....Then it can move in a vacuum....... For since the velocity along the arc HI, or along the emerging tangent HN, may be able to be considered equal, and the body in the same moment of time in which it may describe HN by the vi insitu, by the uniform force of gravity, with the resistance ignored which here may be considered as zero, falls through the height NI; the nascent arc HI, that the body with the forces taken together will describe, can be taken as the arc of a parabola, the diameter of which is HC, the tangent HN with the ordinates parallel, and NI parallel and equal to the abscissa to which the equal ordinate HN may correspond. Whereby the latus rectum of this parabola will be

2HNNI , by the geometry of the parabola, or (by Lemma VII of Book I),

2 1oo QQoo QQHINI Roo R

+ += = , with the negligible terms to be ignored. And if that body hence may be moving in a vacuum, it will describe this parabola. ...That is, so that, etc. For because the resistance is to constant gravity as 3 1S QQ+ to

4RR, the resistance will be as 3 14

S QQRR+ . But the velocity is as HI

t , and the square of that

as 2

22; and is HI

tHI oo QQoo+ , neglecting smaller quantities, t2 is indeed as NI, or as

Roo (from the demonstration) ; and thus the square of the velocity is as 1 QQR+ . Whereby

the density of the medium will be as ( )3 14 1

S QQR QQ

++

, and from the given number 34 , as

( )1

1 1S QQ SR QQ R QQ

++ +

= . Back to Newton.]



Corol. 1. If the tangent HN may be produced to the side then it may cross the ordinate AF at some point T: HT

AC will be equal to 1 QQ+ , and thus can be written for 1 QQ+ in the above. n which ratio the resistance will be to gravity as 3 to 4S HT RR AC× × , the velocity will be as HT

AC R and the density of the medium will be as S AC

R HT×× .

[ HT

AC will be equal.... From the points H and I the perpendiculars HS and IR may be sent to AF and CH, and on account of the similar triangles IRH, HST, there will be HT to HS as

AC to HI or CD, and thus 1 1o QQHT HIAC CD o QQ .+= = = + ]

Corol. 2. And hence, if the curved line PFHQ may be defined by a relation between the base or the abscissa AC and the applied ordinate CH, as is customary; and the value of the applied ordinate may be resolved into a converging series : the problem may be solved expeditely by the first terms of the series, as in the following examples. Example1. Let the semicircular line PFH be described on the diameter PQ, and the density of the medium may be required which may act so that the projectile may be moving along this line. The diameter PQ may be bisected at A ; call AQ, n; AC, a; CH, e; and CD, o : and there will be DI2 or 2 2 2AQ AD nn aa ao oo,− = − − − or 2ee ao oo− − , and with the root extracted by our method, there becomes

3 3 3

3 3 52 2 2 2etcao oo aaoo ao a o

e e e e eDI e .= − − − − − − Here there may be

written nn for ee aa+ , and there emerges 3

3 52 2etcao nnoo anno

e e eDI e .= − − − −

A series of this kind may be separated into successive terms in this manner. I call the term the first, in which the infinitely small quantity o does not appear , the second, in which that quantity is of one dimension ; the third, in which it arises of dimension two ; the fourth, in which it is of the third, and thus indefinitely. And the first term, which here is e, always will denote the length of the ordinate CH standing at the start of the variable quantity o. The second term, which here is ao

e , will denote the difference e between CH and DN, that is, the small line MN, which is cut off by completing the parallelogram HCDM, and thus may always determine the position of the tangent HN [note one below]; as in this case by taking = MN ao a

HM eo e= . The third term 32nnoo

e, which here

will the small line IN, which lies between the tangent and the curve, and thus determines the angle of contact IHN or the curvature that the curved line has at H [note two below] . If that small line IN is of finite magnitude, it will be designated by the third term together with the following terms



indefinitely. But if that small line may be diminished indefinitely, the following terms arise infinitely smaller than the third, and thus may be able to be ignored. The fourth term determines the variation of the curvature, the fifth the variation of the variation, and thus henceforth. From which along the way the use of these series in the solution of problems is apparent, and not to be dismissed, which depend on tangents and the curvature of curves. The series

3

3 52 2etcao nnoo anno

e e ee .− − − − now may be established, with the series

3 etcP Qo Roo So .− − − − and from which by writing

3 52 2 and a nn ann

e e ee, , , , for P, Q, R and S and by writing 1 aa

ee+

for 1 QQ+ or ne , the density of the medium will be

produced as ane , that is (on account of n given), as a

e or ACCH ,

that is, as that length of the tangent HT, which may be terminated on the radius AF standing normally on PQ itself: and the resistance will be to gravity as 3a to 2n, that is, as 3AC to the diameter PQ of the circle: moreover the velocity will be as CH . Whereby if a body may leave from the position F with the correct velocity along a line parallel to PQ itself, and the density of the medium at the individual places H shall be as the length of the tangent HT, and the resistance also at some place H shall be to the force of gravity as 3AC to PQ that body will describe the quadrant of a circle FHQ. Q.E.I. But if the same body may be proceeding from the same place P, along a line perpendicular to PQ, and it may begin to move in the arc of a circle PFQ, it shall be necessary to take AC or a on the opposite side of the centre A, and therefore the sign of this must change and it is required to write – a for + a. With which done the density of the medium may appear as a

e− . But a negative density, that is, which accelerates the motion of bodies, is not allowed by nature : and therefore cannot happen naturally, so that the body by ascending from P may describe the quadrant of a circle PF. Towards this effect the body must be impelled to accelerate by the medium, and not to be impeded by resistance, [thus only one quadrant of the circle can be considered; either the left-hand quadrant for the initially vertically rising body, or the right-hand part for the initially horizontally projected descending body.]. [L. & J. Note one : The tangent HN may be produced so that it crosses the diameter AQ at T; and because of the similarity of the triangles HMN, TCH, there will be CT HM

HC MN= . Now truly in general there is and HM o MN Qo= = , and Q is the coefficient of the second term of the general series for any curve (from the demonstration of Prop. X) ; whereby if there is taken 1CT

HC Q= , the subtangent CT will be found.

S & J note two : Let O be the centre of curvature of the curve FHQ osculating at H; OH, OI the radii, HPI the chord of the arc HI, NP the radii, HPI the chord of the arc HI, NP the small arc of the circle with centre H and radius HN described. The two triangles IPN,



IMH will be similar, on account of the right angle at P and M, and the common angle at I; and therefore and hence NI HM NIHI

HM NP HI NP .×= = The measure of the angle NHI, that the tangent HN makes with the subtended arc HPI, is half the arc HI, and to the angle at the centre HOI the measure is the whole arc HI (from the nature of the circle) ; from which NP or ( ) 1

2: or :HM NIHI HN HI HI HO× = , and thus the radius of osculation

3

2HI

HM NIHO .×= And because (from the demonstration of Prop. X),

1 ; ; and HI o QQ HM o NI Roo= + = = ; there will be ( )321

2QQ

RHO .+= But the contact angle and the curvature of the curved line FHQ in H is as the radius of osculation HO inversely, that is, as

( )32

21

RQQ

.+

Whereby that angle, or the curvature at H will be

determined, by the given second and third terms of the series into which the value of the applied ordinate is resolved.] Example 2. The line PFQ shall be a parabola, having the axis AF perpendicular to the horizontal PQ, and the density of the medium is required, that enables the projectile to be moving along this curve. From the nature of the parabola [see note], the rectangle PDQ is equal to the rectangle under the ordinate DI and some given right line ; that is, if that right line may be called b; PC, a; PQ, c; CH, e; & CD, o ; the rectangle

by or 2 a o c a o ac aa ao co oo+ − − − − + − is equal to the rectangle b by DI, and thus DI equals 2ac aa c a oo

b b bo .− −+ − Now the second term of this series is required to be

written, 2c ab o− for Qo, likewise the third term for oo

b Roo . Since indeed there are no more terms, the coefficient S of the fourth term must vanish, and therefore the quantity

1S

R QQ+, to which the density of the medium is proportional, will be zero. Therefore the

projectile does not move in a parabola for any density of the medium, as Galileo showed at one time. Q. E. I. [L. & J. Note : From the point I to the axis of the parabola FA a perpendicular IR may be sent, and let the latus rectum axis be equal to b; from the nature of the parabola, there will be : 2 2 2 and b FR RI AD b FA AQ .× = = × = Whereby

( )( )2 2 or or b FR b FA b RA b DI AQ AD AQ AD AQ AD PD DQ.× − × × × = − = + − = ×]



Example. 3. The line AGK shall be a hyperbola, having the asymptote NX perpendicular to the horizontal plane AK, and the density of the medium is sought, which may enable the projectile to move along this line. MX shall be the other asymptote, crossing the applied ordinate produced DG at V; and from the nature of the hyperbola, the rectangle XV by VG may be given [from the nature of the hyperbola : see de Hyper. Th. 4; Apoll.]. Moreover the ratio DN

VX , and therefore also the rectangle DN by VG may be given [from similar triangles.] That shall be bb, and on completing the parallelogram DNXZ; there may be called : BN, a; BD, o; NX, c; and the given ratio VZ to ZX or DN may be put equal to m

n . And there will be DN equal to a o− , VG equal to bba o− , VZ equal to m

n a o− ,

and GD or NX –VZ –VG equal to the same term m m bbn n a oc a o −− + − . The term bb

a o− may be resolved into the converging series

3 43 etcbb bb bb bb

a aa a ao oo o .+ + + , and there becomes GD

equal to 3 42 3 etcm bb m bb bb bb

n a n aa a ac a o o o o .− − + − − −

The second term of this series m bbn aao o− is required to

be taken for Qo [as this series is equal to the series 3 etcP Qo Roo So .− − − − , and so the terms may be

equated ]; the third with the sign changed 32bb

ao for

2Ro and the fourth also with the sign changed 43bb

ao

for 3So , the coefficients of these 3 4and m bb bb bbn aa a a

,−

are required to be written in the above rules for Q, R & S. With which done the density of the medium will be as [the numerator and denominator in 4

bba

:]

4

4 4

4

12 213

or bba

bb mm mbb b mm mbb baa aann naa nn n aaaa+ − + + − +

that is, [note one] if in VZ there may be taken VY equal to VG, as 1

XY . For if aa and 42mm mbb b

nn n aaaa − + are squares of XZ and ZY themselves. Moreover the resistance is found in the ratio to gravity that 3 has to 2XY YG [note two]; and the velocity is that, [note three], by which a body may proceed in the parabola, with vertex G, the diameter DG, and by having the latus rectum

2XYVG . And thus put in place that the densities of the medium at

the individual places G shall be inversely as the distances XY, and that the resistance at some place G shall be to gravity as 3XY to 2YG; and the body sent from that place A, with the correct velocity, will describe that hyperbola AGK. Q. E. I.



[L. & J. Note one : For indeed and bb bb m ma o a n nVG , VG a o a−= = = − = , where BD or o

vanishes. Whereby bb ma nVY VZ ZY a− = = − ; and because ZX or

2 2 2 and DN a, YX YZ ZX= = + , there becomes 42 2 ;mm mbb b

nn n aaYX aa aa= + − + and thus

the density of the medium as 1XY .

Note two : The resistance is to gravity in the ratio 3 14

S QQRR+ , that is, as

2 4 4

4 4 63 2 41 to bb mm mb b b

nn naaa a a× + − + , or on dividing by 5

bba

, as

2 4 2

22 43 to mm mb b b

nn n aaaa aa+ − + , or as 3 to 4 2 . XY VG YG=

Note three : The latus rectum of this parabola is

42 42 24

3

11mm mbb b mm mbb bnn naa a nn n aa

bb bbaa

aa aaQQ YXR VG

+ − + + − ++ = = = . But the velocity is as

1 QQR+ and thus as YX

VG.]

Example 4. It may be considered generally, that the line AGK shall be a hyperbola, with centre X, asymptotes MX and NX described by that rule, that with the rectangle constructed XZDN the side of which ZD may cut the hyperbola in G and its asymptote in V, VG were inversely as some power nDN of ZX or of DN, the index of which is the number n [note one]: and the density of the medium is sought, by which the projectile may progress along this curve. For BN, BD, NX there may be written A, O, C respectively, and let or =VZ d

XZ DN e , and

VG is equal to nbb

DN, and DN equals bb d

eA OA O, VG , VZ A O

−− = = − , and GD or

NX VZ VG− − equals nd d bbe e A O

C A O−

− + − . This term nbb

A O− may be resolved into an

infinite series [i.e. by the binomial expansion of ( ) nA O −− :] 3

1 2 32 33 2

2 6etcn n n n

bb nbb nn n n nn nA A A A

O bbO bbO .+ + ++ + ++ + + and GD equals

3

1 2 32 33 2

2 6etcn n n n

d bb d nbb nn n n nn ne eA A A A

C A O O bbO bbO .+ + ++ + + + +− − + − − − The second term of this

series 1nd nbbe A

O O+− is required to be taken for Qo, the third 22

2 nnn nA

bbO++ + for 2Ro , the

fourth 3

333 2

6 nn nn n

AbbO+

+ + + for 3So . And thence the density of the medium 1S

R QQ+, in place

of some G, shall be 42 2 2

2

23

dd dnbb nnbee n neA A

nA A A

+

+ − +[note two] , and thus if in VZ , VY may be taken

equal to n VG× , that density is inversely as XY. Indeed 2A and 42 2

2dd dnbb nnbee n neA A

A A− + are the squares of XZ and ZY [note three]. But the resistance in



the same place G [note four] shall be to gravity as 3 to 4XYAS RR× , that is, as XY to

2 22

nn nn VG++ × . And the velocity in that place itself is, by which the projected body may

progress in a parabola, vertex G, diameter GD [note five] and with giving the latus rectum 21 2 or QQ XY

R nn n VG+

+ ×. Q.E.l.

[L. & J. Note one: But hence the hyperbola, while it is produced, the lines XM, XN also produced approach continually, and it is evident that these can only touch at an infinite distance. Note two : For it is found that 2S n

R SA+= , and

4

1 2 221 1 n n

dd dnbb nnbee eA A

QQ + ++ = + − + ; and thus, on account

of the given number

2 422

2 13 1

is as dd dnbbA n bee n neA A

n SR QQ AA AA

, .++ + − +

Note three : For by Hypoth. and n

d nbb de eA

XZ DN A ZY VY VZ n VG A A= = = − = × − = −

; or n

d nbbe A

ZY VZ VY A= − = − , because YV is greater or less

than VZ. Whereby since there shall be 2 2 2XY XZ ZY= + , the density will be as 1XY .

Note four : Because from the demonstration, 31 there will be 3 1 S XYXY

A AQQ S QQ ×= + + = , and thence the resistance to gravity will

be as 3 to 4 S XYA RR,× or ( )2 4

2 34

3as to but 4 nnn n bRR A

S AXY , RR A +

+ ×× × = and

( ) ( ) 2

32

23 n

nn n n bA

S +

+ × += , and hence ( )( )

22 2 2 243 22 n

nn n b nn nRRAS nn A

VG+ × +++ ×

= = × , on account of bbnA

VG .=

Whereby the resistance is to gravity as 2 22 to nn n

nXY VG++ × .

Note five : Indeed there is 2

2 1XYA

QQ,= + and hence ( ) ( )2 21 2 2nQQ XY A XY

R nn n bb nn n VG+ ×

+ × + ×= = , on

account of nbbA

VG .= From which the velocity which is as 1 QQR+ , will be as XY

VG, on

account of the given number 2nn n+ . End of notes.]

Scholium.

By the same reasoning that gave rise to the density of the medium as S ACR HT×× in the first

corollary, if the resistance may be put as some power nV of the velocity V the density of

the medium will be produced as ( )42

1n

nS ACHT

R−

−× . [note one] And therefore if the curve can



be found by that law, so that the ratio may be given 42

nS

R− to ( ) 1nHT

AC−

, or

( )2

41 to 1n

nSR

QQ−

−+ : the body will be moving on this curve in a uniform medium with

resistance that shall be as the power of the velocity nV . But we may return to simpler curves. Because the motion shall not be parabolic unless in a non-resisting medium, with the motion here described in hyperbolas indeed it shall be through a constant resistance ; it is evident that the line, that a projectile will describe in a uniformly resisting medium, approaches closer to these hyperbolas than to a parabola [note two]. Certainly that line is of the hyperbolic kind, but which is described more about the vertex. Truly there is not much difference between this and that, why may not they be put to use with no inconvenience in place of those in practical matters. And perhaps these will soon be more useful, more accurate than the hyperbola and likewise better arranged. Truly in use thus they may follow. The parallelogram XYGT may be completed [note 3], and the right line GT touches the hyperbola in G, and thus the density of the medium at G is inversely as the tangent GT,

and the velocity likewise as GTqGV , moreover the resistance is to the force of gravity as

GT to 2 22

nn nn GV++ × .

Therefore if a body projected from the place A along the right line AH may describe the hyperbola AGK, and AH produced may meet the asymptote SNX in H, and with AI drawn parallel to the same may meet the other asymptote MX in I [note 4] : the density of

the medium at A will be inversely as AH, and the velocity of the body as 2AH

AI , and the

resistance at the same place as AH to 2 22

nn nn AI++ × . From which the following rules may

be produced.

[L. & J. Note one : If indeed there were 42

nS

R− to ( ) 1nHT

AC−

in the ratio a to b, there will be

( ) 1

4 412 2

1 and

n

n nn

nS a S AC aHTb AC b

R R HT

−

− −−

− ×

×= × = , and

therefore uniform. But by Corollary 1. Prop. X, 1HT

AC QQ= + : whereby if there were given the

ratio 42

nS

R− to ( ) 1nHT

AC−

, also the ratio of the squares

will be given : ( )2

41 to 1n

nSR

QQ−

−+ , and

conversely.



Note two : Since yet from these it cannot be agreed completely, that from these hyperbolas the density of the medium shall be inversely proportional to the variable right line XY, and besides it may not be shown satisfactorily that the curve, which the projectile will describe in the uniform medium in the hypothesis of the resistance proportional to the square of the velocity, to have a vertical asymptote as XN : since especially in this hypothesis of the resistance the horizontal distance described by the motion present, separated from gravity, may emerge infinite (by Cor.1, Prop.V.). Yet indeed hyperbolas can be found in which for that small part of the curve AGK, which in the course of practical matters is necessary, the right line XY shall be as approximately constant, and therefore the density of the medium as approximately uniform; from which it comes about that these curves in practice shall not be inconvenient to use. Note three : From the point G draw the ordinate BF through B, and from the point T to the ordinate DG there shall be sent the perpendiculars GR and TS, and let GT be a tangent at G. There will be = = GSFR FR

RG BD ST , on account of the similar triangles FRG, GST. But FR is Qo or

1 , is and nnbbO d

eAO BD O, ST ZX A+ − = = . Whereby there

will be 1 1 to 1, or to as or . n n

nbbO d nbb d GSe e ZXA A

A A, A+ +− − Therefore

ZY GS= ; and therefore the tangent GT is equal and parallel to the right line YX. But from the demonstration the density of the medium at G is inversely as the

tangent GT, the velocity at that place as 2GT

GV , and

the resistance to gravity as 2 22 to .nn n

nGT GV++ ×

Note four : With the point G coinciding with the point A. the tangent GT agrees with the tangent AH, and the right line VG with AI, and hence the density of the medium at A is inversely as the AH. End of notes.] Rule 1. If both the density of the medium as well as the velocity remain the same by which some body may be projected from A, and the angle NAH may be changed; the lengths AH, AI, HX will remain. And thus if these lengths may be found in some case, thereupon the hyperbola from some given angle NAH can be conveniently found. [L. & J. note on Rule 1 : With the index n of the hyperbola remaining the same and with the density of the medium at A, the length of the tangent remains AH which is inversely



proportional to the density. With which remaining velocity the body is projected from the place A, the

line2AH

AI remains which is as the velocity; and thus since given there shall be AH and AI given. On account of the parallels GT, YX, there is TX GY GV VY GV n GV= = + = + × (Example 4), and because with the point G coinciding with the point A, there becomes

and ; there will be GV AI , TX HX HX AI n AI .= = = + ×Whereby on account of the given quantities AI and n, also HX is given. From which if these lengths AH, AI, and HX may be found in some case, the hyperbola henceforth may be conveniently found from some given angle NAH. Indeed with these given, the points A, H, and I are given. Through H the line XHN is drawn vertical to the horizontal AN, and the point N will be given; and because HX is given, the point X is also given; indeed with the two points X and I given, the right line XIM is given with the point M by which the horizontal line MN may be cut. From which with some right line drawn VD normal to the horizontal AN, and if on that there may be taken =

n

nVG ANAI DN

, or as n

nXIXV

, the point G will be given on

the trajectory AGK. For indeed, by example 4, some ordinate VG to another ordinate IA , i.e. VG

IA is as n

nANDN

, or as n

nXIXV

.]

Rule 2. If both the angle NAB, as well as the density of the medium may remain the same at A, and the velocity may be changed by which some body is projected; the length AH will remain, and AI will be changed inversely in the square ratio of the velocity. [L. & J. note on Rule 2 : With the density of the medium serving its purpose at A, the length of the tangent AH will be maintained, which is inversely as the density. And

because the velocity at A is as 2AH

AI , and the square of the velocity as 2AH

AI , that is, as 1AI , on account of AH given; AI will be inversely proportional to the square of the

velocity.] Rule 3. If both the angle NAH , as well as the velocity of the body at A, and the accelerating gravity may be maintained, and the proportion of the resistance at A may be increased a little to gravity by some ratio ; the proportion AH to AI will be increased in the same ratio, with the latus rectum of the aforementioned parabola remaining, and proportional to that length

2AHAI : and therefore AH may be diminished in the same ratio,

and AI may be diminished in that ratio squared. Truly the ratio of the resistance to the weight may be increased, when either the specific gravity shall be smaller under an equal



magnitude [in its ratio to the resistance] or the density of the medium increased, or the resistance from the diminished magnitude, is diminished in a smaller ratio than the weight [is diminished]. [L. & J. note on Rule 3 : With the velocity of the body and with the acceleration of gravity given at the point A, the length

2AHAI is giving both by the square of the velocity as well as

by the latus rectum of the proportional parabola (Example 4). But the motive resistance, if it may be called that, to the motive gravity, is as 2 2

2 to nn nnAH AI++ × (Example 4).

Whereby if the proportion of the motive resistance at A to the motive gravity may be increased in some ratio, the proportion of 2 2

2 to nn nnAH AI++ × , or, on account of the given

number 2 22

nn nn++ , will increase AH

AI in the same proportion, and because the length2AH

AI is

constant, and hence 21 is as , and is as AHAI AH AI AH , it is necessary that AH may be

decreased in the ratio by which AHAI may be increased, and so that AI may be reduced in

that ratio squared.] Rule 4. Because the density of the medium near the vertex of the hyperbola is greater than at the place A; so that it may have a lesser density, the ratio of the smallest tangent GT to the tangent AH must be found, and the density at A to be augmented in a slightly greater ratio than half the sum of these tangents to the minimum tangent GT. [L. & J. note on Rule 4 : Because the density at some place G is inversely as the tangent GT, which is less near the vertex of the hyperbola than at the place A; it is evident that the density of the medium near the vertex of the hyperbola is greater than at the position A. The density at the place A may be called K, at the place G through which the minimal tangent GT may be drawn, may be called B ; and there will be

2 2 and hence , and GT AHK BGT GT AHK K B

B AH K GT K GT .++

++= = = But 2K B+ must be the mean density,

if the tangent AH becomes the maximum of all, and thus so that GT is the minimum of all; and thus, so that the density of the medium may be had as nearly uniform, it may be

required to increase the density at A in the ratio of half the sum of the tangents 2GT AH+

to the minimum tangent GT. Truly because the tangent AH is not the maximum of all, but other tangents drawn to the parts of the curve towards K are greater; the density at A is to

be increased in a ratio a little greater than half the sum 2GT AH+

to GT, so that the medium may be considered to be almost uniform. And from this agreement errors arising from that because the medium at the place A may be supposed denser, they may be almost corrected by other errors which arise from that because at G the medium may be placed rarer than for ratio of the curve AGK.] Rule 5. If the lengths AH, AI, may be given and the figure AGK is required to be described : produce HN to X, so that there shall be HX to AI as 1n + to 1, and with centre X and with



the asymptotes MX, NX a hyperbola may be described through the point A, by that rule, so that there shall be AI to some VG as to n nXV XI . [L. & J. note on Rule 5 : If the lengths AH, AI with the angle HAN may be given, and the figure AGK may be described : from the point H to the horizontal AN send the perpendicular HN ; produce HN to X, so that ( )1HX n AI= + × , as we have shown before for Rule 1, and with centre X and asymptotes MX, NX the hyperbola is described through the point A, by that law, so that

n

nXVAI

VG XI= , for some VG : for by example 4,

n n

n nVG AN XIAI DN XV

= = .]

Rule 6. From which the greater the number n is, from that the more accurate these hyperbolas shall be described in the ascent of the body from A, and the less accurate in the descent of this body to K; and conversely. The conic hyperbola maintains a mean ratio, and is simpler than the other curves. Therefore if the hyperbola shall be of this kind, and the point K is sought, where the projected body falls on some line AN passing through the point A: AN produced may cross the asymptotes MX, NX at M and N, and NK may be taken equal to AM. [L. & J. Extended Note on Rule 6 : Since the larger the number n becomes, there the greater these hyperbolas in the ascent of the body from A approach to trajectories described in a uniform medium, and there they are the less accurate in the descent to K; and conversely. For since the greater the number n, there the smaller the tangent GT, which is inversely proportional to the density, in the ascent of the body from A it may be varied; and there the more it may be changed in the descent to K, certainly the density of the medium shall be given at the A with the angle of projection HAN, and the quantity

2nAH+ proportional to the density at A will be given, by Example 4, and thus there the

tangent AH will be longer as the number n becomes greater; and because from the given angle HAN, the kind of right angled triangle may be given, and thus the ratio of the sides AH, AN, HN also are given, it is evident that with the increase in AH or in the number n, also the sides AN and HN may increase. From the demonstration in Example 4, with the body ascending, the square of the tangent GT, ( )22 2GT DN ZV nVG ,= + − and with the

body descending the square is ( )22 2GT DN nVG ZV .= + − From the nature of the

hyperbola AGK, and thus n n

n nDN nAI ANAI

VGAN DNnVG .×= = From the demonstration of the 1st

rule, ( ) ( )1 and thus 1HX n AI NX HN n AI= + × = + + × , and NX AI HN nAI .− = + But

on account of the similar triangles XZV, MNX, MAI, ( )or ZX DN MN MAZV NX AI= = , and

separating, DN AN ANZV NX AI HN nAI− += = ; from which there becomes DN HN nAI DN

ANZV .× + ×=

Whereby in the ascent of the body, ( )22 2 n

nDN HN nAI DN nAI AN

AN DNGT DN × + × ×= + − , and in the



descent

( )22 2 n

nnAI AN DN HN nAI DN

ANDNGT DN × × + ×= + − .

Now truly if the number n were large enough in the ascent, the lines AH, AN, HN both in the ascent and in the descent of the body are longer, and in the ascent from A, DN is almost equal to AN, in the descent indeed DN may be permitted to be a little less than AN. From which in the ascent from A there is nearly

n

nnAI AN

DNnAI× = , and

thus

( )22 2 2 2DN HNANGT DN DN HN×= + = + nearly. But 2 2 2AH AN HN= + : whereby the

ratio GT to AH in the ascent of the body from A is almost one of equality, while the number n may be supposed large enough, and hence the density does not vary very much ; in the descent indeed to K, DN becomes as some small amount with respect to the given AN, and thus the quantity

n

nnAI AN

DN× will increase markedly, and hence the tangent GT is

changed greatly when the number n is large. The opposite happens, if that number shall be exceedingly small. Again since the number n can be any integer or fraction, and in the hyperbolic conic n shall be equal to 1, which just as the medium may maintain a place among all the whole and fractional numbers, it is shown well enough that the conical hyperbola maintains a mean ratio between all the greater and lesser hyperbolas, and because it is simpler than the others, the true trajectory of the projectile in the medium can be given. Therefore if the hyperbola AGK shall be of this kind, and the point K is sought where the projected body strikes some right line, horizontal or oblique to the horizontal passing through the point A : AN produced meets the asymptotes MX, NX at M and N, and NK may be taken equal to AM, and the point K will be found, by Theorem I, Conics, Apoll. End of note.] Rule 7. And hence the method set out may be clear for determining this hyperbola from phenomena. Two similar and equal bodies may be projected, with the same velocity, at different angles HAK, hAk, and incident in the horizontal plane at K & k; and the proportion AK to Ak may be noted. Let this be as d to e. Then with some perpendicular AI length erected, [note one] assume some length AH to Ah, and thence deduce graphically the lengths AK, Ak, by rule 6. If the ratio AK to Ak shall be the same as with the ratio d to e, with the length AH had been correctly assumed [note two]. But if less take on the infinite line SM a length SM equal to the assumed AH, and erect the perpendicular MN equal to the difference of the ratios dAK

Ak e− drawn on some given line. By a similar



method from several assumed lengths AB several points N may be found, and through all by drawing a regular curved line NNXN, cutting the line SMMM in X. Finally AH may be assumed equal to the abscissa SX, and thence with the length AK found anew; and the lengths, which shall be to the assumed length, which shall be the assumed length AI and hence finally AH, as the length AK known by trial and error to the final length found AK, truly will be these lengths AI and AH [note three], that it was necessary to find. And also truly with these given the resistance of the medium at the place, clearly which shall be to the force of gravity as AH to 2AI. But the density of the medium is required to be increased by rule 4 and the resistance found in this manner, if it may be augmented in the same manner, and it becomes more accurate. [L. & J. Note one : For with the tangent AH given, both in magnitude and position, the vertical HN may be given together with the point N; and because also AI is assumed, also there is 2 by the demonstraton of Rule 1, because 1; HX AI n= = the centre X of the hyperbola is given, and thence on account of the given point I, the other asymptote XIM is given with the point M on the horizontal line MN; and by taking NK equal to the given MA, the point K is given, and hence the length AK will be obtained. And the other length Ak may be found in the same way. Note two : With the density of the medium given at A with the velocity of the body projected at differing angles HAK, hAk, the perpendicular AI remains, and the tangent AH is equal to the tangent Ah, by the 1st Rule. With the tangent AH and the angle HAK given, the hyperbola AGK can be described by the 6th Rule and the preceding note, and thus it is given both in kind and magnitude. From which finally the angle HAK and the ratio of the tangents HA to AI, the kind of hyperbola will be given finally, that is, all the hyperbolas will be similar that may be described from these two given. Whereby if in hyperbola AGK, which may be supposed to be described in the diagram, the tangent assumed AH shall be to the perpendicular AI, as the tangent of the hyperbola (that the body projected with an angle equal to HAK will describe in the resisting medium), is to its own perpendicular AI; the hyperbola AGK described on the page will be similar to the hyperbola which is described in the medium. And by the same argument the other hyperbola, whose amplitude is Ak, and tangent Ah, with the perpendicular AI remaining, will be similar to that hyperbola which the body described projected at an angle equal to hAk , in the second experiment. From which therefore, on account of the similitude of the figures described on the page and in the resisting medium, the amplitudes AK, Ak will be between themselves as the homologous amplitudes of the hyperbolas which were described in the experiments, that is dAK

Ak e= . Note three : For since the abscissa SM shall be assumed to be equal to the length AH, and the difference of the ratios dAK

Ak e− may be shown be the ordinate MN; where there

becomes and hence 0SM SX MN= = , also 0dAKAk e− = , and thus dAK

Ak e= , and SX indeed is equal to AH, by the preceding note. Thus if from the given perpendicular AI and the true length found AH with the angle HAN required to be found, as above, with the length AK ; on account of the similar figures in the resisting medium and described on the



page, the length AK found by experiment and the length AK finally found on the page, will be as the length AH in the resisting medium to the length AH drawn on the figure, and also as the perpendicular AI in the resisting medium assumed in the diagram. With which found, it will be possible to describe a hyperbola equal and similar to the hyperbola the body describes in the resisting medium. End of notes.] Rule 8. With the lengths AH, HX found; if now it may be wished to put in place the position of the line AH, along which the projectile sent with some given velocity, it will fall on some point K: at the points A and K the right lines AC, KF may be erected to the horizontal, of which AC tends downwards, and may be equated to AI or 1

2 HX , [by Rule 5, as 1n = .] .With asymptotes AK, KF a hyperbola may be described, the conjugate of which may pass through the point C, and with centre A and with and radius AH a circle may be described cutting that hyperbola at the point H; and the projectile sent along the right line AH will fall at the point K. Q. E. I. For the point H, on account of the given length AH, is located somewhere on the circle described. CH may be drawn passing through AK and KF, the one in E, the other in F; and on account of the parallel lines CH, MX and the equal lines AC, AI, AE equals AM, and therefore also equals KN [note one]. But =CE FH

AE KN , and therefore CE and FH are equal. Therefore the point H falls on the hyperbola described with the asymptotes AK, KF, the conjugate of which passes through the point C, and thus is found at the common intersection of this hyperbola and of the described circle. Q. E. D. Moreover it is to be noted that this operation thus may be had itself, whether with the right line AKN shall be parallel to the horizontal, or inclined at some angle to the horizontal : each produce two angles NAH, NAH from the two intersections H, h ; and because it suffices to describe the circle mechanically once in practice, then apply the endless ruler CH to the point C thus, so that the part FH of this, intersected by the circle and the right line FK, shall be equal to the part of this CE situated between the point C and the right line AK. [note 2] What has been said about hyperbolas may also be applied to parabolas. For if XAGK may designate a parabola that the right line XV may touch at the vertex X, and let the applied ordinates be IA, VG so that any powers of the abscissas XI, XV : n nXI , XV may be drawn XT, GT, AH, of which XT shall be parallel to VG, and GT, AH may touch the parabola at G and A: and the body projected with the correct velocity from some place A, along the right line AH produced, describes this parabola, only if the density of the



medium, at individual places G, shall be inversely as the tangent GT. But the velocity at G will be that by which a projectile may go on, in a space without resistance, in a conical parabola having the vertex G, diameter VG produced upwards, and the latus rectum

22GTnn n VG− ×

. And the resistance

at G will be to the force of gravity as GT to 2 22

nn nn VG−− × .

From which if NAK may designate a horizontal straight line, and with the density of the medium then remaining at A, while the velocity by which the body may be projected at some angle NAH may be changed; the lengths AH, AI, HX, will remain, and thence the vertex X of the parabola, and the position of the right line XI, and on taking VG to AI as to n nXV XI , all the points of the parabola G will be given, through which the projectile will pass. [Extended final note.] [L. & J. Note one : For if we suppose H to be the point sought, through which it is required to draw the line AH, by the construction there will be HX equal and parallel to IC, and thus CH parallel to IX or MX, and hence the triangles CAE, IAM similar and thus since there shall be CA AI= , by construction, also there will be AE MA KN= = , by Th. I Conics, Apoll. But on account of the similar triangles CAE, HNE, and on account of the parallel lines KF, NH, there is CE EH FH

AE EN KN= = ; and hence the point H falls on the hyperbola, again by Th.1. Note two : Since the point H may be determined by the intersection of a circle with the hyperbola, from the demonstration, and the circle can cut the hyperbola in two points, from the two points of intersection H, h, two angles are produced, or there are two positions of the tangent AH, along which the projectile sent with a given velocity falls on the point K. Extended final note : The line VG may be produced so that it cuts the horizontal line NK in D, and the line XZ parallel to the horizontal in Z. For BN, BD, NX there may be written A, O, c. respectively ; and M shall be the intersection of the lines XV, NK; and XN to NM, or on account of the similar triangles XNM, VZX , or

VZ dZX DN e= ; and thus

( ) and deDN A O, VZ A O .= + = × + Indeed because

or , in the given ratio nVG VX XN

XZ DN NMXV= ; also there will be as nVG DN . Therefore there

may be put ( ) ( ) ( )( )2 2 3 31 1 1 21 2 1 2 3 etc

n n nn nnA O n n A O n n n A ODN nA OAbb bb bb bb . bb . . bbVG .

− −−+ − − −= = = + + + + , and there will be



( )

( ) ( )( )2 2 3 31 1 1 21 2 1 2 3 etc

n

n nnn

A Ode bb

n n A O n n n A Od d nA OAe bb e bb . bb . . bb

GD VZ NZ VG A O c

A c O .− −−

+

− − −

= − − = × + − −

= − − + − − − −

Whereby there will be

( ) ( )( )2 31 1 1 22 6, and

n nn n n A n n n AnA O dbb e bb bbQ ;R S .

− −− − − −= − = = The ordinate Bg may be drawn through the point B, to which there may be sent the perpendicular Gr from G, and XY shall be equal and parallel to the tangent GT; and on account of the similar triangles Grg and XZY,

2 22

2 2 2Gr DNXZGg XY GT

= = ; but there is

( )2 2 2 2 and thus 1Gr O , rg QQOO, Gg OO QQ= = = × + ; whereby since also there shall be BN or DN A= , there will be

( )2 1 1GT AA QQ ,GT A QQ.= × + = + Per Corollary 1, Prop. X, the density of the

medium at the place G is as S AR GT×× and from the demonstration, 2

3S nR A

−= , and thus S AR GT×× is

as 23nGT− ; whereby, on account of the given number 2

3nGT− , the density varies inversely as

the tangent GT. The velocity at G, by Prop. X, is that, since by which the projectile may go forwards, in a space without resistance, in the parabolic conic having the vertex G, the diameter GD, and the latus rectum 1 QQ

R+ ; and thus since there shall be

( ) ( )2 2 2

21 2 2

nAbb

QQ GT GT GTR nn n VGA R nn n+

− ×− ×= = = → , from the demonstration, the latus rectum of the

parabola will be ( )22GT

nn n VG− ×. The resistance at G, by Cor. I, Prop. X, is to the force of

gravity 433 to 4 that is, as to RR A

SS GT RR A GT ×× × ; but ( ) ( ) ( )2 2 3 2 3

42

24 and 3n nnn n A nn n n A

bbbRR A , S

− −− × − × −× = = × , and thus

2 2 243 2 2

nnn n nn nRR A AS n bb n VG− −×

− −= × = × . Therefore the resistance will be to gravity, as GT to

2 22

nn nn VG−− × . The velocity at the place G, by Prop. X, is as ( )

21 2QQ GTR nn n VG ,+

− ×= and thus

on account of the given number 2nn n− , as GT

VG.

Therefore when the body is at A, the resistance of the medium is as 1AH , and the

velocity is as AHAI

; from which with both the density of the medium at A, as well as the

velocity with which the body is projected, and with some change in the angle NAH, AH will remain, and AH

AI, and hence AI. Again because



( )2 2 2 2 2 1ZY XY XZ GT DN AA QQ AA AAQQ= − = − = × + − = , and thus nnA d

bb eZY Q A A nVG VZ= × = − = − , and ZY VZ VY nVG+ = = , there will be in place of A, and hence Iy n AI , Ay XH nAI AI .= × = = − Whereby with AI remaining, also HX will remain, on account of the given number 1n − . The lengths AH, AI and hence HX may be found using the 7th Rule for the hyperbola made; and thence the point H will be given, by which if there is drawn THX perpendicular to the horizontal, with XH given, the position of the line XI will be given, and on taking

n

nVG XVIA XI

= , all the points G of the

parabola will be given, through which the projectile will pass. The most elegant of problems concerning the finding of trajectories that a body will describe in a medium with the resistance close to the square of the velocity was passed over by Newton in his Principia. The matter was completely resolved later by the most distinguished of mathematicians, Johan Bernoulli, Hermann, and Euler, who found analytically the trajectory described in some medium that had a resistance as some power of the velocity. End of extended note.]


Book II Section III. Translated and Annotated by Ian Bruce. Page 517

SECTION III. Concerning the motion of bodies in which it is resisted partially in the ratio of the velocity and partially as the square ratio of the same.

PROPOSITION XI. THEOREM VIII. If for a body resisted partially in the ratio of the velocity, partially in the ratio of the square of the velocity, and likewise that may be moved only by the force of inertia in a similar medium : and moreover the times may be taken in an arithmetic progression ; magnitudes inversely proportional to the velocities increased by a certain amount, shall be in a geometric progression. With centre C, with the rectangular asymptotes CADd and CH, the hyperbola BEe may be described, and AB, DE, dc shall be parallel to the asymptote CH. The points A, G may be given on the asymptote CD: And if the time may be put in place by the hyperbolic area ABED increasing uniformly ; I say that the velocity can be shown by the length DF, the reciprocal of which GD together with the reciprocal of the given CG shall produce the length CD increasing in a geometric progression. For the small area DEed given shall be as the smallest increment of the time, and Dd will be inversely as DE, and thus directly as CD. [For if

then tDEDE Dd t Dd CDΔΔ× = = ∝ .] But the decrement of 1

GD , which (by Lemma III of

this) is 2Dd

GD, will be as 2 2 or CD CG GD

GD GD+ , that is, as 2

1 CGGD GD

+ . Therefore in the time ABED

increased uniformly by the addition of the given small part EDde, 1GD decreases in the

same ratio with the velocity. For the decrease of the velocity is as the resistance, that is (by hypothesis) as the sum of two quantities, one of which is as the velocity, the other as the square of the velocity, and the decrement of 1

GD is as the sum of the quantities

21 and CG

GD GD, the former of which is 1

GD and the latter 2CGGD

is as 21

GD: hence 1

GD is as

the velocity, on account of the analogous decrement. And if the quantity GD, inversely proportional to 1

GD itself, may be increased by the qiven quantity CG ; the sum CD, with the time ABED increasing uniformly, increases in a geometric progression. Q. E. D. Corol. 1. Therefore if, with the points A, and G given, the time may be shown by the hyperbolic area ABED, the velocity is able to be shown by the reciprocal of GD itself. Corol. 2. But with GA to GD taken as the reciprocal of the velocity from the start, to the reciprocal of the velocity at the end of some time ABED, the point G may be found. Moreover with that found, the velocity from some other given time is able to be found.



[We follow Brougham and Routh, p. 205., changing their notation a little: The resistance is represented by 2kR kv vα= + per unit mass, and the equation of motion,

in the presence of gravity g, becomes 2dv dv kdt dxv g kv vα= = − − . Thus, in the absence of

gravity, we have 2dv kdt kv vα= − − , which we can write in the form :

( )0

0 0

00

1 11 1

1 1 1 1 11

integrate :

or , and

v v

kt ktv

v

vdv dv k v ktv v v v

v v

dt; ln ln ln ln .

e e

α α

αα α

α

α α α α α

α α

+ + + + +

++

− = − − = − = −

= + = +

Thus when the times are in an arithmetical progression, quantities associated with the reciprocals of the velocity with an added given quantity are in a geometric progression. We can proceed to a demonstration of the next proposition here, again with zero gravity:

( ) hence dv k dv kdx vv dxα α αα += − + = − , giving ( )0

kxv v e αα α −+ = + . In this case, if the

distances are augmented by regular intervals, the velocities with an added constant are increased in a geometric progression. Proposition XIV finds the velocities that correspond to uniform increases in the distance gone, in the case of gravity being present.]

PROPOSITION XII. THEOREM IX. With the same in place, I say that if the distances described may be taken in an arithmetic progression, the velocities increased by a certain given amount will be in a geometric progression. The point R may be given on the asymptote CD, and with the perpendicular RS erected, which meets the hyperbola at S, a description of the distances may be shown by the hyperbolic area RSED ; and the velocity will be as the length GD, which with the given CG puts together the length CD decreasing in a geometric progression, while meanwhile the distance RSED may be increased in an arithmetic progression. And indeed on account of the given increment of the distance EDde, the linelet Dd, which is the decrement of GD itself, will be inversely as ED, and thus directly as CD, that is, as the sum of the same GD and of the given length CG. But the decrement of the velocity in a time inversely proportional to itself, in which a small amount of distance DdeE is described, is jointly as the resistance and the time, that is, directly as the sum of the two quantities, of which the one is as the velocity, the other is as the square of the velocity, and inversely as the velocity, and thus directly as the sum of the two quantities, with one of which given, the other is as the velocity. Therefore the decrement of the velocity as well as of the line GD, is as the given quantity, and the decreasing quantity jointly; and will always be analogous decreasing quantities ; without doubt the velocity and the line GD. Q.E.D.



Corol. 1. If the velocity may be shown by the length GD, the distance described will be as the area of the hyperbola DESR. Corol. 2. And if some point R may be assumed, the point G may be found by taking GR to GD, as the velocity is from the start, to the velocity after some described distance RSED. Moreover with the point G found, the distance is given from the velocity, and conversely. Corol. 3. From which since (by Prop. XI.) the velocity may be given from the given time, and by that proposition the distance may be given from the given velocity, the distance may be given from the given time, and conversely.

PROPOSITION XIII. THEOREM X.

Because on putting in place a body attracted uniformly downwards by gravity, it may ascend or descent along a right line; and because it may be resisted in part in the ratio of the velocity and in part in the same ratio squared : I say that, if parallel right lines may be drawn parallel to the diameters through the ends of the conjugate diameters of a circle and hyperbola, and the velocities shall be as the segments of certain parallels drawn from a given point ; the times will be as the sectors of the areas, with right lines drawn to the ends of the segments cut off : and conversely. Case I. In the first place we may consider the body rising, and with centre D and with some radius DB, the quadrant of a circle BETF is described, and through the end of the radius DB the indefinite line BAP may be drawn, parallel to the radius DF. On that the point A may be given, and the segment AP may be taken proportional to the velocity. And since the one part of the resistance shall be as the velocity and the other part as the velocity squared ; the whole resistance shall be as

2 2AP BA.AP+ ; DA and DP may be joined cutting the circle at E and T, and gravity may be shown by DA2 thus so that gravity shall be to the resistance as DA2 to 2 2AP BA.AP+ : and the time of the whole ascent shall be as the sector of the circle EDT. [Note that the physical problem in this case has been replaced by purely geometrical one. In this case the velocity of a point along the line BP corresponds to the rate at which the area of the triangle PAD is swept out in the quadrant of the circle, the radius of which is

g ] Indeed DVQ may be drawn, and marking off the moment PQ of the velocity AP, and the moment DTV of the sector DET corresponding to the given moment of the time , and that decrement PQ of the velocity will be as the sum of the forces of gravity DA2 and of the resistance 2 2AP BA.AP+ , that is (by Prop. 12. Book 2. Euclid Elem.) as DP2.



[From elementary geometry, we have ( ) ( )22 2 2 2 2 2 22 below.kDP DB BA AP DB BA DA AP BA.AP g kv vα= + + = + = + + ∝ + + ]

Therefore the area DPQ, itself proportional to PQ, is as DP2; [essentially 2dv k

dt g kv vα= − − − as below.]

and the area DTV, which is to the area DPQ as 2 2to DT DP , is as DT2. Therefore the area EDT decreases uniformly in the manner of the future time, by the subtraction of the given small elements DTV, and therefore is proportional to the time of the whole ascent. Q.E.D. [Following Brougham & Routh, in the general case with gravity present for the falling body, as discussed below by Newton, we have, 2dv dv k

dt dxv g kv vα= = − − giving

2 21 or g gk k

dv dv kv v v v

kdt dtαα

αα+ − + −= − = − , and thence

( )2 22 4

gk

dv kv

dtαα α α+ − −

= − , which we may

write as( )2 2

2

dv kv c

dtα α+ −

= − , which may be written as : ( ) ( )2 22

dv dv kcv c v c

dtα α α+ − + +− = − , and

which hence may be integrated to give : ( )( )( )( )

( )( )( )( )

2 2

0 02 22

v c v c ktcv c v c

ln lnα α

α α α+ − + +

+ − + +− = − and

( )( )( )( )

( )( )( )( )

0 02 2 2 2 2

0 02 2 2 22 or

ktc

v c v c v c v cktcv c v c v c v c

ln eα α α α

αα α α αα

−+ + + − + − + +

+ − + + + + + −= = ,

thus enabling t to be found in terms of v, and by solving the equation, vice versa. On the other hand, if the body is rising, as Newton considers above, we have to solve

2dv dv kdt dxv g kv vα= = − − − , giving : 2 21 or g g

k k

dv dv kv v v v

kdt dtαα

αα+ + + += − = − , or

( )2 22 4

gk

dv kv

dtαα α α+ + −

= − ; we now have the problem whether 4 4 or g gk k

α α> < ; if 4gk

α> then

we have the equation, ( )

2

2 22

24 where gdv k

kv bdt , b

α

α αα+ +

= − = − . Now the solution becomes :

( ) ( )02 21 1v v bb btan tan ktα α

α+ +− −− = − . ]

Case 2. If the velocity of the body in the ascent may be expressed by the length AP as before, and the resistance may be put to be as

2 2AP BA AP+ × , and if the force of gravity shall be less than may be possible to express by DA2 ; BD is taken of this length so that 2 2AB BD− shall be proportional to gravity, and let DF be perpendicular and equal to DB itself, and through



the vertex F the hyperbola FTVE may be described, of which DB & DF shall be the conjugate semi diameters, which will cut DA in E, and DP, DQ in T and V; the total time of the ascent will be as the sector TDE of the hyperbola. For the decrement made in the velocity PQ in a given small time interval, is as the sum of the resistance 2 2AP BA AP+ × and of gravity 2 2AB BD− that is, as 2 2BP BD− . Moreover the area DTV to the area DPQ, is as DT2 to DP2; and thus, if DF may be sent to the perpendicular GT, as GT2 or 2 2GD DF− to BD2, and as GD2 to BP2, and separately, as DF2 to 2 2BP BD− . Whereby since the area DPQ shall be as PQ that is, as 2 2BP BD− ; the area DTV will be as DF2 given. Therefore the area EDT is described uniformly in individual equal elements of time, by subtraction of small parts from just as many given parts of the given DTV, and therefore is proportional to the time. Q. E. D. Case 3. Let AP be the velocity of the body in the descent of the body, and 2 2AP BA AP+ × the resistance, and

2 2BD AB− the force of gravity, with a right angle DBA present. And if from the centre D, with the principle vertex B, the rectangular hyperbola BETV may be described cutting DA, DP and DQ produced at E, T and V; DET will be to a sector of this hyperbola as the total time of descent. For the increment of the velocity PQ, and proportional to that the area DPQ, is as the excess of gravity over the resistance, that is, as 2 2 22BD AB BA AP AP− − × − or

2 2BD BP− . [As above, ( )2kdv g kv v dtα∝ − − .]

And the ratio of the areas of the triangles 2

2=DTV DTDPQ DP

ΔΔ ,

[ As and DTV PD QD sin PD Q TDV TD VD sin PD Q;Δ Δ∝ × × ∝ × × since the triangles are evanescent, the relevant sides are equal, and thus in the same ratio.], and thus

2 2 22

2 2 2=DTV GT GD BDDTDPQ DP BP BP

ΔΔ

−= = , [For if 2 2 2 2 2 2 then y x b GD GT BD− = − = ]

and thus as 2

2GDBD

, and separately as 2

2 2BD

BD BP− [For we can write

2 2

2GD BD

BP− +

− and to this

may be added the corresponding terms of another equal ratio: in this case 2 2 2

2 2GD BD GD

BD BP− + +

+ −].

Whereby since the area DPQ shall be as 2 2BD BP− , the area DTV will be as the given BD2. Therefore the area EDT increases uniformly with the individual increments of the time, by adding just as many of the given increments DTV, and therefore is proportional to the time of the descent. Q.E.D. Corol. If with the centre D the arc At may be drawn similar to the arc ET with the radius DA drawn through the vertex A subtending the angle ADT: the velocity AP will be to the velocity, that the body in the time EDT, in a distance without resistance on ascending may lose, or in descending may acquire, as the area of the triangle DAP to the



area of the sector DAt; and thus may be given from a given time. For the velocity, in a non-resisting medium, is proportional to the time and thus to this sector; in a medium with resistance it is as the triangle, and in each medium, when that is a minimum, it approaches the ratio of equality, for the customary sectors and triangles. [The elemental triangles coincide at the start if there is no resistance, and do so finally when the resistance vanishes.]

Scholium. Also it may be possible to show the case in the ascent of the body, when the force of gravity is less than what may be able to show by DA2 or 2 2AB BD+ , and greater than what can be shown by 2 2AB BD− , and must be shown by AB2. But I will hurry on to other matters.

PROPOSITION XIV. THEOREM XI. With everything the same in place, I say that the distance described in ascending or descending, is as the difference of the area put in place in that time, and of a certain other area which may be augmented or diminished in an arithmetic progression; if the forces from the resistance and from gravity added together may be taken in a geometric progression.

AC may be taken (in the three final figures) proportional to gravity, and AK to the resistance. Moreover they may be taken in the same part of the point A if the body descends, otherwise in the contrary part, Ab may be erected which shall be to DB as DB2 to 4BA AC× : and the hyperbola bN described to the rectangular asymptotes CK and CH, and with KN to be perpendicular to CK, the area AbNK will be increased or diminished in an arithmetical progression, while the forces CK are taken in a geometric progression. Therefore I say that the distance of the body from the maximum height of this shall be as the excess area AbNK above the area DET.



For since AK shall be as the resistance, that is, as 2 2AP BA AP+ × ; some given quantity Z may be assumed, and AK may be put equal to

2 2AP BA APZ

+ × ; and (by Lemma II

of this section) the moment KL of this AK will be equal to 2 2 2 or AP PQ BA PQ BP PQZ Z

× + × ×

and the moment KLON of the area AbNK equals 2BP PQ LOZ

× × or 3

2BP PQ BD

Z CK AB× ×× × .

[For, by hypothesis from Theorem IV, there is 2

4 and by construction, LO CA Ab DBAb CK DB BA AC, ×= = , and thus

2 3

4 4 and hence LO DB DBDB BA CK BA CKLO× ×= = ,

322

BP PQ LO BP PQ DBZ Z CK AB

× × × ×× ×= .]

Case 1. Now if the body ascends, and gravity shall be as 2 2AB BD+ , with the circle BET present (in the first figure), the line AC which is proportional to gravity, will be

2 2AB BDZ+ , or 2 2 22AP BA AP AB BD+ × + + will be or AK Z AC Z CK Z× + × × ; and thus

the area DTV will be to the area 2 2 as or to DPQ DT VB CK Z× . Case 2. But if the body ascends, and gravity shall be as 2 2AB BD− , the line AC (in the second figure) will be

2 2AB BDZ− and DT2 will be to DP2 as DF2 or DB2 to

2 2AB BD− or 2 2 22AP BA AP AB BD+ × + − , that is, as AK Z AC Z× + × or CK Z× . And thus the area DTV will be to the area VPQ as VB2 to CK Z× . Case 3. And by the same argument, if the body descends, and therefore gravity shall be as 2 2BD AB− , and the line AC (in the third figure) may be equal to

2 2BD ABZ− , the area

DTV will be to the area DPQ as DB2 to CK Z× : as above. Therefore since these areas shall always be in this ratio ; if for the area DTV, by which the moment of the time may always be shown equal to itself, there may be written some rectangle requiring to be determined, for example BD m× , the area will be DPQ, that is, 12 BD PQ× to BD m× as CK Z× to BD2. And thus there may be 3PQ BD× equal to 2BD m CK Z× × × , and the moment KLON of the area AbNK found above shall be BP BD m

AB× × . The moment, DTV or BD m× , of the area DET may be taken away, and

AP BD mAB× × will remain. Therefore the difference of the moments, that is, the moment of the

difference of the areas, equals AP BD mAB× × ; and therefore on account of the given BD m

AB× , as

the velocity AP, that is, as the moment of the distance that the body will describe either by ascending or descending. And thus the difference of the areas and that distance shall be proportional to the moments, either increasing or decreasing, and likewise either arising or vanishing. Q. E. D. [Following Brougham & Routh as above, in the general case with gravity present for the falling body, as discussed above by Newton, we have, 2dv dv k

dt dxv g kv vα= = − − giving



( )( )( )( )

( )( )( )( )

0 02 2 2 2 2

0 02 2 2 22 or

ktc

v c v c v c v cktcv c v c v c v c

ln eα α α α

αα α α αα

−+ + + − + − + +

+ − + + + + + −= = , thus enabling t to be found in terms of v,

and by solving the equation, vice versa. The equation 2dv kdxv g kv vα= − − for the falling

body [case 3] can now be solved : ( ) ( )2 2

2 4g

k

vdv kdxv αα α α+ − +

= − ; setting 22

4g

kc αα= + , we find :

( )( )( )

( ) ( )

2 21 12 2 22 2 22 2 2

2 2 2

and d v c dvvdv kdx kdx

v c v c v c

α

α α α

αα α

+ −

+ − + − + −= − − = − ; leading to

( ) ( )2

2

2 2 22 2

v c kxc v cln v c ln C

α

αα α

α+ ++ −

⎞⎛ + − + = −⎜ ⎟⎝ ⎠

, where C corresponds to the l.h.s. when

00 and x v v= = . On the other hand, if the body is rising, as Newton considers above, we have to solve

2dv dv kdt dxv g kv vα= = − − − , giving : ( ) ( )02 21 1v v b

b btan tan ktα α

α+ +− −− = − , for the velocity at

time t. In this case, in a similar manner, if g is made negative above, and 4kg α> , then the

quantity c2 becomes imaginary [case 1]; however, on putting 22

4g

kb α α= − , the integral

becomes ( ) ( )22 2 1 2

2v kx

b bln v b tan Cαα α

α+−⎞⎛ + + − = −⎜ ⎟

⎝ ⎠, with a result similar to the above if

c2 is still positive, and thus 4kg α< [case 2].]

Cor. If the length may be called M, which arises by applying the area DET to the line BD ; and some other length V may be taken in this same ratio to the length M, that the line DA has to the line DE: the distance that the body will describe in the whole ascent or descent in a medium with resistance, will be to the distance that the body in a medium without resistance can describe by falling from rest in the same time, as the difference of the aforesaid areas to

2BD VAB× : and thus is given from the time. For the distance in this non-

resisting medium in the square ratio of the time, or as 2V ; and on account of BD and AB given as

2BD VAB× . This area is equal to the area

2 2

2DA BD M

DE AB× ××

, and the moment of M itself is

m; and therefore the moment of this area is 2

22DA BD M m

DE AB× × ×

× . But this moment is to the

moment of the difference of the aforesaid areas DET and AbNK, viz. to AP BD mAB× × as

2

2DA BD M

DE× × to 1

2 BD AP× , or as 2

2DADE

by DET to DAP; and thus, when the areas DET and

DAP shall be as small as possible, in the ratio of equality. Therefore the area 2BD V

AB× , and

the difference of the areas DET and AbNK, when all these areas shall be as small as possible, have equal moments; and thus are equal. From which since the velocities, and therefore also the distances in each medium descending from the start or ascending to the end described in the same time approach equality ; and thus they shall be in turn then as the area

2BD VAB× , and the difference of the areas DET and AbNK; and therefore since the



distance in the non-resisting medium always shall be as 2BD V

AB× , and the distance in the

resisting medium always shall be as the difference of the areas DET and AbNK : it is necessary that the distances in each medium, described in some equal moments of time, shall be in turn as that area

2BD VAB× , and the difference of the areas DET and AbNK.

Q.E.D.

Scholium. The resistance of spherical bodies in fluids arises in part from the tenacity, partially from the friction, and partially from the density of the medium. And that part of the resistance which arises from the density of the fluid we have said to be in the square ratio of the speed; the other part, which arises from the tenacity of the fluid, is uniform, or as the moment of the time: and thus now may be allowed to go to the motion of the body, by which it is resisted partially by a uniform force or in the ratio of the moments of time, and partially in the ratio of the square of the velocity. But it suffices to have revealed the approach regarding this observation in Propositions VIII. & IX. which preceded, and the corollaries of these. Certainly for the same ascent of the body with uniform resistance, which arises from the weight of this, it is possible to substitute a uniform resistance, which arises from the tenacity of the medium, when the body may be moving by the inertial force alone , and with the body ascending in a straight line it is allowed to add this uniform resistance to the force of gravity , and to subtract the same, when the body falls along as straight line. Also it may be allowed to go on to the motion of bodies, in which the resistance is partially uniform, and partially in the ratio of the velocity, and partially in the ratio of the square of the velocities. And I have shown the way in the preceding Propositions XIII and XIV, in which also uniform resistance, which arises from the tenacity of the medium can be substituted for the force of gravity, or since the same as before may be added to it. But I hurry on to other matters.


Book II Section IV. Translated and Annotated by Ian Bruce. Page 532

SECTION IV.

Concerning the circular motion of bodies in resisting media.

LEMMA III. Let PQR be a spiral which may cut all the radii SP, SQ, SR, &c. at equal angles. The right line PT may be drawn touching the same at some point P, and may cut the radius SQ at T; and with the perpendiculars PO and QO erected concurrent at O, SO may be joined. I say that if the points P & Q approach each other in turn an coincide, the angle PSO becomes right, and the final ratio of the rectangle 2TQ PS× to PQ2 will be the ratio of equality. For indeed from the right angles OPQ and OQR the equal angles SPQ and SQR may be taken away, and the equal angles OPS and OQS remain. Therefore the circle which passes through the points O, S, P will also pass through the point Q. The points P and Q may coalesce and this circle at the point of coalescing PQ will touch the spiral, and thus it will cut the right line OP perpendicularly. Therefore OP becomes the diameter of this circle, and the angle OSP the right angle in the semicircle. Q.E.D. The perpendiculars QD, SE may be sent to OP, and the ultimate ratios of the lines will be of this kind: = =TQ TS PS

PD PE PE , or 22POPS ; likewise 2= PQPD

PQ PO ; and from the disturbed

equation, 2=TQ PQPQ PS . From which PQ2 shall be equal to 2TQ PS× . Q.E.D.

[Note from L. & J. : Because the lines PT, DQ, ES normal to PO are parallel, by Prop. X, Book VI, Euclid's Elements , there will be = =TQ TS PS

PD PE PE , and on account of the similar

triangles PSO, PES, 2 22 2 and thus TQPS PO PO PO

PE PS PS PD PS= = = . Now because the radii OP and OQ are perpendicular to the vanishing arc PQ, the point O is the centre, PO the radius, and 2PO the diameter of the osculating circle to the spiral at P. PQ the arc or chord of this circle, and thus the abscissa PD is to the chord PQ as PQ to the diameter 2PO. Whereby the result follows.]



PROPOSITION XV. THEOREM XII.

If the density of the medium at individual places shall be reciprocally as the distances of the places from the motionless centre, and the centripetal force shall be as the square of the density: I say that the body can rotate in a spiral, all the radii of which drawn from that centre may intersect at a given angle. Everything may be put in place as in the above lemma, and SQ may be produced to V, so that SV may be equal to SP. At any time, in the resisting medium, the body may describe that minimum arc PQ, and in twice the time double that minimum arc PR; and the decrements of these arcs arising from the resistance, or the amounts taken from the arcs which that may be described in the same times without resistance, will in turn be as the squares of the times in which they are generated. [Thus, in modern terms, we may say that the decrease in the elemental arc length sΔ due to the small resistive deceleration a, in the increment of the time tΔ , is given by 2 21

2 or s a t s tΔ Δ Δ Δ= × ∝ .] : And thus the decrement of the arc PQ is a quarter part of the decrement of the arc PR. From which also, if the area PSQ thus may be taken equal to an area QSr, the decrement of the arc PQ will be equal to half the decrement of the linule Rr [see note one below]; and thus the force of resistance and the centripetal force are in turn as the small lines 1

2 Rr and TQ , which they produce at

the same time. [If the incremental angle T PQΔθ = and v is the tangential velocity corresponding to PT, then in the time tΔ , the tangent line or vector rotates through this angle, and the change in velocity is vΔθ , corresponding to TQ, from which the centripetal acceleration is proportional to tv Δθ

Δ , towards the centre.] Because the centripetal force, by which the body is acted on at P, is reciprocally as SP2 [from the Proposition] and (by Lem. X, Book I.) the line element TQ, which may be generated by that force, is in a ratio put together from the ratio of this force and the ratio of the time squared in which the arc PQ will be described (for I ignore the resistance in this case as infinitely less than the centripetal force), it follows that 2TQ SP× , that is 21

2 PQ SP× (by the most recent

lemma), will be in the square ratio of the time, and thus the time t PQ SPΔ ∝ × ; and the velocity of the body, by which the arc PQ may be described in that time, will be as

1 or PQPQ SP SP

,×

that is, inversely in the square root ratio of SP.

[We may reiterate this as follows : 2 and cTQ F TQ tΔ∝ ∝ hence 2cTQ F tΔ∝ ; but from

the nature of the spiral as shown in the lemma above, 2

2PQ

PSTQ = and hence 2 2 22 2

c

PQ PQ PSF PS PSt PQ PSΔ ××∝ ∝ = × and t PQ PSΔ ∝ × as required.]



And by a like argument, the velocity by which the arc QR will be described is inversely as the square root ratio of SQ or 1

SQ , [for we already have the arc PQ described by a

velocity as 1SP

; see note two below]. But these arcs PQ and QR are as the describing

velocities in turn, that is, in the ratio , or as SQ SQSP SP SQ

×

; and on account of the equal

angles SPQ and SQr [from the nature of the spiral], and the equal areas given PSQ and QSr, arc

arcPQ SQ SQQr SP SP SQ×

= = . The differences of the following proportions may be taken,

and there arises: 12

arc arc

PQ SQ SQRr VQSP SP SQ− ×

= =

[Recall that SP SV= by def., and 2 and hence SQ SP VQ SP SQ SP SP VQ= − × = − × ;

hence on extracting the square root, there becomes 2

12 8 etcVQ

SP SQ SP VQ SP .× = − − − The other terms past the second can be ignored, because with P and Q coinciding, they vanish with respect to VQ, and thus there will be 1 1

2 2 and hence SP SQ SP VQ VQ SP SP SQ,× = − = − × , giving the ultimate ratio when P

and Q coincide : 12

SQ SQVQSP SP SQ− ×

→ ].

For with the points P and Q coinciding, the ultimate ratio is equal to 12

SP SP SQVQ

− × .

Because the decrement of the arc PQ, arising from the resistance or from 2 Rr× , is as the resistance and the square of the time jointly, the resistance will be as 2

RrPQ SP×

. But there

was [in the limit] 12

arc arc

PQ SQRr VQ= , and thence 2

RrPQ SP×

shall be as 12VQ

PQ SP SQ× × , or as 12

2OS

OP SP×.

For with the points P and Q coinciding, SP and SQ coincide, and the angle PVQ shall be right ; and on account of the similar triangles PVQ and PSO, 1 1

2 2

PQ OPVQ OS= . Therefore

2OS

OP SP×is to the resistance, in the ratio of the density at P and in the square ratio of the

velocity jointly. The square ratio of the velocity may be taken, clearly the ratio 1SP , and

the density of the medium at P will remain as OSOP SP× . The spiral may be given, and on

account of the given ratio OSOP , the density of the medium at P will be as 1

SP . Therefore in a medium the density of which is inversely as the distance from the centre SP, a body is able to revolve on this spiral. Q.E.D. [L. & J. note one: The body with that velocity which it may have at the place P, in equal times may describe as minimal the arcs Pq, qv in a non– resisting medium (the hatched lines), and the arcs PQ, QR in a medium with resistance, and from the demonstration above there will be had 4Qq Rv= , but these areas PSq and qSv are equal, by Prop. I, Book I, and thus on account of the given equal areas PSQ and QSr, by hypothesis also PSq PSQ− or the area QSq equals or qSv QSr rSv QSq− − , and hence the area rSv is



equal to 2QSq : but with the perpendiculars ST and St send from the centre S to the tangents QT and rt drawn through the points Q and r, the vanishing area QSq is 12 ST Qq× , and the area rSv is 1

2 St rv× . Whereby 12 is equal to ST Qq St rv× × , and with

the points P and v merging together, there becomes St ST= and thus 1

2Qq rv= , and 2Qq rv= . Therefore since above we have found 4 there becomes 4 2or 2 Qq Rv Qq Qq,

Qq Rv rv Rr,= −

= − =

and thus 12Qq Rr= . And thus in the same

time in which the resistance generates the decrement Qq, or 1

2 Rr , the centripetal force, by which the body is drawn back from the tangent PT – see previous diagram ,as there is a difference in the labeling from the extra diagram here – to the point Q of the arc PQ, generates the decrement TQ, and thus the force of resistance is to the centripetal force as 1

2 Rr to TQ, (by Cor. 4, Lem. X), and thus all these may be generally obtained, whatever were the curve PQR, whose properties we have not yet used, nor the centripetal force, the resistance, nor the velocity of the body.] [L. & J. note two: For since from the demonstration and PQ QR PQ Qr

SQ SQ SPSP SQ×= = , there

will also be Qr QRSP SP SQ

,×

= from which there will be

( ) and hence Qr QR RrPQ PQ SQSQ RrSP SP SQ SP SP SQ

.− =

− × − ×= = ]

[Brougham & Routh note using analytical methods, p. 213: The equiangular spiral, by definition, has the property that the tangent at any point makes contact with the radius vector at the same angle always; call this angle α . Let r be the radius, v the velocity along the spiral1, the central force P, and kv2 the resistance at any point on the spiral; thus, Newton's unusual density and force descriptions are not adhered to in this description to follow – we note 'unusual' in the sense 'non physical', and of course it was adopted by Newton to render his methods of analysis – being a combination of intuition, geometry, and his early form of calculus – more tractable. The equation satisfied at any point is clearly 2dv

dt kv P cosα= − − ; but on all continuous plane

curves, drdt v cosα= and hence

212

dv dv dr dv dvdt dr dt dr drv cos cosα α= = = ; hence the above

equation becomes 2 22 2dv k

dr cos v Pα+ = − , and this is a general equation for all plane curves. The equation giving the motion perpendicular to the arc under these circumstances is well-known in terms of the local radius of curvature R :

2vR P sinα= ; but for the



equiangular spiral, the radius of curvature is rsinα (consult triangle POS in the first

diagram above); hence we have 2v Pr= ; if we substitute this in above equation 2 22 2dv k

dr cos v Pα+ = − , we obtain ( ) 2 2d Pr kdr cos Pr Pα+ = − , giving

( ) ( )2 23 and 3d P dr kd r kr drP r cos cos rln P Cα α= − − = − +∫ . Now, if the force varies inversely as the

distance, we have 2rP μ= then we have 1

2cos D

r rk α= − = ; that is, the density varies

inversely as the distance, and the negative sign indicates that the body is approaching the centre of force, or that the angle α is greater than a right angle.] Corol. 1. That velocity at any place P is always the same, with which a body in a non-resisting medium can rotate in a circle, at the same distance from the centre SP.

[This also follows from 2v Pr= , and when 2rP μ= we have rv μ= .]

Corol. 2. The density of the medium, if the distance SP may be given, is as OSOP , if that

distance is not given, as OSOP SP× : From thence the spiral can be adapted to any density of

the medium. [This also follows from 1

2cos D

r rk α= − = , where 2cosD α= − . For, if k and r are given, then

α can be found. ] Corol. 3. The force of resistance at any place P, is to the centripetal force at the same place as 1

2 OS to OP. For these forces are in turn as 12 Rr and TQ or as

21 14 2 and VQ PQ PQ

SQ SP× , that is, as 1

2VQ and PQ, or 12 OS and OP. Therefore for a given spiral

the proportion of the resistance to the centripetal force is given, and in turn from that given proportion the spiral is given. Corol. 4. And thus the body is unable to rotate in a spiral except when the force of resistance is less than half of the centripetal force. Make the resistance equal to half of the centripetal force, and the spiral agrees with the right line PS, and in accordance with this the body may fall along the right line to the centre with that velocity, which shall be to the velocity, as we have proven in the above in the parabolic case (Theorem X, Book I) able to fall in a non resisting medium, in the square root ratio of one to two. And here the descent times will be inversely as the velocities, and thus may be given. [This also follows from 2

cosD α= − , where we observe that the magnitude 12D ≤ ; when

the angleα is zero, the spiral degenerates into the right line SP . Corol. 5. And because at equal distances from the centre the velocity is the same in the spiral PQR and along the right line SP, and the length of the spiral to the length of the right line PS is in a given ratio, evidently in the ratio OP to OS ; the descent time in the



spiral will be in the same given ratio to the descent time along the line SP, and hence is given. Corol. 6. If from the centre S with any two given radii, two circles may be described ; and with these two circles remaining, the angle that the spiral maintains with the radius PS may be changed in some manner : the number of revolutions that the body can complete within the circumferences of the circles, by going along the spiral from circumference to circumference, is as PS

OS or as the tangent of that angle that the spiral maintains with the

radius PS; truly the time of the same revolutions is as OPOS , that is, as the secant of the

same angle, or also inversely as the density of the medium.

[For if drdt rv cos .cosμα α= = , then dr rdr

v cos cosdt α μ α= = , and thus the time to go from a

distance r1 to a distance r2 is given by : ( )3 32 2

2 12

3 cosT r r

μ α= − ; if the angleα is almost

right, then the time is very long, and if 0α = , then the same formula T' gives the time of descent to any part of the radius vector ; hence we may write T '

cosT α= ; and the number of revolutions also may be found, for if θ be the angle the radius r makes with some fixed line, then dr

rd tanθ α= , which can be found from the first diagram above in triangle

PQV, where dr TV= , etc., hence 2

12 1

rrln tanθ θ α− = , and the number of revolutions will

be 2

1 2r tanrln α

π ; this includes corollaries 5 & 6.]

Corol. 7. If a body in a medium, the density of which is inversely as the distance of the locations from the centre, has made a revolution along some curve AEB about the centre, and may have cut the first radius AS at A and with the same angle at B as the first, and with a velocity which was inversely in the square root ratio of its distance from the centre to its first velocity at A (that is, so that AS is to the mean proportional between AS and BS) that body goes on to make innumerable similar revolutions BFC, CGD, &c., and from the intersections it may separate the radius AS into the parts AS, BS, CS, DS, &c. in continued proportions. Truly the times of the revolutions will be directly as the perimeters of the orbits AEB, BFC, CGD, &c., and inversely with the starting velocities at A, B, C ; that is, as

3 3 32 2 2AS ,BS ,CS . And the total

time, in which the body may arrive at the centre, will be to the time of the first revolution, as the sum of all the continued proportionals

3 3 32 2 2AS ,BS ,CS , going off to infinite, to the

first term32AS ; that is, as that first term

32AS to the difference of the two first terms



3 32 2AS BS− , or as 2

3 to AS AB as an approximation. From which that whole time is found expeditiously. Corol. 8. In addition from these it is possible also to deduce the motion of bodies in mediums, the density of which is not uniform, or which may observe some other designated law. With centre S, with the radii in continued proportions SA, S B, SC, &c. describe as many circles, and put in place the time of the revolutions between the perimeters of any two of these circles, in the medium which we have treated, to be to the time of revolution between the same in the proposed medium, as the mean density of the proposed medium between these circles, to the density of the medium which we have considered, taken approximately as the mean density between the same circles: But also the secant of the angle by which the determined spiral, in the medium we have treated, may cut the radius AS, to be in the same ratio to the secant of the angle by which the new spiral may cut the radius in the proposed medium; also so that the tangents of the same angles thus to be approximately the number of all the revolutions between the same two circles. If these are made everywhere between two circles, the motion will be continued for all the circles. And with this agreed upon we can imagine without difficulty in what manners and times bodies in some medium ought to rotate regularly. Corol. 9. And although eccentric motions may be completed in spirals approaching the form of ovals; yet be considering the individual revolutions of these to stand apart in turn with the same radius, and to approach the centre by the same steps with the above spiral described, we will understand also how the motions of bodies may be completed in spirals of this kind.

PROPOSITION XVI. THEOREM XIII. If the density of the medium at individual places shall be inversely as the distances from a fixed centre, and the centripetal force shall be as some power of the same distance : I say that the body can rotate in a spiral that cuts all the radii drawn from the centre at a given angle. This may be demonstrated by the same method as the above proposition. For if the centripetal force at P shall be inversely as any power 1nSP + of the distance SP, of which the index is 1n + : it may be deduced as above that the time, in which the body may describe some arc PQ will be as

12 nPQ PS× ; and the resistance

at P will be as 2 nRr

PQ SP×, [these result follow at once

by regarding the curve locally as circular], or as 121

nn VQ

PQ SP SQ− ×× ×

, and thus as 12

11

nn OS

OP SP +

− ××

, that is,

from 121 n OSOP

− × given, reciprocally as 1nSP + . And therefore, since the velocity shall be

inversely as 12 nSP , the density at P will be inversely as SP.



Corol. 1. The resistance is to the centripetal force as 121 n OS− × to OP.

Corol.2. If the centripetal force shall be inversely as SP3, there will be 121 0n− = ; and

thus the resistance, and the density of the medium will be zero, as in proposition nine of the first book. Corol. 3. If the centripetal force shall be inversely as some power of the radius SP of which the index is greater than the number 3, the positive resistance will be changed into a negative resistance.

Scholium. Besides this proposition and the above ones, which consider unequally dense mediums, they are required to be understood concerning the motion of very small bodies, so that the greater density from one side of the body to the other may not be required to be considered. I suppose also that the resistance to be proportional to the density, with all other things being equal. From which in media, in which the force of resistance is not as the density, the density in that must be increased or diminished to such an extent that the excess resistance may be removed or the deficiency may be supplied.

PROPOSITION XVII. PROBLEM IV. To find both the centripetal force and the resistance of the medium, by which the body in

a given spiral can rotate with a given law of the velocity. Let PQR be that given spiral. From the velocity, by which the body runs through that minimum arc PQ, the time will be given, and the force will be given from the altitude TQ, which is as the centripetal force and the square of the time. Then from the difference RSr of the areas PSQ and QSR completed in equal small intervals of time, the retardation of the body is given, and from the retardation the resistance may be found and the density of the medium.

PROPOSITION XVIII. PROBLEM V. With a given law of the centripetal force, to find the density of the medium at individual

places, that the body will describe in the given spiral. From the centripetal force the velocity is required to be found at individual places, then from the retardation of the velocity the density of the medium may be found, as in the above proposition. Truly I have uncovered the method required to treat this problem in proposition ten of this section and the second lemma ; and I do not wish to detain the reader with lengthy inquires into complexities of this kind. Now other matters are required to be added towards the progression of bodies by forces, and from the density and resistance of the mediums, in which the motion up to this point has been treated, and may be completed from these relations.


Book II Section V. Translated and Annotated by Ian Bruce. Page 545

SECTION V.

Concerning the hydrostatic density and compression of fluids.

The Definition of a Fluid.

A fluid is any body, the parts of which yield to any force inflicted, and by yielding the parts may move easily among themselves.

PROPOSITION XIX. THEOREM XIV. All the parts of a homogeneous and motionless fluid, that is confined and pressed together on all sides in some motionless vessel (with the consideration of condensation, gravity, and of all centripetal forces disregarded) are pressed together equally on all sides, and without any motion arising from that pressure, remain in their places. Case 1. A fluid may be confined to a spherical vessel ABC and may be uniformly compressed on all sides : I say that no part of the same will be moved by that compression. For if some part D may be moved, it is necessary that all the parts of this kind, in place at the same distance from the centre, likewise may be moved in a similar motion; and thus because this pressure is the same and equal generally, and the motion of the whole may be supposed excluded, unless that which may arise from the pressure. And all the parts cannot approach closer towards the centre, unless the fluid may be condensed at the centre ; contrary to the hypothesis. The parts are unable to recede further from the centre, unless the fluid may be condensing towards the circumference; also contrary to the hypothesis. They are unable to move in any direction that maintains their distance from the centre, for by the same reason parts will be moving in the opposite direction, but the same part cannot be moving in opposite directions at the same time. Therefore no part of the fluid will be moving from its place. Q.E.D. Case 2. I say now, that all the spherical parts of this fluid are compressed equally in every direction. For let EF be a spherical part of the fluid, if this may not be pressed equally on all sides, the lesser pressure may be increased, so that then the sphere may be pressed upon equally; and the parts of this, by the first case, will remain in their places. But before the pressure increase they would remain in their own places, by the same first case, and with the addition of the new pressure they will be moved from their places, from the definition of a fluid. Which two explanations contradict each other. Therefore it was said incorrectly that the sphere EF may not be pressed on equally in all directions. Q.E.D. Case 3. I say besides that the pressure on the different parts of the spheres shall be equal. For the parts of the spheres in contact press on each other equally at the point of contact, by the third law of motion. And also, by the second case, they are pressed on by the same



force on all sides. Therefore any two non contiguous spherical parts, because an intermediate spherical part can touch each, will be pressed on by the same force. Q.E.D. Case 4. Now I say that all parts of a fluid are pressed on equally everywhere. For any two parts can be in contact with spherical parts at any points, and these spherical parts press there equally : by the third case 3. And in turn they are pressed equally by the others, by the third law of motion. Q.E.D. Case 5. Therefore since any part of the fluid GHI shall be enclosed by the remaining fluid as in the vessel, and pressed upon equally on all sides; moreover the parts of this shall press mutually on each other equally and shall be at rest among themselves, it is evident that every part GHI of the fluid, that is pressed on all sides equally, that all the parts press on each other equally, and are at rest between themselves. Q.E.D. Case 6. Therefore if that fluid may be retained in a non rigid vessel, and may not be pressed on equally on all sides, the same fluid will withdraw from the stronger pressure, by the definition of fluidity. Case: 7. And thus in a rigid vase the fluid cannot sustain a stronger pressure from one side than from the other, but concedes to the same [pressure], and that in an instant of time, because the side of the rigid vessel does not follow the yielding liquid. But on yielding the opposite side will be acted on, and thus the pressure inclines towards equality in each direction. And because the fluid, that first is trying to recede from the part with the greater pressure, may be prevented by the resistance of the vessel on the opposite side, the pressure will be reduced to equality on both sides, in a moment of time, without movement from the place : and at once the parts of the fluid, by the fifth case, may press on each other equally, and will rest between themselves. Q.E.D. Corol. From which neither can the motions of the parts of the fluids amongst themselves change, by a pressure inflicted somewhere on the external surface, unless in so far as either the shape of the surface itself may be changed somewhere, or else all the parts of the fluid may slide over each other with more or less difficulty on account of pressing on each other more or less intensely.



PROPOSITION XX. THEOREM XV.

If the individual parts of a homogeneous spherical fluid, and at equal distances from the centre, concentric with the bottom sphere, gravitate towards the centre of the incumbent whole ; the innermost sphere may sustain the weight of a cylinder, the base of which is equal to the surface of the bottom sphere, and the height the same as that of the incumbent fluid. Let DHM be the bottom surface, and AEI the upper surface of the fluid. The fluid may be separated into innumerable spherical surfaces BFK, CGL by shells of equal thickness ; and consider the force of gravity to act only on the upper surface of each shell, and the forces to be equal on equal parts of all of the surfaces. Therefore the upper surface AE is pressed by the simple force of its own weight, and by which all the parts of the upper shell and the second surface BFK (by Prop. XIX.) may be pressed, by its own measure. Besides the second surface BFK is pressed by the force of its own weight, which added to the first force makes the pressure double. The third surface CGL is acted on by this pressure, according to its measure, and in addition to its own weight, that is, by triple the pressure. And similarly the fourth surface is acted on by four times the pressure, the fifth by five times, and thus so forth. Therefore the pressure by which any surface may be acted on, is not as the solid quantity of the fluid acting, but as the number of shells as far as the largest of the fluid, and is equal to the weight of the lowest shell multiplied by the number of shells : that is, to the weight of the solid the final ratio of which to the cylinder determined becomes one of equality (only if the number of shells may be increased and the thickness may be diminished indefinitely, thus so that the action of gravity from the lowest surface to the highest may be returned continually). Therefore the lowest surface will sustain the weight of the determined cylinder. Q.E.D. And by a similar argument it is apparent the proposition, when the force of gravity decreases in some assigned ratio of the distance from the centre, and so that where the fluid rarer upwards, denser downwards. Q.E.D. Corol. 1. Therefore the bottom is not acted on by the whole weight of the incumbent fluid, but only sustains that part of the weight which will be described in the proposition ; with the rest of the weight supported by an arched figure of fluid. Corol. 2. But the size of the pressure is always the same at equal distances from the centre, whether the surface pressed shall be parallel to the horizontal or the vertical or oblique, or if the fluid, continued upwards from the pressed surface, rises perpendicularly along a right line, or may creep along sideways through turning cavities and channels, and these regular or very irregular, wide or most narrow. And in these circumstances no change in the pressure is to be deduced, by applying the demonstration of this theorem to the individual cases of fluids.



Corol. 3. It is also deduced from the same demonstration (by Prop. XIX.) that no parts of the weight of the fluid, from the pressure of the incumbent weight, acquire any motion between themselves, but only if the motion that may arise from condensation may be excluded. Corol. 4. And therefore if another body of the same specific gravity, which shall be free from condensation [i.e. the immersed body cannot be condensed or change its density by contracting], may be submerged in this fluid, that will acquire no motion from the pressure of the incumbent weight : it will neither descent nor ascent, nor will it be forced to change its shape. If it is spherical, it will remain spherical, not withstanding the pressure; if it is square, it will remain square; and whether it shall be either soft or more fluid ; whether it may swim around freely in the fluid, or rest on the bottom. Indeed any internal part of the fluid has the ratio of the submerged body, and the ratio of every magnitude of this kind, of the figures and of the specific gravities of the submerged bodies, is the same. If a submerged body may be liquefied with the weight preserved and may adopt the form of the fluid ; this, if at first it may have ascended or descended or adopted a new figure from the pressure, also now it may ascend or descend or may be forced to form a new figure : and that is the case because the weight and the other causes of motion remain. But (by case 5. Prop. XIX.) it may now be at rest and it may retain the shape. And therefore as at first. Corol. 5. Hence a body which has a slightly greater specific gravity than the fluid will subside, and that which has a lesser specific gravity will rise, and both the motion and a change of the shape may follow, as much as the excess or deficiency of the weight can bring about. In as much as that excess or defect ratio may give an impulse, by which the body may be forced into equilibrium with parts of the fluid in place elsewhere, and can be compared with the excess or defect of the weight on either plate of the scales. Corol. 6. Therefore the weight of bodies constituted in fluids is two-fold : the one the true and absolute weight, the other the apparent, common and comparative weight. The absolute weight is the whole force by which the body tends downwards: the relative and common weight is the excess of the weight by which the body tends more downwards than the ambient fluid. The parts of all fluids and bodies of the first kind gravitate by weight in their own places : and thus the weights joined together compose the weight of the whole body. For any weight, it is the whole that one is allowed to experience, as in vessels full of liquids, and the weight of the whole is equal to the weight of all the parts, and thus may be composed from the same. With weights of the other kind, bodies do not gravitate at their places, that is, they are not weighed down gathered among themselves, but mutually endeavour to impede the descending of bodies, remaining in their own places, and thus as if there were no gravity. Which weights present in air and that do not weigh down, may not be concluded to be the common [or apparent] weight. Which weights that do weigh down together may be concluded to be the common weights, as long as they may not be sustained by the weight of the air. Common weights are nothing other than the excess of the true weight over the weight of the air. And with the



commonly called levities [i.e. bodies that rise under everyday conditions; thus, at this time according to L. & J., the terms specific weight or gravity designated the ratio of weight to volume for a substance that sinks in a given medium, usually water, whereas specific levity or lightness similarly designates a substance that rises in a given medium], which are negative weights, and by yielding to the weightiness of the air, they desire to rise higher. They are comparative levities, not true ones, because they descent in a vacuum. And thus in water, bodies which on account of greater or lesser weight descent or ascend, are comparatively and apparently weights or levities, and the apparent and comparative weight or levity of these is the excess or deficiency by which the true weight either exceeds the weight of the water or by that is exceeded. Which truly neither descent by weighing down, nor ascending by not yielding to weighing down, even if they may increase the total weight by their true weight, yet comparatively in water and in the common sense they may not gravitate in water. For the demonstration of these cases is similar. Corol.7. These things which are demonstrated from weights may be obtained from any other centripetal forces. Corol. 8. Hence if a medium, in which some body may be moving, may be acted on either by gravity proper, or by some other centripetal force, and the body may be acted on more strongly by that same force, the difference of the forces is that motive force, that we have considered in the preceding propositions as the centripetal force. But if the body may be acted on by that lighter force, the difference of the forces must be had for the centrifugal force. Corol. 9. But since fluids pressing included bodies shall not change the external shapes of these figures, it is apparent above (by corollary of Prop. XIX) that they will not change the position of the internal parts between themselves : and hence, if animals may be immersed, and the sensation of all the parts may arise from the motion ; neither will fluids harm these immersed bodies, nor will they excite any sensation, unless at this point they are able to condense these bodies by compression. And the account is the same of any system of bodies pressed together by the surrounding fluid. All the parts of the system will be agitated by the same motion, as if they were put in place in a vacuum, and may only retain their comparative weight, unless as far as either the fluid may resist the motion of these parts a little, or according to some sticking together that may come about by compression.



PROPOSITION XXI. THEOREM XVI. Let the density of a certain fluid be proportional to the compression, and the parts of this may be drawn downwards by a centripetal force inversely proportional to the distances of these from the centre : I say that, if the distances may be taken in continued proportion, the densities of the fluid at the same distances also will be in continued proportion. ATV may describe the spherical bottom on which the fluid will lie, S the centre, SA, SB, SC, SD, SE, SF, &c. the distances in continued proportion. The perpendiculars AH, BI, CK, DL, EM, FN, &c. may be erected which shall be as the densities of the medium at the places A, B, C, D, E, F ; and the specific gravities at the same places will be as

CKAH BIAS BS CS , , , &c.

[i.e. the local specific gravity density

distance from local density acc. of gravity S∝ × ∝ ,] or, which likewise is thus, as CKAH BI

AB B C CD, , ,&c. Consider first these weights to be uniformly continued well enough from A to B, from B to C, from C to D, &c. by the steps of the decrease at the points B, C, D, &c. And these weights [or specific gravities] multiplied by the heights AB, BC, CD, &c. will make the pressures AH, BI, CK, etc. by which the bottom ATV (as in Theorem XV.) is acted on. Therefore the small element A will sustain all the pressures AB, BI, CK, DL, going of indefinitely, and also the element B all the pressures except AH; and the element C all except the first two AB, BI ; and thus henceforth : and thus the density AH of the first element A is to the density B of the second element as the sum of all AH BI CK DL+ + + to infinity, to the sum of all BI CK DL+ + , &c. And the second density BI of the element B is to the density CK of the third C, as the sum of all BI CK DL+ + , &c. to the sum of all CK DL+ , &c. Therefore these sums are proportional to their differences AH, BI, CK, &c., and thus are continued proportionals (by Lem. I of this section) and hence the proportional differences AH, BI, CK, &c. from the sums, are also continued proportionals. Whereby since the densities at the places A, B, C, &c. shall be as AH, BI, CK, &c. these are also continued proportions. It may advance by a leap, and from the equation with the distances SA, SC, SE in continued proportions, the densities AH, CK, EM will be in continued proportions. And by the same argument, with the distances in some continued proportions SA, SD, SG, the densities AH, DL, GO will be in continued proportions. Now the points A, B, C, D, E, &c. may coalesce, so that therefore the progression of specific gravities from the bottom A to the top of the fluid may be returned continually, and in some continued proportions in the distances SA, SD, SG, the densities AH, DL, GO, also present as continued proportions, will also remain continued proportionals. Q.E.D.



Corol. Hence if the density of the fluid may be given at two places, for example, A and E, it is possible to deduce its density at some other place Q. With centre S, with the rectangular asymptotes SQ and SX a hyperbola may be described cutting the perpendiculars AH, EM, QT at a, e, q, and so the perpendiculars HX, MY, TZ, sent to the asymptote SX, at h, m and t. The area YmtZ is made to the given area YmhX as the given area EeqQ to the given area EeaA; and the line Zt produced will cut the proportional line of the density QT. For if the lines SA, SE, SQ are continued proportionals, the areas EeqQ and EeaA are equal, and thence the areas YmtZ and XhmY are also equal from these proportionals, and the lines SX, SY, SZ, that is AH, EM, QT are continued proportionals, as required. And if the lines SA, SE, SQ maintain some other order in the continued series of proportions, the lines AH, EM, QT, on account of the proportional hyperbolic areas, will maintain the same order in another series of quantities in continued proportion. [Note from L. & J. : Indeed the hyperbolic areas EeaA and QqaA are the logarithms of the lines SE and SQ, and equally the areas YmtZ and XhtZ are the logarithms of the lines SY and SX; but since the areas YmtZ and XhtZ shall be by construction proportional to the areas EeaA and QqaA , these areas YmtZ and XhtZ by the theory of logarithms can become the logarithms of the lines SE and SQ; therefore since the same quantities shall be able to be the logarithms both of the quantities SE and SQ, as well as of the quantities SY and SX, it is required that these quantities SE, SY and SQ, SX will occupy corresponding places in the geometric progressions to which they belong.]

PROPOSITION XXII. THEOREM XVII. Let the density of a certain fluid be proportional to the compression, and the parts of this may be drawn downwards by a weight inversely proportional to the squares of the distances from the centre: I say that, if the distances are taken in a harmonic progression, the density of the fluid at these distances are in a geometric progression. Let S designate the centre, and SA, SB, SC, SD, SE the distances in a geometric progression. The perpendiculars AH, BI, CK, etc. may be erected which shall be as the densities of the fluid at the places A, B, C, D, E, etc. and the specific gravities of this at the same places will be as 2 2 2

CKAH BISA SB SC

, , , &c.

Suppose these gravities to continue uniformly, the first from A to B, the second from B to C, the third from C to D, etc. And these multiplied by the heights AB, BC, CD, DE, etc. or, as it is likewise, by the distances SA, SB, SC, etc. proportional to these heights, make the exponents of the pressure



CKAH BISA SB SC, , etc. Whereby since the densities shall be as the sum of these pressures, the

differences of the densities AH BI , BI CK ,− − etc. will be as the sum of the differences CKAH BI

SA SB SC, , , etc. With centre S, with the asymptotes SA and Sx some hyperbola may be described, which will cut the perpendiculars AH, BI, CK, etc. at a, b, c, etc. and so also the perpendiculars sent to the asymptote Sx, Ht, Iu, Kw in h, i, k; and the differences of the densities tu, uw, etc. will be as AH BI

SA SB, , etc. And the rectangles tu th,uw ui,× × etc., or tp,

uq, etc. as AH th BI uiSA SB, ,× × etc. That is , as Aa, Bb, etc. For there is, from the nature of the

hyperbola, SA to AH or St, as th to Aa, and thus AH thSA× is equal to Aa. And by similar

reasoning, BI uiSB× is equal to Bb, etc. Moreover Aa, Bb, Cc, etc. are in continued

proportion, and therefore proportional to the differences of these : Aa Bb, Bb Cc− − , etc.; and thus the rectangles tp, uq, etc. are also proportional to these differences, and as with the sums of the differences or Aa Cc Aa Dd− − , the sums of the rectangles tp+uq or tp uq wr+ + . Let there be a number of terms of this kind, and the sum of all the differences, such as Aa Ff− , will be proportional to the sum of all the rectangles, such as zthn. The number of terms may be increased and the distances between the points A, B, C, etc. diminished indefinitely, and these rectangles emerge equal to the area of the hyperbola zthn, and thus the difference Aa Ff− is proportional to this area. Now there may be assumed some distances, such as SA, SD, SF in a harmonic progression, and the differences Aa Dd , Dd Ff− − will be equal; and therefore with these differences proportional to the area , thlx and xlnz will be equal to each other, and the densities St, Sx, Sz, that is, AH, DL, FN, continued proportionals. Q.E.D. Corol. Hence if some two densities of the fluid may be given, for example AH and BI, the area thiu will be given, corresponding to the difference tu of these ; and thence the density FN may be found at some height SF, on taking the area thnz to that given area thiu, the difference Aa Ff− is to the difference Aa Bb− . [L. & J. Note : Truly since the area thiu is to the area thnz as the logarithm of the line St or AH to the logarithm of the line Sz or FN, the density FN can be found from a table of logarithms. And conversely, with the density FN given, the height SF may be found : for by the above proposition, Aa – Ff will be given, and thence Ff will be given, from which

SA AnFfFS ×= .]

Scholium. By a similar argument it is possible to prove, that if the particular gravity of a fluid be diminished in the triple ratio of the distances from the centre, and the reciprocals of the squares of the distances SA, SB, SC, etc. (clearly

3 3 3

2 2 2, ,SA SA SASA SB SC

) may be assumed in an

arithmetic progression ; the densities AH, BI, CK, etc. will be in a geometric progression. And if the gravity may be diminished in the quadruple ratio of the distances, and the inverse cubes of the distances are taken in an arithmetic progression (such as



4 4 4

3 3 3, ,SA SA SASA SB SC

, etc.) ; the densities AH, BI, CK, etc. will be in a geometric progression.

And thus ad infinitum. Again if the gravity of a particular fluid shall be the same at all distances, and the distances shall be in an arithmetic progression, the densities will be in a geometric progression, as Edmund Halley, the most distinguished of men, has found. If the gravity shall be as the distance, and the squares of the distances shall be in an arithmetic progression, the densities will be in a geometric progression. And thus indefinitely. These thus are had themselves where with the compression of the fluid the density of condensation is as the force of compression, or just as because the distance occupied by a fluid is inversely as this force. Other laws of condensation can be devised, so that the cube of the compressing force shall be as the fourth power of the density, or the same triplicate ratio of the force with the quadruple ratio of the density. In which case, if the gravity is inversely as the square of the distance : from the centre, the density will be inversely as the cube of the distance. It may be devised that the cube of the compressing force shall be as the fifth of the density, and if gravity is inversely as the square of the distance, the density shall be inversely as the three on two ratio of the distance. [See the final L & J. notes 177 & 178 below.] It may be devised that the compressing force shall be in the duplicate ratio of the density, and gravity inversely in the duplicate ratio of the distance as the distance. It would be tedious to run through all the cases. Finally it is agreed by experiment that the density of the air shall be as the compressing force, either accurately or at least approximately : and therefore the density of the air in the earth's atmosphere is as the weight of all the incumbent air, that is, as the height of mercury in a barometer. [Extended note 177 from L. & J. : Let the centripetal force of some fluid be as some power of the distance, the index of which is n; S may designate the centre, and SA, SB, SC, CD, SE may designate the distances in a geometric progression. The perpendiculars AH, BI, CK, etc., may be erected which shall be as the fluid densities at the places A, B, C, D, E, etc. And the specific gravities of this at the same places will be etcn n n

CKAH BISA SB SC

, , , .

Suppose these gravities to be continued uniformly first from A to B, second from B to C, third for C to D, etc. And thus multiplied by the heights AB, BC, CD, DE, etc., or what amounts to the same, by the distances AS, SB, SC, etc. , by these proportional heights, puts in place the terms of the pressure 1 1 1 etcn n n

CKAH BISA SB SC

, , , .− − − Whereby since the densities shall

be as the sum of these pressures, the differences of the densities etcAH BI , BI CK , .− − will be as the differences of the sums 1 1 1 etcn n n

CKAH BISA SB SC

, , , .− − − made by the same

constructions, as above in Prop. XXII., and the difference of the densities tv, uw, etc. will be as 1 1 1 etcn n n

CKAH BISA SB SC

, , , .− − − and the rectangles etctv th,uw ui, .× × , or tp, uq, etc. as

1 11 1etc that is, as , ,etcn n

n nAH th BI uiSA SB

, , ., Aa Bb .− −− −× × For indeed, by Theorem IV by Hypothesis,

equals AH th, SA An,× × and Aa inversely as SA, or directly as 1SA , and thus

11 as n

nAH thSA

SA Aa Aa ,−−× × × or as 1nAa − since there shall be 1SA Aa× = , and by a similar



argument 11 as etcn

nBI uiSB

Bb , .−−× ; but Aa, Bb, Cc, etc., and thus 1 1 1 etcn n nAa ,Bb ,Cc , .− − − are

continued proportionals, and therefore with the differences of these 1 1 1 1 etcn n n nAa Bb ,Bb Cc , .− − − −− − proportionals, and thus with these differences the

proportionals are the rectangles tp, uq, etc. and as with the sum of the differences 1 1 1 1or n n n nAa Cc , Aa Dd− − − −− − the sums of the rectangles or tp uq, tp uq wr.+ + +

There will be several terms of this kind, and the sum of all the differences; for example 1 1n nAa Ff ,− −− will be the sum of all the rectangles, for example zthn, with the

proportionals. The number of terms may be increased, and the separations of the points A, B, C, etc., diminished indefinitely, and these rectangles become equal to the hyperbolic areas zthn, and thus the difference 1 1n nAa Ff− −− is proportional to this area. Now there may be taken powers of any distances you please, such as SA, SD, SF,

1 1 1 etcn n nSA ,SD ,SF , .− − − in a harmonic progression, and thus the reciprocals of these

1 1 11 1 11 1 1 etc or n n n

n n nSA SD SF

, , , . Aa ,Dd ,Ff− − −− − − are in an arithmetic progression, and the

differences 1 1 1 1n n n nAa Dd ,Dd Ff− − − −− − will be equal; and therefore the differences from these proportional areas thlx, xlnz are equal to each other, and the densities St, Sx, Sz, that is, AH, DL, FN are continued proportionals. Whereby if the weights of the particles of fluid may be diminished in some multiple of the distances, the exponent of which shall be n, and of the powers 1 1 1 etcn n nSA ,SB ,SC , .− − − , the reciprocals (evidently

1 1 1 etcn n n

n n nSA SA SA

SA SB SC, , , .− − − , in which SA has been given) may be taken in an arithmetic

progression; the densities AH, BI, CK, etc., will be in a geometric progression. And if therefore in place of n there may be written the numbers 3,4,5,6, etc., indefinitely; and again there may be written 0,-1, -2, -3, etc., indefinitely, the truth of the scholii in the hypothesis of the proportional density for the compressing force. But when

0n = , or when the gravity of a particular fluid is the same at all distances, there is

1 1

n n

n nSA SA

SA SBSA, SB− −= = , and thus if the distances may be taken in an arithmetical

progression, the densities will be in a geometric progression, and thus the distances are as the logarithms of the densities, because with increased distances in the arithmetical progression, the densities will decrease in a geometric progression. Because truly it is agreed by experiment, that the density of air, with all else being equal and mainly with the same degree of heat remaining, shall be as the force of compression either accurately or perhaps as an approximation in air that we are able to expose by experiments, but the force on the air being pressed below, with all else being equal, shall be equal to the weight of all the incumbent air, and thus proportional to the height of mercury in a barometer, and besides the weight of particles of air, perhaps at small distances from the surface of the earth, will be agreed upon to be constant, it is apparent that, with all else equal, the density of the air at small distances of this kind, we are able to measure with logarithms. Consult the work of Wolfe, Elementis Aerometriae, Book II of Phoronomiae by Hermann, and section 10 of Hydrodynamicae by Daniel Bernoulli.]



PROPOSITION XXIII. THEOREM XVIII. If the density of a fluid composed from particles mutually repelling each other shall be as the compression, the centrifugal forces of the particles are inversely proportional to their distances from their centres. And in turn, the small particles with the forces which are inversely proportional to the distances of their centres mutually repelling each other comprise an elastic fluid, the density of which is proportional to the compression. The fluid may be understood to be enclosed within the cubical space ACE, then on compression to be reduced into a smaller cubical volume ace; and of the particles, maintaining a similar position in each case between each other, the distances will be as the cube roots of the sides AB, ab ; and the densities of the mediums inversely as the containing volumes 3 3 and AB ab . On the side of the greater cube in the plane ABCD, the square DP may be taken equal to the side in the lesser cube db; and from hypothesis, the pressure, by which the square DP presses on the enclosed fluid, will be to the pressure, by which that square db presses on the enclosed fluid, as the medium densities in turn, that is, as 3 3 to ab AB . But the pressure, by which the square DB presses on the enclosed fluid, is to the pressure, by which DP likewise presses on the fluid, as the square DB to the square DP, that is, as 2 2 to AB ab . Therefore, from the equality, the pressure by which the square DB presses on the fluid, is to the pressure by which the square db presses on the fluid, as ab to AB. With the planes FGH and fgh drawn through the middles of the cubes, the fluid may be separated into two parts, and these press on each other mutually by the same forces, by which they were pressed upon by the planes AC and ac, that is, in the proportion ab to AB: and thus the centrifugal forces, by which these pressures may be sustained, are in the same ratio. On account of the same number of particles and the similar situation in each cube, the forces which all the particles exercise on the planes FGH, fgh generally, are as the forces which the individual particles exert on the individual particles. Therefore the forces, which the individuals exert on the individuals following the plane FGH in the larger cube, are to the forces, which the individuals exert on the individuals following the plane fgh in the minor cube, ab to AB, that is, inversely as the separations of the particles in turn. Q.E.D. And vice versa, if the forces of the individual particles are inversely as the separations, that is, inversely as the cube root of the sides AB, ab ; the sum of the forces will be in the same ratio, and the pressures of the sides DB, db will be as the sum of the forces ; and the pressure on the square DP will be to the pressure of the side DB as ab2 to AB2. And, from the equation, the pressure of the square DP to the pressure of the side db as ab3 to AB3, that is, the force of compression to the force of compression as the density to the density. Q.E.D. [L. & J. Note : D and d shall be the separations of the particles in the volumes of the cubes ACE and ace which are as AB and ab, the centrifugal forces of the same inversely



as Dn and dn, the densities of the fluids E and e, and the compressing forces shall be as 2 2

3 3 and n n

E e .+ +

For since the sum of the forces which all the forces exert at the same time on the sides DB and db shall be as the forces of the individual particles, the sum of these forces will be inversely as Dn and dn, or as and n nab AB directly; and the pressure of the square DP to the pressure of the square db , as ab2 to AB2 ; from which equation the pressure of the square DP to the pressure of the square db, that is, the compressing force in the space ACE to the compressing force in the space ace, as 2 2 to n nab AB+ + . But the densities, either are as E to e, or as 3 3 to ab AB , and thus

2 23 3 2 2 to as to

n n n nE e ab AB+ + + + .

Whereby the compressing forces are as 2 2

3 3 and n n

E e .+ +

And conversely, etc.]

Scholium. By a similar argument, if the centrifugal forces of the particles shall be in the inverse square ratio of the distances between the centres, the cubes of the compressing forces are as the fourth powers of the densities. If the centrifugal forces shall be in the triplicate or quadruplicate ratio of the distances, the cubes of the compressing forces will be as the squares cubes or the cubed cubes of the densities. And generally, if D may be put for the distance, and E for the density of the compressed fluid, and the centrifugal forces shall be inversely as some power of the distance D, the index of which is the number n, the compressing forces will be as the cube roots of the power 2nE + , the index of which is

2n + : and conversely. Truly all these are to be understood concerning the centrifugal forces of the particles which are restricted to nearby particles, but do not spread out far beyond. We have an example with magnetic bodies. The attractive force of these may be restricted almost to nearby bodies of the same kind close to themselves. The magnetic force may be drawn together by interposing a sheet of iron, and may be almost be restricted to the iron. For bodies beyond are not attracted by the magnet as much as the iron. In the same manner, if particles repel other particles of the same kind close to themselves, but exert no force on more remote particles, fluids may be composed from particles of this kind as has been acted on in this proposition. Because if the force of particles of this kind may be propagated to infinity, there would be a need for a greater force to equal the condensation of a greater quantity of fluid. Or indeed the physical question is, that it may be agreed that an elastic fluid may consist of a particles mutually repelling each other. We have shown our property of fluids mathematically from particles of this kind agreeing mathematically, so that we may offer an opening to philosophers towards treating that question. [Extended note 178 from L. & J. : It will be better to treat the general formula, from which the individual cases will be elicited in a more pleasing manner. Therefore with the same which have been put in place above, let the variable distance SC x= , the height CD dx= , the density CK y= , the total compressing force at the place C v= , the force of gravity at



the same place g= ; and the specific gravity at the same place will be as gy, and with this multiplied by the vanishing height CD or dx there may be prepared the moment of the pressure gydx dv= − . But take the fluxion to be negative, because by increasing the distance x, the incumbent weight v may decrease. Let the gravity g be as 1

mx, the density y as the compressing force to the power nv , and

thus 1 a ny s v , and on taking the fluxions

11 a n

nn y dy s dv

−. In place of g and dv these values

may be substituted into the equation gydx dv= − , and there arises 1 1 21 1 or

n nn n

m mydx dx

n nx xy dy y dy

− −= − = − . Truly in these equations we have set out no

equalities, but only proportionalities, and thus we have ignored the given coefficients. If there is put 1n = in the final equation, that is, the density of the proportional compressing force, will be m

dy dxy x= − . The quantities 1

1mx − may be taken in arithmetic

progression; and the fluxions of these, or the differentials arising ( )1m

m dxx−− , and thus m

dxx

−

also will be constants, and therefore the quantities dyy also given; and hence the

proportional densities y from its differentials dy, will be continued proportionals, by Lemma II, Book II. If under the same hypothesis there may be put 1m = , there shall be dy dxy x= − ; hence if there may be taken the constant quantities dx

x , or the distances x in a

geometric progression, the quantities dyy will also be constant, and thus the densities y are

in a geometric progression. In short as has been demonstrated in Prop. XXI and Prop. XXII, and the first scholium of this section. On taking the fluents, the equation

1 2 1 11 1 11 1 will become

n nn n

mmdx

n n mxy dy y x Q,

− − −− −= − = − + in which it is apparent that that

there cannot be 1; 1; or 0m n n= = = , and Q is a constant. But in order that the value of the constant Q may be determined, initially the height SF must be defined, where the density y vanishes; Now if that height is finite and it may be said to be a= , on putting

0y = , there will be had 111 and hencem

mQ a ,−−= −

1 1 1 1 111 1 1

n m m m mn x a a x

n m my ,− − − − −− −

− − −× = − = in

which equation 1 nn− must be a positive number less than one, so that with increasing

distances x, the densities y decrease, and conversely. If the height SF at which the density y vanishes may be supposed to be infinite, the constant Q will be equal to zero, and hence the equation

1 11 11 1

nn m

n my x .− −

− −= For if in the equation 1 1 11

1 1n m m

n x an my ,

− − −−− −× = y may be put

equal to zero and x becomes infinite, the constant quantity a will be infinite, contrary to the hypothesis. Now indeed if gravity is inversely as the square of the distance, that is if 2m = , the equation

1 11 11 1

nn m

n my x− −

− −= changes into 11 1

1n

nn xy

−

− = , from which 1

1nnx

y−

∝ . It may be

supposed that the cube of the compressing force shall be as the quadruple power of the



density, or 344 3 3

4 and thus and hence y v y v , n∝ ∝ = , and there will be 31

1 1nn xx

y−

∝ =

[there is a misprint here in the L. & J. text], and hence the density y varies inversely as x3, or the density varies inversely as the cube of the distance. Again, the cases mentioned by Newton,

355 3 3

5 and thus and hence y v y v , n∝ ∝ = , and there will be 32

1

xy ∝ ; again,

122 1

2 and thus and hence y v y v , n∝ ∝ = , gives y inversely proportional to x. See Monumenta Academiae Regiae Scientiarum (1716), where P. Varignon has treated this material. ]


Book II Section VI. Translated and Annotated by Ian Bruce. Page 567

SECTION VI.

Concerning the motion and resistance of simple pendulums.

PROPOSITION XXIV. THEOREM XIX. The quantities of matter in suspended bodies, the centres of oscillation of which are equidistant from the centre of suspension, are in a ratio compounded from the ratio of the weights and in the square ratio of the times of the oscillations in a vacuum. [In this proposition and corollaries, the suspended bodies which are to be compared, are supposed to be oscillating on cycloidal arcs or on the arcs of exceedingly large circles. In addition, the acceleration of gravity is not assumed constant, but varies from place to place between the pendulums, and the force hence likewise on account of the gravitational forces differing, while the inertial masses in turn give rise to differing accelerations. (From an L. & J. note). This section follows on from Section10, Book I, which the reader may wish to consult. We may also note here the use of the word funipendulus by Newton in describing what we call a simple pendulum, which is a made up Latin word, describing a body hanging from a rope or cord, and implied able to swing freely, or in this case for the cord to wrap around cycloidal cusps; the involute of the cycloid described being a similar inverted cycloid. A vertical cycloid arc supports simple harmonic motion of any amplitude for a particle travelling along such a curve under the influence of gravity, as can be worked out from the parametric equations of the curve, found in most older books on mechanics. (We note however, that the motion of a bead on a smooth wire of this shape has different boundary conditions from a pendulum swinging and suspended between the inverted arcs of a cycloid: as in the latter case the length of the pendulum is taken customarily as twice the diameter of the circle that generates the cycloid, so that the bob never comes into contact with the cycloid walls, only the string.) Thus the arc length is proportional to the (– ve) acceleration, and so to the force acting on the mass. These questions were taken up and solved analytically by Euler in Book II of his Mechanica, Chapter 3, §545 – §601; see e.g. the translations by this writer. We will give here in addition, the partial analytic solutions of Brougham and Routh and from modern sources; the relevant notes of Leseur & Janquier are used occasionally to augment Newton's arguments.] For the velocity, that a given force in a given material, can generate in a given time, is as the force and the time directly, and as the [quantity of] matter indirectly. So that the greater the force or the greater the time or the less the matter, from that a greater velocity will be generated. Because that is evident from the second law of motion. Now truly if the pendulums are of the same length, the motive forces in places at equal distances from the perpendicular are as the weights : and thus if two bodies describe equal arcs by oscillating, and these arcs may be divided into equal parts ; since the times in which the bodies describe the corresponding parts of arcs shall be as the times of the whole oscillations, the velocities in turn, in the corresponding parts of the oscillations, shall be as the motive forces and the times of the whole oscillations directly, and inversely as the quantities of matter : and thus the quantities of matter are as the [motive] forces and the



times of oscillations directly and inversely as the velocities. But the velocities are inversely as the times, and thus the times directly and the velocities inversely together are as the squares of the times, and therefore the quantities of matter are as the motive forces and the squares of the times, that is, as the weights and the squares of the times. Q.E.D.

[Thus, in an obvious notation, 2

2m w tM W T∝ × .]

Corol. 1. Thus if the times shall be equal, the quantities of matter in the individual bodies shall be as the weights. Corol. 2. If the weight shall be equal, the quantities of matter shall be as the squares of the times. Corol. 3. If the quantities of matter shall be equal, the weights shall be inversely as the squares of the times. Corol.4. From which since the squares of the times, with all else being equal, shall be as the lengths of the pendulums, and if both the times and the quantities of matter are equal, the weights shall be as the lengths of the pendulums. Corol. 5. And generally, the quantity of matter of the pendulum is as the weight and the square of the time directly, and inversely as the length of the pendulum. Corol. 6. But also in a non resisting medium, the quantity of matter in the pendulum is as the comparative weight, the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in some heavy medium, as I have explained above; and thus likewise performs in such a non-resisting medium as the absolute weight in a vacuum. Corol. 7. And hence an account may be clear both of comparing bodies between each other, as far as the quantity of matter in each, as well as comparing the weights of the same bodies in different places, and requiring an understanding of the variation of gravity. But with the most accurate experiments performed I have always found that the quantity of matter in individual bodies are proportional to the weights of these.

PROPOSITIO XXV. THEOREMA XX.

Bodies suspended as simple pendulums in which, with some medium present that resists in the ratio of instants of time, and suspended bodies which may be moving without resistance in a medium of the same specific gravity, complete oscillations on a cycloid in the same time, and describe proportional parts of the arcs in the same time. Let AB be the arc of a cycloid, that the body D will describe by oscillating in some time in a non resisting medium. The same may be bisected in C, thus so that C shall be the lowest point of this ; and the accelerative force by which the body may be urged on if in some place D or d or E shall be as the length of the arc CD or Cd or CE. That force may be shown by the same arc, and since the resistance shall be as a moment of time [which is taken to mean that the resistance is constant], and it may be given thus, the same may be shown by the part CO of the given arc of the cycloid, and the arc Od may be taken in the ratio to the



arc CD that the arc CB has to the arc CB: and the force by which the body may be acted on at d in the resisting medium, since it shall be the excess of the force Cd over the resistance CO, may be shown by the arc CD, and thus it will be to the force, by which the body may be acted on in the medium without resistance at the place D, as the arc Od to the arc CD; and therefore also at the place B as the arc OB to the arc CB. Therefore if two bodies, D, d may depart from the place B, and may be acted on by these forces: since the forces from the beginning shall be as the arcs CB and OB, the first velocities and the first arc will be described in the same ratio. Let these arcs be BD and Bd, and the remaining arcs CD and Od will be in the same ratio. Hence the forces will remain proportional to CD and Od themselves in the same ratio and from the start, and therefore the bodies may traverse the arcs to be likewise described in the same ratio. Therefore the forces and the velocities and the remaining arcs CD and Od will always be as the whole arcs CB and OB, and therefore these remaining arcs will be described likewise. Whereby the two bodies D and d will arrive at the places C and O at the same time, indeed the one at the place C in the non-resisting medium, and the other at the place O in the resisting medium. But since the velocities at C and O shall be as the arcs CB and OB; these arcs, which the bodies likewise describe by going further, will be in the same ratio. Let these be CE and Oe. The force by which the body D may be retarded in the non-resisting medium at E is as CE, and the force by which the body d may be retarded in the resisting medium at e is as the sum of the force Ce, and of the resistance CO, that is as Oe; and thus the forces, by which the bodies may be retarded, are as the arcs CB and OB proportional to the arcs CE, Oe; and hence the velocities, retarded in that given ratio, remain in that same given ratio. Therefore the velocities and the arcs described by the same are always in that given ratio of the arcs CB and OB; and therefore if the whole arcs AB and aB may be taken in the same ratio, the bodies D and d likewise describe these arcs, and all the motion may be let go from the places A and a at the same time. Therefore all the oscillations are isochronous, and any parts of arcs BD and Bd, or BE and Be which are described at the same time shall be proportional to the whole arcs BA and Ba. Q.E.D. Corol. Therefore the most rapid motion does not fall on the lowest point C, but is found at that point O, by which the whole arc described aB may be bisected. And the body at once by progressing to a, may be retarded by the same steps by which before it was accelerated in its descent from aB to O. [This proposition follows at once from the isochronous nature of the oscillations, being independent of the amplitude; thus, the role of the resistance is merely to enable the system to pass through successive isochronous oscillations of diminishing amplitudes : on the assumption that there is no dependence of the time of oscillation on the resistance, which is certainly the case for damping of this kind. For a modern analysis; see, e.g. Chorlton, Textbook of Dynamics, p.96, V.N.: Let l be twice the radius of the generating circle, s the arc length from C, v the velocity, and take



the mass of the particle as unity. Let 2gl f be the constant resistance. We note that the

intrinsic equation of the cycloid can be written as 2s l sinψ= , where ψ is the angle the tangent to the cycloid makes at this point, in the sense of increasing s. The motive force

on the body can be written as ( ) ( )22

2 22 2 2 or d s fg g gd sl l ldt dts f s f−+ = = − − ; this is the

standard differential equation for S.H.M. about a point distant f to the right of the lowest

point, with the angular frequency 2gl , and we can immediately write down

( ) ( )0 2gls t s f cos t f= − + for the first half cycle of the motion starting from s0 and

reaching the maximum tangential speed at f, where the attracting force becomes zero; however, when the particle comes to rest after completing the arc, the sign of the

resistance changes, and a new equation can now be set up ( ) ( )2

2 2d s f g

ldts f+ = − + for the

other half of the swing, and solved for these conditions; the outcome being amplitudes of swings decreasing in an arithmetic progression, as an arc f is removed in each half swing while the frequency and hence the period of the oscillation is unchanged, as Newton has shown. B. & R. have incorrectly used l rather than 2l in their derivations.]

PROPOSITION XXVI. THEOREM XXI. The oscillations of simple pendulums on a cycloid, which are resisted in the ratio of the velocities, are isochronous. For if two bodies, at equal distances from the centre of suspension, by oscillating describe unequal arcs, and the velocities in the corresponding parts of the arcs shall be in turn as the total arcs ; the resistances proportional to the velocities, also in turn they will be as the same arcs. Hence if with the motive forces arising from gravity, which shall be as the same arcs, these resistances may be taken away or added on, the differences or the sums in turn will be in the same ratio of the arcs : and since the increments or decrements of the velocities shall be as these differences or sums, the velocities always will be as the whole arcs : Therefore the velocities, if in any case they shall be as the whole arcs, always will remain in the same ratio. But at the start of the motion, when the bodies begin to descend and to describe these whole arcs, the forces, since they shall be proportional to the arcs, will generate velocities proportional to the arcs. Therefore the velocities always will be as the whole arcs described, and therefore these arcs may be described in the same times. Q.E.D. [In this case analytically, the force equation can be written as

2

2 22 0gd s dsdt ldt

k s+ + = ; in the

case of under damped motion, the particle oscillates with a reduced frequency, depending on the size of the damping coefficient k, and decays exponentially, or the amplitudes are in a geometric progression . Again, this is a standard equation that can be found in the appropriate texts.]



PROPOSITION XXVII. THEOREM XXII.

If for funicular bodies there is resistance in the square ratio of the velocities, the differences between the times of the oscillations in a resisting medium and the times of the oscillations in a non-resisting medium of the same specific gravity, will be nearly proportional to the arcs described. For with equal pendulums in the resisting medium, unequal arcs A and B may be described ; and the resistance of the body in the arc A to the resistance of the body in the corresponding arc B will be in the square ratio of the velocities, that is, as A2 to B2, as an approximation. If the resistance in the arc B should be to the resistance in the arc A as AB to A2 ; the times in the arcs A and B become equal, by the above Proposition. [As the resistances are again in the ratio of B to A.] And thus the resistances A2 in the arc A, or AB in the arc B, brings about the excess of the time in the arc A above the time in the non-resisting medium, and the resistance B2 brings about an excess of the time in the arc B above the time in the non-resisting medium. But these excesses are as the effecting forces AB and BB as an approximation, that is, as the arcs A and B. Q.E.D. Corol 1. Hence from the times of the oscillations made, in the resisting medium, in unequal arcs, the times of the oscillations are able to become known in the non-resisting medium of the same specific gravity. For the differences of the times will be to the excess of the time in the smaller arc above the time in the non-resisting medium, as the difference of the arcs to the minor arc. Corol. 2. The shorter oscillations are more isochronous, and the shortest may be carried out in the same times as in the non-resisting medium, approximately. Truly of these which are made in the greater arcs, the times then become a little greater, and therefore so that the resistance in the descent of the body in which time it may be produced, shall be greater for the ratio of the length to be described in the descent, than the resistance in the subsequent ascent in which the time is shortened. Moreover the time of the oscillations both of the shorter as well as of the longer may be seen never to be produced by the motion of the medium. For with the bodies slowed there is a little less resistance, for the ratio of the velocities, and with the accelerations a little more than from these which are progressing uniformly : and thus because the medium, from that motion it has acquired from the bodies, may be progressing in the same direction, in the first case is disturbed more, in the second less; and hence may agree more or less with the motion of the body. Therefore with descending pendulums the resistance will be greater, and in the ascent less than with the ratio of the velocities, and the time is increased from each cause.


Book II Section VI. Translated and Annotated by Ian Bruce. Page 572 PROPOSITION XXVIII. THEOREM XXIII.

If, with a simple pendulum oscillating in the cycloid, it may be resisted in the ratio of the moments of time [i.e. constant resistive forces], the resistance of this to the force of gravity will be as the excess of the whole descending arc described over the whole ascending arc subsequently described, to the length of the pendulum doubled. Let BC designate the descended arc described, Ca the ascended arc described, and Aa the difference of the arcs : and with the demonstrations and constructions which were put in place in Proposition XXV, the force will be, by which the oscillating body may be urged at some location D, to the force of resistance as the arc CD to the arc CO, which is half of that difference Aa. And thus the force, by which the oscillating body may be acted on in the beginning or at highest point of the cycloid, that is, the force of gravity, will be to the resistance as the arc of the cycloid between the highest point of that and the lowest point C to the arc CO; that is (if the arcs may be doubled): as the whole arc of the cycloid, or twice the length of the pendulum, to the arc Aa. Q. E. D.

PROPOSITION XXIX. PROBLEM VI. For a body put in place oscillating on a cycloid resisting in the square ratio of the velocity, to find the resistance at individual places. Let Ba be the whole arc described in an oscillation, and let C be the lowest point of the cycloid, and CZ the half arc of the whole cycloid, equal to the length of the pendulum ; and the resistance may be sought at some place D. The indefinite right line OQ may be cut at the points O, S, P, Q, by that rule, so that (if the perpendiculars OK, ST, PI, QE may be erected and with centre O and with the asymptotes OK, OQ , the hyperbola TIGE may be described cutting the perpendiculars ST, PI, QE in T, I & E, and through the point I , KF may be drawn parallel to the asymptote OQ meeting the asymptote OK in K, and the perpendiculars ST and QE in L and F), the hyperbolic area PIEQ to the hyperbolic area PITS shall be as the arc BC described by the descent of the body to the arc Ca described by the ascent, and the area IEF to the area ILT as OQ to OS. Then the hyperbolic area PINM may be cut by the perpendicular MN which shall be to the hyperbolic area PIEQ as the arc CZ to the arc BC described in the descent. And if the hyperbolic area PIGR may be cut by the perpendicular RG, which shall be to the area PIEQ as some arc CD to the arc BC described in the whole descent, the resistance at the point D to the force of gravity, shall be as the area OROQ IEF IGH× − to the area PINM.



For since the forces arising from gravity by which the body may be acted on at the places Z, B, D, a, shall be as the arcs CZ, CB, CD, Ca, and these arcs shall be as the areas PINM, PIEQ, PIGR, PITS; then both the arcs as well as the forces may be represented by these areas respectively. Let Dd above be as the minimum distance described in the descent of the body, and likewise there may be shown by the area RGgr taken as the minimum from the parallels RG, rg, and rg may be produced to h, so that GHhg and RGgr shall be the decrements of the areas IGH and PIGR during the same time. And the increment or Rr Rr

OQ OQGHhg IEF , Rr HG IEF− × − of the area OROQ IEF IGH× − , will be

to the decrement RGgr of the area PIGR, or to or RrOQOR HG IEF OR GR OP PI× − × × ,

that is (on account of the equality of OR HG, OR HR OR GR, ORHK OPIK , PIHR & PIGR IGH× × − × − + ),

as to OROQPIGR IGH IEF OPIK+ − × . Therefore if the area OR

OQ IEF IGH× − may be

called Y, and the decrement RGgr of the area PIGR may be given, the increment of the area Y will be as PIGR–Y. But if V may designate the force arising from gravity, proportional to the arc CD described, by which the body may be acted on at V, and R may be put in place for the resistance; V R− will be the whole force by which the body is acted on at D. And thus the increment of the velocity is as V R− and that small element of the time in which it has been made jointly. But also the velocity itself shall be as the increment of the distance described directly and inversely to the same element of the time. From which , since the resistance by hypothesis shall be as the square of the velocity, the increment of the resistance, (by Lemma II) will be as the velocity and the increment of the velocity jointly, that is, as the moment of the distance and V R− together ; and thus, if the moment of the distance may be given, as V R− ; that is, if for a given force V it may be expressed by writing PIGR, and the resistance R may be expressed by some other area Z, as PIGR Z− . Therefore the area PIGR uniformly decreasing by the removal of the given moments, the areas Y increase in the ratio PIGR Y− , and the area Z in the ratio PIGR Z− . And therefore if the areas Y and Z may be taken together and they shall be equal from the beginning, these by the addition of equal moments will be able to go on equal, and likewise from the equal moments at once the decreases vanish together. And in turn, if they both begin and vanish together, they will have equal moments and they will always be equal: hence that is the case because if the resistance Z may be increased, the velocity together with the that arc Ca, which will be described in the ascent of the arc, will be diminished, and at the point at which all the motion together with the resistance may cease by approaching closer to the point C, the resistance may vanish more rapidly than the area Y. And conversely it will arise when the resistance is diminished. Now truly the area Z begins and is definite when the resistance is zero, that is, in the first place the motion of the arc CD may be equal to the motion of the arc CB everywhere and the right line RG begins on the right line QE, & in the end the motion everywhere of the arc CD may be equal to the arc Ca and RG falls on the right line ST. And the area Y or OROQ IEF IGH× − begins and is defined where it is zero, and thus and OR

OQ IEF IGH× are



equal everywhere : that is (by the construction) where the right line RG successively begins on the right lines QE and ST. And hence these areas begin and vanish together, and therefore they are always equal. Therefore the area OR

OQ IEF IGH× − is equal to the area

Z, by which the resistance is expressed, and therefore it is to that area PINM through which gravity is expressed, as the resistance to the weight. Q.E.D. Corol. I. Therefore the resistance at the lowest place C is to the force of gravity, as the area OP

OQ IEF× to the area PINM.

Corol: 2. But the maximum shall come about, when the area PIHR is to the area IEF as OR to OQ. For in that case the moment of this (without doubt PIGR Y− ) becomes zero. Corol. 3. Hence also the velocity at individual places may become known, clearly which is in the square root ratio of the resistance, and the motion from the beginning itself may be equal to the velocity of the body on the same cycloid without resistance of oscillation. Subsequently, on account of the difficult calculation by which the resistance and the velocity are required to be found by this proposition, it has been considered to attach the following proposition. [It is convenient to give here the Brougham & Routh derivation, starting from

2 2dvdsv s kvω= − − in an obvious notation, where 2

2glω = ; initially the particle is moving

in the direction of the increasing arc; when it returns, the damping factor must change sign. B & R point out that the displacement cannot be written finitely as a function of time; however, it is possible to find the velocity of the particle at any point on the arc using an integrating factor. Put the equation in the form :

2 22 2 2 2 2 2 22 2 or 2 2ks ks ksdv dvds dskv s e ke v e sω ω+ = − + = − , giving

( )2 22 2 2 2 2 22 or 2

ksd v e ks ks ksds e s v e e sdsω ω= − = − ∫ , which in turn gives :

2 2 22 2 2 2 2 2 122

ksks ks ks ks ek k kv e e sds e s e ds C sω ωω ⎡ ⎤ ⎡ ⎤= − = − − = − −⎣ ⎦⎣ ⎦∫ ∫ . The constant C may

be found by putting 2

22

2 when 0; in which case

kv V s C V ω= = = − ; if the damping is

small, this finite expression can be expanded out approximately. These writers dismiss Newton's geometrical solution for the velocity as, ' of no value except as a matter of curiosity'. We follow on with B. & R.'s analysis for deducing the law of the resistance from experiments with a pendulum, before examining Newton's general formulation of the problem in the next two propositions, for a small disturbance produced in the oscillation of any kind whatever. Let the quantities s, t, ω , etc., have the same meaning as before, and let f be some small disturbance acting along the tangent of the motion of the particle. The equation of motion will then be, on adapting the original terminology slightly to more modern :



2

22d s

dts f .ω+ =

If 0f = , the motion will be given by ( ) ( ); s Asin t v A cos tω ϕ ω ω ϕ= + = + ; we assume that these are the equations of motion when f is not zero, and in which case A and ϕ are functions of the time t (Camb. Phil. Trans. 1826), were the second equation is the differential of the first: ( ) ( )( ) ( )ddA

dt dtsin t Acos t A cos tϕω ϕ ω ϕ ω ω ω ϕ+ + + + = + ; or

( ) ( ) 0ddA

dt dtsin t Acos t ϕω ϕ ω ϕ+ + + = ; and since these equations satisfy the original equation of motion, we have

( ) ( ) ( ) ( )2 2 or ddv dAdt dt dts f , cos t A sin t Asin t fϕω ω ω ϕ ω ω ω ϕ ω ω ϕ+ = + − + + + + = ;

leading to : ( ) ( )d fdA

dt dtcos t A sin tϕωω ϕ ω ϕ+ − + = .

Solving these equations, we have :

( ) ( ) and f d fdAdt dt A.cos t .sin tϕ

ω ωω ϕ ω ϕ= + = − + ] These equations, when solved, will give the changes in the arc and the time, produced by the cause f. If f is very small, then so are the variations of the amplitude A and of the phase ϕ , and

so can be ignored when multiplied by f, as quantities proportional to 2f arise. Hence, if A' and ϕ ' are the new values of A and ϕ ,

( ) ( )1 1 and AA' A f cos t dt ' f sin t dtω ωω ϕ ϕ ϕ ω ϕ− = + − = − +∫ ∫ ;

from which we learn that if f consists of two disturbing causes, the total disturbance will be equal almost to the sum of the two disturbances separately. Suppose that mf kv= , i.e. the resistance is proportional to the mth power of the velocity, then the velocity in moving from the lowest point is :

( ) ( ) and hence m m mv A cos t f kA cos tω ω ϕ ω ω ϕ= + = + ;

on substituting in the above integrals, and integrating between the limits

2 2 and t tπ πω ϕ ω ϕ+ = − + = , we have ( )( )

( )( )( )2 4 ...2

1 1 3 ....... , and 0m m mm mm m mA' A k . A 'α ω ϕ ϕ− −−+ − −

− = − − =



where if is odd, and 2 if is even. m mα π α= = Hence on ignoring second order quantities, since the phase is preserved, the time of the oscillations is unchanged, and the arcs decrease continually, and the difference between the arc described in the descent and that described on the subsequent ascent will be proportional to the same power of the arc that expresses the power of the velocity for the velocity, all else being unchanged. Thus the law governing the resistance can be found.]

PROPOSITION XXX. THEOREM XXIV.

If the right line aB shall be equal to the arc of the cycloid that the body will describe by oscillating, and at the individual points of this D the perpendiculars DK may be erected, which shall be to the length of the pendulum as the resistance of the body at the corresponding points of the arc to the force of gravity : I say that the difference between the arc described in the whole descent and the arc described in the subsequent whole ascent multiplied by the half sum of the same arcs, will be equal to the area BKa occupied by all the perpendiculars DK. For both the arc of the cycloid may be expressed in a whole oscillation, described by that right line itself equal to aB, as well as the arc which may be described in a vacuum by the length AB. AB may be bisected in C, and the point C will represent the lowest point of the cycloid, and CD will be the force arising from gravity, by which the body at D may be urged along the tangent to the cycloid, and it will have that ratio to the length of the pendulum that the force at D had to the force of gravity. Therefore that force may be expressed by the [arc] length CD, and the force of gravity expressed by the length of the pendulum, and if DK may be taken on DE, in that ratio to the length of the pendulum that the resistance has to the weight, DK will be expressing the resistance.



[It is convenient to interrupt Newton's discourse here, by noting that the intrinsic equation of the inverted cycloid is 4s a sinψ= , where ψ is the tangent angle to the horizontal at

the point P in question where the bob is present, and at this point the force along the tangent is 4

mgsamg sinψ = , where 4a is the length of the pendulum and also the length of

the complete arc of the cycloid, where the generating circle has radius a. Thus the force along the tangent at this point to the force of gravity is as the arc length to the length of the pendulum, as stated by Newton. In turn, the resistance f is to mg as the arc DE is to the whole arc 4a. Thus as above, all the tangential forces are represented by corresponding arcs, while the tangential velocity at this point is 4s a cosψ ψ= . In the diagram below, the radii CA and CB are of length 4a, and ratios are taken for which the quantity ψ is the same for the damped and undamped cases.] The semicircle BEeA may be put in place with centre C and with the radius CA or CB. Moreover, the body may describe the distance Dd in a minimum time, and with the perpendiculars DE and de erected, meeting the circumference at E and e, these will be as the velocities which the body in a vacuum, by descending from the point B, may acquire at the places D and d. This is apparent (by Prop. LII. Book I.) And thus these velocities may be expressed by these perpendiculars DE and de; and the velocity DF as that acquired at D by [the body] falling from B in the resisting medium. And if, with centre C and with radius CF, a circle FfM may be described crossing the right lines de and AB at f and M, M will be the place to which it would ascend henceforth without further resistance, and df the velocity that it may acquire at d. From which also if Fg may indicate the moment of the velocity that the body D, by describing the distance taken as minimum Dd, loses from the resistance of the medium ; and CN may be taken equal to Cg: N will be the place to which the body henceforth would ascend without resistance, and MN will be the decrease in the ascent arising from the loss of that velocity. The perpendicular Fm may be sent to df, and the decrement Fg of the velocity DF arising



from the resistance DK, will be to the increment of this same velocity fm arising from the force CD, as the generating force DK to the generated force CD. But also on account of the similar triangles Fmf, Fhg, FDC: fm is to Fm or Dd as CD is to DF ; and from the equality, Fg is to Dd as DK to DF . Likewise Fh is to Fg as DF to CF; and from the rearranged equation, Fh or MN to Dd is as DK to CF or CM; and thus the sum of all the terms MN CM× will be equal to the sum of all the terms Dd DK× . At the moveable point M a rectangular coordinate may always be understood to be erected equal to the indeterminate CM, which may be drawn by a continued motion along the whole length Aa; and the trapezium described by that motion or equal to this rectangle 1

2Aa aB× will be equal to the sum of all the products MN CM× , and thus to the sum of all Dd DK× , that is, to the area BKVTa. Q.E.D. [We see now that Newton's diagram above considers the arc length lost due to the friction in a half cycle: in the one hand it is the sum of all the decrements of the arc given by the

sum of Dd DK× , and on the other hand it is the difference of the initial and final arcs in the half oscillation, given by the sum of MN CM× . We return now to B. & R., who give a simple explanation of the final

formulas produced by Newton geometrically in the following corollary : Let the straight line aB be drawn equal to the arc of the cycloid described by the oscillating body, and at each point D draw the perpendicular DK equal to the fraction 2

1ω

of the resistance at D.

Let 0 1 and a a be the arcs described in the descent and subsequent ascent , then the area

under the curve aKB is ( ) 1 0+ 1 0 2 a aa a− . This can be proven readily, for the equation of

motion is 2dvdsv s f ,ω+ = for some resistance f; and we have

11

00

2 2 2 2 then on integrating, 2 ds a

advds a

a

v s R, v s fω ω+

+

−−

⎡ ⎤+ = + =⎣ ⎦ ∫ . Now, at the limits of

integration, the speed v is zero, and hence ( )1

0

2 2 21 0 2

a

a

a a fdsω+

−

− = ∫ , or ( )1

2

0

2 211 02

afds

a

a aω

+

−

− = ∫

as required. We have now to find the nature of the resistance that describes the curve aKB. As previously, we have ( ) ( ) and s Asin t v A cos tω ϕ ω ω ϕ= + = + . Now, if y is the ordinate

and the resistance is of the form mkv , then DK or y becomes equal to ( ) ( )2

2m m m m m mky A cos t kA cos tω

ω ω ϕ ω ω ϕ−= + = + , while the distance x from the

initial geometric centre + half the differences of the ascending and descending arcs, 1 02

a a− ,



gives an average location about which the oscillation takes place : thus, at least approximately, ( )1 0

2a ax Asin tω ϕ−+ = + .

Now, if the resistance varies as the velocity, then ( )1 02

a ax Asin tω ϕ−+ = + and

( )ky Acos tω ω ϕ= + , and on eliminating t, we have approximately,

( ) ( )1 02 2 2

2a a y

kx Aω−+ + = , which is the equation of an ellipse, if we consider A as being

time independent. (There is a misprint in the original equation here.) Again, if the resistance varies with the velocity squared, we have

( )1 02

a ax Asin tω ϕ−+ = + as before, while ( )2 2y kA cos tω ϕ= + , and on eliminating t, we

have : ( )1 02 2

2a a y

kx A−+ + = , which is the equation of a parabola, again on treating A as

constant. ] Corol. Hence from the law of the resistance and the difference of the arcs Ca and CB the difference Aa can always be deduced for the proportion of the resistance to the weight approximately. For if the resistance DK shall be uniform, the rectangular figure BKTa will be under Ba and DK; and thence the rectangle under 1

2 Ba and Aa will be equal to the rectangle under

Ba and DK, and DK will be equal to 12 Aa . Whereby since DK shall be expressing the

resistance, and the length of the pendulum an expression of the weight, the resistance to the weight will be as 1

2 Aa to the length of the pendulum ; everything has been shown as in Prop. XXVIII. If the resistance shall be as the velocity, the figure BKTa will approximate to an ellipse. For if the body, in the non-resisting medium, may describe by oscillating the whole length BA, the velocity at some place D may be as the applied ordinate DE of the circle with diameter AB. Therefore with Ba in the resisting medium, and BA in the non resisting medium, they may be described in around equal times ; and thus the velocities at the individual points of Ba, shall be approximately to the velocities at the corresponding points of the length BA, as Ba is to BA; the velocity at the point D in the resisting medium will be as the applied ordinate Ba of the circle or ellipse described on the diameter ; and thus the figure BKVTa will be approximately an ellipse. Since the resistance may be supposed proportional to the velocity, OV shall be an expression of the resistance at the middle point O; and the ellipse BRVSa, with centre O, described with semi-axes OB and OV, and the figure BKVTa, equal to the rectangle Aa BO× , will be approximately equal. Therefore Aa BO× is to OV BO× as the area of this ellipse to OV BO× : that is, Aa to OV as the area of the semicircle to the square of the radius, or approximately as 11 to 7: And therefore 7

11 Aa shall be to the length of the pendulum as the resistance of the oscillating body at O to the weight of the same. But if the resistance DK shall be in the square ratio of the velocity, the figure BKVTa will be nearly a parabola having the vertex V and the axis OV, and thus will be



approximately equal to the rectangle under 23 Ba and OV. Therefore the rectangle under

12 Ba and Aa is equal to the rectangle under 2

3 Ba and OV, and thus OV equals 14 Aa : and

therefore the resistance of the oscillating body at O to the weight of this is as 34 Aa to the

length of the pendulum. And I think that these conclusions have been taken care of abundantly enough in practical matters. For since the ellipse or parabola BRVSa may agree with the figure BKVTa at the mid-point V, this if either the part BRV or VSa exceeds that figure, it will be deficient from that figure by the same for the other part, and thus it will be approximately equal to the same.

PROPOSITION XXXI. THEOREM XXV. If the resistance in the individual proportional parts of the arcs of an oscillating body described may be augmented or diminished in a given ratio ; the difference between the arc described in the descent and the arc subsequently described in the ascent, will be increased or diminished in the same ratio. For that difference arises from the retardation of the pendulum by the resistance of the medium, and thus is proportional to that retarding resistance. In the above proposition the rectangle under the right line 1

2 aB and the difference Aa of these arcs CB and Ca was equal to the area BKTa. And that area, if the length aB may remain, may be increased or decreased in the ratio of the applied ordinates DK [i.e. the y ordinate]; that is, in the ratio of the resistance, and thus is as the length aB and the resistance jointly. And hence the rectangle under Aa and 1

2 aB is as aB and the resistance jointly, and therefore Aa is as the resistance. Q.E.D. Corol. I. From which if the resistance shall be as the velocity, the difference of the arcs in the same medium will be as the total arc described , and conversely. Corol. 2. If the resistance shall be in the square ratio of the velocity, that difference will be in the square ratio of the whole arc, and conversely. Corol. 3. And generally, if the resistance shall be in the cubic or some other power of the velocity, the difference will be in that same ratio of the total arc, and conversely. Corol. 4. And if the resistance shall be partially in the simple ratio of the velocity, and partially in the same square ratio, the difference will be partially in the ratio of the whole arc and partially in the ratio of the whole arc squared, and conversely. The law will be the same both for the ratio of the resistance for the velocity, and which is also the law of that difference for the length of the arc. Corol. 5. And thus if with a pendulum successively describing unequal arcs, the ratio of the increment or decrement of this difference for the length of the described arc can be



found and also the ratio may be come upon of the increment and decrement of the resistance for a larger or smaller velocity.

General Scholium. From these Propositions, through the oscillations of pendulums in mediums of any kinds, it is permitted to find the resistance of the mediums. Indeed I have investigated the resistance of the air by the following experiments. A wooden sphere weighing 7

2257

ounces Avoirdupois, made with a diameter of 786 London inches, I have hung from a

hook securely enough by a thin thread, thus so that between the hook and the centre of oscillation of the sphere should be 1

210 feet. I noted a point on the thread ten feet and one inch distance from the centre of suspension, and in the region of that point I have put in place a ruler separated into inches, with the aid of which I could observe the lengths of the arcs described by the pendulum. Then I have counted the oscillations in which the sphere lost an eighth part of its motion. If the pendulum may be drawn away from the perpendicular to a distance of two inches, and then it may be sent off, thus so that in its whole descent it may describe an arc of two inches, and in the first whole oscillation, composed from the descent and the subsequent ascent, an arc of almost four inches : the same lost an eighth part of its motion in 164 oscillations, thus so that in its final ascent it described an arc of one and three quarter inches. If the first arc described an arc of four inches; it lost an eighth part of the motion in 121 oscillations, thus so that in the final ascent it described an arc of 1

23 inches. If the first descent described an arc of eight, sixteen, thirty two or of sixty four inches , then an eighth part of the motion was lost in

1 1 22 2 369 35 18 , 9, , , oscillations respectively. Therefore the difference between the first arc

descended and the final arc ascended, in the first, second, third, forth, fifth and sixth case was of 1 1

4 2 1 2 4 8, , , , , inches respectively. These differences may be divided in each case by the number of oscillations, and into one mean oscillation, by which the arc of

1 14 23 , 7 ,15 30 60 120, , , inches were described, the difference of the arcs described in the

descent and subsequent ascent [i.e. the difference in the amplitude per half oscillation in modern terms] will be 81 1 1 4 24

656 242 69 71 37 29, , , , , parts of an inch respectively. But these are approximately in the square ratio of the arcs described in the larger oscillations, truly in the smaller a little greater than in this ratio ; and therefore (by Corol. 2. Prop. XXXI. of this book) the resistance of the sphere, when it may be moving faster, is in the square ratio of the velocity as an approximation; when it moves slower, a little greater than in this ratio. Now V may designate the maximum velocity in some oscillation, and let A, B, C be given quantities, and we may suppose that the difference of the arcs shall be

32 2AV BV CV+ + . Since the maximum velocities in the cycloid shall be as half of the arcs

described in the oscillation, truly in the circle as half chords of these arcs, and thus with equal arcs the velocities shall be greater in the cycloid than in the circle, in the ratio of half the arcs to the same chords ; but the times in the circle shall be greater than in the cycloid in the inverse ratio of the velocity, it is apparent the differences of the arcs (which



are as the resistance and the square of the time jointly) to be approximately the same, in each curve : for these differences must be increased in the cycloid, together with the resistance, in around the square ratio of the arc to the chord, on account of the velocity increased in that simple ratio ; and to be diminished, together with the square of the time, in the same square ratio. And thus so that it comes about by reduction to the cycloid, that the same differences of the arcs are required to be taken which were observed in the circle, indeed the greatest velocities are required to be put in place analogous either with the halve or whole arcs, that is to the numbers 1

2 1 2 4 8 16, , , , , . Therefore we may write in the second, fourth and sixth case the numbers 1, 4 and 16 for V; and in the second case

there will be produced the difference of the arcs 12

121 A B C= + + ; 12

235 4 8 16A B C= + +

in the fourth case ; and 23

89 16 64 256 A B C= + + in the sixth case. And from these

equations, by placing together and reducing analytically, there shall be 0 0000916 0 0010847 0 0029558A , , B , , & C ,= = = . Therefore the difference of the arcs

is as 32 20 0000916 0 0010847 0 0029558, V , V , V+ + ; and therefore since (by the corollary

of Proposition XXX applied to this case) the resistance of the sphere in the average arc described in the oscillation, where the velocity is V, shall be to the weight of this as

32 27 7 3

11 10 4AV BV CV+ + to the length of the pendulum ; [for this theorem asserts that at any point in the oscillation, the resistive force f shall be to the weight of the bob mg as the ordinate DK to the length of the pendulum l : fDK

l mg= ;

we have shown above that the area under the curve is ( ) 1 0+ 1 0 2 a aa a− ; and for the case of

an approximate ellipse, half the area of such an ellipse is 112 7ab abπ , where the semi-

minor axis b is the maximum resistance fm , and the semi-major axis a is 2aB ; hence

( ) ( ) ( )1 0 1 0+ + 7 7 7111 0 1 0 1 02 7 2 11 11 11 or a a a aaB

m m aBa a f f a a a a AV− − = − = ; Newton gives a similar argument in the corollary above for the elliptic case and also for the parabolic case depending on the square of the velocity, while the middle term is an approximate average of both cases.] If the numbers found may be written with A, B and C, the resistance of the sphere to its weight becomes as

32 20 0000583 0 0007593 0 0022169, V , V , V+ + to the length of the

pendulum between the centre of suspension and the ruler, that is, to 121 inches. From which since V may be appointed equal to 1 in the second case, 4 in the fourth, and 16 in the sixth: the resistance will be to the weight in the second case as 0,0030345 to 121, in the fourth as 0,041748 to 121, and in the sixth as 0,61705 to 121.



The arc that the noted point on the string described in the sixth case was 23

8 5299120 or 119− inches. And therefore since the radius must be 121 inches, and the

length of the pendulum between the point of suspension and the centre of the sphere must be 126 inches, the arc that the centre of sphere described was 3

31124 inches. Because the maximum velocity of the body, on account of the resistance of the air, does not fall on the lowest point of the arc described, but is situated almost at the centre of the arc : this motion will be almost the same if the sphere described the half arc of 3

6262 inches in a non-resisting in the whole descent in a cycloid, to which we have reduced the motion of the pendulum above: and therefore the velocity will be equal to that velocity that the sphere may acquire, by falling perpendicularly, and in that case describing an arc equal to the versed sine of that height. [Note from L. & J. : The body by oscillating may describe the arc Ba in the resisting medium, and the arc BA in the non-resisting medium; let C be the lowest point of the cycloid; O, the midpoint of the arc Ba; and the arc CD shall be equal to the arc BO : the maximum velocity acquired in the descent of the body in the resisting medium in the arc BO is to the maximum velocity acquired in the arc BC in the resisting medium as the arc BO, to the arc BC. But if the body by falling from the location D in the non-resisting medium may describe the arc DC, also its velocity at C acquired by the descent through the arc DC, in the same place in the descent through the arc BC will be as the arc CD, or equally, BO, to the arc BC. Therefore the velocity in the resisting medium acquired by the resisting medium at O is equal to the velocity that the body falling in the non-resisting medium by the arc DC BO= may have at C; and therefore that velocity is equal to that velocity that the body may be able to acquire by falling perpendicularly in the non-resisting medium, and in that case by describing its own height FC equal to the versed sine of the arc CH. Now P shall be the point of suspension, PC the length of the pendulum; SDC the half cycloid; SG and DF normals to PC, and CHGC the circle with the diameter GC described cutting the secant DF in H. The chord CH may be joined, and the arc of the cycloid 2 2SD GC CH= − , and the arc 2SG GC= and thus the arc 2DC CH= . But moreover, from the nature of the circle,

( )24 22 or and hence or CF CH CF CH DC

CH CG CG PCCH DC ,= = ; that is, the versed sine CF is to the arc

CD, as the same arc to double the length of the pendulum.] But that versed sine in the cycloid is to the arc itself 3

6262 as the same arc to a length double the length of the pendulum 252, and therefore equal to 15,278 inches. Whereby that velocity is what a body itself may be able to acquire, and in its own case by falling through a distance of 15,278 inches. Therefore with such a velocity the sphere is subject



to a resistance, which shall be to its weight as 0,61705 to 121, or (if that part of the resistance only may be considered which is as the square ratio of the velocity) as 0,56752 to 121. But I have found by a hydrostatic experiment that the weight of this wooden sphere to be equal to a water sphere of the same size as 55 to 97: and therefore since 121 shall be in the same ratio to 213,4, the resistance of the prepared sphere with the velocity of progression to the weight itself is as 0,56752 to 213,4 that is, as 1 to 1

50376 . From which since the weight of the water sphere, in which time the sphere continued uniformly with the velocity may describe a length of 30,556 inches, may be able to generate all that velocity by the sphere falling, it is evident that the force of the resistance continued uniformly in the same time may be able to remove a velocity in the smaller ratio 1 to

150376 , that is, the part 1

50

1376 of the whole velocity. And therefore in which time the

sphere, with that uniform velocity continued, may be able to describe a length of half its own diameter, or of 1

163 inches, and it may lose the 13542 part of its motion.

I was also counting the number of oscillations in which the pendulum lost the fourth part of its motion. In the following table the top numbers indicate the lengths of the arcs described in the first arc, expressed in inches and parts of inches : the middle numbers indicate the length of the arc described in the final ascent ; and at the lowest level stand the number of oscillations. I have described the experiment as more accurate than in which only the eighth part was lost. Anyone who wishes may test the calculation.

First descent 2 4 8 16 32 64 Final ascent 1

21 3 6 12 24 48 Number of osc. 374 272 1

2162 1283 2

341 2322

Later I suspended a leaden sphere with a diameter of 2 inches, and with a weight of

1426 ounces Avoirdupois by the same thread, thus so that the interval between the centre

of the sphere and the point of suspension should be 1210 , and I counted the number of

oscillations in which a given part of the motion was lost. The beginning of the following tables shows the number of oscillations in which an eighth part of the whole motion had ceased; the second the number of oscillations in which a quarter part of the same oscillations had been lost.

First descent 1 2 4 8 16 32 64 Final ascent 7

8 74 1

23 7 14 28 56 Number of osc. 226 228 193 140 1

290 53 30



First descent 1 2 4 8 16 32 64 Final ascent 3

4 121 3 6 12 24 48

Number of osc. 510 518 420 318 204 121 70 In the first table by selecting from the third, fifth, and seventh observations, and by expressing the maximum velocity for these observations particularly by the numbers 1, 4, 16 respectively, and generally by the quantity V as above: from the third observation

12

193 A B C= + + will arise, in the fifth 12

290 4 8 16A B C= + + , and in the seventh

830 16 64 256A B C= + + . Truly these reduced equations give

0 0014 0 000297 0 000879A , , B , , C ,= = = . From thence the resistance of motion of the sphere arises with the velocity V in that ratio to its weight of 1

426 ounces, that it has 32 20 0009 0 000208 0 000659, V , V , V+ + to the length of the pendulum of 121 inches. And

if we may consider only that part of the resistance which is in the square ratio of the velocity, this will be to the weight of the sphere as 20 000659, V to 121 inches. But this part of the resistance in the first experiment was to the weight of the wooden sphere of

272257 ounces as 0 002217 to 121, V : and thence the resistance of the wooden sphere to

the resistance of the leaden sphere (with the same velocities of these), as 7 122 457 by 0 002217 to 26 by 0 000659, , , that is, as 1

37 to 1. The diameters of the two

sphere were of 786 and 2 inches, and the squares of these are in turn as 1

447 and 4, or 151611 and 1 approximately. Therefore the resistances of these equally moving spheres

were in a smaller ratio than the double of the diameters. But we have not yet considered the resistance of the thread, which certainly by the size it was, and that ought to be taken away from the resistance of the pendulums found. This I was not able to define accurately, but yet I found it to be greater than the third part of the resistance of the smaller pendulum; and then I have learned that the resistance of the spheres, without the resistance of the thread, are almost in the square ratio of the diameters. For the ratio

1 1 13 3 37 to 1− − , or 1

210 to 1 is not far removed from the ratio of the diameters 151611

squared to 1. Since the resistance of the thread shall be of less concern in the larger spheres, I have also tested an experiment with a sphere the diameter of which was 3

418 inches. The length of the pendulum between the point of suspension and the centre of oscillation was

12122 inches, between the point of suspension and the knot on the string 1

2109 inches. The first arc of the pendulum in the descent from the node described 32 inches. The arc in the ascend after five oscillations from the same knot described 28 dig. The sum of the arcs or the whole arc described in the oscillation from the centre was 60 inches. The difference of the arcs was 4 inches. The tenth part of this or the mean difference between the descent and the ascent the oscillation was 2

5 of an inch. So that as the radius 12109 to the radius



12122 , thus the whole arc 60 inches described by the knot in the oscillation from the

centre, to the whole arc 1867 inches described in the oscillation from the centre of the

sphere, and thus the difference 25 in. to the new difference 0,4475 in. If the length of the

pendulum, with the length of the arc described remaining, were increased in the ratio 126 to 1

2122 ; the time of the oscillation may be increased and the velocity of the pendulum may be diminished in that square root ratio, the true difference of the descent and subsequent ascent of the arcs 0,4475 may remain. Then if the arc described may be increased in the ratio 3 1

31 8124 to 67 , the difference 0,4475 itself may be increased in that ratio squared, and thus 1,5295 will arise. Thus these may themselves be considered, under the hypothesis that the resistance of the pendulum may be in the square ratio of the velocity. Hence if the pendulum may describe the whole arc of 3

31124 inches, and the length of this between the point of suspension and the centre of oscillation should be126 inches, the difference of the arcs described in the descent and subsequent ascent should be 1,5295 inches. And this difference taken by the weight of the pendulum, which was 208 inches, gives 318,136. Again when the above pendulum constructed from the wooden sphere with the centre of oscillation, that was 126 inches from the point of suspension, described a whole arc of 3

31124 inches, the difference of the descending and ascending

arcs was 23

126 8121 9 by , which taken by the weight of the sphere, which was 7

2257 ounces,

produced 49,396. Moreover I multiplied these differences into the weights of the spheres so that I could find the resistances of these. For the differences are arising from the resistances, and they are as the resistance directly and the weights inversely. Therefore the resistances are as the numbers 318,136 and 49,396. But the part of the resistance of the small sphere, which is in the square ratio of the velocity, was to the whole resistance as 0,56752 to 0,61675, that is, as 45,453 to 49,396 ; and the part of the resistance of the greater sphere may be almost equal to the whole resistance itself, and thus these parts are as 318,136 to 45,453 approximately, that is, as 7 to 1. But the diameters of the spheres

3 74 818 and 6 ; and the squares of these 9 17

16 64351 and 47 are as 7,438 to 1, that is, approximately as the resistances of the spheres 7 to 1. The difference of the ratios in not much greater, than what can arise from the resistance of the string. Therefore these parts which are of the resistances, which are with equal spheres, as the squares of the velocities; are also, with equal velocities, as the squares of the diameters of the spheres. Of the other spheres, with which I have used in these experiments, the greatest was not perfectly spherical, and therefore in the calculation reported here I have ignored certain details for the sake of brevity ; not being too concerned in the accuracy of the calculation in a not very satisfactory experiment. And thus I might have chosen, since the demonstration of empty space may depend on these, so that experiments might be tried with several greater and more accurate spheres. If spheres may be taken in geometric proportion, e.g. the diameters of which shall be 4, 8, 16, 32 inches ; from the progression of the experiments it may be deduced what must come about from still larger spheres. Now indeed for comparing the resistances of different fluids between each other I have tried the following. I have prepared a wooden box four feet long, with width and height of



one foot. This I filled with spring water without the lid, and I have arranged so that immersed pendulums may be moving by oscillating in the water medium. Moreover a leaden sphere with a weight of 1

6166 ounces, and with a diameter of 583 inches is moved

as we have described in the following table, it may be seen with the length of the pendulum from the point of suspension to a certain marked point on the string of 126 inches, but to the centre of the oscillation of 3

8134 inches.

First descent arc described by point marked on the string, in inches

64 32 16 8 4 2 1 12 1

4

Final ascent arc described, in inches 48 24 12 6 3 121 3

4 38 5

16 Difference of the arcs, proportional to the motion lost, in inches.

16 8 4 2 1 12 1

4 18 1

16

Number of oscillations in water. 2960 1

31 3 7 1411 2

312 1313

Number of oscillations in air. 1285 287 535

In the experiment of the fourth column, an equal motion with 535 oscillations in air, and 1

31 in water were lost. The oscillations in air were indeed a little faster than in water. But if the oscillations in water were accelerated in that ratio so that the motion of the pendulums in each medium were made with equal velocities, the same number 1

31 of oscillations would remain in water, from which the same motion was lost as at first, on account of the increased resistance and likewise the square of the time diminished in the same ratio squared. Therefore with equal velocities of the pendulums equal motions have been lost, with 535 oscillations in the air and with 1

31 oscillations in water; and thus the

resistance of the pendulum in water is to its resistance in air as 535 to 131 . This is the

proportion of the whole resistance in the case of the fourth column. Now 2AV CV+ may designate the difference of the arcs described in the descent and in the subsequent ascent by the sphere in air I set in motion with the maximum velocity V ; and since the maximum velocity in the case of the fourth column shall be to the maximum velocity in the case of the first column, as 1 to 8; and that difference of the arcs in the case of the fourth column to the difference in the case of the first column as

12

162535 85 to or as 1

285 to 4280: we may write in these cases 1 and 8 for the velocities, and

1285 and 4280 for the differences of the arcs, and there becomes 1

285A C+ = ; and 8 64 4280A C+ = or 8 535A C+ = ; and thence by the reduction of the equation there comes about 31 2

2 4 77 449 , 64 , and 21C C A= = = : and thus the resistance since it shall b

as 27 311 4AV CV+ , it will become as 26 9

11 5613 48V V+ . Whereby in the case of the fourth column when the velocity was 1, the whole resistance is to its proportional part with the square of the velocity, as 6 9 912

11 56 17 5613 48 or 61 to 48 + ; and thus the resistance of the pendulum in water is to that part of the resistance in air, which is proportional to the square of the velocity, and which alone in the more rapid motions come to be considered,



as 91217 5661 to 48 and 535 to 1

51 taken jointly, that is, as 571 to 1. If the whole string of the pendulum should be immersed in water, its resistance would be greater still ; and thus that resistance of the pendulum oscillating in water, which is proportional to the square of the velocity, and which alone comes to be considered in more rapidly moving bodies, shall be to the resistance of the same whole pendulum, oscillating in air with the same velocity, as around 850 to 1, that is, as the density of water to the density of air approximately. In this calculation that part of the resistance of the pendulum in the water must also be taken, which might be as the square of the velocity, but (which may be considered as a surprise) the resistance in water may be increased in a ratio greater than the square. With the cause of this matter requiring to be investigated, I came upon this [explanation], that the area should be exceedingly narrow for the size of the sphere of the pendulum, and the motion of the water going before was being impeded exceedingly by its own narrowness. For if the sphere of a pendulum, the diameter of which was one inch, were immersed, the resistance was increased approximately in the square ratio of the velocity. I tested that by the construction of a pendulum with two spheres, the lower and smaller of which could oscillate in water, the upper and greater being attached just above the water, and by oscillating in the air, might aid the motion of the pendulum and render it more long lasting. Moreover the experiments set up in this manner may themselves be had as described in the following table.

Arc described in the first descent. 16 8 4 2 1 12 1

4 Arc described in the final ascent. 12 6 3 1

21 34 3

8 516

Diff. of the arcs, prop.to motion lost. 4 2 1 12 1

4 18 1

16 Number of oscillations. 3

83 126 1

1212 1521 34 53 1

562 By requiring the resistances of mediums to be compared between each other, I also arranged that iron pendulums could oscillate in quicksilver. With the length of the string of the iron pendulum to be nearly three feet, and the diameter of the sphere of the pendulum to be a third of an inch. But another leaden sphere was fixed to the string near the surface of the mercury of such a size that the motion of the pendulum could continue for some time. Then a vessel, that could hold almost three pounds of quicksilver, I filled alternately with quicksilver and ordinary water, so that with a pendulum oscillating successively in each fluid, I was able to find the proportion of the resistances : and the resistance of quicksilver to water was produced as around 13 or 14 to 1 : that is, as the density of quicksilver to the density of water. When I used an iron sphere a little larger, for example with a diameter that should be around 1 2

3 3or of an inch, the resistance of the quick silver was produced in that ratio to the resistance of water, that the number 12 or 10 has to 1 roughly. But the first experiment is more trustworthy, because that vessel in these final experiments was exceedingly narrow for the size of the sphere immersed. With the sphere enlarged, the vessel also is required to be enlarged. I set up certain experiments of this kind in larger vessels and with liquids both of melted metals as well as with certain others to be repeated with hot as well as with cold liquids: but there was not time to prove everything, and now from the description it may be clear enough how the resistance of



bodies moving rapidly may be approximately proportional to the density of the fluids in which they are moving. I do not say accurately. For more tenacious fluids, on a par with the density, without doubt are more resistant than the more liquid, such as cold oil rather than hot oil, cold rather than hot rainwater, water rather than spirit of wine [i.e. brandy]. Truly with liquids, which are fluid enough to the senses, as with air, water either sweet or salty, with spirits , spirits of wine, of turpentine and salts, with oil freed from its dregs by distillation and by heating, and with oil of vitriol and with mercury, and with liquefied metals, and others which there may be, which are both fluid so that they may conserve a long time impressed motion of agitation, and on being poured and running away freely may be resolved into drops, I have no reason to doubt that the rule already laid down may prevail accurately enough : especially if experiments with larger pendulous bodies with faster motions may be put in place. Finally since it shall be the opinion of some, of a certain must subtle ethereal medium extending afar, that may permeate all the pores of all bodies freely; moreover the resistance must originate from such a medium flowing through the pores of bodies : So that I can test the resistance that we experience from the motions of bodies, whether the whole shall be on the external surface of these, or the resistance may be perceived also from the various internal parts near the surfaces, I have thought out a test for such. With a string eleven feet long firmly attached by a steel hook, I hung up a round wooden box by means of a steel ring, towards constructing a pendulum of the aforestated length. The upper concave edge of the hook was sharpened, so that the ring resting on its upper arc could move freely on the sharp edge. Moreover the string was attached to the lower arc. The pendulum thus constituted I drew away from the vertical to a distance of around six feet, and that along a plane perpendicular to the plane of the sharpened hook, lest the ring, with the pendulum oscillating, might slip beyond the side of the sharpened edge of the hook. For the point of the suspension, in which the ring touches the hook, must remain motionless. Therefore I marked the place accurately, to which I drew the pendulum aside, then with the pendulum sent off I noted the three other places to which it returned at the end of the first, second, and third oscillation. I repeated this more often, so that I could find these places most accurately. Then I filled the box with lead and with metal weights which were at hand. But first I weighed the empty box, together with the part of the string that was tied around the box and with half of the remaining part which was being stretched between the hook and the pendulum box. For the extended string always acts with half its weight on the pendulum when drawn aside from the perpendicular. To this weight I added the weight of the air that the box contained. And the whole weight was as if a 78th part of the weight full of the metal. Then because with the box filled with metal, with the weight itself by stretching the string, the length of the pendulum must be increased, I shortened the string so that now the length of the pendulum oscillating should be the same as at first. Then with the pendulum drawn aside and released from the first marked place, I counted around 77 oscillations, then the box returned to the second marked place, and just as many one after the other until the box returned to the third marked place, and again with just as many until the box returned to reach its fourth marked place. From which I conclude that the whole resistance of the full box did not have a greater proportion to the resistance of the empty box than 78 to 77. For if the resistance of both should be equal, the full box on account of its inertial force , which was



78 times greater than the inertial force of the empty box, the motion of its oscillations must be maintained for such a longer time, and thus always with 78 oscillations completed to have returned to that marked place. But it returned to the same after 77 complete oscillations. Therefore A may designate the resistance of the box on the external surface, and B the resistance of the empty box from the inside parts; and if the resistances of the bodies of the same velocities in the inside parts shall be as the matter, or the number of small parts by which it is resisted : 78 B will be the resistance of the full box from the inside parts : and thus the whole resistance A B+ of the empty box will be to the total resistance of the full box 78A B+ as 77 to 78, and separately A B+ to 77B, as 77 to 1, and thence A B+ to B as 77 77× to 1, and separately A to B as 5918 to 1. Therefore the resistance of the empty box from the interior parts is more than 5000 times less than the same resistance of this from the external surface. Thus truly we may question that hypothesis that the greater part of the resistance of the full box may arise from no other hidden cause than from the action alone of some subtle fluid enclosed in the metal. This experiment I have recalled from memory. For the page, in which I have described that a little, has become lost. From which certain fractional parts of the numbers, which have disappeared from memory, I have been compelled to omit. For there is not time to try everything anew. In the first place, since I was using an infirm hook, the full box was being slowed down more. On seeking the reason, to be found that a weak hook yielded to the weight of the box, and in the oscillations of this was being bent by all the parts giving way. Therefore I prepared a strong hook, so that the point of suspension might remain fixed, and then everything thus came about as we have described above. [It seems likely that the extra friction at the loaded knife edge gave the slightly different results.]


Book II Section VII. Translated and Annotated by Ian Bruce. Page 607

SECTION VII.

Concerning the motion of fluids & the resistance of projectiles.

PROPOSITION XXXII. THEOREM XXVI.

If two similar systems of bodies may consist of an equal number of particles, and the corresponding particles shall be similar and proportional, with an individual in one system to an individual in the other, and situated similarly to each other, and in turn they may have a given ratio of density to each other, and they may begin to move similarly between each other in proportion to the time (these amongst themselves which are in one system and those amongst themselves which are in the other system), and if those which are in the same system may not touch each other, except at instants of reflection, nor attract each other, nor repel each other, except by accelerative forces which shall be inversely as the corresponding diameters of the particles and directly as the squares of the velocities : I say that these particles of the systems may in proportional times go on moving similarly among themselves. [In this proposition, Newton sets out his mechanical view of the world, essentially following the geometrical lines of Euclidean geometry : the motions of particles proceed along straight lines in time; groups of particles or bodies may describe geometrical figures under the influence of action at a distance forces ; collisions between particles and bodies also are geometrical in form, in that they have outcomes known from calculation given the in-going boundary conditions ; thus, the whole motion of a system of particles and bodies proceeds like some giant piece of clockwork, for ever; thus, such a system could be traced back to some starting configuration in the past, as Laplace was to observe later. The formulas introduced here in the notes correspond to these in Ch. IV, B. & R.; these authors have changed the order of presenting Newton's work at this point.] I say that similar bodies, situated similarly, move similarly amongst themselves in proportional times, the situations of which in turn at the end of these times shall be similar always: for example, the particles of one system may be compared with the corresponding particles of the other system. From which the times shall be proportional, in which the parts of similar and proportional figures will be described by corresponding particles [in the given system]. Therefore if there shall be two systems of this kind, on account of the similitude of the starting motions, they will proceed to move similarly, until finally they may meet each other. For if they are not disturbed by forces, they may be progressing uniformly on right lines by the first law of motion. If they may be mutually disturbed by some forces, and these forces shall be inversely as the diameters of the corresponding particles and directly as the velocity squared, because the situations of the particles are similar and the forces proportional, the whole forces by which the corresponding particles are agitated, composed from individual agitating forces (by the corollary of the second law), will have similar determinations, and likewise as if the forces may be considered acting between the centres of similarly situated particles ; and those total forces in turn will be as the individual composing forces, that is, inversely as the diameters of the



corresponding particles and as square of the velocities directly: and therefore the forces act so that the particles go on to describe corresponding figures. Thus these themselves may be had (by Corol.1 & 8, Prop. IV. Book. I.) only if the centres may be at rest. But if the centres may be moving, because on account of the similitude of the translations, the particles of the systems remain in their places ; similar changes may be produced in the figures which the particles describe. Therefore the motions of the corresponding particles are similar until their first meeting, and therefore there are similar meetings and reflections, and from that (by that shown) the motions among themselves again are similar until they next meet, and thus henceforth indefinitely. Q.E.D. Corol.1. Hence if any two [large] bodies, which shall be similar and similarly put in place corresponding to the particles of the systems, similarly may begin to move between themselves in proportional times, and the magnitudes of these bodies shall be as the densities and in turn as the magnitudes and densities of the corresponding particles: these bodies may likewise proceed in proportional times and to be moving similarly. For the ratio of each of the greater parts of the system and of the particles is the same. Corol. 2. And if all similar and similarly placed parts of systems are at rest relative to each other: and two parts of these, which shall be larger than the rest, and may correspond between each other mutually in each system, may begin to move with some motion along similar lines similarly situated : these may excite similar motions in the remaining parts of the systems, and may go on to move similarly between themselves in proportional times ; and thus to describe distances proportional to their own diameters.

PROPOSITION XXXIII. THEOREM XXVII. With the same in place, I say that the greater parts of the system are resisted in the ratio composed from the square of their velocities and the square ratio of the diameters and in the ratio of the densities of the parts of the system. For the resistance arises partially from the centripetal or centrifugal forces by which the particles of the systems mutually disturb each other, partially from the coming together and reflections of the particles, and of the greater parts [present in the systems]. Moreover, the resistances of the first kind in turn shall be as the whole motive forces from which they arise, that is, the accelerations and quantities of matter in the corresponding parts ; that is (by hypothesis) as the squares of the velocities directly and inversely as the distances of the corresponding particles, and directly as the quantities of matter in the corresponding parts : [We may write these resistive forces algebraically, in an obvious notation, in the form:

2 3 2 2vd d v d .ρ ρ× = × × ]

and thus since the distances of one system of particles shall be to the corresponding distances of the other particles, as the diameter of the particle or of the parts in the first system to the diameter of the particles, or to the corresponding parts in the other, and the quantities of matter shall be as the densities of the parts and the cubes of the diameters;



the resistances shall be in turn as the squares of the velocities and the squares of the diameters of the parts of the systems. Q.E.D. The resistances of the second kind are as the number of corresponding reflections and the forces taken together. But the number of reflections are in turn as the velocities of the corresponding parts directly, and inversely as the distances between the reflections of these. And the forces of the reflections are as the velocities and the magnitudes and densities of the corresponding parts jointly ; that is, as the velocities and the cubes of the diameters and the densities of the parts. And from all these ratios jointly, the resistance of the corresponding parts are in turn as the squares of the velocities and the squares of the diameters and the densities of the parts jointly. Q. E. D. [In the second case, the resistances can be written as the frequency of the collisions by the momentum change per collision. The frequency is given by v

d , where d is the particle separation and v the velocity; the momentum change per collision is proportional to

3v dρ× × , giving same result 3 2 2vd v d v dρ ρ× × = × × as above.]

Corol. 1. Therefore if these systems shall be two elastic fluids after the manner of air, and the parts of these may be at rest within themselves : moreover two similar bodies may be projected along certain lines put in place in some manner, similarly put in place among these parts, as long as the magnitude and density shall be proportional, and moreover the accelerative forces, by which the particles of the fluid disturb each other, shall be inversely as the diameters of the projected bodies, and directly as the squares of the velocities: then these bodies will excite motions in the fluids proportional to the times, and describe similar distances with their diameters proportional to these. Corol. 2. Therefore in the same fluid a swift projectile suffers a resistance, which is in the square ratio of the velocity approximately. For if the forces, by which distant particles mutually agitate each other, may be increased in the square ratio of the velocity, the resistance will be in the same ratio squared accurately ; and thus in a medium, the parts of which in turn are disturbed by no forces with the distances, the resistance is accurately in the square ratio of the velocity. Therefore let there be three mediums A, B, C with similar and equal parts consistently set out regularly along equal distances. The parts of the mediums A and B may mutually repel each other by forces which shall be in turn as T and V, and these parts of the medium C shall be completely free from forces of this kind. And if four equal bodies D, E, F, G may be moving in these mediums, the first two D and E in the first two mediums A and B, and the other two F and G in the third medium C.

[i.e. D is in medium A, E is in medium B; F and G are in medium C.] The velocity of the body D to the velocity of the body E, and the velocity of the body F to the velocity of the body G shall be in the square root ratio of the forces T to the forces V [i.e. vel.of vel.of

vel.of vel.of D F TE G V= = ; recall that the resistances are as the square of the velocities

and inversely as the diameters of the particles, which latter are equal in this case, and hence]



: the resistance of the body D will be to the resistance of the body E, and the resistance of the body F to the resistance of the body G, in the square ratio of the velocities, and therefore the resistance of the body D will be to the resistance of the body F as the resistance of the body E to the resistance of the body G. [ i.e. res.of res.of

res.of res.of and if vel.of vel.of then vel.of vel.of D F TE G V D F E G.= = = = ]

Let the bodies D & F have equal velocities, as with the bodies E & G; and on increasing the velocities of the bodies D and F in some ratio, and by reducing the forces of the particles of the medium B in the same ratio, the medium B will approach to the form and condition of the medium C as it pleases, and on that account the resistances of the equal bodies and of the bodies moving equally E & G in these mediums, may always approach to equality, thus so that the difference may emerge finally less than any given amount. Hence since the resistances of the bodies D and F shall be in turn as the resistances of the bodies E and G, these similarly approach to the ratio of equality. Therefore the resistances of the bodies D and F are approximately equal, when they are moving fastest : and therefore since the resistance of the body F shall be in the square ratio of the velocity, the resistance of the body D will be approximately in the same ratio. [Thus, if a body is moving much faster than the particles of the medium, they can be considered at rest, and all the collisions occur with particles as if at rest, for which the v2 formula holds accurately.] Corol. 3. The resistance of the fastest moving body in some elastic fluid is almost the same as if the parts of the fluid were without their centrifugal forces, and with these not mutually repelling each other : but only if the elastic force of the fluid may arise from the centrifugal forces of the particles, and the velocity shall be so great that the forces have not enough time to act. Corol. 4. Hence since the resistances of similar and equally swift bodies, in a medium of which the distant parts do not fly apart mutually, shall be as the squares of the diameters ; also the resistances of the fastest and equally speedy bodies in an elastic fluid are as the squares of the diameters approximately. Corol: 5. And since similar bodies, equal and equally swift, in mediums of the same densities, the particles of which do not mutually fly apart, these particles either shall be more plentiful and smaller, or fewer and larger, may impinge on equal quantities of matter in equal times, may impress an equal quantity of motion, and in turn (by the third law of motion) from the same they may experience an equal reaction, that is, they are resisted equally : it is evident also that the resistances of the same density of elastic fluids shall be approximately the same, when they are moving the quickest, whether that fluid shall consist of grosser particles, or be constituted from the most subtle of all. From the most subtle resistance the speed of the fastest projectiles is not much diminished. Corol. 6. All these thus may themselves come about in fluids, the elastic force of which has its origin in the centrifugal forces of the particles. But if that may arise otherwise, or as from the expansion of the particles in the manner of wool or the branches of trees, or



from any other cause whatever, by which the motions of the particles among themselves are rendered less free, on account of smaller fluidity of the medium, it will be greater than in the above corollaries. [We now need to consider the effect of the individual shapes of bodies on the resistance. It is necessary to assume that the density of the fluid particles is much rarer than of the solid body, and each fluid particle can deliver its blow independently without interference from other particles of fluid]

PROPOSITION XXXIV. THEOREM XXVIII. If a sphere and a cylinder be described with equal diameters moving with equal velocity, in a rarefied medium with equal particles, and with these being placed freely at equal distances along the direction of the axis of the cylinder : the resistance of the sphere will be half the resistance of the cylinder. For because the action of the medium on the same body is the same (by corol.5 of the laws) whether the body may be moving in a medium at rest, or the particles of the medium may strike the body at rest with the same velocity : we may consider the body as being at rest, and we may consider by which impetus it may be urged by the moving medium. Therefore ABKI may designate the spherical body described with centre C and with radius CA, and medium particles may be incident on that spherical body with a given velocity, along right lines parallel to AC itself : and let FB be a right line of this kind. On that LB may be taken equal to the radius CB, and BD may be drawn a tangent to the sphere at B. The perpendiculars BE and LD may be sent to the lines KC and BD, and the force by which a particle of the medium, by being incident obliquely along the right line FB, strikes the sphere at B, will be to the force by which the particle may strike the same cylinder ONGQ at b, with the axis ACI described perpendicularly around the sphere, as LD to LB or as BE to BC [i. e. as the cosine of the angle θ ; thus, the normal component of the force striking the

body on a small planar area A at that point varies as 2 2Av cos θ× , where the average value of 2cos θ must be calculated. Since only the component of this force along the

direction of motion is required, the resistance is as 2 3Av cos θ× . Thus for a spherical

surface, the force on an annulus is proportional to 2 3 22 r sin cos v dπ θ θ θ× , which integrates to give 2 2

42 rv rπ × ; which is half the value for a great circle; below we show how S. & J. tackle this problem geometrically.] Again the effectiveness of this force to the movement of the sphere along its direction FB or AC, is to the same effectiveness of the sphere moving along the direction of its determination, that is, along the direction BC by which the sphere may be urged directly as BE to BC. And with the ratios taken jointly, the effectiveness of a particle on the sphere along the oblique right line FB, to the same being moved along the direction of its



incidence, is to effectiveness of the same particle incident on the cylinder along the same right line perpendicularly, to that required circle to be moved in the same direction, as BE squared to BC squared. Whereby if on bE, which is perpendicular to the circular base of the cylinder NAO and equal to the radius AC, bH may be taken equal to .BEquad

CB ; bH will be to bE as the effectiveness of the force of the particle on the sphere to the effectiveness of the force of the particle on the cylinder. And therefore the solid that may be occupied by all the right lines bH will be to the solid which may be occupied by all the right lines bE, as the effect of all the particles on the sphere to the effect of all the particles on the cylinder. But the first solid is a paraboloid described with vertex C, axis CA and with the latus rectum CA , and the latter solid is the circumscribed cylinder to the paraboloid : and it is to be noted that the paraboloid shall be half of the circumscribed cylinder. Therefore the total force of the medium on the sphere is twice as small as the same total force on the cylinder. And therefore if the particles of the medium are at rest, and both the cylinder and the sphere may be moving with equal velocities, the resistance of the sphere will be less than the resistance of the cylinder by a factor of two. Q.E.D. [L. & S. notes : If on all the points of the right line NA perpendiculars may be erected as bH and bE, and let NHC be that curve the point H always touches, and the line KC the locus of all the points E; the solid that may be occupied by all the perpendiculars bH, drawn through the whole base of the cylinder, will be equal to a conoid or to the solid figure which may be generated by the rotation made of the plane figures NHCA about the axis CA, and the solid that may be formed from all the right lines bE will be the cylinder described by the rotation of the rectangle AK made about the same axis. Since, by construction, there shall be

2 2 2 2 and thus BECBbH bH CB BE BC CE= × = = − , and from the nature of the circle,

BC CA KC= = , and thus ( )2 2 2 and or ,BE KC CE bH CB, KC EH KC= − × − × or 2 2 2KC KC EH KC CE ,− × = − and thus 2;KC EH CE× = but if the point H may be

drawn to CA, the perpendicular ordinate, this must be equal to CE, and from CA there may be cut the part equal to EH. Whereby the rectangle under the given abscissa and the given line KC or CA, is equal to the square of the ordinate the perpendicular CA; thence the curve CHN, by Theorem I (de parabola ; Apollonius) is a parabola the vertex of which is C, the axis CA, and the latus rectum CA. The paraboloid or solid generated from the rotation of the parabola CHN, about the axis CA is half of the circumscribed cylinder, which is produced from the rotation of the rectangle AK about the line CA. Through the moveable point P, the normal PM is erected to the axis CA, cutting the parabola in H, and the right line KN in M; and in the rotation of the whole figure about the axis CA, the lines PH and PM describe circles, which will be among themselves as the squares of the radii PH and IM, or, from the nature of the



parabola and on account of PM AN= , as the abscissae CP and CA. Now the point P is drawn with the vertical PHM through the whole altitude CA, and the solid arising from the rotation of the figure CHN will be to the cylinder arising from the rotation of the rectangle CKNA, as the sum of all the circles which the moving line PH will describe by rotating, to the sum of all the circles which the right line PM will describe, that is, as the sum of all CP , to the sum of all CA. On the line AN there may be taken AR equal to AC, CR is joined cutting PH in L, and the perpendicular RQ is erected to AR, cutting PM in V; since there shall always be and PL CP, PC CA,= = the sum of all CP, or PL, by the whole altitude CA, is the isosceles triangle CRA, and the sum of all CA, or PV, by the same altitude CA, is the square CARQ; therefore since the triangle CRA, shall be half of the square CARQ, the paraboloid is also half of the circumscribed cylinder.]

Scholium. By the same method other figures may be compared between themselves as far as concerns resistance, and these found which are more suited to continue their motion in resisting mediums. So that if to the circular base CEBH, which will be described with centre O, with radius OC, and with the altitude OD, the conic frustum CBGF shall be constructed, which of all constructed with the same base and height and their axis following towards the direction of progression D shall be resisting the least: bisect the height OD in Q and produce OQ to S so that there shall be QS equal to QC, and S will be the vertex of the cone of which the frustum is sought. [Following B & R, if the left hand base of the cone has radius r, the right hand base has radius r h tanθ− , and

the resistance of the curved surface will be proportional to ( )22 2sin r r h tanθ θ− − , while

that of front circle will be proportional to ( )2r h tanθ− , on omitting other constant factors.

The total resistance can be seen to be 2 2 22r rh sin h sinθ θ− + ; to find the turning point, differentiate w.r.t. h and we find 22h sin cos r r sinθ θ θ= − ; if x is the whole height of the cone, then this equation becomes :

22

2 2 2 22 2 2

2 42 and xr h hrr x r x

h r r x hx r x r+ +

× = − × ∴ − = = + + , leading to the above

construction.] From which by the way, since the angle CSB shall always be acute, it follows that if the solid ADBE made may be generated by the rotation of the elliptic or oval figure ADBE about the axis AB, and the generating figure may be touched by the three right lines FG, GH, HI at the points F, B and I, by that rule so that GH shall be perpendicular to the axis at the point of



contact B, and FG, HI may contain with the same GH the angles FGB and BHI of 1350 of the solid, that may be generated by rotation of the figure ADFGHIE about the same axis AB, resists less than the first solid; but only if each may be progressing along the direction of its axis AB, and each end B goes in front. Indeed which proposition I consider to be of use in the future construction of ships. But if the figure DNFG shall be a curve of this kind, so that if the perpendicular NM may be sent from some point N of this curve to the axis AB, and from some given point G the right line GR may be drawn which shall be parallel to the tangent of the figure at N, and may cut the axis produced at R, MN will be to GR as GR3 to 24BR GB× ; the solid that is described by the revolution made of the figure about the axis AB, in the aforementioned rare medium by moving from A towards B, will be resisted less than some other encircled solid described with the same length and breadth.

PROPOSITION XXXV. PROBLEM VII.

If a rare medium may consist of the smallest equal particles at rest and in turn to be placed at equal distances freely : to find the resistance of a sphere progressing uniformly in this medium. Case 1. A cylinder described with the same diameter and altitude may be understood to be progressing with the same velocity along the length of its axis in the same medium. And we may consider that the particles of the medium, on which the sphere or cylinder is incident, may recoil with the greatest force of reflection. [Thus, elastic collisions offer the greatest resistance to motion, and completely inelastic collisions the lest resistance, in this model. Thus, for a cylinder of length l, the time to describe half the axis will be

2212 and the acceleration giving a velocity in this time is l v

v lv ; hence the resistance is

simply 22Av ρ , where A is the base area of the cylinder and ρ the velocity.] And since the resistance of the sphere (by the latest Proposition) shall be half as great as the resistance of the cylinder, and the [volume of the] sphere shall be to the [volume of the] cylinder as two to three, and the cylinder by being incident on the particles perpendicularly, and these by being maximally reflected, may impart twice their velocity : the cylinder progressing uniformly in that time will describe half the length of its axis, will communicate the motion to the particles, which shall be to the whole motion of the cylinder as the density of the medium to the density of the cylinder ; and the sphere, in which time the whole length of its diameter will be described in progressing uniformly, will share the same motion with the particles ; and in that time in which it will describe a motion two thirds part of its diameter, that it will communicate to the motion of the particles, which shall be to the whole motion of the sphere as the density of the medium to the density of the sphere. And therefore the sphere suffers a resistance, which shall be to the force by which the whole motion of this may be taken away or generated in which time the two thirds part of its diameter will be described by progressing uniformly, as the density of the medium to the density of the sphere.



Case 2. We may consider that the particles of the medium incident on the sphere or cylinder are not reflected, and by being incident perpendicularly on the particles the cylinder will simple communicate its velocity, and thus the resistance experienced is half as great as in the previous case, and the resistance of the sphere will also be half as great as before. Case 3. We may consider that the particles of the medium are reflected neither maximally nor not at all, but they may circle from the sphere in some manner in between, and the resistance of the sphere will be in the same mean ratio between the resistance in the first case and the resistance in the second case. Q.E.I. Corol. 1. Hence if the sphere and the particles shall be indefinitely hard, and therefore destitute of all elasticity and all force of reflection : the resistance of the sphere will be to the force by which the whole of that motion may be removed or generated, in the time that the sphere can describe four thirds parts of its diameter, as the density of the medium to the density of the sphere. Corol. 2. The resistance of the sphere, with all else being equal, is in the square ratio of the velocity. Corol. 3. The resistance of the sphere, with all else being equal, is in the square ratio of the diameter. Corol. 4 The resistance of the sphere, with all else being equal, is as the density of the medium. Corol. 5 The resistance of the sphere is in a ratio which is composed from the square ratio of the velocity and the square ratio of the diameter and in the ratio of the density of the medium. Corol. 6. And the motion of the sphere with its resistance may be explained thus. Let AB be the time in which the sphere can lose all its motion by a uniformly continued resistance. The perpendiculars AD, BC may be erected to AB. And BC shall be that total motion, and through the point C a hyperbola CF may be described with the asymptotes AD and AB ; AB may be produced to some point E. The perpendicular EF may be erected crossing the hyperbola at F. The parallelogram CBEG may be completed, and AF may be drawn crossing BC at H. And if the sphere in some time BE, continued uniformly by its own initial motion BC, in a non-resisting medium may describe the distance CBEG shown by the area of the parallelogram, likewise in the resisting medium it describes the distance CBEF shown by the hyperbolic area, and the motion of this at the end of the time may be shown by the ordinate EF of the hyperbola, with the part of its motion EG lost. And the resistance of this at the end of any time may be expressed by the length BH, with the part CH lost to resistance. All these are apparent by Corol.1.& 3, Prop. V. Book II. [Thus, using customary integration, if a body is dropped from rest ,with the resistance proportional to the velocity squared given by 2

rF kmv= , and g' is the apparent

acceleration of gravity, with u the terminal velocity given by 2g' ku= , then at any instant,


Book II Section VII. Translated and Annotated by Ian Bruce. Page 616 2

2 222 2 and 2g' g'dv dv

dx dxu uv g' v v g'= − + = ; this requires the integration factor

22g'

ux

e , and

hence 2 2

2 22 2g' g'

u ux xd

dx v e g' e= , giving2

22 2 1g'

ux

v u e− ⎞⎛

= − ⎟⎜⎝ ⎠

;

again, ( ) 2 22 2 1 1 and 2dv dv

dt u v u vu vk u v ukdt− +−

= − = + = giving 2u vu vln ukt C+− = + ; from

which in turn 2

211

kut kut kut

kut kut kute e ee e e

v u u u tanhkut− −

− −− −+ +

= = = ; hence

( ) ( )( ) ( )

21 1 1

2 21 1 1

1

1 1 2 ]

kut kut

kut kutdp kut kut kut kutdx e e

dt k p k ke ekut kut

k k k

u x ln e e lne e

ut ln e C ut ln e ln .

−

−− −−

+− −

= = ∴ = + = +

= + + + = + + −

Corol. 7. Hence if the sphere may lose its whole motion M in the time T by a resistance R continued uniformly: the same sphere in the time t in the resisting medium, by a resistance R decreasing as the square of the speed, may lose a part tM

t T+ of its motion M,

with the part TMt T+ remaining ; and it will describe a distance that shall be to the distance

described by the uniform motion M in the same time t, as the logarithm of the number t T

t+ multiplied by the number 2,30258092994 is to the number t

T , as that hyperbolic area BCFE is in this proportion to the rectangle BCGE . [It seems appropriate to give here the geometrical explanations of L. & J. for Cor. 7, which although they are rather longer than the modern calculus approach, may yield some clues as to Newton's manner of thinking at the time: And the resistance at the end of this time, etc. The resistance, at the beginning of the motion when the velocity is BC, may be shown by the same line BC, and because the resistances are as the squares of the velocities, and BC shall be to FE as the velocity from the start of the motion to the velocity at the end of the time BE, to FE2, as BC to the line which may show the resistance at the end of the time BE, and thus this line is equal to

2FEBC : that is

2 20

2t t

R BC BC FEtR R BCFE

R= = ∴ = . Now, from the nature of the hyperbola, we have

AB BC AE FE× = × , and in addition from the similar triangles, ABH & AEF, BC AE FEFE AB HB= = , and hence

2FEBCHB = . Whereby the right line HB will show the resistance

at the end of the time BE, and hence the part of the right line CH may show the part of the resistance that has been lost since the start of the motion, shown by the line BC. Thus proceeding, the part of the motion M remaining at the end of the time t may be called m, and since and hence T tT AB AE

t BE T AB+= = , and besides CBM AE

m FE AB= = , there will



be T t MT m+ = , from which there may be had MT

T tm += , and thence the part of the motion M

lost is MtMTT t T tM + +− = as required.

Proceeding further, the parts of the axis and ordinates of hyperbola may be called AB a,BC b,BE x,AE a x;= = = = + and from the nature of the hyperbola, as

aba xFE y += = , the element of the area CFEB will be abdx

a x+ , and the area CFEB itself, is

equal to dxa xab +∫ , which integral (fluent) thus is to be summed so that it may vanish when

0x = , but the integral dxa x+∫ thus taken is the logarithm (called L. in this work, but which

we shall call ln) of the number a xa+ , chosen from the logistic curve the subtangent of

which is unity (see the additional material for Section 1 of Book 2 : essentially the definition of the exponential function, or antilogarithm curve in these days, being that curve, which on finding the gradient at the point x, y, gives a right angled triangle under the tangent, for which the tangent of the angle with the x axis is y/1 ; or dy/dx = y), or what amounts to the same thing, from the hyperbola of which the power is one; if indeed there may be put 0x = , the number a x

a+ becomes equal to 1, and thus ( ) 0a x

aln .+ =

Whereby the area BCFE ( )a xxab ln += × ; truly the rectangle BCGE bx= . Therefore the

hyperbolic area BCFE is to the rectangle ACGE as ( ):a xxab ln bx+× , that is , as

( ):a x xx aln + . Indeed here we have and a x T t x t

a T a T+ += = ; whereby the hyperbolic area

BCFE is to the rectangle BCGE , as ( ) to T t tT Tln + . Therefore it remains to find the

logarithm of the number T tT+ ; by the logarithmic curve of which the subtangent is one.

Again the logarithms of different kinds of the same number are between themselves in a given ratio, and the number 2,302585092994 is the logarithm of the number ten in the species of logarithms the subtangent of which is unity (i.e. natural logarithms) and the logarithm of the number ten taken in tables is 1 0000000 1, = ; thus as the logarithm of the number T t

T+ taken in tables to the logarithm of the same number taken in logarithms the

subtangent of which is unity, or in the hyperbola the power of which is 1; therefore the logarithm sought , if the logarithm of the number T t

T+ taken from tables may be

multiplied by the number 2,302585092994. ]

Scholium. In this proposition I have set out the resistance and retardation of spherical projectiles in non continuous mediums, and I have shown that this resistance shall be to the force by which the whole motion of the sphere shall be removed or generated in the time in which the sphere may describe two thirds parts of the diameter, with a uniformly continued velocity, as the density of the medium to the density of the sphere, but only if the sphere and the particles of the medium shall be completely elastic and they may be influenced by the maximum force of reflection: and that this force shall be half as great when the sphere and the particles of the medium are infinitely hard, and evidently without any force of



reflection. But in continuous mediums such as water, hot oil, and quicksilver, in which the sphere is not incident at once on all the fluid particles generating resistance, but presses only to the nearby particles and these press on other particles, and these in turn still others, the resistance is hence twice as small. Certainly a sphere in mediums of this kind of the most fluid, experience a resistance which is to the force by which the whole motion of this may either be removed or generated in the time, with that motion continued uniformly, eight third parts of its diameter will be described, as the density of the medium to the density of the sphere. That which we will try to show in the following medium.

PROPOSITION XXXVI. PROBLEM VIII. To define the motion of water flowing from a hole made at the bottom of a cylindrical vessel. [An excellent introduction to the history of this complex problem can be found in : Die Werke von Daniel Bernoulli : Gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli (Basel; Boston; Birkhauser, 1982-2002; ed. David Speiser.) Vol. I, page 200. Early studies on the outflow of water from vessels, by G.K. Mikhailov (tr. from Russian into English by Rainer Radok.); this includes an independent modern view of Newton's contributions.] Let ACDB be the cylindrical vessel, AB the upper opening of this, CD the base parallel to the horizontal, EF a circular hole in the middle of the base, G the centre of the hole, and GH the axis of the cylinder perpendicular to the horizontal. And imagine a cylinder of ice APQB to be of the same width as the cavity of the vessel, and to have the same axis, and to be descending with a constant uniform motion, and the parts of this that first touch the surface AB become liquid, and converted into water to flow by their own weight into the vessel, and the head of flowing water formed by falling ABNFEM passes through the hole EF, and likewise to be filled equally. [One might wonder initially why Newton considered this rather odd way of obtaining a constant head of water; perhaps it was just a domestic problem that intrigued him..... In any case, there are fundamental defects in Newton's approach, which does not agree with experiment for the motion of the water within the vessel near the walls; thus the idea of a funnel or an ice funnel through which the water flowed was completely erroneous, if that is what Newton had in mind; however, he may well have realized that the original problem was too hard to solve, so that he decided to solve an easier problem, involving the ice funnel, for which he was able to calculate the shape, and thus have a far better idea of the flow of water through such a shape, as we show below in an L & S derivation. In addition, in the first edition, the initial contraction or waisting of the stream of water emerging – the vena contracta - had been ignored totally – to be fixed up in the second edition after complaints from the Bernoulli camp; otherwise, Newton may have considered water to be far stickier or to have a much greater viscosity than it really does, as his approach might well describe the fall of a sticky liquid such as honey or oil through



the hole at the bottom of a vessel. For in truth all the particles of water in the vessel descent slowly with no horizontal motion until the level of the hole is reached, at which stage there is considerable horizontal motion of a complex nature. Did Newton not bother to observe what actually happens, or could not see because he used wooded vessels ? It certainly would be atypical of his methods in investigating natural phenomena, such as the decomposition of light into its spectrum, if this were the case. As it was, his investigations described here provided a foundation for further inquiries, which had actually started some time previously, as the above reference sets out; in particular, his lines drawn in the fluid are essentially the lines of force or flow lines along which the particles travel. The hydrodynamics text of Daniel Bernoulli a little later gave improved explanations of the phenomena involved, following his [Bernoulli's] Principle, which made use of Leibniz's vis viva idea : the missing half was to create havoc in understanding kinetic energy for at least 100 years! So the complete idea of laminar flow had to await the concept of energy conservation before it could be placed on a firmer footing. The question of the shape of the vessel and the nature of its surface more or less dictates whether or not turbulent flow will occur near the hole; thus, this is still a very difficult problem to solve, and Newton may have been content to give approximate answers only by solving another problem.] As indeed there shall be a constant velocity of the ice descending and with the adjoining water [formed] in that case describing the circle AB, so that the water by falling can acquire an altitude IH, and both IH and HG may be placed along the same direction, and through the point I the right line KL may be drawn parallel to the horizontal, and meeting the sides of the ice at K & L. And the velocity of the water flowing through the hole EF, will be as that the water, by falling from I, is able to acquire in its own case by describing the height IG . And thus by a theorem of Galileo, [really Torricelli] IG will be to IH in the square ratio of the velocity flowing from the orifice to the velocity of the water in the circle AB, that is, in the square ratio of the circle AB to the circle EF; for these circles are inversely as the velocities of the water which through themselves may adequately be passed in the same time and in an equal amount. [Thus, the continuity equation of fluid flow is called upon, and also the conversion of potential into kinetic energy, in modern terms, of any small particle of the fluid.] Here we are concerned with the velocity of the water that is disturbed horizontally : the motion parallel to the horizontal by which the parts of water falling may approach in turn will not be considered here, since it does not arise from gravity, nor may the motion arising from gravity perpendicular to the horizontal change. Indeed we may suppose that the parts of the water adhere just a little, and by it cohesion in falling approach each other through a motion parallel to the horizontal, so that they may form only a single stream downwards and may not be divided into several: but we shall not consider here the motion parallel to the horizontal arising from that cohesion. Case 1. Consider now the whole cavity in the vessel, in the circulation of the falling water ABNFEM, to be filled with ice, so that the water may only pass through the ice as by a funnel. And if the water may hardly touch the ice, or what amounts to the same thing, if yet it may touch and on account of its great smoothness may slide freely and without any resistance, the water may run down through the opening EF with the same velocity as



at first, and the whole weight of the column of water ABNFEM may be pressing on the flow of this being produced as at first, and the bottom of the vessel will sustain the weight of the surrounding column of ice. Now the ice in the vessel may liquefy; and the efflux of the water will remain the same as at first. It will not be less, because ice dissolved in water is trying to descend : nor greater, because ice dissolved in water cannot descend unless by impeding the descent of water descending equally to itself. The same force must generate the velocity of the flowing water. But the aperture at the bottom of the vessel, on account of the oblique motion of the particles of water flowing out, must be a little greater than at first. For the particles of water do not all pass through the opening perpendicularly, but flow together from each side of the vessel and converge on the opening, and they pass through with oblique motions, and in the course of their downwards motion they conspire in the stream of water by bursting forth, which is a little smaller below the hole than at the hole itself, with the diameter of this present to the diameter of the hole as 5 to 6, or 1

25 to 126 as an approximation, but

only if I have measured the diameters correctly. Certainly I had obtained a thin plane sheet pierced by a hole in the middle, with the diameter of the circular hole present of five eights of an inch. And so that the stream of water bursting forth might not be accelerated by falling and by the acceleration rendered narrower, thus I fastened this sheet not onto the base but onto the side of the vessel, so that that stream may emerge along a line parallel to the horizontal. Then when the vessel should be full of water, I uncovered the hole so that the water could flow out; and the diameter of the stream produced of

th 2140 of

an inch as accurately as could be measured, at a distance around half an inch from the opening. Therefore the diameter of the circle of this hole to the diameter of the stream was as 25 to 21 approximately. Therefore the water by passing through the hole, converges on all sides, and after it has flowed out from the vessel, is rendered narrower by converging, and it is accelerated by the attenuation on arriving at a distance of half an inch from the hole, and with that narrower at that distance it shall be faster than at the hole itself in the ratio 25 25 to 21 21× × or approximately as 17 to 12, that is around the ratio of the square root of 2 to 1. [Thus, from the continuity equation, 1 1 2 2A v A v× = × , and the ratio of the velocities

2 1

1 2

625441 1 42 2v A

v A . ...= = = , where A1 and A2 are the areas of cross-section.]

Indeed it is agreed by experiment that the quantity of water, which flows out in a given time through the circular hole made in the bottom of the vessel, shall be with the predicted velocity, not only through that hole, but it must flow also in the same time through the circular hole, the diameter of which is to the diameter of this hole as 21 to 25. And thus that water flowing has a velocity downwards at this [actual] hole describing approximately by the falling of a weight through half the height of the water at rest in the vessel. [Translator's italics and underlining here and below.] But after it has escaped from the vessel, it may be accelerated by converging until it arrives at a distance from the



opening nearly equal to the diameter of the hole, and it will have acquired a velocity greater nearly in the ratio of the square root of two, than certainly in this case it can acquire a velocity described approximately by a weight falling the whole height of the water at rest in the vessel. Therefore in the following the diameter of the stream may be designated by that smaller hole that we have called EF. And another superior plane VW is understood to be drawn parallel to the plane of the opening EF at a distance approximately equal to the diameter and with a greater hole ST bored through, through which certainly the stream falls, which completely fills the lower hole EF, and thus the diameter of this shall be to the diameter of the lower opening approximately as 25 to 21. For thus the stream will be able to cross over perpendicularly from the lower opening ; and the quantity of water flowing out, for the size of this hole, will be as the solution of the problem postulated approximately. Indeed the distance, that is enclosed by the two planes and by the stream falling, can be considered as the bottom of the vessel. But so that the solution of the problem shall be simpler and more mathematical, it may be agreed to take only the lower plane for the base of the vessel, and to imagine that the water either flowing through the ice or through the funnel, and escaping from the vessel at the opening made in the lower plane EF , perpetually maintains its motion, and the ice remains at rest. In the following therefore let ST be the diameter of the circular opening Z described through which the stream flowed from the vessel when all the water in the vessel is fluid. And EF shall be the diameter of the opening through which by falling the stream may adequately pass through, either the water may exit from the vessel through that upper opening ST, or it may fall through the middle of the ice in the vessel as if it were through a funnel. And let the diameter of the upper opening ST be to the diameter of the lower EF as around 25 to 21, and the distance of the perpendicular between the planes of the openings shall be equal to the diameter of the smaller opening EF. And the velocity of the water escaping from the vessel by the opening ST there will be in the opening itself as the velocity that a body can acquire by falling from a height of half IZ : but the velocity of each stream by falling in the opening EF there will be as a body would acquire in falling from the whole height IG. [Thus we have 2 2 2v u gs= + , which essentially is an energy conservation equation where unit mass is falling ; Newton considers that the motion of the water through EF is not vertically downwards, and the average velocity downwards corresponds to the water falling a height equal to half the height of the column; however, the column narrows, really by surface tension forces, so that the area of cross-section diminishes by 2 , and at this point the water is considered to be flowing downwards only, and a drop can be considered to have fallen the whole height of the column, approximately; the extra distance being ignored. It is convenient here to consider the actual shape of the ice funnel proposed by Newton: as developed by L. & J. in Note 272 : With these items in place, the geometrical figure of the cataract is readily defined. Let MN cut the axis IG at P; and because the altitude IP is in the square ratio of the velocity at P, truly this velocity is inversely as the circle MN, and thence the circle MN is in the square ratio of the radius MP, and thus IP or the abscissa is in the inverse fourth power of the radius or of the ordinate MP, or 4

1MP

IP ∝ , and 4MP IP× a given quantity. (Thus,



the Galilean proportionality 2h v∝ is combined with the continuity equation

2 421 1 1 to give v A r

A h v∝ ∝ ∝ ∝ )

Therefore the curve EMA is a hyperbola of the fourth order, having the asymptotes IG, IK, to which convexity it turns towards. The arc EMA and the asymptotes IK may be produced indefinitely towards the parts X, and the figure EAXXIG describes the cataract, rotated around the asymptote or the axis IG, produced indefinitely to the parts X, x ; truly the figure EMAHG will be generated, that part of the cataract which is contained within the vessel ABDC.] Case 2. If the opening EF shall not be in the middle of the base of the vessel, but the base may be perforated in some manner : the water may flow out with the same velocity as before, but only if the hole shall have the same size. For indeed a weight descends in a longer time through the same depth along an oblique line than along the perpendicular line, but in falling it acquires the same velocity in each case, as Galileo has shown. Case 3. The velocity of water is the same flowing out from a hole in the side of the vessel. For if the opening shall be small, so that the interval between the surfaces AB and KL may be considered to vanish, and the jet of water springing forth horizontally will form the a parabolic figure: from the latus rectum of this parabola it may be deduced, what that velocity of the water flowing out from the water at rest in the vessel shall be, as a body may be able to acquire by falling with a height HG or IG. Certainly with that experiment done I found that, if the height of the water at rest above the opening should be twenty inches and the height of the opening above a plane parallel to the horizontal should also be twenty inches, the stream of water streaming out would fall on that plane at a distance of around 37 inches from the perpendicular that may be taken in that plane from the opening. For without resistance, the jet would have been incident in that plane at a distance of 40 inches, with the latus rectum of the parabolic jet arising of 80 inches. [Note from L & S : A drop of water from the location D, may gush out along some horizontal direction DT with that velocity which it can acquire by falling through the (half the height of the vessel) BD, and being borne by the resistance of the medium, may describe the parabola DNZ, the vertex of which D, the tangent DT, and diameter DH or with the vertical line BD produced, the abscissa DH may be taken equal to the height BD, and the ordinate HZ may be drawn, which will be parallel to the tangent DT; and in that time taken for the drop of water to fall through the height BD or DH under gravity, it will describe the length HZ of BD or DH squared. The latus rectum DNZ of the parabola pertaining to the diameter DH



is 2HZ

DH , and thus since there shall be 2 2HZ DH BD,= = the latus rectum is 4BD:( i.e. 2 4y ax= ). Therefore the height BD that the water must

describe by falling as the velocity it may acquire with that bursting out from the place D, is the fourth part of the latus rectum pertaining to the diameter DH of the parabola DNZ.] Case 4. Truly the water flowing out, also if it were carried upwards, emerges with the same velocity. For the small jet of water streaming out rises in a perpendicular motion to the height GH or GI of the water at rest in the vessel, except in as far as its ascent may be impeded a little by air resistance ; and hence that flows out with the velocity that it may be able to acquire by falling from that height. Each particle of the water at rest is pressed equally on all sides (by Prop. XIX, Book. 2.) and by conceding to the pressure to all the parts there is imparted an equal impetus, either it may fall through a hole in the bottom of the vessel, or may flow out horizontally through a hole in the side of this, or it may go out along a pipe and thence ascent through a small hole made in the upper part of the pipe. An the velocity by which the water flows is that, as we have assigned in this proposition, not only is it deduced from reasoning, but also it is shown by the well known experiments now described. Case 5. The velocity of the water emerging is the same whether the shape of the opening D shall be circular or square or triangular or some other equal to the circular shape. For the velocity of the water emerging does not depend on the figure of the opening but arises from the height of this below the plane KL. Case 6. If the lower part of the vessel ABDC may be immersed in still water, and the height of the still water above the bottom of the vessel shall be GR: the velocity with which the water in the vessel may flow out through the hole EF into the still water, will be as that which the water can acquire describing by falling in that case the height IR. For the weight of all the water in the vessel which is below the surface of the still water, will be sustained in equilibrium by the weight of the still water, and thus the motion of the water descending in the vessel will be accelerated less. In this case it will also be apparent by experiments, evidently by measuring the time in which the water flowed out. Corol. 1. Hence if the height of the water CA may be produced to K, so that there shall be AK to CK in the square ratio of the area of the opening made in some part of the bottom, to the area of the circle AB : the velocity of the water flowing out will be equal to the velocity that the water is able to acquire by falling and by describing the height KC in that case. [Thus, we apply the continuity equation to a circle of water at AB, and to the water at the opening EF: The speed acquired by the water falling twice the height AK may be called vA , and the speed at EF may be called vE, then

2 2 4 4 and A Ev AB v EF AK AB GI EF× = × × = × .]



Corol. 2. And the force, by which the whole motion of the water streaming forth can be generated, is equal to the weight of the cylindrical column of water, the base of which is the opening EF, and the height 2GI or 2CK. For the water rushing out is able to acquire its emergence velocity, to which in time that column may be equated, by its own weight falling from a height of GI. [Thus, the force is taken to be this weight of water in modern terms 2 2EF g GIπ ρ× × × . The actual mechanism for the acceleration of the water in the opening to form the jet emerging is not explained, but we are to accept the final speed of emergence in modern terms, correcting for the vena contracta, as being

2g GI× .] Corol. 3. The weight of all the water in the vessel ABDC is to the part of the weight, which is used in the out flowing of the water, as the sum of the circles AB and EF to the double of the circle EF. For if IO is the mean proportional between IH and IG; and the water emerging through the opening EF, in the time that a drop falling from I may be able to describe the height IG, will be equal to a cylinder the base of which is the circle EF and the height is 2IG, that is, to a cylinder whose base is the circle AB and height is 2IO, for the circle EF is to the circle AB in the square root ratio of the height IH to the height IG, that is, in the simple ratio of the mean proportional IO to the height IG: and in which time the drop by falling from I can describe the height IH, the water passing out will be equal to the cylinder of which the base is the circle AB and the height is 2IH : in which time the drop by falling from I through H to G will describe the difference of the heights HG, the water emerging, that is, all the water in the figure ABNFEM will be equal to the difference of the cylinders, that is, to a cylinder of which the base is AB and the height 2HO. And therefore the total water in the vessel ABDC is to the total water fallen in the figure ABNFEM as HG to 2HO, that is, as HO OG+ to 2HO, or IH IO+ to 2IH. But the weight of all the water in the figure ABNFEM is expended in the out flowing of the water: and hence the weight of all the water in the vessel is to the part of the weight which is implemented in the out flowing of the water, as IH IO+ to 2IH, and thus as the sum of the circles EF and AB to twice the circle EF. [Enlarged note from L. & J. : Thus, the same velocity of the efflux arises from the whole circle AB and the height 2IO as from the circle EF and the height 2IG, or what amounts to the same, cylinders which have these equal volumes. Again, the same amount of water passes through the circles AF and EF in the same time, and the amount of water passing through AB will be in the time that a drop falls through the height IH, equal to the volume of a cylinder of water of which the base is the circle AB and the height 2IH. We may add to this, inverted, that E

A

vArea AB IGArea EF v IH

= = .



Now, if or IG IOIO IHIO IG IH= × = then E

A

vArea AB IG IG IOArea EF v IO IHIH

= = = = . Thus, in the

time a drop falls the distance IH, the jet emits a volume of water equal to 2the area AB IH× , due to the waist narrowing; then by simple proportion, in the time a

drop falling from I to G via H describes the difference of the heights GH, that is the whole volume of water in the ice funnel ABNFEM, the jet will emit a volume equal to the difference of the (whole) cylinders, that is, a cylinder with base AB and height 2HO. Hence, the total amount of water in the vessel is to the water escaped through the ice funnel, as HG to 2HO. The volume of water contained in the vessel ABDC is equal to the capacity of the vessel or cylinder the base of which is the circle AB, and the height HG; and therefore the total water in the vessel ABCE, is to the total water falling in the solid ABNFEM, as HG to 2HO, that is, as 2

HO OGHO+ , and because by hypothesis:

2 2

22 2 2 2 2

,

there is = = .]

Area EF IO IO IH HOIHArea AB IO IG IG IO OG

Area AB Area EFHG HO OG IH IO AB EFHO HO IH Area EF EF

−−

++ + +× ×

= = = =

= =

Corol. 4. And hence the weight of all the water in the vessel ABDC is to the part of the weight that the base of the vessel sustains, as the sum of the circles AB and EF to the difference of the same circles. [The weight of all the water in the vessel ABCD shall be P, the part of that weight which is involved in the efflux of the water shall be p, and hence the part P p− of the whole weight or clearly equal to the difference of the circles CD and EF which is sustained by the bottom of vessel and is not involved in the efflux. And, by Cor. 3, there will be

2 2 2 2

2 2 22 and hence .P AB EF P AB EF

p P pEF AB EF+ +

− −= = ]

Corol. 5. And the part of the weight that the base of the vessel sustains, is to the other part of the weight, which is implemented in the out flowing of the water, as the difference of the circles AB and EF to twice the smaller circle EF, or as the area of the base to twice the aperture. [ Since

2 2 2 2

2 22 2, also .P pP AB EF AB EF

p pEF EF−+ −= = ]

Corol. 6. But the part of the weight, by which the base only may be acted on, is to the whole weight of the water, which will press perpendicularly on the base, as the circle AB to the sum of the circles AB and EF, or as the circle AB to the excess of twice the circle AB above the base. For the part of the weight, by which the base may be acted on alone, is to the weight of the whole water in the vessel, as the difference of the circles AB and EF to the sum of the same circles by Cor. 4 : [As above,

2 2 2 2 2

2 2 2 22

2 2 and hence P p pP AB EF AB EF EF

p p PEF EF AB EF−+ − ×

× += × = × , giving

2 2

2 2P p AB EF

P AB EF− −

+= and

2 2 22

2 2 2 2 or pPP AB EF AB

P p P pAB EF AB EF−+

− −− −= = and

22 2

2 2

p pP PP p ABP p P P AB EF− −−− +

× = = ]



and the weight of all the water in the vessel is to the weight of all the water which presses normally on the base, as the circle AB to the difference of the circles AB and EF. And thus from the rearranged equation, the part of the weight, by which the base alone is acted on, is to the weight of all the water, which presses normally on the base, as the circle AB to the sum of the circles AB and EF, or the excess of twice the circle AB above the base. Corol. 7. If a small circle PQ may be located in the middle of the hole EF described with centre G and parallel to the horizontal : the weight of water which that circle shall sustain, is greater than the weight of a third part of the cylinder of water the base of which that circle and the height is GH. For let ABNFEM be the cataract or column of water falling having the axis GH as above, and as it is understood all the water in the vessel has become frozen, as long as the most immediate and fastest may not be required in the circulation of the cataract above the circle. And let PHQ be the column of water frozen above the circle, having the vertex H and the height GH. And consider the cataract both to fall by its own total weight, and not in the least to encumber or press upon PHQ, but to slide past freely and without friction, except perhaps at the vertex of the ice itself from which the cataract itself may begin to fall in the cavity. And in whatever manner the frozen water in the circuit of the cataract AMEC, BNFD is convex on the internal surface AME, BNF falling towards the cataract, thus also this column PHQ will be convex towards the stream, and therefore greater than for a cone the base of which is that circle PQ and with the height GH, that is, greater than a third part of the cylinder with the same base and with the height described. But the circle may sustain the weight of that column, that is, the weight which is greater than the weight of the cone or a third part of the cylinder. [Thus, the weight of water sustained by the small circle is greater than the weight of the cone, given by 1

3 Ch. gρ in an obvious notation with C the area of the small circle, while in the following, the weight is again less than the weight of a semi-spheroid, given by 2

3 Ch. gρ . If the circle is very small, then the weight supported can be taken as he

arithmetic mean, that is 12 Ch. gρ (Cor. 8 & 9 following.) ; this leads on in the following

to an examination of the frictional force on a horizontal circle in a stream.] Corol. 8. The weight of water that a very small circle PQ can sustain, may be seen to be less than two thirds of the cylinder of water the base of which is that circle and the altitude is HG. For with everything now remaining in place, it may be understood to have described half of a spheroid, the base of which is that circle and the semi axis or height is HG. And this figure will be equal to two third parts of that cylinder and it may be understood that the circle may sustain the weight of this frozen column of water PHQ. For as the motion of the water shall be mainly straight [down], the surface of that column may meet externally with the base PQ in some acute angle, therefore as with the water falling it always will be accelerated, and on account of the acceleration it shall become narrower, and since that angle shall be less than a right angle, this column will be laid within the half spheroid in the lower parts of this. Truly the same upwards will be acute or pointed, lest the horizontal motion of the water at the vertex of the spheroid shall be infinitely faster than the motion of this towards the horizontal. And so that the smaller the circle PQ becomes the more acute the vertex of the column becomes to that ; and with the tiny circle diminished indefinitely, the angle PHQ will be diminished indefinitely, and therefore that



column will be placed within the half spheroid. Therefore that column is less than the half spheroid, or than two thirds parts of the cylinder of which the base is that small circle, and with the altitude GH. But the small circle sustains the force equal to the weight of the water of this column, since the weight of the water around is employed in the outflow of this water. Corol 9. The weight of water that a very small circle PQ may sustain, is equal to the weight of the cylinder of water the base of which is that circle and the height is 1

2 GH approximately. For this weight is the arithmetical mean between the weights of the cone and of the aforementioned hemisphere. But if that circle shall not be very small, but may be increased until it may equal the opening EF ; here it will be the weight of all the water itself overhanging perpendicularly, that is, the weight of the cylinder of water of which the base is that circle and the height is GH. Corol. 10. And (as far as I know) the weight that the little circle sustains is always to the weight of the cylinder of water, the base of which is that little circle and of which the height is 1

2 GH , as EF2 to 12

2 2EF PQ− , or as the area of the circle EF to the excess of this circle above half the area of this circle of this circle PQ, as an approximation. [L & S note : For this supposition satisfies the above requirements : For if p shall be the weight of water sustained by the small circle, and P the weight of the cylinder of water sustained by the small circle and with the height GH; and if there is put

2121 1

2 2 2 212

2 2 2: : then P EFEF PQ

p P EF EF PQ p .×−

= − = But the quantity 2

2 22P EF

EF PQ×−

is always

greater than 13 P , as it may satisfy Cor. 7, and likewise it can be shown always to be less

than 23 P .

B. & R. proceed here as follows: the weight of the water on the little circle is 12P Ch. gρ= , from the above average, if the area of the circle is very small; however, it is

comparable to the cross-sectional area B of the pipe EF, we may assume this weight takes the form 1 C

BP Ch gβ

α ρ−= , then when C is very small with respect to B, this reverts to the

above result, so that 12β = , and when C B= , the weight supported is that of a cylinder

with base C and height h, so that also 12α = ; hence, while C is less than 2

B , the expression 1

212

BB CP Ch gρ−= × makes the weight lie between the given limits above.

There is an analogy between the force due to water running out through a hole, and the resistance experienced by a body travelling at a constant rate through a medium. The resistance Newton has in mind here is of the dynamic kind, due to the particles of the medium rebounding from the moving surface, rather than due to the adhering nature of the medium. If u is the velocity at the surface, v that at the orifice, A the area of the cross-section of the base, B and C that of the orifice and the little circle PQ, and h the height of the cylinder. Then ( )2 2 2 and v u gh v B C uA= + − = ; the former we may now think of as an energy conservation equation, while the latter is a continuity equation. Now if A is



much greater than B, then 2 2v gh= and the weight acting on the small circle is 12P Ch gα ρ= × , where 1 when 0C

Bα → → , in any manner. 24

CvP α ρ∴ = × × . Now let the opening of the pipe EFST be closed, and let the small circle ascend with such a velocity that the relative motion of the circle and fluid compelled to flow past it is the same as before; hence the weight or force acting on the small circle will be the same as before; now the velocity of the fluid will be Cv

B C− and that of the plane moving through it Bv

B C− . If we imagine that B is infinitely greater than C, then the resistance of a plane

moving in still water with a velocity v will be 24

CvP ρ∴ = × . The resistance depends only on the area of the great circle, and according to this, a sphere. a spheroid, and a cylinder will offer the same resistance. As many commentators have already indicated, there are many flaws in these arguments; the principal one being that water does not flow out of a hole in a vessel as Newton had envisaged! Nevertheless, the arguments are intriguing, and there may be situations where a fluid obeys these rules. In any case, Newton went about this analysis in order that he could examine the resistance to the motions of projectiles, falling bodies, and pendulums, in resisting mediums, that follow.]

LEMMA IV. The resistance of a cylinder, which is progressing along its own length uniformly, does not change if the length of this may be increased or diminished ; and thus it is the same as the resistance of a circle described with the same diameter, and with the same velocity of progression along a right line perpendicular to its plane. For the sides of the cylinder by the motion of this are minimally opposed : and the cylinder, with the length of this diminished indefinitely, is turned into a circle. [Thus Newton solves the simpler problem where the resistance is purely dynamic, and ignores viscous effects in mediums, where the length would be important.]

PROPOSITION XXXVII. THEOREM XXIX. The resistance of a cylinder which arises from the magnitude of the transverse section, which is progressing uniformly along its length in a fluid compressed infinitely and non-elastic, is to the force by which the whole motion of this, while meanwhile it describes four times its length may be either removed or generated, as the density of the medium to the density of the cylinder approximately. For if the vessel ABDC may touch the surface of the still water with its base CD, and water may flow from this vessel by the cylindrical pipe EFTS perpendicular to the horizontal into still water, and moreover the small circle PQ may be located parallel to the horizontal somewhere in the middle of the pipe, and CA may be produced to K, so that AK shall be to CK in the squared ratio that the excess of the opening of the pipe EF over the circle PQ has to the circle AB: it is evident (by Cases 5 and 6, & Cor. 1. Prop XXXVI) that the velocity of the water passing through the annular space between the small circle



and the side of the vessel, that will be as that velocity the water can acquire by falling, and in that case by describing the height KC or IG. And (by Corol. X, Prop. XXXVI.) if the width of the vessel is infinite, so that the small line HI may become vanishing and the altitudes IG and HG are equal: the force of the water flowing down on the circle will be to the weight of the cylinder whose base is that small circle and the height is 1

2 IG , as

EF2 to 12

2 2EF PQ− approximately. For the force of the water, by flowing with a uniform velocity through the whole pipe, will be the same at the location of the circle PQ as in any part of the pipe. Now the openings of the pipe EF and ST may be closed, and the small circle may ascend in the fluid compressed on all sides and by itself ascending it may force the water above to fall through the annular space between the small circle and the side of the pipe : and the velocity of the ascending small circle will be to the velocity of the water descending as the difference of the circles EF and PQ to the circle PQ, and the velocity of the ascending little circle to the sum of the velocities, that is, to the relative velocity of the water descending which flows past the ascending circle, as the difference of the circles EF and PQ to the circle EF, or as 2 2 2 to EF PQ EF− .

[For : ( )2 2 2asc. desc.v PQ v EF PQ× = × − and

2 2 2 2 2 2 22

2 2 2 2 2, andasc . desc . asc . rel . desc . asc . asc .

desc . desc . desc . rel . desc . rel .

v v v v v v vEF PQ EF PQ PQ EF PQEFv v v v v vPQ PQ PQ EF EF

+− − −= = = × = = × = ]

Let that relative velocity be equal to the velocity, which has been shown above to pass through the same annular space while the small circle meanwhile may remain at rest, that is, to the velocity that the water can acquire by falling and in that case by describing the altitude 1G : and the force of the water in the ascending small circle will be the same as before (by the rule of Corol. V.) that is, the resistance of the ascending circle will be to the weight of the cylinder of water whose base is that small circle and the height is 1

2 IG , as 2EF to 1

22 2EF PQ− approximately. But the velocity of the small circle will be to the

velocity that the water acquires by falling and in that case be describing the altitude IG, as 2 2 2 to EF PQ EF− . The size of the pipe may be increased indefinitely : and these ratios between

2 2 2 and EF PQ EF− , and between 12

2 2 2 and EF EF PQ− finally will approach to ratios of equality [, as EF >>PQ]. And therefore the velocity of the small circle will now be that as can be acquired by the water descending in that case from the altitude described IG, and truly the resistance of this will emerge equal to the weight of the cylinder whose base is that small circle and the altitude is half of the altitude IG, from which the cylinder must fall as the velocity of the ascending small circle may acquire ; and the cylinder with this velocity, in the time of falling, will describe four times its own length. [As the force is proportional to the velocity squared, then the cylinder must fall a distance



( )122 4IG IG× = × × .] But the resistance of the cylinder progressing along its length with

this velocity, is the same as the resistance of the small circle (by Lemma IV.) and thus is equal to the force by which the motion of this, as long as the quadruple of its length shall be described, is able to be generated approximately. If the length of the cylinder may be increased or diminished : so that the motion of this and the time in which it will describe four times its length, will be increased or diminished in the same ratio; and thus that force will not be changed, by which the increased or diminished motion, equally increased or decreased in the time, may be able to be generated or taken away, and thus even now is equal to the resistance of the cylinder, and since that too remains unchanged by Lemma IV. If the density of the cylinder may be increased or diminished : so that the motion of this, and the force by which the motion can be generated or removed in the same time, will be increased or diminished in the same ratio. And thus the resistance of the cylinder of any kind will be to the force by which the whole of its motion, as the fourfold of its length meanwhile will be described, may be generated or removed, as the density of the medium to the density of the cylinder approximately. Q.E.D. But a fluid must be pressed together so that it shall be continuous, truly it must be continuous and not elastic so that all the pressure, which arises from the compression of this, may be propagated in an instant, and in the motion all the parts of the body acted on equally may not change the resistance. Certainly the pressure, which arises from the motion of the body, is impeded in the motion of the parts of the fluid being generated and may create the resistance. But the pressure which arises from the compression of the fluid, however strong it may be, if it may be propagated instantaneously, shall generate no motion in the parts of the fluid, and in general it will lead to no change of the motion ; and thus the resistance shall neither be increased or diminished. Certainly the reaction of the fluid, which arises from the compression of this, cannot be stronger in the rear parts of the of the moving body than in the front parts, and thus the resistance described in this proposition cannot be diminished : and cannot be stronger in the front parts than in the latter parts, but only if the propagation of this may be infinitely faster than the motion of the compressed body. But it will only be infinitely faster and propagate instantaneously if the fluid shall be continuous and not elastic. Corol. 1. The resistances of cylinders, which are progressing uniformly along their lengths in infinitely continuous mediums, are in a ratio composed from the square of the ratio of the velocities and in the square ratio of the diameters and in the ratio of the densities of the mediums. Corol. 2. If the width of the pipe may not be increased indefinitely, but the cylinder may be progressing along its length in an enclosed medium at rest, and meanwhile its axis may coincide with the axis of the pipe : the resistance of this will be to the force by which the whole motion of this, in the time it will describe the quadruple of its length, either generated or removed, in a ratio which is composed from 1

22 2 2 to EF EF PQ− once

[this factor as explained above, see Cor. 10, expressed the force on the small circle



compared to the force on the cylinder with the same small circular base and height 12 IG

as equal to the ratio 12

2 2 2 to EF EF PQ− , and to the ratio 2 2 2 and EF EF PQ− squared [i.e. the velocity squared], and in the ratio of density of the medium to the density of the cylinder.] Corol. 3. With the same in place, and since the length L shall be to the quadruple of the length of the cylinder in a ratio which is composed from the ratio 1

22 2EF PQ− to EF2

once, and in the ratio 2 2 EF PQ− to EF2 squared : the resistance of the cylinder will be to the force, either to be taken away or generated, by which the whole motion of this, while the length L meanwhile will be described, as the density of the medium to the density of the cylinder.

Scholium.

In this proposition we have investigated the resistance which arises from the magnitude of the transverse section of the cylinder only, with the part of the resistance ignored which may arise from the obliquity of the motions. For just as in the first case of Proposition XXXVI, the obliquity of the motions by which the parts of the water in the vessel certainly converged on the hole EF, impeding the efflux of this water through the hole : thus in this proposition, obliquity of the motion, in which the parts of the water compressed by the front part of the cylinder concede to the compression and diverge on every side, retards the transition of these from the places at its anterior end by circulating to the posterior parts of the cylinder, and effects that the fluid will be moved together at a greater distance and will increase the resistance, and that almost in the ratio by which the efflux of the water from the vessel may be diminished, that is, in around the ratio 25 to 21. squared. And in the same manner, as in the first case of that proposition, we effected that the parts of the water be transferred perpendicularly and with maximum abundance through the hole EF, by putting all that water in a vessel in which the circulation of the stream was frozen, and the oblique motion of this was not used and remained in place without motion : thus in this proposition, so that the obliquity of the motion may be removed, and the parts of the water may allow the passage proceeding most easily by the cylinder in the shortest time with the motion directed mainly forwards, and only the resistance may remain, which arises from the magnitude of the transverse section, and which cannot be diminished except by diminishing the diameter of the cylinder. It is considered that the parts of the fluid, the motions of which are both useless and create resistance, may remain at rest among themselves at each end of the cylinder, and may stick together and be joined to the cylinder. Let ABCD be a rectangle, and both AE and BE shall be two parabolic arcs described with the axis AB, moreover with the latus rectum which shall be to the distance HG, with a cylinder being described by falling while it acquired its velocity, as HG to 12 AB . Also CF and DF shall be two other parabolic arcs, with the axis CD and with the latus rectum which shall be four times the first latus rectum described ; and by the rotation of the figure around the axes EF a solid may be generated the middle part of which shall



be the cylinder ABDC on which we are acting, and the end parts ABE and CDF may contain parts of the fluid at rest amongst themselves and acting on the body as two rigid solid parts, which adhere to the cylinder at each ends as head and tail. And the resistance of the solid EACFDB, following the length of its axis FE progressing in the direction of E, will be as nearly as we have described in this proposition, that is, which has that ratio to the force which the whole motion of the cylinder, while the whole length 4AC meanwhile may be described by that motion continued uniformly, that may be taken away or generated, that the density of the fluid has to the density of the cylinder approximately. And the resistance of this force cannot be less than in the ratio 2 to 3, by Corol. 7, Prop. XXXVI.

LEMMA V. If a cylinder, a sphere, and a spheroid, the widths of which are equal, thus may be placed successively in the middle of a cylindrical pipe so that their axes may coincide with the axis of the pipe: these bodies may equally impede the flow of the water through the pipe. For the spaces through which the water passes between the pipe and the cylinder, the sphere, and the spheroid are equal: and the water passes through equal spaces equally. Thus these themselves are had by hypothesis, because all the water above the cylinder, sphere, or spheroid may be frozen, of which the fluidity is not required for the swiftest passage of the water, as I have explained in Corol. VII, Prop. XXXVl.

LEMMA VI. With the same in place, the aforementioned bodies may be acted on equally by the water flowing in the pipe. It is apparent by Lemma V and the third law of motion. Certainly the water and the bodies act on each equally and mutually.

LEMMA VII. If the water may be at rest in the pipe, and these bodies may be carried in opposite directions with equal velocities, then the resistances of these are equal to each other. This is agreed upon from the above lemma, for the relative motions remain between each other.

Scholium. The ratio is the same of all convex and rounded bodies, the axes of which coincide with the axis of the pipe. Any differences can arise from the greater or lesser resistance, but in these lemmas we suppose the bodies to be very smooth, and the stickiness and friction of the medium to be zero, and because the parts of the fluid, which by their oblique motions and superfluous flow of water through the pipe, are able to disturb, impede, and to retard, are at rest among themselves as if they were restricted by ice, and may be attached to the front and rear parts of bodies, just as I have shown in the scholium



of the preceding proposition. For in the following the smallest resistance of all that round bodies can have is acting, with the greatest transverse cross section described. Bodies swimming on fluids, where they may be moving in a straight direction, effect that the fluid may ascend at the front parts and subside at the back parts, especially if the figures shall be obtuse , and therefore they may feel the resistance a little greater than if they were with sharp heads and tails. And bodies moving in elastic fluids, if they shall be obtuse before and aft, they may compress the fluid a little more at the front part and be a more relaxed at the rear part ; and therefore they experience a little more resistance than if they were sharp at the head and tail. But we do not work with elastic fluids in these lemmas and propositions, but with non elastic ones ; not with sitting on the fluid, but with deeply immersed in it. And when the resistance of bodies in non elastic fluids becomes known, this resistance will be increased by a small amount in elastic fluids, such as air, as on the surfaces of fluids at rest, such as seas and marshes.

PROPOSITION XXXVIII. THEOREM XXX. The resistance of a sphere progressing in an infinitely compressed and non elastic fluid is approximately to the force by which the whole motion, in which time a three eights part of its diameter will be described, either taken away or generated, as the density of the fluid to the density of the sphere. For the sphere is to the circumscribed cylinder as two is to three ; and therefore that force, which may be able to remove all the motion, while the cylinder meanwhile may describe a length of four diameters, all the motion of the sphere meanwhile may be taken away while the sphere describes two third parts of this length, that is, eight third parts of its own diameter. But the resistance of the cylinder is to this force approximately as the density of the fluid to the density of the cylinder or sphere by Prop. XXXVII, and the resistance of the sphere is equal to the resistance of the cylinder by Lem. V, VI, VII. Q.E.D. Corol. I. The resistances of spheres, in infinitely compressed mediums, are in a ratio that is composed from the square ratio of the velocities, and in the square ratios of the diameters, and in the ratio of the densities of the mediums. Corol. 2. The maximum velocity by which a sphere, by a force to be compared with its own weight, is able to descend in a resisting fluid, is that which the sphere likewise can acquire, by the same weight, by falling without resistance and in that case by describing a distance which shall be to four thirds parts of its diameter as the density of the sphere to the density of the fluid. For the sphere in the time of its own case, with the velocity of falling acquired, describes a distance which will be as eight thirds of its diameter, as the density of the sphere to the density of the fluid ; and the force of this weight generating this motion, will be to the force which may generate the same motion, in that time the sphere will describe eight thirds of its diameter with the same velocity, as the density of the fluid to the density of the sphere : and thus by this proposition, the force of the weight will be equal to the force of the resistance, and therefore the sphere cannot accelerate. Corol. 3. With both the density of the sphere and its velocity at the beginning of the motion given, and as with the density of the compressed fluid at rest in which the sphere



is moving; both the velocity of the sphere and the resistance of this sphere may be given at any time, as well as the distance described by that, by Corol. VII. Prop. XXXV. Corol. 4. A sphere in a compressed fluid at rest with the same density by moving, the half part of its motion will be lost as it describes a length of two of its diameters, by the same Corol. VII.

PROPOSITION XXXIX. THEOREM XXXI.

The resistance of a sphere, progressing uniformly through the compressed fluid in a closed pipe, is to the force, by which the whole of this motion, while it will describe meanwhile the eight third parts of its diameter, either may be generated or removed, in a ratio which is composed from the ratio of the openings of the pipe to the excess of the opening over half the great circle of the sphere, and in the ratio doubled of the opening to the excess of this ratio over the great circle of the sphere, and with the density of the fluid to the density of the sphere approximately. This is apparent by Corol. 2. Prop. XXXVII, and indeed the demonstration proceeds as in the preceding proposition.

Scholium.

In the two most recent demonstrations (in the same manner as in Lem. V.) I suppose that all the water which precedes the sphere may be turned to ice, and its fluidity increases the resistance of the sphere. If all that water may become liquid, the resistance will be increased a small amount. But that increase will be small in these propositions and can be ignored, provided that the whole convex surface of the sphere almost serves to be made of ice.

PROPOSITION XL. PROBLEM IX. To find the resistance by phenomena, of a sphere progressing in a most compressed fluid

medium. Let A be the weight of the sphere in a vacuum, B its weight in a resisting medium, D the diameter of the sphere, F the distance which shall be to 4

3 D as the density of the sphere to the density of the medium, that is, as A to A B− , G the time by which the sphere with the weight B by falling without resistance will describe the distance F, and H the velocity that the sphere acquires in this case itself. And H will be the maximum velocity by which the sphere, by its weight B, can fall in the resisting medium, by Corol. 2. Prop. XXXVIII, and the resistance that the sphere is allowed falling with that velocity, will be equal to the weight B of this: truly the resistance that is experienced with any velocity, will be to the weight B in the square ratio of the velocity of this to that maximum velocity H, by Corol, I. Prop. XXXVIII. This is the resistance that arose from the inertia of the matter of the fluid. Indeed that which arises from the elasticity, the tenacity, and from the friction of the parts of this, thus will be investigated. A sphere may be sent off so that it may descend in the fluid by its own weight B ; and P shall be the descent time, and that may be had in seconds, if the time G may be given in



seconds. The absolute number N may be found which agrees with the logarithm 20 4342944819 PG, , and let L be the logarithm of the number 1N

N+ ; and the velocity

acquired by falling will be 11

NN H−+ , but the height described will be

2 1 3862943611 4 605170186PFG , F , LF− + . If the fluid may be of sufficient depth, the term

4 605170186, LF can be ignored; and 2 1 3862943611PFG , F− will describe the height

approximately. These are shown by ninth proposition of the second book and its corollary, from the hypothesis that the sphere experiences no other resistance except for that which arises from the inertia of the matter. For if another resistance may be experienced above, the descent will be slower, and from the retardation the amount of this resistance may become known. In order that the velocities of a body falling in a medium may become known more easily, I have composed the following table, the first column of which may denote the times of the descent, the second shows the velocities acquired by falling with the maximum velocity present 100000000, the third shows the distances described by falling in these times, with the distance 2F that the body will describe in the time G with the maximum velocity, and the fourth shows the distances in the same times described with the maximum velocity. The numbers in the fourth column are 2P

G , and by subtracting the number 1 3862944 4 6051702 , , L− , the numbers in the third column are found, and these numbers are multiplied by the distance F so that the distances described by falling may be obtained. To these above a fifth column is added, which contains the distances described in the same times by a body falling in a vacuum, to be compared with the force of its own weight B. . Times

P. Velocities

falling in fluid. Distances

described falling in fluid.

Distances described at the

maximum velocity.

Distances described falling

in a vacuum. 0,001G 29

3099999 0 000001, F 0,002F 0,000001F 0,01G 999967 0,0001F 0,02F 0,0001F 0,1G 9966799 0,0099834F 0,2F 0,01F 0,2G 19737532 0,0397361F 0,4F 0,04F 0,3G 29131261 0,0886815F 0,6F 0,09F 0,4G 37994896 0,1559070F 0,8F 0,16F 0,5G 46211716 0,2402290F 1,0F 0,25F 0,6G 53704957 0,3402706F 1,2F 0,36F 0,7G 60436778 0,4545405F 1,4F 0,49F 0,8G 66403677 0,5815071F 1,6F 0,64 F 0,9G 71629787 0,7196609F 1,8F 0,81F 1G 7615 9416 0,8675617F 2F 1F 2G 96402758 2,65 00055F 4F 4F 3G 99505475 4,6186570F 6F 9F 4G 99932930 6,6143765F 8F 16F 5G 99990920 8,6137964F 10F 25F



6G 99998771 10,6137179F 12F 36F 7G 99999834 12,6137073F 14F 49F 8G 99999980 14,6137059F 16F 64F 9G 99999997 16,6137057F 18F 81F 10G 99999999 3

5 18,6137056F 20F 100F

[Note 284. L. & J. p. 699. So that the demonstration of these things which Newton has presented may be understood easily, some of the matters which he demonstrated in Propositions VIII & IX in Section 2 are to be recalled. Let CH and AB be perpendicular right lines to a given right line AC, indeed with CH infinite, and with

14BA AC= . With centre C and with the asymptotes

CH, CA, a hyperbola BNS may be described through the point B taking AC, AP, AK in continued proportion, and the right line KN is drawn parallel to AB. And if a heavy body may fall from rest in a medium that resists in the square ratio of the velocity, the area ABNK may represent the distance described by the body in falling; and the velocity of the body acquired in this case will be able to be represented by the line AP, and its maximum velocity by the given line AC (by Cor.1 & 2 Prop. VIII). Now BA may be produced to D so that AD AC= , DC may be joined, and with centre D, with the asymptote DC, and with the the principal vertex A, another hyperbola ATZ may be described, which the line DP produced may cut in T, and the line DQ becomes infinitely near to the line DP itself at V; and the vanishing sector

PDQ ACCKDTV ×= , and the sector ATD represents the time in which the falling body

describes the distance ABNK, and by which it acquires the velocity AP (by Case 2, Prop. IX). Truly the distance that the body will describe falling in some time ATD, will be to the distance that the body can describe by progressing uniformly in the same time with the maximum velocity AC, as the area ABNK to the area ATD (by Cor.1. Prop. IX), and the time in which the body by falling in the resisting medium acquires the velocity AP, will be to the time in which the maximum velocity AC in non-resisting medium by the force of its weight by falling in comparison may acquire, as the sector ATD to the triangle ADC (by Cor.5. Prop. IX). Note 285. L. & J. With these presumed, there may be called

14AC AD a,AB a,AP x,PQ dx.= = = = = The hyperbola SNB with the origin at A, is given

by 14a

x ay . += ; while the hyperbola ZTA, with the origin at D, is given by 2 2 2y x a− = .

Because AC APAP AK= ,

2 2 2x a xa aAK ,CK −= = , the triangle 1

2PDQ adx= , and the sector 3 2 21 1 1

2 4 42 2a dx a dx a dxPDQ AC

CK a x a xa xDTV ×

+ −−= = = + ;



[Thus, in modern terms, from the equation ( )2 20

dvdt k v v= − , where 0v represents the

terminal velocity from 20g kv= , on separating the terms and integrating, we have using

different variables, ( )02 2

0 0 000 0

; 2v v vdv dv dv

v v v v v vv vkdt kv t ln +

+ − −−= = + =∫ .] ; hence, on taking the

integral, the sector ATD representing the time is obtained 1 1 14 4 4

2 2 2 a xa xATD a L.a x a L.a x a L. +−= + − − = , to which quantity nothing is required to be

added or subtracted, because there shall be 0 and 0ATD , x= = , vanishing as required. LO may be drawn parallel to KN and infinitely close to it; and since there shall be

2xaAK = , and (by Theorem IV by hypothesis)

3122 2

2 and aCA AB xdxCK aa x

KL×−

= = , the

differential of the area ABNK will be 21

22 2a xdx

a x−= , and with the integrations made, the area

corresponding to the distance 14

2 2 2ABNK Q a L.a x= − − ; indeed because the area ABNK

vanishes when 0x = , the constant Q will become 14

2 2a L.a , and the final area 2

1 1 14 4 4 2 2

2 2 2 2 2 2 aa x

ABNK a L.a a L.a x a L.−

= − − = × .

[In this case, from the equation ( )2 20

dvdsv k v v= − on taking the downwards direction as

positive, this gives 201

22 2 2 20 0 0 0

2 2 and kvvdv dv dv

v v v v v v v vk ds; s ln

− − + −= − = =∫ ∫ ∫ ∫ , with an obvious

change of variables, recalling that 20kv g= , where the terminal velocity is v0, so that k has

the dimensions of time. Back to L. & J.] Again the time P in which the body, by falling in the resisting medium, acquires the velocity of the line AP, or the proportional x is to the time G in which the maximum velocity H can be acquired by the force of its weight, by the weight B falling in comparison without resistance, as the sector ATD to the triangle ADC, that is,

214 1

2212

a xa xa L. a xP

G a xaL.

+− +

−= = .

[For without resistance, the weight B falls under gravity g alone, and in time T it will reach the terminal velocity v0, so that 02

0 0 and vTv gT kv g= = = ; hence

( ) ( )0

0 0

2 1 1 12 2 or ; v va a v

G ka kv v v ka a vka g G t ln P ln+ +− −= = = = → = ; hence 2 a v a xP

G a v a xL. L.+ +− −= = . ]

Whereby there will be 2 a xPG a xL. +

−= , with this logarithm taken with the logistic of which

the subtangent is one. On account of which if the logarithm of the numbers a xa x+− may be

taken from tables, it is done by multiplying by the number 2,302585093 [ 10ln= ], as in Cor. 7 Prop. XXXV, and there will be had 2 2 302585093 a xP

G a x, L. +−= , and thus on dividing

1. by 2,3025..... the number 20 4342944819 PG, × is the logarithm of the number a x

a x+− [to

base 10]. And thus if the absolute number N is sought from tables which agree with the



logarithm 20 4342944819 PG, × , there will be a x

a xN +−= , and thus ( )1

1a N

Nx .−+= But AC to AP

or a to x, shall be as the maximum velocity H to the velocity acquired by falling. Whereby this velocity will be 1

1xH Na N H−

+= × , just as Newton found. The distance the sphere will describe in the time P, progressing uniformly with the maximum velocity H, is to the distance 2F that it can traverse with the same velocity H in the time G, as the time P to the time G (5. Lib. I), and therefore that distance is 2PF

G . The height S that the

sphere will describe in the time P by falling in a resisting medium, is to the distance 2PFG ,

as the area ABNK to the sector ATD, that is, as 2 2 2

1 14 42 2 2

2 2 to or to a a x a x a xa x a xa x a

a L. a L. L. L.+ − +− −−

× × , but from above, a xa xN +−= , and

( )11

a NNx −+= , and hence ( ) ( )2 2

2

2 2 21 1

4 4N N Na

Na x N+ × +

−= = , and if logarithms are taken in the

logistic of which the subtangent is one (i.e. natural logs) 12 2 4a x NP

G a x N L. L.N L. L.+ +−= = + − ; and hence

2

2 2

1

1

2 4 1 22

: 2 4:

1 :1 1 4:1 :NN

a a x Na x Na x

L. L. G N G PFL.N P N P G

L. L. L.N L. L. L.N

L. L. S .+

+ +−−

− +

= + −

= + = + − =

Whereby the altitude 12 4 2 NPF

G NS FL. FL. += − + But if we wish to use log. tables, these are multiplied by the number 2,302585092994 or 2,302585093. Here the number may be called M, the logarithm of the number 4 taken from the tables Q, and the logarithm of the number 1N

N+ shall be L; and there will be

2 2PFGS MQF MLF= − + . But there is 2 4 605170186M , ,= and Q in common tables of

logarithms is 0,60206; or more accurately 0,602059991333, and thus 1 3862943611MQ ,= approximately. Whereby the height S, that the sphere describes in

the time P by falling in the resisting medium, is 2 1 3862943911 4 605170186PFG , F , LF− + ,

as Newton defined. If the distance S that the sphere may fall were so great, that the term 4 605170186, LF could be ignored; then L shall be the logarithm of the number 1N

N+ ,

where N shall be a number so large, or where the number 1NN+ may be almost equal to

one, the logarithm L vanishes approximately. But, if the maximum velocity may be called H, and the velocity V of the sphere is acquired in that time P , there is

and thus a H V a xHV x H V a x N+ +

− −= = = , and when the distance S is large enough, there becomes

V H= approximately, and hence or H VH V N+− a number large enough, as is evident from the

above table; hence the proposition is shown. ]

Scholium.



In order that I might investigate the resistance of fluids by experiments, I prepared a square wooden vessel, with length and width of nine inches inside of English feet, with a depth of nine and a half feet, the same I filled with rainwater; and with spheres formed from wax with lead enclosed, I noted the times of descent of the spheres, with the descent in the height being 112 inches. A volume of an English cubic foot contains 76 pounds Avoirdupois of rainwater, and of this a cubic inch contains 19

36 ounces of weight or 13253

grains ; and a sphere of water of diameter one inch contains 132,645 grains in the medium of air, or 132,8 grains Avoirdupois in a vacuum ; and any other sphere is as the excess of this weight in a vacuum over its weight in water. Expt. 1. A sphere, the weight of which was 1

4156 grains in air and 77 grains in water, described the whole height of 112 digits in a time of 4 seconds. And with the experiment repeated, the sphere again fell in the same time of 4 seconds. The weight of the sphere in a vacuum is 13

38156 grains and the excess of this over the weight of the sphere in water is 13

3879 grains. From which the diameter of the sphere produced is 0,84224 parts of an inch. But as that excess is to the weight of the sphere in a vacuum, hence the density of the water to the density of the sphere, and thus 8/3 parts of the diameter of the sphere (viz. 2,24597 inches) to the distance 2F, that hence will be 4,4156 inches. The sphere by falling in a vacuum in the time of one second with its whole weight of 13

38156 grains, will describe 13193 inches; and with a weight of 77 grains, in the

same time, by falling without resistance in water will describe 95,219 inches; and in the time G, which shall be to one second in the square root ratio of the distance F, or 2,2128 inches to 95,219 inches, it will describe 2,2128 inches, and it will be able to acquire that maximum velocity H to descend in water. Therefore the time G is 0,15244 seconds. And in this time G, with that maximum velocity H, the sphere will describe a distance 2F of 4,4256 inches; and thus in the time of four seconds it will describe a distance of 1l6,1245 inches. The distance 1,3862944F or 3,0676 inches may be taken away, and there will remain a distance of 113,0569 inches that the sphere by falling in water, in the widest vessel, will describe in a time of four seconds. This distance, on account of the aforementioned narrow wooden vessel, ought to be lessened in a ratio that is composed from the square root ratio of the opening of the vessel to the excess of this opening over the greatest semicircle of the sphere, and from the simple of the same opening to the excess of this over the great circle of the sphere, that is, in the ratio 1 to 0,9914. With which done, it will give a distance of 112,08 inches, which the sphere by falling in water in this wooden vessel in a time of four seconds must describe approximately. Indeed by experiment it has described 112 inches. Expt. 2. Three equal spheres, of which the weights themselves were 1

376 grains in air and 1

165 grains in water, were released successively ; and each one fell in a time of 15 seconds, in each case by describing a height of 112 inches. By entering into the computation they produced a weight of the sphere in a vacuum of

51276 grains, the excess of the weight of 17

4871 grains over the weight in water, of a sphere

of diameter 0,81296 inches, 83 parts of this diameter 2,16789 inches; the distance 2F

2,3217 inches; the distance that the sphere with a weight of 1165 grains in a time of 1

second may describe by falling 11,808 inches without resistance, and the time G 0,301056



seconds. Therefore the sphere, with that maximum velocity it can describe in water by the weight of the force 1

165 grains, in the time 0,301056 seconds will describe the distance 2,3217 inches and in the time 15 a distance of 115,678 inches. The distance 1,3862944F or 1,609 inches may be subtracted and the distance114,069 inches will remain which the sphere must be able to describe by falling in the same time in the widest vessel. Therefore the narrowness of our vessel must take away a distance of around 0,895 inches. And thus there will remain a distance of 113,174 inches which the sphere by falling in this vessel, ought to describe in 15 seconds by the theorem approximately. Truly it describes 112 inches by experiment. The difference is insignificant. Expt. 3. Three equal spheres, whose weights were separately 121 grains in air and 1 grain in water, were successively dropped ; and they were falling in water describing heights of 112 inches in the times 46, 47, and 50 seconds. By the theorem these spheres should fall in a time around 40 seconds. Because they have fallen slower, whether for a smaller part of the resistance arising from the force of inertia in slowing the motions, or it is required to attribute a resistance that arises to other causes ; perhaps to some bubbles adhering to the sphere, or to the evaporation of the wax either by the heat or warmth of the season or by dropping the sphere by hand, or even by unknown errors in weighing the spheres in water, I am unsure. And thus the weight of the sphere in water must be of several grains, so that the experiment may be rendered certain and trustworthy. Expt. 4. The experiments described so far I had began so that I could investigate the resistance of fluids, before the theory in the nearby preceding propositions set out by me was known. Afterwards, so that I could examine the theory found, I prepares a wooden vessel with an internal width of 2

38 inches, with a depth of 1315 feet. Then I made four

spheres from wax with lead inside, the individual ones weighing 14139 grains in air and

187 grains in water. And these I released so that I could measure the falling times in water

by a pendulum, oscillating in half seconds. The spheres, when they were being weighed and afterwards were cold, and they remained cold for some time ; because heat evaporated the wax, and by the evaporation diminished the weight of the sphere in water, and the evaporated wax is not at once restored to the former density by cold. Before they fell, they were thoroughly immersed in water; lest with the weight from some parts standing clear from the water might accelerate the descent from the start. And when immersed they become completely still, they were being dropped most cautiously, lest they might accept some impulse from the hand on being dropped. Moreover they fell in the successive times of oscillation 1 1

2 247 48, , 50 and 51, describing a height of 15 feet and 2 inches. But the weather was now a little colder than when the spheres were weighed, and thus I repeated the experiment on another day, and the spheres were falling in the times of 49, 1

249 , 50 and 53 oscillations, and on a third attempt with the times of 1249 , 50, 51 and 53 oscillations. And with the experiment taken more often, the spheres

fell mainly from the times of the oscillations 1249 and 50. When falling slower, I suspect

to be retarded by striking with the sides of the vessel. Now the computation by the theorem being entered into, they produce the weight of the sphere in a vacuum of 2

5139 grains. An excess of this weight over the weight of the



sphere in water of 1140132 grains. The diameter of the sphere is 0,99868 inches. The 8

3 parts of the diameter 2,66315 inches. The interval 2F becomes 2,8066 inches. The distance which a sphere with a weight of 1

87 grains describes in a time of one second by falling without resistance 9,88164 inches. And the time G 0, 376843 seconds. Therefore the sphere, with the maximum velocity by which it can descent in water by a force of a weight of 1

87 , in a time 0,376843 seconds will describe a distance 2,8066 inches, and in a time of 1 second, a distance of 7,44766 inches, and in the time of 25 seconds or of 50 oscillations a distance of 186,1915 inches [in these days an oscillation was the motion of a pendulum from one side to the other, or half the modern period]. The distance 1,386294F, or 1,9454 inches may be taken away, and there will remain 184,2461 inches which the sphere in the same time in the widest vessel. On account of the narrowness of our vessel, this distance may be diminished in a ratio which is composed from the square root ratio of the opening of the vessel and the excess of this opening over the great semicircle of the sphere, and to the simple ratio of this same orifice to its excess over a great circle of the sphere ; and the distance 181,86 inches will be had, which the sphere ought to describe in this vessel in the time of 50 approximately by the theorem. In truth it may describe a distance of 182 inches in a time of 1

249 or 50 oscillations by experiment. Expt. 5. Four spheres with a weight of 1

8154 grains in air and 1221 grains in water are

dropped often, falling in a time of 1 12 228 29 29, , and 30 oscillations, and occasionally of

31, 32 and 33 oscillations, describing heights of 15 feet and 2 inches. By the theorem they ought to fall in a time of 29 approximately. Expt. 6. Five spheres with a weight of 3

8212 grains in air and 1279 grains in water were

dropped a number of times, they were falling in the times of 1215 , 16, 17 and 18

oscillations, describing heights of 15 feet and 2 inches. By the theorem they ought to fall in a time of approximately15 oscillations. Expt.7. Four spheres weigh 3

8293 grains in air and 7835 grains in water were dropped a

number of times, they were falling in the times 1 12 229 30 30, , , 31, 32 and 33 oscillations,

describing heights of 15 feet and 2 inches. By the theorem they ought to fall in a time of approximately 28 oscillations. The cause requiring to be investigated why of spheres of the same weight and magnitude, some may fall faster or slower, I fell upon this ; because the spheres, when they were being first released and they were beginning to fall, were turning about the centres, with the side that was perhaps the heavier to be the first to descend, and by generating a motion of oscillation. For by its oscillations the sphere could communicate more motion to the water, than if it were descending without oscillations ; and by communicating, it lost a part of its proper motion by which it ought to descend: and by a greater or smaller oscillation, it may be retarded more or less. Truly indeed the sphere always departed from its side that descended by the oscillation, and by receding approached the sides of vessel and occasionally struck the sides. And this oscillation was stronger in heavier spheres, and with the larger disturbed more water. On which account, so that the oscillation of the spheres could be made less, I constructed nine spheres from wax and lead, I put in place the lead on some side of the sphere close to the surface of this ; and the sphere thus dropped, so that the heavier side, as long as that could be done,



should be the lowest from the beginning of the descent. Thus the oscillations were made much less than at first, and the spheres fell in less unequal times, as in the following experiments. Expt. 8. Four spheres, with a weight of 139 grains in air and 1

26 in water, were dropped a number of times, they fell in times of not more than 52 oscillations, not many less than 50, and the most from a time of around 51 oscillations, describing a height of 182 inches. By the theorem they ought to fall in a time of approximately 52 oscillations. Expt. 9. Four spheres, with a weight of 1

4273 in air and 14140 in water, were dropped a

number of times, they fell in times of not fewer than 12 oscillations, not of much more than 13, describing a height of 182 inches. By the theorem they ought to fall in a time of approximately 1

311 oscillations. Expt. 10. Four spheres, with a weight of 384 in air and 1

2119 in water, were dropped a number of times, they fell in the times of 3 1

4 217 , 18 18 and 19, , , oscillations, describing a height of 1

2181 inches. And when they fell in the time of 19 oscillations, I heard only a few strike the side of the vessel before they arrived at the bottom. By the theorem they ought to fall in a time of approximately 5

815 oscillations. Expt. 11. Three equal spheres, with weights of 48 grains in air and 29

303 grains in water, were dropped often, and they fell in times of 1 1

2 243 44 44, , , 45 and 46 oscillations, and for the greater part, from 44 and 45 , describing a height of 1

2182 approximately. By the theorem they ought to fall in a time of approximately 5

946 oscillations. Expt. 12. Three equal spheres, with weights of 141 grains in air and 3

84 grains in water, were dropped a number of times, they dropped in times of 61, 62, 63, 64 and 65 oscillations, describing a height of 182 inches. By the theorem they ought to fall in a time of approximately 3

464 oscillations. By these experiments it is clear that, when the spheres fell slowly, as in the second, fourth, fifth, eighth, eleventh and twelfth experiments, the falling times were correctly shown by theory; but when the spheres fell faster, as in the sixth, ninth, and tenth experiments, the resistance stood out a little more than in the square of the velocity. For the spheres during falling oscillate a little, and this oscillation in the lighter and slower falling spheres quickly ceases, on account of the lightness of the motion ; but in the heavier and greater, on account of the strength the motion the oscillations may endure a long time, and cannot be confined until after several oscillations in the surrounding water. Truly the swifter spheres, there may be pressed on less by the fluid on their rear parts ; and if the velocity may be constantly increased, they will leave finally a vacuum in the space behind, unless likewise the compression of the fluid may be increased. But the compression of the fluid must be increased in the square ratio of the velocity (by Prop. XXXII. & XXXIII.), so that the resistance shall be in the same square ratio. Because this may not be, the faster spheres are pressed a little less from behind, and from the deficiency of this pressure, the resistance of these shall be a little greater than in the square ratio of the velocities. Therefore the theory agrees with the phenomena of bodies falling in water, it remains that we examine the phenomena of bodies falling in air.



Expt.13. From the top of St. Paul's Church, in the town of London, in the month of June, 1710, two glass spheres were dropped simultaneously, the one full of mercury, and the other air ; and falling they described a height of 220 English feet. A wooden table was suspended at one end by an iron rod, at the other it rested on a wooden peg, and the two spheres set on this table were dropped at the same time, by removing the peg with the help of an iron wire sent as far as the ground so that the table supported only by the iron rod could rotate about the same, and at the same instant by pulling on that wire a pendulum could start oscillating in seconds. The diameters and weights of the spheres and the times of falling are shown in the following table.

The spheres full of mercury. The spheres full of air.

Weights in grains.

Diameters in inches

Times of falling in seconds

Weights ingrains.

Diameters in

inches.

Times of falling in seconds

908 0,8. 4 510 5,1 8 12

983 0,8 4 – 642 5,2 8 866 0,8 4 599 5,1 8 747 0,75 4+ 5I5 5,0 8 1

4 808 0,75 4 483 5,0 8 1

2 784 0,75 4+ 641 5,2 8 Besides the observed times must be corrected. For the mercury spheres (from Galilio's theory) describe 257 English feet in four seconds, and 220 feet in only 3" 41'" . [We will use Newton's notation henceforth; thus 41

603 41 means 3 seconds " '" ]. Certainly the

wooden table, with the peg removed, was turning slower than suitable, and by its slowness in rotation impeded the descent of the spheres from the start. For the spheres were resting on the table near its middle, and indeed they were a little nearer to the axis of this than to the peg. And hence the falling times were prolonged around 18'" , and now must be corrected by taking that small amount, especially with the larger spheres which were resting a little longer on the rotating table on account of the size of the diameters. With which done the times, in which the 6 larger spheres fell, became 8" 12'" , 7" 42'" , 7" 42'", 7" 57'", 8" 12'" , and 7" 42'". Therefore the fifth of the spheres full of air, constructed with a diameter of five inches and with a weight of 483 grains, fell in a time of 8" 12'" , in describing a height of 220 feet. The weight of water equal to this sphere is 16600 grains; and the weight of air equal to the same [volume] is 16600

800 grains or 31019 grains and thus the weight of the sphere in a

vacuum is 310502 grains and this weight is to the weight of the air in the sphere, as

3 310 10502 to 19 , and thus there shall be 2F to 8

3 parts of the diameter of the sphere, that is, to 1

313 inches. From which 2F produces 28 feet 11 inches. The sphere by falling in a vacuum, with its whole weight 3

10502 grains, will describe in a time of one second 13193

inches as above, and with a weight of 483 grains it will describe 185,905 inches, and with



the same weight of 483 grains also in a vacuum it will describe a distance F or 14 feet 125

inches in a time of 57" 58"" , and with that maximum velocity it can acquire by falling in air. With this velocity the sphere, in the time 8" 12'", describes the distance 245feet and

135 inches. Take away 1,3863F or 20 feet 1

20 inches and 225 feet 5 inches shall remain. Therefore the sphere, in the time 8" 12'" must describe this distance by falling according to the theory. Truly the distance described will be 220 feet by experiment. The difference is negligible. By similar computations applied also to the remaining spheres filled with air, I have put together the following table.

Weights of the spheres (grains).

Diameters (inches).

Time to fall 220 feet.

(sec.)

Distance describe by

theory.

Excess.

510 5,1 8" 12'" 226 ft. 11 in. 6 ft. 11 in. 642 5,2 7 42 230 9 7 10 599 5,1 7 42 227 10 7 10 515 5 7 57 224 5 4 5 483 5 8 12 225 5 5 5 641 5,2 7 42 230 7 10 7 Expt. 14. In the year 1719. in the month of July, Dr. Desaguliers took in hand experiments of this kind again, by forming pigs bladders into spherical shapes with the aid of a concave wooden sphere, which wetted were forced to be filled with air ; and these were then dried and removed. By dropping from a higher place in the same holy place from the arch of the copula, namely from a height of 272 feet; and at the same moment of time by dropping also a leaden sphere, whose weight was around two pounds avoirdupois. And meanwhile someone standing in the upper part of the dome, when the spheres were dropped, was noting the whole time of falling, and others standing on the ground were noting the difference of the times between the case of the leaden sphere and of the bladder. Moreover the times were measured by pendulums oscillating at the half second. And of these who were standing on the ground one had a clock which vibrated with a sound in a individual quarter seconds, another had a different machine skillfully constructed also with a pendulum that vibrated four times per second. And one of those present who were at the top of the church had a similar machine. And these instruments thus were formed, so that the motion of these could either be started or stopped as it pleased. Moreover the leaden sphere fell in a time of around 1

44 seconds. And by adding this time to the aforementioned difference of the times, the total time could be deduced in which the bladder fell. The times, in which the five bladders fell after the case of the first leaden sphere, were 3 3 5 3 3

4 4 8 4 414 12 14 17 and 16", ", ", ", " , and the following in turn 31 1

2 4 414 12 14 19 and 16", ", ", ", " . There may be added 144 " , certainly the time in which the

leaden sphere fell, and the total time, in which the five bladders fell, were in the first place 7 18 819 17 18 22 and 21", ", ", ", " ; and in the second place, 3 1 1 1

4 2 4 418 18 18 23 and 21", ", ", ", " . Moreover the times noted from the top of the church , were in the first turn

3 3 51 14 4 4 8 819 17 18 22 and 21", ", ", ", " ; and in the second turn 5 3 1

8 8 419 18 18 24 and 21", ", ", ", " .



The rest of the bladders did not always fall straight down, but sometimes were flying about, and hence thus they were moving to and fro while falling. And from these motions the times of falling were extended and sometimes increased by as much as half a second, sometimes by a whole second. But the second and fourth fell more straight in the first turn; and the first and third in the second turn. The fifth bladder was more wrinkled and by it wrinkles somewhat retarded. The diameters of the bladders I deduced from their circumferences measured twice by a fine thread passed around. And I have brought together the theory with the experiments in the following table, by assuming the density of air to be to the density of water as 1 to 860, and by computing the distances which the spheres must describe by falling by theory. The weights of the bladders (grains).

Diameters (inches).

Times required to fall from a height of 272 feet.

Distance described in the same times by theory.

Difference between theory & expt.

128 5, 28 19" 271ft. 11 in. – 0ft. 1in. 156 5, 19 17 272 1

20 + 0 120

12137 5, 3 1

218 272 7 + 0 7 1297 5 26 22 277 4 + 5 4 1899 5 1

821 282 0 + 10 0 Therefore the resistances of nearly all the spheres moving both in air and in water are shown correctly from our theory, and is proportional to the densities of the fluids, with equal velocities and magnitudes of the spheres. In the scholium, which has been added to the sixth section, we have shown by experiments with pendulums that the resistances of the motions of equal and equal moving spheres in air, water, and in quicksilver are as the densities of the fluids. Here we have shown the same more accurately from experiments with bodies falling in air and in water. For the individual oscillations of pendulums always move the fluid in a direction opposite to the direction of the returning swing, and the resistance that arises from that motion, and as the resistance of the thread by which it was being suspended, the total resistance of the pendulum were rendered greater than the resistance produced by a body falling. And also by the experiments with pendulums set out there in the scholium, a sphere of the same density as water, by describing a length of half its diameter in air, ought to lose the 1

3342th part of its motion. But by the theory I have set out in this seventh

section and I have confirmed from the experiments with falling bodies, the same sphere by describing the same length, ought to lose only the 1

4586 th part, supposing the density of water shall be put to the density of air as 860 to 1. Therefore greater resistances were produced by the experiments with pendulums (on account of the reasons now described) than by the experiments with falling spheres, and that approximately in the ratio of 4 to 3. Yet since the resistance of pendulums in air, water, and in quicksilver may be likewise increased by like causes, the proportion of the resistance in these mediums, both by the experiments with pendulums, as well as by the experiments with falling spheres,



may be demonstrated well enough. And thence it can be concluded that the resistances of the motions of bodies in any of the most free of fluids, with all else being equal, are as the densities of the fluids. Thus with these established, now it is permitted [to find] what part of its motion will be lost by any sphere, projected in some fluid, in some given time approximately. D shall be the diameter of the sphere, and V its initial velocity, and T the time, in which the sphere with the velocity V in a vacuum may describe a distance, which shall be to the distance 83 D as the density of the sphere to the density of the fluid : and the sphere projected into the fluid, at some other time t, will lose the tV

T t+ part of its velocity, with the part TVT t+

remaining, and a distance described, which shall be to the distance described with the uniform speed V in the same time, as the logarithm of the number T t

T+ multiplies by the

number 2,302585093 is to the number tT by Corol. VII, Prop. XXXV. In the slower

motions the resistance can be a little less, because that figure of the sphere shall be a little more suited to the motion than the figure of a cylinder described of the same diameter. In motions with greater velocities the resistance can be a little greater, because since the elasticity and the compression of the fluid may not be increased in the square ratio of the velocity. But I will not dwell on trifling details of this kind here. And although air, water, quicksilver and like fluids, by the indefinite division of parts, may become more subtle and be made infinitely fluid mediums ; yet they may offer no less resistance to projected spheres. For the resistance, by which it was acted on in the preceding propositions, arises from the inertia of the matter, and the essential inertia of matter in bodies is always in proportion to the quantity of matter. By the division of the parts of the fluid, the resistance which arises from the tenacity and the friction of the parts can indeed be diminished : but the quantity of matter through the divisions of the parts of this is not diminished; and with the quantity of matter remaining, the inertial force of this remains, to which the resistance, by which this is acted on, is always proportional. In order that this resistance may be diminished, the quantity of matter must be diminished in the interval through which the body is moving. And because the celestial spaces, through which the spheres of the planets and comets to all parts freely and without any diminution of the motion may be considered to be moving perpetually, they are free of all corporal fluid, if perhaps rare vapours and the trajectories of light ray be excepted. Certainly projectiles excite motions in fluids by passing through them, and this motion arises from the excess of the pressure of the fluid on the anterior parts of the projectile over the pressure on the posterior parts of this, and cannot be less in infinite fluid mediums than in air, water, and quicksilver for the density of the matter in each. But this excess pressure, from its amount, not only may excite motion in the fluid, but also act on the projectile and to retard its motion : and therefore the resistance in any fluid is to the motion excited in the fluid by a projectile, cannot be less in the most subtle aether than to the density of that aether, as it is in air, water, and quicksilver to the densities of these fluids.


Book II Section VIII. Translated and Annotated by Ian Bruce. Page 672

SECTION VIII.

Concerning motion propagated by fluids.

PROPOSITION XLI. THEOREM XXXII.

A pressure is not propagated by a fluid along straight lines, except when the particles of the fluid lie in a straight line.

If the particles a, b, c, d, e may lie on a right line, some pressure can be propagated directly from a to e ; but the particle e will act on the sideways placed particles f and g in an oblique manner, and these particles f and g will not sustain the inflected pressure, unless they are supported by the more distant particles h and k ; but provided they are supported, they press on supporting particles, and these cannot support the pressure unless they are supported by the further particles t and m, and they press on these; and thus henceforth indefinitely. Therefore the pressure, that in the first place is propagated to the particles which do not lie along the direction, will begin to branch out, and to be propagated indefinitely obliquely; and after the pressure begins to be propagated obliquely, if it should be incident on more distant particles, which do not lie along the same direction, it will branch out again ; and just as often as it is incident on particles that are not accurately in line, so this will happen. Q.E.D.

Corol. If some part of the pressure, propagated from some part of the fluid, may be intercepted by some obstacle, the remaining part, by which it is not intercepted, will branch out in the space behind the obstacle, as that can also be demonstrated thus. From the point A the pressure can be propagated in every direction, and thus if it were able to made along right lines, and all that pressure may be intercepted by the obstacle NBCK perforated in BC, except the conical part APQ, which passes through the circular opening



BC. The cone APQ may be separated into frustums by the transverse planes de, fg, hi ; and while being propagated within the cone ABC, the pressure may act on the further conical frustum degf at the surface de, and this frustum may act on the nearby frustum fgih at the surface fg, and that frustum acts on the third frustum, and thus henceforth indefinitely; it is evident (by the third law of motion) that the first frustum defg by the reaction with the second frustum fghi, will be acting and by pressing with as great a pressure on the surface fg, as may act and press on that second surface. Therefore the frustum degf is compressed on both sides between the cone Ade and the frustum fhig, and therefore (by Corol. VI. Prop. XIX.) is unable to maintain its shape, unless it shall be compressed by the same force on both sides. Therefore by the same impulse by which it is pressed on the surfaces de and fg, it will try to go to the sides df and eg; and there (since it shall not be rigid, but always a fluid) it will run and expand out, unless surrounding fluid may be near. Therefore by trying to depart, it presses with the same impulse both on the surrounding fluid on the sides df and eg, as well as on the frustum fghi : and on that account none the less the pressure will be propagated from the sides df and eg into the spaces NO and KL and thus hence, so that it may be propagated from the surface fg towards PQ. Q.E.D.

PROPOSITION XLII. THEOREM XXXIII.

All motion propagated by a fluid diverges from rectilinear motion in an immoveable space.

Case1. The motion may be propagated from the point A through the hole BC, and it may go, if it can happen, into the conical space BCQP, along right lines diverging from the point A. And in the first place we may suppose that this motion shall itself be of waves on the surface of water at rest. And let de, fg, hi, kl, &c. be the highest parts of individual waves [crests], with just as many distinct intermediate valleys [troughs] in turn. Therefore because the water in the crests of the waves is higher than in the fluid in the stationary parts LK, NO, the same may flow away from the crest terms e, g, i, l, etc. d, f, b, k, etc.



hence from there towards KL and NO: and since it is lower in the troughs of the waves than in the motionless parts of the waves KL, NO; the same may flow from the motionless parts into the troughs of the waves. By flowing from the first crest of the waves, hence it will be expanding out into the troughs behind that and may propagate towards KL and NO. And because the motion of the waves from A towards PQ shall be by a continued flowing away of the crests into the nearest troughs, and thus the speed of crests shall not be faster than the speed of the troughs ; and hence the descent of the water from there towards KL and NO must be going with the same velocity; hence the expansion of the waves from there towards KL and NO will be propagated with the same velocity by which these waves are progressing from A towards PQ . And hence the total space from there towards KL and NO is occupied by an abundance of spreading crests rfgr, shis, tklt, vmnv, &c.. Q.E.D. Thus these can be tested by anyone for themselves in any still water. Case 2. We may now suppose that de, fg, hi, kl, mn may designate pulses propagating successively from the point A through an elastic medium. The pulses may be considered

to be propagated by successive condensations and rarefactions of the medium, thus so that the most dense part of each pulse may occupy a spherical surface described about the centre A, and equal successive intervals may fall between the pulses. Moreover the lines de, fg, hi, kl, etc. may designate the densest parts of the pulses, propagating through the hole BC. And because the medium here is more dense than in any quarter towards the spaces KL and NO, thus it will expand itself both towards these spaces KL and NO situated on each side, as well as towards the intervals of more rare pulses; and with that understood they will share in the same motion, the rarer always emerge from the region of the [quiet] intervals and the denser from the region of the [denser] pulses. And because the progressive motion of the pulses arises from the continual relaxation of the more dense parts towards the antecedent rarer intervals ; and the pulses with almost the same speed must relax in the quiet parts of the medium from the different quarters KL and NO ; these pulses expand themselves out on each side with almost the same speed into the unmoved spaces KL, NO, by which they may be propagated directly from the centre A ; and thus they occupy the whole space KLON. Q.E.D. We experience this with sounds, which are heard either with a mountain in between, or they expand on themselves being admitted into a room by the window, to be heard in all the corners, not only as reflected



from the opposite walls, but propagated directly from the window as well, as much as can be judged from the senses. Case 3. Finally we may suppose that a motion of any kind may be propagated from A through the opening BC : and because the propagation itself does not happen, unless in so far as the parts of the medium closer to the centre A may urge the more distant parts into some excitation, and the parts which are acted on by the fluid, and thus which recede in every direction where they act with a lesser pressure: all the same recede towards the resting parts of the medium, both to the sides KL and NO, as well as to the anterior parts PQ, and with that agreed upon the motion of all, that first has passed through the opening BC, will begin to expand, and thence at last from a beginning at the centre, is propagated directly in every direction. Q. E. D.

PROPOSITION XLIII. THEOREM XXXIV.

Any body vibrating in an elastic medium will propagate the motion of pulses in every direction; truly in a non-elastic medium it will excite a circular motion. Case 1. For the parts of a vibrating body by alternately coming and going in turn, in its going will act on and propel the nearby parts of the medium themselves, and on being acted on the same will be compressed and condensed; then by its return the compressed parts themselves recede and expand. Therefore the parts of the medium by the vibrations of the body will in turn go and return, in the image of the vibrating parts of the body : and by that account the parts of this body will disturb these parts of the medium, these by similar agitated vibrations will disturb parts nearby to themselves, and these similarly agitated will disturb the further parts, and thus henceforth indefinitely. And in whatever manner the first parts of the medium by going may be condensed and are relaxed on return, thus the remaining parts as often as they shall go will be condensed, and just as often as they return they expand themselves. And therefore not all will go and return at the same time (for since in turn by maintaining determined distances, they would not be rarefied and condensed in turn) but in turn by approaching where they are condensed, and by receding where they are rarefied, some of these will be going while others will be returning ; and that alternately in turn indefinitely. Moreover the parts going and by so going being condensed, are the pulses on account of their progressive motion, by which they strike obstacles, and therefore the pulses will be propagated successively from any vibrating body along a line ; and thus at around equal distances in turn, on account of the equal intervals of time, in which the body by vibrating excites the individual pulses. And whenever the vibrations of the body may go and return along a certain determined direction, and the pulses thence propagated by the medium still themselves expand laterally, by the preceding proposition; and they will be propagated in both directions from that vibrating body as from a common centre, along almost spherical and concentric surfaces. We have some sort of example of this from waves, which if they are excited by a waving finger, not only do they go to and fro in the direction of the motion of the finger, but, in the manner of concentric circles, they will surround the finger at once and be



propagated in all directions. For the weight of the waves takes the place of the elastic force. Case 2. For if the medium shall not be elastic : because the parts of this are unable to be compressed to condense from the vibrations of the shaking parts of the body, the motion will be propagated at once to the parts where the medium can go most easily, that is, to the parts that the shaking body otherwise may leave empty behind it. It is likewise the case with a body projected in some medium. A medium by making way for projectiles, does not recede indefinitely, but by going in a circle, it will go to the space which the body has left behind. Therefore as often as a vibrating body will go into some part, the medium by giving way goes by a circle into the parts that the body has abandoned ; and as often as the body is returned to the first place, the medium will be repelled from there and will return to its first place. And in whatever manner the vibrating body may not be rigid, but flexible in some manner; if yet it may have a given magnitude that remains, because the medium is unable to become disturbed by the vibrations, in what other way can it affect the medium, so that the medium, by receding from the parts where it is pressed, will always go round to the parts which may yield to the same. Q. E. D. Corol. Therefore people must be wandering in their minds who believe that the disturbance of the parts of a flame be conducive to a pressure propagating along right lines, through the surrounding medium. A pressure of this kind cannot be derived from the agitation of the flame alone, but from the expansion of the whole medium.

PROPOSITION 44. THEOREM 35. If water alternately rises and falls in turn in [uniform] pipes with upright legs KL and MN ; and moreover, if a pendulum is made of which the length between the point of suspension V and the centre of oscillation P is equal to half the length of the water in the pipe : then I say that the water rises and falls in time with the oscillations of the pendulum.

I measure the length of the water along the axes of the pipe and legs, with the same equal height of these ; and I ignore the resistance of the water which arises from the friction with the pipes. AB and CD designate the mean height of the water in both legs ; and when the water in leg KL has risen to the height EF, the water in leg MN has fallen to the height GH. Also, let P be the body of the pendulum, VP the thread, V the point of suspension, RPQS is the cycloid that the pendulum describes, of which P is the lowest point, and the arc PQ is equal to the height AE. The force, by which the motion of the water is either accelerated or decelerated in turn, is the excess of the weight of water in the one leg above the weight in the other; thus, when the water in KL has risen to EF, and in the other leg fallen to GH, that force is twice the weight of water EABF, and therefore is to the total weight of water as AE is to VP or PQ or PR. Also, the force by which the weight P at some place Q is accelerated or decelerated in the cycloid (from the corol. to prop 51.) is to the total force as this distance PQ from the lowest place P, is to the length



ψP

Q

V

s

4a

of the cycloid PR. Whereby the equal intervals of the water and the pendulum AE and PQ describing the motive forces are as the weights to be moved; and thus, if the water and the pendulum are initially at rest, these forces will move the same equally in the same time, and are effective in order that the reciprocal motions can go and return at the same time. Q. E. D.

Corollary 1. Therefore all the oscillations for the rise and fall of the water in turn are isochronous, whether they are made stronger or weaker [i. e. the period of oscillation is independent of the amplitude.]

Corollary 2. If the whole length of the water in the pipes is 916 Parisian feet, then the

water descends in a time of one second, and rises in the time of one second, and thus henceforth in alternate turns indefinitely. Likewise, the pendulum of length 18

13 is oscillating with a time of one second.

Corollary 3. Moreover with the length of the water increased or diminished, the time of reciprocation is increased or diminished in the ratio of the square root of the length.

[The Manometer as a S. H. M. Oscillator : In modern terms, if A is the cross-sectional

area of the pipe, l the length of water in the pipe, and ρ the density of the water, then if x is the extension AE of one arm of the manometer from the equilibrium level AB, and - x or DH is the depression of the other level, or vice versa, then the mass of water accelerated is Alρ, while the unbalanced force is 2ρAgx; hence, from Newton's Second Law of motion,

xxg/l-x or AgxxAl 2)2(,2 ωρρ −==−= . Hence, the period of the oscillation is given by

.2 2/g

lT π= In the case where the period of the whole oscillation is 2 seconds, note that

Newton has a habit of referring to half periods - in his example 1 second - then the length of water is approximately 2 m.

The Inverted Cycloidal Pendulum as a S. H. M. Oscillator : We will save some time by merely quoting the formula for the length of arc s of an

inverted cycloid, which is of course a rectifiable curve - and hence was part of its fascination for early workers - in terms of the tangent angle ψ at some point Q:

,sin4 ψas = where 4a is the length of the thread of the equivalent simple pendulum VP ( the cycloid can be considered as generated by a point on a circle of radius a rolling along a horizontal line at a vertical distance 2a above the x- axis.) An

equivalent S. H. M. is a small bead of mass m to slide on a wire in the shape of the inverted cycloid without friction. Thus, the unbalanced force on the bead due to gravity acting down the slope at Q is ,sinψmg and we can set .sinψmgvm −= That is,

.sin 42

2sg a

gdt

sd −=−= ψ Hence, setting 4a = l/2 insures equality of the periods, and both

motions are independent of the amplitude, though Newton has set these or the half periods



equal in his experiment as this was presumably more expedient : ,14

2/==

al

TT

pendulum

manometer on

setting 4a = l/2. Note that one does not have to contend with the factors 2π and g, on taking a ratio in this way, and the whole or half periods can be used with impunity.

PROPOSITION XLIV. THEOREMA XXXVI.

The velocity of waves varies as the square root of the wavelength. This theorem follows from the construction of the following proposition.

PROPOSITION XLVI. PROBLEM X.

To find the speed of waves. A pendulum is set up, the length of which between the point of suspension and the

centre of oscillation, is equal to the length of the waves : and during the time the

pendulum performs single oscillations, by advancing the same amount, the waves progress to almost their own width.

The transverse width of the waves measured I call the length of the waves, which lies between either the deepest valleys or the highest peaks. ABCDEF designates the surface of the still water, with successive waves rising and falling; A, C, E, &c. are the peaks of the waves, & B, D, F, &c. are the intervening valleys. And since the motion of the waves is by the water successively ascending and descending, thus the parts A, C, E, &c. of this surface which now are the highest, soon will become the lowest; & the driving force of the motion, by which the highest parts will descend & the lowest parts ascend, is the weight of the elevated water ; this alternate rising and falling is analogous to the reciprocal motion of the water in the pipes, and the same laws governing the time will be observed : & therefore (by prop. XLIV) if the distances between the peaks of the waves A, C, E & and the troughs B, D, F are equal to twice the length of the pendulum, then the highest parts A, C, E, in the time of one oscillation avoid the troughs, & in the time of another oscillation have ascended again. [Recall that the manometer always has a peak and a trough for the maximum displacements, and therefore corresponds to half a wavelength.] Therefore between the passage of individual waves there will be the time of two oscillations; that is, the wave describes its own width in the time that pendulum oscillates twice ; but the pendulum that oscillates in time with the wave is four times as long, and thus oscillates once, equal in time with the length of the waves. Q.E.I.

Corol. 1. Therefore waves which are 1813 feet long, progress a distance equal to their

own width in a time of one second (around 1 m/s); and thus in a time of one minute from

A

B

C

D

E

F



the start run through a distance of 31183 feet, & in the space of an hour 11000 ft

approximately. Corol. 2. And the speed of the long waves is increased or diminished in the ratio of the

square root of the width. From the hypothesis, thus waves are considered to have part or the water either

ascending straight up or descending straight down (as in the manometer) ; but the up and down motion of the water shall more truly be described in a circle, and likewise I emphasize that the time derived from this proposition can only be defined approximately.

[ Notes : There may be some confusion as to what Newton means by the time of an

oscillation - the word itself just means a swing, of course. However, in the experiment the pendulum bob is released as a peak of the wave train passes, and reaches the position of the peak of the passing wave again at the end of its forward swing, which occupies at this instant the position of the preceding peak at the start of the pendulum's motion. Newton asserts that the pendulum which achieves this synchronous behavior has the same length as the distance between the peaks. The previous experiment with the manometer tube, which is a sort of standing wave generator of water waves with a 'free end', has a wave or pulse that travels a distance set to some half wavelength by the length l (for small amplitudes) in the time the pendulum completes its forward motion, as the water in the legs interchange peaks and troughs. Now, by analyzing the s. h. m. of the pendulum and the manometer, we find the periods are equal when 4a = l/2, or a half-wavelength l corresponds to a pendulum twice as long as that used at present, and the period needed for a whole wavelength is thus four times as long as the original pendulum. The original pendulum is l/2 or λ/4, so that the reasoning is correct : a pendulum of length λ is required to be synchronous with the waves. However, the period of such a simple pendulum is 2 seconds, and hence we conclude that Newton is talking about single swings when he considers oscillations of 1 second.

As mentioned in Cor. 2, there is augmentation or diminution of the waves as they proceed, as they do not all travel with the same speed, and dispersion is taking place. Hence, it is more appropriate to consider the group velocity of the pulses of waves, rather than the phase velocity - which cannot be measured in any case - and the group velocity is responsible for the transfer of energy down the channel. Thus, from his experimental measurements, Newton had observed that the length of the pendulum λ executed its forward swing in a time T given by g

λπ , for which smgTv /1~/ 1 λλ π== as λ ~ 1m and

g ~10m/s2. This quantity we would identify as the group velocity . We cannot read much more into Newton's experiments, as he has not furnished details of the physical dimensions of the channel and pipes apart from the total length of the axis, factors upon which the rate of transmission depends. Nevertheless, the main ideas are essentially correct, and at the time, he was a man in a hurry, and he had sown the seeds for further development.]



HI

K

O

S

P

NML n

m

l h

ik

PROP. XLVII. THEOR. XXXVII.

For pulses propagating through the fluid, the individual particles of the fluid are oscillating in the shortest reciprocal motion, always accelerating and decelerating according to the law of the pendulum. Let AB, BC, and CD, &c. designate the positions of equally spaced successive pulses [i. e. such as progressive sound waves of a given wavelength λ of AB.] ; the motion of the pulses is propagated from A towards B along a line ABC in the region ; E, F, G are three physical points of the quiescent medium on the line AC, situated at equal distances from each other; Ee, Ff, Gg are equal lengths in turn [of the maximum amplitudes] through which in short time intervals, by the individual reciprocal motions, these points E, F, and G move to and fro ; ε, ϕ, γ are some intermediate locations of the same points in the medium; EF and FG are small physical sections or incremental parts of the medium placed between these points, & which in succession are translated into the positions εϕ and ϕγ, & then ef and fg. The line PS is drawn equal to the line Ee [in the lower diagram]. PS is bisected in O, and with centre O & length OP, a small circle SIPi is described. In this circle, the whole circumference represents the time of one complete vibration, together with its proportional intermediate parts. Thus, in order that some time such as PH or PHSh can be compared with the time of the complete oscillation, if a perpendicular HL or hl is dropped on PS, then Eε is taken to be equal PL or Pl, at this instant the physical point E is to be found at ε in the moving fluid. According to the law of the pendulum, any point E in the fluid moves from the equilibrium value E to the maximum displacement e through ε, & returns to E again through ε,

where each vibration has the same degrees of acceleration and retardation [at intermediate points], so that the oscillation is completed in step with the oscillation of a pendulum [i.e. any particle such as E executes s. h. m. from its equilibrium point in the fluid; the actual words in Newton's explanation have been augmented occasionally to reinforce the reader's understanding, as Latin is a little skimpy at times]. This must be the case since the individual physical points of the medium are

C

D

B

F E G

fe

g

Ω

ε φ γ



disturbed in this way by such a motion [as in the analogous case of the water waves]. Hence we establish a medium in which such a motion is produced in some manner, and we observe what may then follow. On the circumference PHSh, the equal arcs HI and IK or hi and ik [of a traveling wave or pulse] are taken in the same ratio to the total circumference as the equal lines EF and FG have to the total length of the pulse interval BC, and the perpendiculars IM and KN or im and kn can be dropped, as the points E, F and G are disturbed in turn by the same motion, & their whole vibrations meanwhile are carried out from the sum of the oscillations as the pulse is transferred from B to C. Thus, if PH or PHSh is the time of the motion starting from the initial point E, then similarly PI or PHSi is the time of the motion starting from the initial point F, and again PK or PHSk is the time of the motion starting from the initial point G [An extended pulse passed through the increments E, F, and G in turn, then the angles are in proportion to the times as the are length s = OP × Δθ = OP × ωΔt , where ω is the angular frequency]. Hence, Eε, Fϕ and Gγ will be respectively equal to the lengths PL, PM and PN themselves in the movement away from equilibrium position , or to Pl, Pm and Pn themselves in the return. From which εγ or EG + Gγ - Eε leads to EG - LN being equal to the incremental pulse width in the movement away from equilibrium. But εγ is the width or the expansion [contraction really] of the part of the medium EG when it is transferred to the location εγ ; & therefore the expansion of that part in the outward motion is to the mean expansion as EG - LN is to EG ; and moreover in the return journey, the ratio is as EG + ln to EG. [EG - LN is the contracted length εγ ; and thus (EG - LN)/EG is 1 - ΔV/V, as Newton goes on to demonstrate. Again, this is needed to make the outgoing contraction into an in going expansion on the return leg of the journey, taken to be (EG + ln)/EG or 1 + ΔV/V]

Whereby the ratio LN to KH shall be as IM to the radius OP [This involves differentiation : see following note.], & KH to EG as the circumference PHShP to BC, i. e., if V is put in place for the radius of another circle with the circumference set equal to the pulse interval BC, then the ratio becomes as [the amplitude] OP to [the wavelength λ or] V ; & from the equality LN to EG as IM to V : the expansion of the part EG or of the physical point F at the location εγ to the mean or quiescent expansion, as that part has in its first position EG, as V - IM to V in going, and as V + im to V on returning. From which the elastic force of the point F at the position εγ is to the mean elastic force of this at the position EG, as IMV −

1 to V1 in going, and on returning truly as imV +

1 ad V1 . And by the

same argument the elastic forces of the physical points E and G on going, are in the ratios V111 toand KNVHLV −− ; and the difference of the forces to the mean or quiescent elastic force

of the medium, as V1 toKNHLKNVHLVVV

KNHL×+×−×−

− . That is, as V1 toVV

KNHL− , or as HL - KN to V, if (on account of the narrow limits of the vibrations) we may suppose HL and KN to be indefinitely smaller quantities than V. Whereby when the quantity V is given, the difference of the forces is as HL - KN, that is as OM (on account of the proportionals HL - KN to HK, & OM to OI or OP, with HK & OP given) ; i.e. if Ff is bisected in Ω, as Ωϕ. And by the same argument the difference of the elastic forces of the physical points ε & γ, in the return of the small physical line εγ is as Ωϕ. But that difference (i.e, the excess of the elastic force at the point ε over the elastic force at point γ) is the force of the medium which is introduced for the small physical line εγ to be accelerated in returning and



HI

K

O

S

P and S are the positions where the amplitude and pressure gradient are at maximumvalues, and the condensation is zero; as the phase angle increases, the condensationgrows to a maximum at 900, while the amplitude and pressure gradient go to zero,

and subsequently these quantities revert again at S to their P conditions. The returnstroke sees the dilatation go through the same sequence.

The left-hand sketches show positive density or pressure changes for the outgoingair, while the right-hand ones show negative changes for the returning air. The

projection of HIK on to the PS axis shows the passage of the condensation along PS,and the returning dilatation, which are a maximum along the diameter at I and i.

NML n

ml

hi

maximum +vecondensation.

increasing +vecondensation.

condensation.maximum -ve

P and S are the maximumdisplacements in the positive/

negative senses. The pulse drawnshows the degree of condensation

transverse to the direction oftransmission, which is really

longitudinal.

(ε)

(φ)

(γ) h

i

k

θΔθΔθ

P

k K

I

H

zero condensation.

zero condensation.

decreasing -vecondensation.

retarded in going; & therefore the acceleration force on the small physical line εγ, is as its distance from the mean position of the vibrations Ω. Hence the time for the straight line motion PI is explained (by prop. XXXVIII. Book. I) ; & the part εγ of the medium is moved according to the prescribed law, that is, by the law for the oscillations of a pendulum : and the reasoning is the same for all the line increments from which the whole medium is composed. Q. E.D.

Corol. Hence it is apparent that the number of pulses propagating is the same as the number of vibrations of the trembling body, without change in their number in progressing. For the incremental line in the medium εγ, when first to its original situation returns remains at rest, and henceforth will not move, except either by the impulse of a trembling or oscillating body, or from the impulse of a pulse which is propagating from such a body, when it sets off a new movement. The medium will therefore be quiescent when the starting pulses from the vibrating body cease to be propagated.

Notes on Prop. 48 : We enlarge on Newton's ideas a little from a modern perspective, but relate to his derivation as much as possible. First, we need to explain his diagram accompanying the trajectory diagram for a ray of sound in one dimension, based on his ideas. According to the diagram opposite, the points P and S represent the positions of the maximum amplitude χ of the s.h.m. associated with the point F. In Newton's words, the time for the motion is spread around the circumference of the circle; in the course of the motion, the matter in the incremental length EFG is physically moved as a whole to some intermediate distance Fϕ relative to the reference frame of still air, and a wave of compression passes through the element to give the incremental length εγ , finally to come to rest again momentarily at f with no compression. The pulse is considered to move from the positive displacement χ of the s. h. m. to the negative displacement in the diagram. Thus, R = OP = χ rotates clockwise. The times of arrival at e, f, and g can also be recorded, as are the times of the return of the air to its instantaneous



position of maximum displacement : the matter that left first returns first, so H → h, etc, in the intermediate section. The actual rest position of the air in the absence of waves is at Ω, while P and S are the points of instantaneous rest for a continuous wave. Newton considers the projection of the maximum compression HIK on to the line PS as the contraction or expansion of the element at the same point of its motion to and fro : thus, there is no compression or condensation at P and S, while the maximum compression/expansion occurs at O, and a wave of compression/ expansion of some lesser amount but in the same ratio passes along the element PS at other times. In addition, by taking ratios at the same out and in positions on the cycle, he is able to proceed without using the bulk modulus, that we now consider as part of the modern theory.

Modern ideas: Initially, we consider the boundary conditions placed on the elemental oscillators. Each incremental length acts as an s.h.m. oscillator, each driven by or coupled to the one before, and driving the next one in a chain of oscillators, which we assume to be in one dimension. Each oscillator has the same amplitude χ, and all vibrate with the same angular frequency ω as the wave, which we assume to be continuous and of a single frequency; also, each increment has an in-built constant phase factor ϕ(x) depending on its location: when t = 0, these phases construct a harmonic wave between the crests of the wavelength λ. Thus, the increment associated with the point E has two components of phase that add/subtract to give the total phase : There is the time related phase of the form (2π/T).t = ωt, and for a given quiescent position, there is the constant distance related phase angle for the oscillator, that we can call e. g. φ(E) = (2π/λ).BE = k.BE. The other equally separated quiescent oscillator points F and G have similar distance phases φ(F) = (2π/λ).BF and φ(G) = (2π/λ).BG associated with them. The passage of the wave is the transmission of regions of constant phase from oscillator to oscillator. This region of constant phase is driven forwards in the mathematical model by an argument of the form kx - ωt = constant, or ωΔt = kΔx, resulting in a phase velocity v = Δx/Δt = ω/k. There are some details of Newton's model that are inconsistent with the modern theory, even at this kinematic level. We have noted already that he describes the air as being at rest at P and S; although this is true, it is not in its quiescent condition as there is a maximum pressure difference across the element here in accordance with s. h. m. principles, and the air element is actually at its true length (i. e. in the absence of the sound wave)when it passes the origin of the oscillation at its maximum speed at the half-way point Ω, when there is no pressure difference across the ends of the element. Recall that for s. h. m. the acceleration is proportional to the negative displacement : hence, there is a maximum acceleration and force at the maximum displacement, and zero acceleration and force at zero displacement. The model succeeds in producing the s. h. m. equation with the correct ω and wave speed ω/k.

Before leaving this narrative, we shall briefly give a modern derivation of the wave equation [following Pain, The Physics of Vibrations and Waves, Ch. 5 (Wiley)]. The motion of an undisturbed infinitesimal element of air of original thickness Δx and unit area under the influence of a sound wave in one dimension is considered. The element as a whole is displaced a distance η, and expanded by an amount (∂η/∂x)dx, as shown in the diagram :



η + dη

η

)dx(dx xη

∂∂+

Px dxxP

xxP ∂

∂+

dx

The increase in the volume is dxx∂∂η , while the

change in the volume per unit volume is x∂∂η or

the dilatation dv/v. The quantity dρ/ρ inverse to the dilitation is known as the condensation. Meanwhile, the net force exerted on the element to the right in the compression or expansion due to the pressure gradient is dx.x

Px∂∂− ; hence, by

Newton's Second Law, 2

2

0dx.ρ dx .

txPx

∂∂

∂∂ =− η . There

is now a need to relate the pressure gradient to the changes in the volume : this is usually done by means of the bulk B modulus for the substance. The change in the volume per unit volume is proportional to the impressed pressure, or dp = - B. dv/v, where B is the constant of proportionality, and the negative sign is necessary as an increase in pressure results in a decrease in the dilatation, or change in volume per unit volume. In the present case, dv/v = x∂

∂η , and hence 2

2.

xxP Bx

∂∂

∂∂ −= η . From

which it follows that 2

2

02

2ρ

txB

∂∂

∂∂ = ηη and 2

2

2

2 2

txc

∂∂

∂∂ = ηη , where c2 = B/ρ0. This is the

conventional wave equation for sound waves in a gas: Newton does not derive this equaton; instead, he derives the equation for the s. h. m. of an elemental section of air, resulting in 2

2 222

tck

∂∂=−=− ηηωη for a sinusoidal motion. Now, compression or expansion

of a gas results in heating or cooling; Newton was unaware of the adiabatic nature of sound waves, and used essentially the isothemal form of B or P/ρ0, rather than the correst adiabatic form γP/ρ0 ; thus his value for c was out by √ γ, where γ is the ratio of the specific heats of the gas at constant pressure to constant volume, and depends on the nature of the gas - internal degrees of freedom, etc. It would take us too far away from Newton's work to consider this matter further, though of course it is developed in books on thermodynamics.

There are inevitably problems associated with understanding what the words actually

mean when comparing Newton's model with the actual model for sound waves that we have briefly outlined above; thus, the word 'expansion' can mean either the volume or the change in volume - the various translations suffer from this ambiguity, and one must proceed with caution. We are not in the business of correcting Newton's model, which would be a great travesty as well as a meaningless exercise, but are merely trying to understand what his thoughts might have been as he developed his ideas.

So we return to Newton's argument: Subsequently, to make further progress with the Proposition, use has to be made of calculus. Initially, two useful ratios are evaluated. Pressure - modified volumes represented by the lengths OL, OM, ON are related to ΔV/V, the dilatation; and the angular phase of the rotating radius OP of the s. h. m. is related to the linear phase of the wave.



Thus, OL, OM, and ON are given by : χcosω(t +δt), χcosωt , and χcosω(t - δt), leading to LN/KH = [χcosω(t +δt) -χcosω(t - δt)]/2χωδt → -sinωt, the limiting value of the ratio; or equivalently, ; 0) as ,sin(

2.)cos()cos(

χϑϑ

ϑχϑϑχϑϑχ IM

KHNLorLN

=→−→+−−

= ΔΔ

ΔΔ

[This first differentiation gives the rate of change of LN or ΔV, the decrease in the volume, and LN/EG = ΔV/V, the fractional change in the volume.] and

)2/(...

πλχ

====V

OPBCcircumwithcircle

PHSbPcircumBC

PHSbPcircumEGKH relates the phases.

[Thus, In the time the line OP turns through a certain small angle, the pulse advances a certain amount along PS. The original circle with radius OP describes the time variation or the phase of the oscillation at some fixed point, while the second circle with radius Vr

= λ/2π, rather than V that we use for the volume for a little while EG describes the displacement variation of the phase of the oscillation at some fixed time. Hence,

πϑ

πχϑχ

λλπχ

2.2

2.2or 2. ΔΔ

====EG

VOP

EGKH

r : thus, a path difference of EG corresponds to a

phase difference of 2.Δθ as required by the coupled oscillators discussed above. Or, v = ω/k = (2Δθ/Δt)/(2π/λ) = (2Δθ/Δt).Vr; hence EG = v.Δt = 2.Δθ . Vr = arc HK.Vr/OP] Since ,..

rr VIM

VOP

OPIM

EGLN

EGKH

KHLN

=== or ]/2

tsin-[ Fatvolumequiescent

at n compressioπλωχεγ

====rV

IMEGLN

VVΔ , it

follows that the excursion compression of EG at εγ to the quiescent volume at F is in the ratio

r

rV

IMVEGLN

EGLNEG

EGVVV −

=−=−

==− 1εγΔ , while the return expansion ratio is :

r

rV

imVEGEG

EGV

VV +=+=

+=

+ ln1lnΔ .

[The common reader may wish to refer to the work by S. Chandrasekhar at this point (Newton's Principia for the Common Reader (1995); Oxford. p. 586) : this author has not attempted, as we have attempted, to actually link up Newton's derivation with modern theory, but has presented this theory from a Newton - friendly point of view. The other authors of interest, Cohen & Whitman: Newton The Principia. (California), and Cajori : Newton's Principia (U. Cal.) have not given any explanation of Newton's theory of sound, and have only presented an English version of the Latin text, with all its vagaries. Cajori in his translation, even goes to the extent of re-arranging the labels on the phase diagram, thus changing something which is correct into something which is incorrect !]

Newton now sets out to construct what is essentially the second order differential equation describing the s. h. m. of an elemental volume of air in the presence of a sound wave of constant frequency.



Now, if the elastic force varies inversely as the expansion or volume, then

IMVV

Fatforceelquiescentatforceel

−=

.. εγ , essentially Boyle's Law, where we revert to Newton's V =

λ/2π rather than Vr. On the return, imV

VFatforceel

atforceel+

=.. εγ . A similar argument applies for

the ratio of the elastic forces (or pressures) acting on the volume increments E and G :

HLVV

Eatforceelatforceel

−=

.. εφ and

KNVV

Gatforceelatforceel

−=

.. φγ respectively. It follows that the difference

of these elastic forces to the mean elastic force, which is the same at E, F, and G, is given by :

VKNHL

VKNHLV

KNVHLVKNHLV

KNVV

HLVV

Fatforceelquiescentatforceelatforceel −

=−

−−−

=−

−−

=−

2)(~

))(()(

... φγεφ .

Note that second order quantities are ignored, as they vanish in the limit. Now, ( ) ( ( ) ( )) , or , as 0 .

2HL KN R sin sin OM OMcos i. e.

HK R OI OPϑ Δϑ ϑ Δϑ ϑ Δϑ

Δϑ− + − −

= → →

Hence:

. ..)2/.(

...2

.. ,or

;.

.cos.

..

ϕϕπλχϑχ

εφφγ

ϑφγεφ

ΩΩΔ

constFatforceelquiescentatforceelatforceel

VOPOMHK

VHK

VKNHL

Fatforceelquiescentatforceelatforceel

−=⎥⎦

⎤⎢⎣

⎡−=

−

==−

=−

Now, λπ

λπεγϑ

λεγ

πϑ 2.~2.2or

22 xΔΔΔ

== . Hence,

. ..)2/(

..).( x. 22

2ϕϕ

πλρ ΩΩΔΔΔ constFatforceelquiescentxx

dxforceeld

dtxd

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡−==

Hence, calling P the pressure or the quiescent elastic force at F, and canceling the Δx which is synonymous with εγ, while setting Ωφ = x, we find that

s.h.m.for required as ,.)/2.(

22

2

22

2xx

kvxP

dtxd ω

λπρ−=−=−= In which case, .ρ

Pv = Newton did

not pursue his differential equation to this logical conclusion for some reason, and was content to note that the equation defined the same kind of motion as the cycloidal pendulum, although he immediately proceeds to use the above formula for the speed. His final proposition is to present an extreme case involving the s. h. m. of a cycloidal pendulum to corroborate his formula ; one gets the impression he was over-impressed with this particular kind of s. h. m., rather than seeing it rather as just another example of this kind of motion, and that he was not entirely convinced with his derivation of this proposition.



PROPOSITION XLVIII. THEOREM XXXVIII.

The speeds of pulses propagating in an elastic fluid are in the ratio composed from the direct proportion of the square root of the elastic force or pressure and the inverse proportion of the square root of the density ; but only if the same elastic force is supposed for the same proportional condensation [ i.e. the gases obey Boyle's Law]. Case. 1. [ Newton's explanation of why pulses or waves of differing intensities travel at the same speed in a medium.]

If the media are homogeneous, & the distances between the pulses in these media are equal amongst themselves, but the motion [i. e. the sound] is more intense in one medium than in the other, then the contractions and expansions of the analogous parts are as the same motions [i.e. one has a larger amplitude than the other]. The proportion of these intensities cannot be measured with accuracy. However, unless the contractions and dilatations are of greatly differing intensities, there will be no sensible error, and thus these can be used [for the measurement of physical quantities] with accuracy. But the elastic motive forces are in the ratio of the contractions & dilations ; & the velocities of the equal parts likewise generated are in the ratio of the forces [as the forces act for the same lengths of time]. Hence equal & corresponding parts of corresponding pulses are coming and going by contracting and dilating in their proportional intervals, with velocities which are in the ratio as the intervals, likewise are carried out : & therefore pulses, which in the time of one oscillation are made to progress a distance of one width, & always follow into the place of the nearest proceeding pulse, on account of the equality of the distances, with equal velocity can proceed in either medium. [We note that the amplitude cancels in the derivation presented in the previous theorem, but see notes below.]

Case 2. [Newton's explanation of why all wavelengths travel at the same speed in the same medium.]

But if the distances between the pulses or the widths are greater in one medium than the other, then we can put corresponding parts in place , and the proportional widths of the individual pulses that come and go can be described [i.e. we can compare the ratio of the wavelengths or pulse widths]: and the contractions and dilations of [each of] these are equal. Thus if the media are homogeneous, then these elastic motive forces by which the reciprocating motion is driven are also equal [i.e. the amplitudes of the pressure fluctuations are the same]. But the masses to be moved by these forces are in the same ratio as the widths of the pulses; & the corresponding wavelengths of the pulses as they come and go are in the same ratio. But the time of one complete reciprocal motion is composed from the ratio of the square root of the mass & the square root of the interval, and thus as the interval. [Thus, the oscillation time T is proportional to the width of the pulses of wavelength λ, or λ α T or 1/f] . But pulses perform their reciprocal motion or s.h.m in a time equal to the passing of one width, that is, the space and the time intervals are proportionals that advance in step, and hence the velocities are equal. [See notes below.]



Case 3. [The ratio of the speeds in differing media.]

Therefore all the pulses travel at the same speed in media with the same density and elastic forces. But if either the density or the elastic force of the medium is increased, since the motive force is increased in the ratio of the elastic force, & the matter to be moved is increased in the ratio of the density ; the time, by which the same motions are driven from the previous situation, will be augmented in the ratio of the square root of the density, and diminished in the square root ratio of the elastic force. And therefore the velocity of the pulses will be in the ratio composed from the ratio of the inverse of the square root of the density & directly in the ratio of the square root of the elastic force. Q.E.D.

This proposition is more apparent from the following construction. Notes on Prop. 48 : Case 1. We are to imagine two media with the same pressure and density, and waves of some wavelength travel at the same speed in each medium. However, pulses with different intensities in the two media are considered. We try to understand why the speeds of the pulses are equal in the two media and independent of the amplitude of the oscillations, using s.h.m. Using the pendulum analogy of s.h.m, the ratio of the elastic motive forces on corresponding elements is the same as the ratio of the amplitudes, and since these forces act for the same lengths of time on the corresponding elements, during which time the pulses move forwards a distance equal to the inter-pulse displacement, then the velocity of propagation is the same. [However, the ratio of the maximum velocities of the elements is proportional to the ratio of the maximum displacements, as there is a greater pressure associated with the more intense pulses. There is hence a distinction to be drawn between the phase velocity and the maximum speed associated with the s.h.m.] Case 2. We are to imagine two media with the same pressure and density, and thus at least waves of one wavelength travel at the same speed in each medium. However, pulses with different widths or wavelengths in the two media are considered, each wavelength of constant width in its medium. We try to understand why the speeds of the pulses are nevertheless equal in the two media using s.h.m ideas. The s.h.m motion of the small incremental widths have the same amplitude in each case, otherwise the waves will have differing amplitudes and intensities. However, the longer wavelength requires more of the elementary oscillators, and the pressure differences across each of the incremental widths is thus less for the longer wavelength, as the total pressure fluctuation is the same for both wavelengths, which supplies the motive force on the element. Newton, however, does not consider the individual elements as such at this stage, but focuses his attention on the ratio of the whole condensed or rarefied pulses at some point, and returns to his pendulum idea of s.h.m. In this case, he considers the period T of an s.h.m to depend on the mass m to be moved, and the force constant k to effect the motion, for which k

mT π2= . By simply adding the number of elementary incremental oscillators, the masses to be moved for the long and short wavelengths λL and λS are in the ratio

L

S

L

Smm

λλ= . Now, regarding the force

constant k, or the spring constant of the air, it can be taken as proportional to the excess



pressure p, and inversely as the wavelength λ; or, λpk = . Hence, the periods for the short

and the long waves are as : .L

S

L

S

L

S

L

S

L

s

L

L

s

s

kk

mm

km

km

L

STT

λλ

λλ

λλ =×=== Or, the frequencies vary

inversely as the ratio of the wavelengths, as required. Case 3. The speed of propagation v of a pulse in a medium with elastic force or pressure p and density ρ is given by ρ

pv = , to be finally proved in the next Proposition. Thus, if

the density is increased from ρ1 to ρ2, for the same pressure, then the associated speeds are in the ratio

1

2

2

1ρρ=v

v ; while if the pressure is increased from p1 to p2, for the same

density, then 2

1

2

1pp

vv = . If both pressure and density increase in the same ratio, then there

is no change in the speed of the pulses. In general, if both pressure and density are allowed to change, but in different ratios, (as for example, for media such as ideal gases at

different temperatures), then 1 1 1 1 2

2 2 2 2 1

v p / pv p / p . ,ρ ρ

ρ ρ= = which is Newton's comment.]

PROPOSITION XLIX. PROBLEM XI.

To find the velocity of pulses for a given density and elastic force of medium.

We can imagine the medium to be compressed by the incumbent weight of the air in the manner of our air ; and let A be the height of the homogeneous medium which is equal to the incumbent weight [from the quiescent point], and the density of which is the same as that of the compressed medium, in which the pulses are propagated. Moreover, a pendulum is considered to be set up, the length of which is A between the point of suspension and the centre of oscillation [thus, the radius of the generating circle of the cycloid is A/2, and the whole arc length is 2A.] : and in the time that the pendulum executes a complete to and fro oscillation, a pulse will travel a distance equal to the circumference of the circle described by the radius A. [Referring back to Prop. 47, and to the phase diagram; note that PS = λ/2.]

For in agreement with what has been stated in proposition XLVII, if some physically narrow region EF is pushed to the limits of the oscillatory motion which are located at P and S, with the vibrations within the space PS to be described by the single element, [thus, the motion of the single increment occupies the whole amplitude or is responsible for the entire s. h. m.] then the elastic force [or pressure] is itself equal to the weight of the air, and it will perform these individual vibrations in the same time that the oscillations can be performed on the cycloid, the whole perimeter of which is equal to the length PS [ = 2A]: since equal forces can push small bodies [of equal mass] through equal distance in the same time. Whereby as the times of oscillations are in the square root ratio of the lengths of pendulums, and the length of the pendulum is equal to half of the whole arc of



EGP

O

S

A/2

A

A/2

P S

O

the cycloid ; the time of one vibration in the air to the time of the oscillation of the pendulum, the length of which is A, is in the ratio of the square root of the length 2

1 PS (or PO) to the length A.

But the elastic force, by which the incremental length EG is forced, present in its extreme places P and S, is (as in the demonstration of proposition XLVII) to the total force of this as HL - KN to V, that is (as the point K thus falls in P) as HK to V: and that total force, that is the incumbent weight by which the incremental line EG is compressed, is to the weight of the elemental line as the altitude A for the weight of the incumbent air to the incremental line of length EG; and thus from the equality, the force by which the incremental line EG is pushed in its locations P and S, is to the weight of the incremental line as HK × A to V × EG, or as PO × A to VV, for HK is to EG as PO to V. Whereby with the times, for which equal bodies are pushed through equal distances, are reciprocally in the square root ratio of the forces [Essentially at2 is constant, or Ft2/m is constant, giving t α 1/√F.], the time of one vibration will be, from the force exerted by the pressure, to the time of the vibration, for the force due to the weight [in the pendulum case], in the square root ratio VV to PO × A, and thus to the time of the oscillation of the pendulum of which the length is A in the square root ratio VV to PO × A and the square root ratio PO to A jointly; that is, in the ratio all together as V to A. [The starting point P or S of the motion is chosen; there is no difference for other points in the motion as the displacement cancels; see the note.] But in the time of one vibration composed from the to and fro motion, the pulse by proceeding makes its own length BC [for λ = 2 × PS. Therefore the time, in which the pulse travels through the distance BC, is to the time of one oscillation composed from the to and fro motion of the pendulum, as V to A, that is, as BC to the circumference of the circle of which the radius is A. Moreover the time, in which the pulse travels through the distance BC, is to the time by which it travels the length of this equal circumference, in the same ratio; and thus for the time of such an oscillation the pulse travels a distance equal to the circumference of this circle. Q.E.D.

Notes on Proposition 49 : Newton initially

considers the height of an atmosphere of uniform density that results in the pressure observed at ground level. We note in passing that this height h is related to the height k of mercury in a barometer according

to the elementary rule kh Hgair ρρ = , or kh airHg )/( ρρ= , or the ratio of the specific gravity of mercury to air times by the length of the mercury column, as you would expect if you is not too concerned about finer details. Newton calls this height A.



Let us set up the oscillating atmosphere envisaged : The height of the homogeneous atmosphere is A, and O is taken as the half-way point. A relatively small segment of air of quiescent length EG is part of the oscillating air mass of amplitude OS = OP. The situation at some intermediate stage ascending is shown at on the left-hand side of the diagram L. The explanation relies on Prop. 47 :

VKNHL

pressurequiescentEGelementonforceunbalanced

Fatforceelquiescentatforceelatforceel −

==−.

.. φγεφ . In the present case, the

element is at an extreme position, in which case KN is zero, and the length HL is

approximately equal to the arc length HK; hence, V

HKpressurequiescent

EGelementonforceUnbalanced→ .

Also, the ratio

.EGA =

×=

EGggA

EGlineofweightPatEGonpressureorforceincumbent

ρρ Hence,

EGA

VHK

EGofweightpressurequiescent

pressurequiescentEGelementonforceUnbalanced

×=× . Again, from Prop.47 :

, ..)2/(

..).(- x. 22

2ϕϕ

πλρ ΩΩΔΔΔ constFatforceelquiescentxx

dxforceeld

dtxd

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡−==

we have the ratio of the unbalanced force to the mass of the element :

... pendulum, for the whilemass;air for the .. : hence ;

.. ./../)(- ./.

2

2

22

2

2222

2

xAg

dtxdx

VAg

dtxd

EGVAgHK

OPVgAOP

VgAxconstx

VgA

dxPdEG

dtxdEG

−=⎥⎦

⎤⎢⎣

⎡−=

××

=

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡−→−=⎥

⎦

⎤⎢⎣

⎡−== ρρρρρ

Thus, the period of oscillation of the air gA

VTair .2

2π= , while the period of the cycloidal

pendulum is given by : gATpen π2. = . Hence. .

2/

2

2

. AAV

AVTT penair π

λ=== Note that

Newton always uses ratios, so there is no need to worry about constant factors such as g and 2π that have to be inserted in absolute measurements and calculations; indeed the use of π as a ratio had not been introduced at this time.

Corollary 1. The velocity of the pulses is the same as that which weighty bodies acquire by falling under the acceleration of gravity, and in their case through half the height A. For the time in this case, for the velocity to be acquired by falling, the pulse travels through the space equal to the whole interval A; and thus the time of the oscillation composed from one coming and going the space traveled through is equal to the circumference of the circle A described. Indeed the ratio of the time to fall to the time of the oscillation is as the radius of the circle to the circumference of the same.

Note on Cor. 1 : For the time for a body to fall a vertical distance A/2 is given by

gATbody = , while the time for the pulse to perform half an oscillation and travel from P to



S is given by gA

VTair .2/

2π= ; hence, .1

2/.:2/:

2====

λππ A

VA

gAV

gATT airbody Also, the

pendulum bob in these times traverses a space in the ratio

.1.22

2/

2

2

. ncecircumferediameter

AA

AAV

AVTT penair ======

πππλ

Corollary 2 : Hence since that height A shall be directly proportional to the elastic force and in inverse proportion to the density of the fluid; the velocity of the pulse will be in the ratio composed from the square root of the inverse of the density and in direct proportion to the square root of the elastic force. Note on Cor. 2 : If the periodic time T for a complete oscillation is inversely proportional to √A, and P = ρgA , then the velocity is proportional to 1/T or √(P/ρ). As Chandrasekhar points out, this theorem was probably added by Newton as he was not entirely satisfied with Prop. 47, which he did not work through to its conclusion. However, this author makes claims for what Newton has done which bear little resemblance to reality - there is a distinct lack of sophisticated mathematical machinery in Newton's work, although the intuitive ideas are there. The mathematical structure describing phase velocity was not in place at the time, and Newton's work presumably set this theory in motion. We may note in passing its use of an extreme amplitude of around 8000m for the height of the isotropic atmosphere to give a known pressure, as Newton sought known numbers to use in his equation as a check, with which one could associate a wave with a period of some 25 seconds, more in the realms of internal gravitational waves in the atmosphere than sound waves.

PROPOSITION L. PROBLEM XII.

To find the lengths of the pulses. The number of vibrations in a given time need to be found for the body which is

exciting the pulses. The distance which the pulses are able to traverse in this time is divided by this number, and the fraction of the length found is the width of one pulse. Q. E. I.

Note : Thus the well-known result for the phase velocity v = fλ comes into being.

Scholium.

The most recent propositions consider the motion of light and of sound. Indeed light is propagated in straight lines, without interaction (by prop. XL & XLII.). Hence, since sounds arise from the vibrations of bodies, they are nothing other than pulses propagated in the air, by prop. XLIII. This is confirmed from the vibrations which they cause in bodies presented to them, but only if they are strong and deep, such as the sounds of small drums. For quicker and shorter vibrations are more difficult to be excited. But it is well known also that any sounds can interact with the strings of musical instruments and excite vibrations. This is also confirmed from the velocity of sound. For since the specific



gravity of rainwater and quicksilver are in turn as 1 to 3213 roughly, and where the height

of mercury in a Barometer reaches a height of 30 English inches, the specific gravity of air and of rainwater are in the ratio 1 to 870 roughly; the specific gravity of air and quicksilver are as 1 to11890. Then since the height of quicksilver is 30 inches, the height of the air in a uniform atmosphere, which our air is subject to in compression, is 356700 inches, or 29725 English feet. This is the height that we have called A in the construction of the above problems. The circumference of the circle described by a radius of 29725 feet is 186768 feet. And since it is well-known that a pendulum 5

139 inches long results in a complete oscillation in a time of two seconds, then a pendulum 29725 feet long or 356700 inches ought to complete a similar oscillation in a time of 4

3190 seconds. Therefore in that time the sound should progress a distance of 186768, and thus in a time of one second sound should travel 979 feet.

In the computation presented here, no account is made for other effects, such as the density of solid particles in the air, through which the sound certainly is propagated. Since the weight of air is to the weight of water as 1 to 870, and salts are nearly twice as dense as water ; if the particles of air are put to be roughly the same density as the particles of water or salt, and the rareness of air arises from the intervals between the particles: then the diameter of the air particles will be as the interval between the centres of the particles, as 1 to 9 or 10 roughly, and to the interval between the particles as 1 to 8 or 9. Hence to the 979 feet in the above calculation, one may add 9

979 or 109 feet roughly, to the distance that sound travels in a time of one second, on account of the density of particles in the air : & thus the distance that sound travels in a time of one second is made to be roughly 1088 feet.

To these you may add the effect of vapours hidden in the air, since they are of a different tone and elastic nature they may or may not participate in the motion of sound that is propagated through the air. But from these quiet sources, the motion is propagated more quickly than by the air alone, and that in the ratio of the square root of the lesser matter. For if the atmosphere is made up from ten parts air and one part of vapor, the speed of sound is faster in the ratio of the square root of 11 to 10, or altogether around the ratio 21 ad 20, than if the sound is propagated by eleven parts of pure air : and thus the motion of the air found above is increased in this ratio. From which the speed of sound is agreed upon to be 1142 feet in one second.

Thus these ought to have an effect in springtime and autumn, when the air is rarefied by the temperate heat and the pressure is increased. In wintertime, when the air is condensed by the cold, and its pressure is lowered, the speed of sound should be less in the square root ratio of the densities, while in summertime in turn, the speed should be increased.

Moreover, it is agreed upon by experiment that the distance gone in a time of one second is more or less London feet 1142, and truly 1070 Parisian feet.

With the speed of sound recognised, the intervals between the pulses can also become known. Certainly Sauveur has found from measurements made in his experiments, that an open pipe, the length of which is more or less five Parisian feet, send forth a sound of the same tone as the sound of strings which are vibrating at a rate of a hundred times in one second. Thus, there are more or less one hundred pulses in a space of 1070 Parisian feet,



which sound travels through in a time of one second ; hence a single pulse takes up a space of 10

710 Parisian feet, that is, around twice the length of the tube. Thus, it is the same for pulses of all lengths from the sounds produced by tubes, for they are equal to twice the lengths of the open ended tubes.

Again since sounds stop with the motion of the vibrating body when we stand nearby, but not for a long time when we stand a long way from the source of the sound, which is apparent from the corollary to proposition XLVII of this book. Moreover why sounds are greatly increased in volume by deep sounding trumpets is apparent from these principles. Indeed the reciprocal motion of all recurring individual pulses is usually increased by the source of the vibration. Moreover, the motion of sound is impeded in trumpets, to be emitted later and louder, and therefore a new individual motion is returned later more loudly. And these are the main phenomena associated with sound.

[Thus, Newton remained unaware of the true source of the error in his analysis; this

was eventually corrected by Laplace when the effect of heat on a gas was much better understood, and the adiabatic form of the gas law was applied, rather than Boyle's Law.]


Book II Section IX. Translated and Annotated by Ian Bruce. Page 704

SECTION IX.

Concerning the circular motion of fluids.

HYPOTHESIS. The resistance, which arises from the deficiency of lubricity of the parts, with all else being equal, is proportional to the velocity, by which the parts of the fluid may be separated from each other.

PROPOSITION LI. THEOREM XXXIX. If an infinitely long solid cylinder may be rotating in an infinite uniform fluid with a given uniform motion about an axis in place , and the fluid may be disturbed into an orbit by the impulse of this alone, and each and every part of the fluid will persevere uniformly in its motion, I say that the periodic times of the parts of the fluid shall be as their distances from the axis of the cylinder. Let AFL be the cylinder driven uniformly about the axis S acting on the orbit, the fluid may be separated by the concentric circles BGM, CHN, DIO, EKP, etc. into innumerable concentric solid orbits of this, with the same thickness. And because the fluid is homogeneous, the impressed forces of the contiguous orbits made between each other will be (by the hypothesis) as the translations of these between each other [i.e. their relative velocities], and as the contiguous surfaces on which impressed forces are made. If the impressed force on some orbit is greater or less from the concave side rather than from the convex side; the stronger impressed force will prevail, and the motion will be accelerated or decelerated, so that it is corrected into the direction of the motion itself, or into the contrary. Hence so that each and every orbit may persevere in its own regular motion, the impressed forces must be equal on each part in turn, and must be made on each in opposing directions. From which since the impressed forces are as the contiguous surfaces and the translations of these one to the other, [Thus, the constant tangential frictional force on an orbit is assumed to vary as the relative velocity between neighbouring surfaces, and as the area of either surface; thus

2 or ddrrel.vel. area const; r r rω δ δ× = ∝ , as the area in this case is that of a cylindrical

surface of unit length, and the relative velocity is the limiting difference in the angular speeds;] the translations will be inversely as the surfaces, that is, inversely as the distance of the surfaces from the axis [i.e., in this case, as the area in contact is proportional to the radius of the orbit]. Moreover, the applied differences of the angular motions about the axis are



as these translations to the distances, or as the translations directly and the distances inversely ; that is, as with the ratios taken together, inversely as the square of the distances. [i.e. the y and x coordinates in the adjoining diagram.] [Thus, according to B. & R., p. 296, we may consider the friction between two adjacent solid cylinders. Suppose and ω ω δω+ are the angular velocities of two consecutive cylinders, and let r be the radius of their common surface, then the relative velocity of the two cylinders is ( ) d

drr r r rωω δω ω δ+ − = ; the internal friction between the rotating fluid cylinders is proportional both to the difference (or differential) of the velocities, and to the common area in contact, which is proportional to the radius in this case, giving the frictional force proportional to 2 d

drr rω δ . Now this quantity must remain constant, as we

have steady state conditions, and hence 2 ddrr ω α= − , for some constant α . This equation

may be integrated to give rαω β= + , where and α β are constants depending on the

angular velocities of the inner and outer cylinders. Hence, from known values of and rω , the constants and α β can be found. This is the integration performed next by Newton, where the outer limit is infinite and so β is zero. In which case r

αω = , and Newton's result follows.] Whereby if, to the parts of the indefinite right line SABCDEQ, the perpendiculars Aa, Bb, cC, Dd, Ee, &c. may be erected inversely proportional to the squares of the individual parts SA, SB, SC, SD, SE, etc., and through the ends of the perpendiculars a hyperbolic [like] curve may be considered to be drawn; the sum of these differences [of the angular speeds] will be, that is, the whole angular motion, as the corresponding sum of the lines Aa, Bb, Cc, Dd, Ee, that is, if for the fluid medium being uniformly put in place, the number of the orbits may be increased and the width may diminished indefinitely, as the analogous hyperbolic areas for these sums AaQ, BbQ, CcQ, DdQ, EeQ, etc. And the times are inversely proportional to the angular motions, also they will be inversely proportional to these areas. Therefore the periodic time of any particular curve D varies inversely as the area DdQ, that is (by the known quadratures of the curves), directly as the distance SD. Q.E.D. Corol. I. Hence the angular motions of the particles of the fluid are inversely as the distances of these from the axis of the cylinder, and the absolute velocities are equal. [That is, the whole mass rotates about the vertical axis with the same tangential velocity, whatever the radius; Newton later acknowledges in a Scholium that this state of affairs is unstable, unless some other condition applies. It is evident that Newton had constructed such a device, but he has not given any details here.] Corol. 2. If the fluid may be contained in a cylindrical vessel of infinite length, and another cylinder may be contained within, but each cylinder may be revolving about their common axis, and the times of the revolutions shall be as the radii of these, and each part



of the fluid may continue in its own motion : the period times of the individual parts will be as the distances of these from the axis of the cylinders.

[Thus, from rαω = we have 1 1

2 2

T rT r= .]

Corol. 3. If some common angular motion may be added to this kind of motion by the cylinder and the fluid, because by this new motion the friction of the mutual parts of the fluid does not change, the motions of the parts among themselves will not be changed. For the translations of the parts from one to the other depend on friction. Any part will persevere in that motion which, with the frictional forces acting in opposite directions on each side, may not be accelerated or retarded more. Corol. 4. From which if all of the angular motion of the external cylinder may be removed from the whole system of the cylinders and fluid, the motion of the fluid will be had for a cylinder at rest. Corol. 5. Therefore if with both the fluid and the exterior cylinder at rest, the interior cylinder may be revolving uniformly; it will communicate a circular motion to the fluid, and gradually it will be propagated through the whole fluid ; nor will the first kind of motion cease to increase until the individual parts may acquire the motion defined by corollary four. Corol.6. And because the fluid tries to propagate its motion more widely at this stage, it will carry the exterior cylinder around by its force around unless strongly retained ; and its motion will be accelerated until the periodic times of both cylinders are equal to each other. Because if the exterior cylinder may be strongly retained, it will try to retard the motion of the fluid ; and unless the interior cylinder be pressed on by some extrinsic force so that motion may be conserved, it will have the effect of gradually slowing that down. All of which can be proven in deep water at rest.

PROPOSITION LII. THEOREM XL. If a solid sphere, in an infinite uniform fluid, may be rotating with a uniform motion about a given axis in place, and the orbiting fluid may be agitated by the impulse of this alone, and moreover every part of the fluid may continue uniformly in its motion : I say that the periodic times of the parts of the fluid will be as the squares of the distances from the centre of the sphere. Case 1. Let AFL be the sphere driven uniformly in obit about the axis S, and the fluid [in a plane normal to the axis] may be separated into innumerable concentric spheres of equal thickness by the concentric circles BGM, CHN, DIO, EKP, &c. Moreover suppose these orbits to be solid, and because the fluid is homogeneous, forces of contact of the orbits made between each other, will be (by hypothesis) as the translations of these from one to the other



[i.e. the relative distance gone per unit time between the shells, or the velocity gradient, in modern terms], and as the surfaces in contact on which the forces act. If the force impressed on some orbit is either greater or less from the concave part as from the convex part; the stronger impressed force will prevail, and the velocity of the orbit will either accelerate or decelerate, as it is returned by its motion in the same or contrary direction. Hence so that each one of the orbits will persevere uniformly in its orbit, the forces impressed from each part must become equal between each other, and are made in opposite directions. Hence since the impressions shall be as the contiguous surfaces and as the translations of these relative to each other, that is, inversely as the square of the distances of the surfaces from the centre. But the differences of the angular motions about the axis are as these applied translations to the distances, or as the translations directly and the distances inversely, that is, with the ratios taken jointly, as the cube of the distances inversely. Whereby if the perpendiculars Aa, Bb, Cc, Dd, Ee, &c. may be erected, the perpendiculars of SA, SB, SC, SD, SE, &c. are themselves inversely proportional to the cubes of the innumerable individual parts of the right line SABCDEQ; the sum of the differences, that is, the whole angular motion will be as the corresponding sum of the lines Aa, Bb, Cc, Dd, Ee : that is (if the number of orbits may be increased and the width may be diminished indefinitely towards uniformly establishing a fluid medium) as the hyperbolic [like] area from these analogous sums AaQ, BbQ, CcQ, DdQ, EeQ, &c. And the periodic times inversely proportional to these angular motions also will be inversely proportional to these areas. Therefore the periodic time of any orbit DIO will be inversely as the area DdQ, that is, from the known quadratures of the curves, directly as the square of the distance SD. That which was wished to show in the first place. [As in the previous proposition, with the surface area now proportional to r2, we may consider the frictional force to become as 3 d

drr rω δ , which is constant, and hence 3 d

drr ω α= − . This may be integrated to give 2rαω β= + , where we consider the angular

integration to be treated as in case 2 below ; as the fluid is infinite, this integration becomes 2r

αω = , as required. ]

Case 2. As many right lines as you wish may be drawn from the centre of the sphere, which may contain a given angle with the axis, mutually greater than each other by a given angle, and from these right lines rotated about the axis, the orbits are to be cut into innumerable rings; and each and every ring will have four rings contiguous to itself, one interior, another exterior, and two lateral ones. There cannot be friction from any inner and outer rings in the motion, except by that acting in equal and opposite directions, just as I have treated in the first case. This is apparent from the demonstration of the first case. [Thus, any cross-section normal to the axis can be treated as in the first case.] And therefore any series of rings going from the sphere along straight lines to infinity, will be moved by the law of the first case, unless they may be impeded by friction from the sides. But in the motion made, according to this law, the friction of the rings on the sides is zero ; and thus no motion may be impeded, so that it may become less by this law. If annuli which are equally distant from the centre, either may be rotating swifter or slower around the poles than around the ecliptic; the slower ones may be accelerated, and the velocities



may be retarded from the friction of the motion, and thus the periodic times always approach towards equality, by the law of the first case. Therefore here the friction does not impede that motion less than following the law of the first case, and therefore that law will be obtained : that is, the times of the individual annuli will be as the squares of the distances of these from the centre of the sphere. Which was the second case to be demonstrated. [Clearly there is a problem with a fluid behaving in this way: for the angular dependence of the motion has not been resolved. This is of course a rather difficult problem, and Newton proceeds to argue his way out of difficulties by considering reduced causes and reduced effects .] Case 3. Now each and every annulus may be divided by transverse sections into innumerable particles, constituting the fluid substance absolutely and uniformly ; and because these sections may not regard the law of circular motion, but only bring about the circulation of the fluid, the circular motion will persevere as at first. All the annuli from these sections as the smallest roughness and the friction force are changed minimally, either will be changed equally, or not be changed equally. And with the proportion of the causes remaining, the proportion of the effect will remain, that is, the proportion of the motions and of the periodic times . Q. E. D. Moreover, since the circular motion, and hence the centrifugal force arising, shall be greater at the ecliptic than at the poles ; some cause will have to be present by which the individual particles may be retained in their circles ; lest the matter, which is at the ecliptic, may not always recede from the centre and from the exterior vortices may move to the poles, and thence by the axis may be returned to the ecliptic by circulating perpetually. Corol. 1. Hence the angular motion of the parts of the fluid around the axis of the sphere, are inversely as the squares of the distances from the centre of the sphere, and the absolute velocities inversely as the same applied squares to the distances from the axis. Corol. 2. If a sphere may be revolving with a given uniform motion around an axis in position in a fluid to be similarly at rest, the motion will be communicated to the fluid in the manner of a vortex, and this motion will be propagated little by little to infinity ; and nor will the first motion cease in the individual parts of the fluid accelerated, as the periodic times of the individual parts shall become as the squares of the distances from the centre of the sphere. Corol. 3. Because the interior parts of the vortices on account of their greater velocity rub against and exert a force on the exterior parts, and perpetually they communicate the motion to these by that action, and these exterior ones likewise transfer the quantity of motion by that action at this stage to other more exterior ones, and by this action they clearly maintain an invariant quantity of motion, it is apparent that the motion is perpetually transferred from the centre to the circumference of the vortex, and may be absorbed by the infinite circumference. The matter between any two concentric spherical surfaces will never be accelerated by the vortex, because there all the motion taken from the interior matter is always transferred to the outer matter. Corol. 4. Hence towards the conservation of the vortex by moving constantly in the same state, some principle of action is required, by which the sphere may always accept the same quantity of motion, that will be impressed on the matter of the vortex. Without



such a kind of principle, it is necessary that the sphere and the interior parts of the vortex, always propagating their motion into the exterior parts, and not receiving any new motion, are slowed down little by little and forced to stop in the orbit. Corol. 5. If another sphere might drift towards this vortex at a certain distance from the centre of this, and meanwhile it may be rotating constantly at an inclination by some force about a given axis ; by its motion it may snatch some fluid from the vortex : and at first this new and very small vortex may be rotating together with the sphere around the centre of the other, and meanwhile its motion will spread out wider, and little by little it will be propagated to infinity, according to the manner of the first vortex. And by the same reasoning, by which the sphere may be seized by the motion of the other sphere, also by its motion it may seize the other sphere, thus so that the two spheres may be revolving about some point between the two, and on account of that mutual circular motion they may fly away from each other, unless they may be restrained by some force. Afterwards if the constantly impressed forces may be stopped, by which the spheres may persevere in their motions, and all may be permitted by the laws of mechanics, the motion of the spheres will become less (on account of the reason I have assigned in Corol. 3. and 4.) and the vortices at last will come to rest. Corol. 6. If several spheres may be revolving constantly in given locations with certain velocities about given axes in position, just as many vortices may be made going off th infinity. For the individual spheres by the same reason, by which some one may propagate its motion to infinity, also they will propagate their motions to infinity, thus so that any part of the infinite fluid will be disturbed by that motion which may result from the actions of all the spheres. From which the vortices will not be defined within certain boundaries, but will extend out a little mutually between themselves, and the spheres by the actions of the vortices will always be moving mutually from their positions, as has been explained in the above corollary ; and nor may it be clear how they may keep their positions among each other, unless retained by some force. Moreover with these forces ceasing, which constantly impressed on the spheres preserve these motions, the matter on account of the reason assigned in the third and fourth corollaries, will relax little by little in the vortices and cease to act. Corol. 7. If the fluid similarly may be enclosed in a spherical vessel, and may be disturbed into a vortex by a sphere present at the centre rotating uniformly, moreover the sphere and the vessel may be revolving around in the same direction, and the periodic times of these shall be as the squares of the radii : the parts of the fluid later will continue in their motions without acceleration or retardation, so that the periodic times of these shall be as the squares of the distances from the centre of the vortex. No other constitution of vortices shall be able to remain. Corol. 8. If the vessel, inclosing the fluid, and the sphere may maintain this motion and in addition may be rotating with a common angular motion about some given axis ; because in this new motion the friction of the parts of the fluid shall not be changed from one to the other, nor will the motion of the parts among themselves be changed. For the translations of the parts amongst themselves depends on the friction. Any part will persevere in that motion, by which it shall not be retarded more by the friction of one side than accelerated more by the friction from the other.



Corol. 9. From which if the vessel may be at rest and the motion of the sphere may be given, the motion of the fluid will be given. For take a plane crossed by the axis of the sphere and to be revolving in a contrary motion ; and suppose the sum of the times of the revolution of this and of the revolutions of the sphere to be to the time of revolution of the sphere, as the square of the radius of the vessel to the square of the radius of the sphere : and the periodic times of the parts of the fluid with respect to its plane will be as the squares of their distances from the centre of the sphere. Corol. 10. Thus if the vessel either may be moving around the same axis with the sphere, or around some different axis with some given velocity, the motion of the fluid will be given. For if the angular motion of the vessel may be taken away from the whole system, all the same motions will remain between themselves which were given at first by Corol. VIII. And this motion will be given by Corol. IX. Corol. 11. If the vessel and the fluid are at rest and the sphere may be revolving with a uniform velocity, the motion will be propagated little by little through the whole fluid in the vessel, and the vessel may be driven around unless strongly detained, nor will the fluid and the vessel cease to accelerate as at first, until the periodic times of these shall be equal to the periodic times of the sphere. Because if the vessel may be detained a little by some force or may be revolving at some constant and uniform motion, the medium may arrive little by little at the state of the motion defined in Corollaries VIII, IX, and X, nor will it persevere in any other state. Then truly if, with these forces ceasing by which the vessel and the sphere will be rotating with a certain motion, the whole motion of the system may be allowed by the laws of mechanics; the vessel and the sphere will act in turn on each other by the fluid medium, nor will their first motion cease to propagate through the fluid between each other, so that the periodic times of these are equal to each other, and the whole system becomes the image of a single solid body revolving at the same time.

Scholium. In all these I suppose the fluid consists of matter of uniform density and fluidity. For such a fluid it is in which the sphere likewise shall be able to propagate by the same motion, in the same time interval, similar and equal motions, always at equal distances to each other, wherever it may be placed in the fluid. Indeed the matter by its own circular motion may try to recede from the axis of the vortex, and therefore presses on all the further matter. From this compression there shall be stronger friction of the further parts and the separation from each other shall be more difficult ; and as a consequence the fluidity of the matter may be diminished. Again if the parts of the fluid are somewhere coarser or larger, the fluidity will be less there, on account of the fewer surfaces into which the parts may be separated from each other in turn. In cases of this kind the deficiency in fluidity either the lubricity or viscosity of the parts I suppose to be restored in some other way or by some other condition. Unless this may be done, the matter where it is less fluid will cohere more and will be slower, and thus the motion is undertaken more slowly and will propagate further than by the account given above. If the shape of the vessel may not be spherical, the particles will be moving along non-circular lines, but by conforming to the same shape as the vessel, and the periodic times will be as the squares of the average distances from the centre approximately. In the parts between the centre and the circumference, where the spaces are wider, the motions will be slower, and



where faster where more restricted, nor yet will the faster particles wish reach the circumference. For the arcs describe smaller curves, and trying to recede from the centre will not be diminished by the decrease of its curvature, as it will be increased by the increase of the velocity. By going from the narrower spaces into the wider they will recede a little more from the centre, but they will be slowed down by this recession, and by approaching later from the wider to the narrower spaces they will be accelerated, and thus the individual particles will be retarded and accelerated in turn for ever. Thus these will be held in a rigid vessel. For the constitution of vortices in an infinite fluid is known by corollary six of this proposition. Moreover I have tried to investigate the properties of vortices in this proposition, so that I might investigate if by some account or other celestial phenomena might be able to be explained by vortices. For the phenomenon is, that the periodic times of revolution of the planets around Jupiter are in the three on two ratio of their distances from the centre of Jupiter, and the same rule prevails with the planets which are revolving around the sun. Moreover these rules prevail as the most accurate on each of the planets, to the extent that astronomical observations can provide at this time. And thus if these planets may be carried by revolving around Jupiter and the sun in vortices, these vortices also will have to be revolving by the same law. Indeed the periodic times of the parts of the vortices will be emerging in the square ratio of the distances from the centre of motion : nor can that ratio be diminished and reduced to the three on two, unless either the matter of the vortex there shall be more fluid that is at a greater distance from the centre, or the resistance, which arises from the deficiency of the lubricity of the parts of the fluid, from the increase in the velocity by which the parts of the fluid may be separated in turn, may be increased in a greater ratio than that by which the velocity may be increased. Of which still neither the one nor the other can be considered to be in agreement. The thicker and less fluid parts will be trying to reach the circumference, except the heavy parts at the centre ; and it is very likely that, even if for the sake of demonstrating such a hypothesis I might propose at the beginning of this section, that the resistance should be proportional to the velocity, yet the resistance shall be in a smaller ratio than that is of the velocity. With which conceded, the periodic times of the parts of the vortex will be in a ratio greater than in the duplicate ratio of the distances from its centre. So that if the vortices (as it is the opinion of some) may be moving faster near the centre, then slower as far as to a certain boundary, then anew faster near the circumference ; certainly neither the three on two ratio [sesquiplicate] nor some other ratio can be obtained and determined with certainty. And thus philosophers will have to consider by what agreement that phenomenon of the three on two ratio may be explained by vortices.

PROPOSITION LIlI. THEOREM XLI.

Bodies, which carried by a vortex return in orbit, are of the same density as the vortex, and are moved by the same law with its parts as far as the velocity and the course determined. For if some very small part of the vortex may be supposed to be congealed, the particles of which or physical points maintain a given situation among themselves : this,



since neither as far as its density, nor as far the insitu force or its figure may be changed, will move by the same law as before: and on the contrary, if a frozen and solid part of the vortex shall be of the same density with the rest of the vortex, and it may be dissolved into the fluid ; this will be moving by the same law as before, unless perhaps particles of this now made fluid may move amongst themselves. Therefore the motion of particles amongst themselves may be ignored, being considered as nothing to the whole progressive motion, and the whole motion will be as before. But the motion likewise will be as the motion of other parts of the vortex equally distant from the centre, therefore so that a solid dissolved in the fluid shall be a part of the vortex just the same as the other parts. Hence a solid, if it shall be of the same density as the matter of the vortex, will be moving with the same motion as the parts themselves, in matter nearby being relatively at rest. If it shall be denser, now it will try more than at first to recede from the centre of the vortex ; and thus that force of the vortex, by which at first it was being held in place in equilibrium in its orbit, now being overcome, it will recede from the centre and describe a spiral on revolving, no more returning in the same orbit. And by the same argument if it shall be rarer, it will approach towards the centre. Therefore it will not return in the same orbit unless it shall be of the same density as the fluid. But in that case it has been shown, that it would be revolving by the same law with the parts of the fluid equally distant from the centre of the vortex. Q.E.D. Corol. 1. Hence a solid which is revolving in a vortex and always returns in the same orbit, is relatively at rest in the fluid in which it floats around. Corol. 2. And if as long as the vortex shall be of a uniform density, the same body can be revolving at any distance from the centre of the vortex.

Scholium. Hence it is clear that the planets cannot be carried by vortex bodies. For the planets following the Copernican [Newton means Kepler here] hypothesis are carried around the sun, revolving in ellipses with the sun at a focus, and with the radii drawn to the sun describing areas proportional to the times. But the parts of a vortex are unable to revolve in such a motion. AD, BE, and CF may designate three orbits described around the sun S, of which the outermost circle CF shall be concentric with the sun, and the aphelions of the two interior shall be A, B and the periheliums D, E. Therefore a body that is revolving in the orbit CF, with the radius drawn to the sun by describing areas proportional to the times, will be moving with a uniform motion. But a body, which is revolving in the orbit BE, will be moving slower at the B and faster at the perihelion E, following the astronomical laws ; yet since following the laws of mechanics the matter of the vortex in the narrower space between A and C ought to be moving faster than in the wider space between D and F; that is, faster at the aphelions than at the perihelion. Which



two disagree amongt themselves. Thus at the start of sign of Virgo, where the aphelion of Mars now turns, the distances between the orbits of Mars and of Venus is to the distance of the same orbits at the beginning of the sign Pisces as three to two approximately, and therefore the matter of the vortex between these orbits at the beginning of Virgo must be in the ratio of three to two. For as the space is narrower through which the same quantity of matter pases in one revolution of the time, it must cross with a greater velocity there. Therefore if the Earth may be carried relatively at rest by that in this celestial matter, and may be revolving in one circle around the sun with it, its velocity at the beginning of Pisces shall be to the velocity of the same at the beginning of Virgo in the ratio of three on two. From which the apparent diurnal motion of the sun at the beginning of Virgo should be as seventy minutes, and at the beginning of Pisces less than forty eight minutes : yet since (shown by experiment) this apparent motion of the motion of the sun shall be greater at the start of Pisces than at the start of Virgo, and therefore the Earth moves faster at the start of Virgo than at the start of Pisces. And thus the hypothesis of vortices disagrees generally with astronomical phenomina, and not only regarding explanations but also regarding the perturbations of the heavenly motions it brings about. On account of which, truly how these motions may be completed in free spaces without vortices, can be understood from book one, and in the system of the world now will be discussed more fully.


Book III Section I. Translated and Annotated by Ian Bruce. Page 722

CONCERNING THE SYSTEM OF THE

WORLD.

BOOK III.

In the preceding books I have treated the principles of [natural] philosophy, yet not in a philosophical manner, but only mathematically, from which clearly it may be possible to argue over philosophical matters. These principles are the laws and conditions of forces and motions, which are considered especially according to philosophy. Yet the same, lest they may appear barren, I have illustrated by certain philosophical scholias, treating those which are general, and in which philosophy especially is considered to lay the foundations, such as the density and resistance of bodies, spaces devoid of bodies, and the motion of light and sound. It remains that we may give some instruction about the constitution of the systems of the world from the same principles. With this in mind I had composed a third book so that it might be read by many. But in which the principles put in place were not understood well enough, they [the readers] barely understood the strength of the consequences, nor were the prejudices cast aside to which they had become accustomed over many years: and therefore lest the matter be drawn into disputes, I have transferred the bulk of this book into propositions, in the mathematical manner, so that from these alone the principles may be read by those who have worked through the earlier principles. Nevertheless since there many propositions occur, which may add excessive delays even for readers versed in mathematics, I do not wish to be the authority that the reader should pursue all of these ; it would suffice for whoever that the definitions, laws of motion and the first three sections of Book I be read with care, then he may transfer to this book on the systems of the world, and he may consult the remaining propositions of the first book here put in place as it pleases. [Introductory Note : The English title of the work being : The Mathematical Principles of Natural Philosophy ; it took a considerable time for Newton's assertion about the predominance of mathematics in describing natural phenomena to be accepted generally, rather than the hand-waving exercises of philosophers, which were then the order of the day. In Newton's view, ideas analogous to those of ancient Greek geometry could be applied to natural phenomena, so that from a carefully chosen set of axioms and laws, successive propositions could be established as in the ancient geometric texts, governing the natural world from simple collisions to the workings of the solar system, which he called ' The World'. Planetary astronomy provided an almost ideal platform on which to establish a coherent theory of dynamics from simple principles, applied to the motions of the planets, where the complications of unwelcome factors such as friction due to air resistance, etc., were absent. The main thrust of such a work must include derivations of Kepler's empirical Laws from the Law of Universal Gravitation. This then was Newton's ambitious project, and the use of geometry was central to the work and given dominance,



though the new and revolutionary calculus – not without its detractors, played a strong supporting role, and actually underpinned the whole edifice that Newton had erected: indeed many of the propositions could not have been established without the use of calculus. Thus, in Newton's scheme, a geometrical point became a particle with a mass or reluctance to move initially or change its state of motion if already moving, its inertia ; interactions of particles that changed the states of rest or of uniform motion of these were ascribed to forces, that acted on the masses either at a distance or by direct contact, obeying the three laws of motion, and the trajectories followed by the particles were represented by lines familiar from geometry ; as were the velocities, accelerations, and the forces themselves, that might be added as the sides of parallelograms; Newton's world was thus one of motion, for which these new ideas of fluxions and fluents were admirably suited for the detailed analysis. A consequence of this time dependent dynamical approach was a disregard for quantities such as energy, that remained conserved with time; and perhaps more surprising, a disregard for the angular form of the laws governing motion, and the extension to bodies of differing shapes, rather than points, though some progress is made in this final book in that regard. Thus Newton's particle mechanics was correct, but incomplete in a number of fundamentals. Leibniz tried to introduce such a time-independent principle via his vis viva notion, which was a close relative of kinetic energy, without the half ; a source of much confusion to workers in the 18th century; in this regard it may be said, and perhaps it is an undue criticism, that Leibnitz was trying to jump onto someone else's bandwagon, so to speak ; supreme logician that he was, Leibnitz was no match for Newton as a natural philosopher. In addition, one needs to bear in mind the cause and effect nature of Newton's arguments, thus, in geometry, the converse of a proposition may be as important and useful as the original proposition; this is not the case in physics : the converse of a proposition may not be physically significant, or the situation may arise from a different cause. In general, the irreversible increase in entropy, the arrow of time as it were, indicates that processes in the macroscopic world generally go in one direction rather than to be reversible, which in turn is the exception rather than the rule. Thus Newton does not necessarily dwell on the converses of his propositions, as one might do in geometry. Finally, in this introductory note, it should be made clear that this work is not an 'elementary' treatment of astronomical calculations, but a rather sophisticated one, in which unfortunately for us, most of the detailed calculations have been omitted by Newton, such as his method of finding the shape of the earth ; instead he plays with the reader at times, telling him at length about relatively trivial matters, such as adventurers sailing to remote places to measure the force of gravity there, which could be briefly encapsulated in a table. Thus, it is advisable to have an understandable text on classical celestial mechanics available to check results, such as that by W. M. Smart (Longmans; 1953) ; occasionally, results derived by Leseur & Janquier are inserted, and references made to the latter part of the book by Chandrasekhar are of interest.]



Required Rules of PHILOSOPHY.

Rule I.

No more causes of natural phenomena must be admitted, than which are both true and needed for the explanation of the phenomena. Certainly philosophers say : Nature does nothing in vain, and more will be in vain from several things in excess than arises from a few. For nature is simple and does not indulge in superfluous causes.

Rule II. Therefore the same causes are to be assigned to natural effects of the same kind, as far as it can come about. Such as breathing in man and beast; the descent of stones in Europe and America; light in a cooking fire and in the sun; reflection of light on the earth and on the planets.

Rule III. The qualities of bodies cannot be extended or remitted, and it is allowed that they be put in place for all bodies, agreed upon by experiment, for they are the universal qualities of bodies requiring to be present. For the qualities of bodies become known only through experiments, and thus such qualities are required to be in general agreement whenever they square up generally with experiments ; and because the qualities cannot be diminished, they cannot be removed. Certainly qualities are not to be rashly fashioned out of dreams contrary to the steady course of experiments, nor is it required to depart from the analogy of nature, since that is simple and always accustomed to be in harmony within itself. The extension of bodies can only become known from the senses, nor can the extension of all bodies be perceived: but because it is agreed upon by all the senses, from that the property of extension can be agreed upon for all bodies. We find out too that many bodies are hard. But the hardness arises from the hardness of the parts, and thence not only of these bodies which are perceived, but we may conclude also of all the others that we cannot perceive, that the indivisible particles are deservedly hard. Again, we gather that all bodies are impenetrable, not from reason but from the senses. The bodies we treat are found to be impenetrable, and thence we conclude that impenetrability is a universal property of bodies. All bodies are moveable, and by certain forces (we call which the forces of inertia) persevere in their motion or rest, and from these we may deduce the properties of observed bodies. The whole extension, hardness, impenetrability, mobility, and the force of inertia arise from the extension, hardness, impenetrability, mobility, and forces of inertia of the parts: and thence we conclude that the smallest parts



of all bodies are to be extended, and to be hard and impenetrable, and to have mobility provided by the forces of inertia. And this is the foundation of all philosophy. Again the divided parts of bodies mutually touching each other can be separated in turn from each other, by phenomena that we know, and it is evident that the indivisible parts may be separated into smaller parts in a ratio from mathematics. Truly it is uncertain whether these distinct parts not yet divided are able to be divided and separated in turn by natural forces. But if in turn it may be agreed by a single experiment that some particles somehow undivided, with a hard and solid body being broken, that a division may be apparent: we would conclude on the strength of this rule, that not only may the separate parts be divided, but also that they may be continued to be divided indefinitely. Finally, if all bodies revolving around with the earth are to exert a weight on the earth, [occasionally we will use the term attract by gravity for Newton's expression] and that for every individual quantity of matter, and the moon by its quantity of matter to attract the earth, and in turn our sea to have an attraction on the moon, and all the planets to mutually attract each other, and in a like manner the attraction of comets to the sun by gravity may be agreed upon generally from astronomical observations : it will be required to be said that by this rule all bodies mutually attract each other by gravitation. For there will be an even stronger argument for the phenomena of universal gravitation, than that for the impenetrability of bodies : concerning which certainly we have no experimental evidence, nor any observations. Yet I am not at all confirming that the force of gravity is essential for bodies. By the force insita I understand only the force of inertia. This is unchangeable; but the force of gravity is diminished on receding from the earth.

Rule IV. In experimental philosophy, propositions are deduced from phenomena by induction, not from contrary hypothesis, for they must be considered as true hypotheses, either accurately or approximately true, from which either more accurate hypotheses may be made available, or the hypothesis be made free from exceptions. This must come about lest the argument of induction be removed by [contrary] hypothesis.



PHENOMENA.

PHENOMENON I.

[Note : In Latin & Greek, the word phaenomena related to an appearance in the sky.] The circumjovial planets, with radii drawn from the centre of Jupiter, describe areas proportional to the times, and the periodic times of these, with respect to the fixed stars at rest, are in the three halves power of their distances from its centre. This is agreed from astronomic observations. The orbits of these planets are not different sensibly from concentric circles around Jupiter, and the motions of these may be taken from uniform circles. Truly astronomers agree that the periods are in the three halves ratio of the radii of their orbits ; and the same is evident from the following table.

The periodic time of the Jovian satellites.

d h d h d h d h1 18 27 34 ; 3 13 13 42 ; 7 3 42 36 ;16 16 32 9. . ' . " . . ' . " . . ' . " . . ' . " .

The distances of the satellites from the centre of Jupiter.

From observations 1 2 3 4 Borelli . 2

35 238 14 2

324 Townley by micrometer 5,52 8,78 13,47 24,72 Cassini by telescope 5 8 13 23 Cassini by satellite eclipse. 2

35 9 236614 3

1625 From the periodic times. 5,667 9,017 14,384 25,299

Jovian radii.

With the best micrometers Master Pound has determined the elongations of the Jovian satellites and the diameter of Jupiter as follows. The maximum elongation of the fourth Jovian satellite from the centre of the sun was taken by a micrometer in a telescope tube 15 feet long, and produced around 8'.16" from the mean distance of Jupiter from the earth. That of the third satellite was taken in a telescope 23 feet long, and produced4'. 42" at the same earth to Jupiter distance. The maximum elongations of the remaining satellites at the same distance of the earth from Jupiter 2'. 56".47"', & 1'. 51".6"' arising from the periodic times. The diameter of Jupiter was often taken from the micrometer in the 123 feet telescope, and reduced according to the mean distance of Jupiter from the sun or the earth, always produced an image less than 40'', at no time less than 38", more often 39". In shorter telescopes this diameter is 40' or 41'. For the light of Jupiter by unequal refraction is dilated to some extent, and this dilation has a smaller ratio to the diameter of Jupiter in the longer and more perfect telescopes than in the shorter and less perfect ones.



The times in which two satellites, the first and the third, will pass across the body of Jupiter, from the beginning of the ingression to the start of the exit, and from the complete ingression to the complete exit, have been observed with the aid of the same longer telescopes. And the diameter of Jupiter at its mean distance from the earth produced 1

837 " by the transition of the first satellite and 3

837 " from the transition of the third. Also the time in which the shadow of the first satellite passed through the body of Jupiter was observed, and thence the diameter of Jupiter at its mean distance from the earth produced around 37". We may assume the diameter of this to be approximately 1

437 "; and the maximum elongations of the first, second, third and fourth satellites were respectively 5,965, 9,494, 15,141, & 26,63 radii of Jupiter.

PHENOMENON II.

The planets around Saturn, with radii drawn to Saturn, describe areas proportional to the times, and the periodic times of these, with the fixed stars, are in the three on two ratio of the distances from the centre of Saturn. Certainly Cassini from his observations has established the distance of these from the centre of Saturn and the periodic times to be of this kind.

The periodic times of the satellites of Saturn.

d h d h d h d h

d h1 21 18 27 ; 2 17 41 22 ; 4 12 25 12 ;15 22 41 14 ;79 7 48 00

. . ' . " . . ' . " . . ' . " . . ' . ". . ' . " .

The distances of the satellites from the centre of Saturn in radii of the ring.

From observations. 19

201 . 132 . 1

23 . 8. 24. From the periodic times. 1,93. 2,47. 3,45 8. 23,35.

The maximum elongation of the fourth satellite from the centre of Saturn gathered from observations is accustomed to be approximately eight radii. But the maximum elongation of this satellite from the centre of Saturn, taken from the best micrometer in the Huygens telescope 123 feet long, has produced a radius of 7

108 radii. And from this observation and from the periodic times, the distances of the satellite from the centre of Saturn in radii of the ring are 2,1. 2,69. 3,75. 8,7. and 25,35. The diameter of Saturn in the same telescope to the diameter of the ring was as 3 to 7, and the diameter of the ring produced on the 28th and 29th days of May of the year 1719 was 43". And thence the diameter of the ring at the mean position of Saturn from the earth is 42", and the diameter of Saturn 18". Thus these are in the longest and best telescopes, because hence the apparent magnitudes of celestial bodies in longer telescopes may have a greater proportion to the dilation of the [rays of] light at the ends of these bodies than in shorter telescopes. If all the stray light may be removed, the diameter of Saturn will remain not greater than 16".



PHENOMENON III.

The first five planets Mercury, Venus, Mars, Jupiter, and Saturn, surround the sun with their orbits. Mercury and Venus may be shown to revolve around the sun from their lunar phases. With the full shape shining when they are situated beyond the sun ; half when away from the direction of the sun; sickle shaped when this side of the sun, and by its disc in the manner of a spot occasionally passing in front of the sun. From Mars also made full near to the sun in conjunction, & gibbous at right angles, it is evident that it goes around the sun. Concerning Jupiter and Saturn the same also may be demonstrated from their full phases : indeed it is evident that these shine with light borrowed from the sun from the shadows of the satellites projected onto themselves.

PHENOMENON IV.

The periodic times of the five primary planets, and either of the sun around the earth or the earth around the sun, with the fixed stars at rest, are in the three on two ratio of the mean distances from the sun. That this ratio was found by Kepler has been acknowledged by everyone. Certainly the periodic times are the same, the same dimensions of the orbits, whether the sun goes around the earth, or the earth may be revolving around the sun. And indeed it is agreed from measurements of the periodic times amongst astronomers. Moreover the magnitudes of all the orbits have been determined most carefully by Kepler and Boulliau from observations: and the mean distances, which correspond to the periodic times, are not sensibly different from the distances which they found, and these among themselves are intermediate mainly, as can be seen from the following table. The periodic times of the planets and of the earth about the sun, with respect to the fixed

stars, in days and decimal parts of days.

10759,275. 4332, 514 686,9785. 365,2565. 224,6176. 87,9692.

The mean distances of the planets and of the earth from the sun.

Following Kepler 951000. 519650. 152350. 100000. 72400. 38806.

Following Boulliau 954198. 522520. 152350. 100000. 72398. 38585. Following the periodic times 954006. 520096. 152369. 100000. 72333 38710.

There is no dispute concerning the distances of Mercury and Venus from the sun, since these may be determined from their elongations from the sun. Also concerning the distances of the outer planets from the sun all the disputation is removed by the eclipses



of the satellites of Jupiter. And indeed the position of the shadow that Jupiter casts may be determined by those eclipses, and with that nominated the heliocentric longitude may be obtained. Moreover from the heliocentric and geocentric longitudes between these taken together the distance of Jupiter may be determined.

PHENOMENON V. The primary planets, with radii drawn to the earth, describe areas not at all proportional to the times; but with radii drawn to the sun, the areas traversed are proportional to the times. For with respect to the earth they are now progressing, now they are stationary, and now also they are regressing : But with respect to the sun they are always progressing, and that with just about a uniform motion, but yet a little faster at the perihelions and a little slower at the aphelions, thus so that there shall be described equal areas. The proposition is the most noteworthy of astronomy, and may be demonstrated to a high degree by the satellites of Jupiter, from which eclipses we have said that the heliocentric longitudes of this planet and the distances from the sun are determined.

PHENOMENON VI. With a radius drawn to the centre of the earth, the moon describes areas proportional to the times. It is evident from the apparent motion of the moon deduced from the apparent diameter. Moreover the motion of the moon is perturbed a little by the force of the sun, but I will ignore the insensible small errors in these phenomena.



PROPOSITIONS.


The forces, by which the circumjovial planets are drawn perpetually from rectilinear motion and are retained in their orbits, with respect to the centre of Jupiter, are inversely as the squares of their distances from the same centre. The first part of the proposition is apparent from the first phenomenon, and the second or third proposition of the first book : and the latter part by the first phenomenon, and the sixth corollary of the fourth proposition of the same book. You may understand the same concerning the planets which accompany the planet Saturn, by the second phenomenon.

PROPOSITION II. THEOREM II. The forces, by which the primary planets are perpetually drawn from rectilinear motion, and retained in their orbits, with respect to the sun, and to be inversely as the square of the distances from its centre. The first part of the preposition is apparent from the fifth proposition, and the second proposition of the first book : and the latter part by the fourth phenomenon, and the fourth proposition of the same book. Moreover this first part of the proposition may be shown most accurately by the resting nature of the aphelions. For the smallest aberration from the square ratio must bring about a noteworthy motion in the individual revolutions (by Corol. I. Prop. XLV. Book. I.) and in more a large motion.

PROPOSITION III. THEOREM III. The force, by which the moon is retained in its orbit, with respect of the earth, is inversely as the square of the distances of the places from the centre of this. The first part of the assertion is apparent from the sixth phenomenon, and the second or third proposition of the first book : and the latter part by the slowest motion of the moon's apogee. For that motion, which in the individual revolutions is only of three degrees and three minutes, can be disregarded. For it is apparent (by Corol. I. Prop. XLV. Book. I.) that if the distance of the moon from the centre of the earth shall be to the radius of the earth as D to 1; the force by which such a motion may arise shall be reciprocally as

42432D , that is, reciprocally as that power of D of which the index is 4

2432 , that is, in a ratio of the distance a little greater than the inverse square, but which approaches by 3

459 times closer to the square than to the cube. Truly it arises from the action of the sun (as will be mentioned later) and therefore here can be ignored. In as much as the action of the sun draws the moon from the earth, is as the distance of the moon from the earth approximately ; and thus (by what were discussed in Corol. 2. Prop. XLV. Book. I.) the action is to the centripetal force of the moon as around 2 to 357,45, or1 to 29

40178 . And with so small a force of the sun, the remaining force by which the moon is retained in its



orbit will be reciprocally as 2D . That which also will be agreed upon more fully by comparing this with the force of gravity, as shall be made in the following proposition. Corol. If the mean centripetal force by which the moon may be retained in orbit may be increased first in the ratio 29 29

40 40177 to 178 , then also in the square ratio of the radius of the earth to the mean distance of the moon from the centre of the earth : the centripetal force will be found of the moon at the surface of the earth, because that force in place descending to the surface of the earth always may be increased in the inverse ratio of the square of the height.

PROPOSITION IV. THEOREM IV. The moon it attracted towards the earth by gravity, and by the force of gravity is drawn always from rectilinear motion, and retained in its orbit. The mean distance of the moon from the earth at syzygies is 59 times the earth radius, according to Ptolemy and most of the astronomers, 60 according to Vendelin and Huygens, 1

360 according to Copernicus, following Street 2560 and according to Tycho

1256 . But Tycho, and all who follow his tables of refraction, by putting in place

refractions of the sun and moon (generally contrary to the nature of light) greater than of the fixed stars, and that by as many as four or five minutes of arc, increased the parallax of the moon by just as many minutes, that is, as if between a 12th or 15th part of the whole parallax. This error may be corrected, and a distance around 1

260 of the earth's radius emerges, almost as that which has be designated by the others. We may assume the mean distance to be 60 of the earth's radius at sygyies ; and the lunar period with respect to the fixed stars to be made up from 27 days, 7 hours, and 43 minutes, as has been established from astronomy ; and the circumference of the earth to be 123249600 Parisian feet, as defined from French measurements : and if the moon may be supposed to be deprived of all its motion and released, so that by being urged by all that force, by which (by the Corol. of Prop. III.) it may be retained in its orbit, it may fall to the earth ; this distance of

11215 Parisian feet it describes in a time of one minute. This is deduced by calculation

accomplished either by Prop. XXXVI of the First Book, or (that returns the same) by the Corollary of the fourth proposition of the same book. For that arc which the moon in a time of one minute, by its mean motion, may describe at a distance of 60 earth radii, is around the versed sine of 1

1215 Parisian feet, or more accurately 15 feet, 1 inch and 491 twelve parts of an inch. From which since that force by acceleration to the earth may

be increased in the inverse square ratio, and thus at the surface of the earth shall be 60 60× times greater than at the surface of the moon ; a body by falling with that force in our region , must describe 1

1260 60 15× × i Parisian feet n an interval of one minute , and in an interval of one second 1

1215 feet, or more accurately 15 feet. 1 inch. and 491 twelfth

parts of an inch. And by the same force weights actually fall on the earth. For the length of a pendulum, in the latitude of Paris for a single second of oscillation, is of three feet and 1

28 lines [i.e. twelfth parts of an inch]; as Huygens observed. And the height, that a weight describes by falling in a time of one second, is to the half length of this pendulum in the square ratio of the circumference of the circle to its diameter (as Huygens also



indicated), [indeed, in modern terms, 12

2 2 2 22 and 4 l l

gs gt t sπ π= = ∴ = × .] and thus is 15

Parisian feet, 1 inch, 791 . lines. And therefore the force by which the moon may be

retained in its orbit, if it were falling onto the surface of the earth, will emerge equal to our force of gravity, and thus (by rules I and II.) is that force itself that we are accustomed to call gravity. For if gravity were different from that, bodies with the two forces taken jointly would be attracted towards the earth with twice the velocity, and would describe by falling a distance of 1

630 Parisian feet: entirely contrary to experience. Here the calculation is founded on the hypothesis that the earth is at rest. For if the earth and the moon may be moving around the sun, and meanwhile also commonly revolving about the common centre of gravity : with the law of gravity remaining, the distance between the centres of the earth and moon in turn will be around 1

260 earth radii ; as will be apparent from beginning the computation. Moreover the computation can be entered into by Prop. LX. Book I.

Scholium. The demonstration of the proposition can be explained more fully thus. If several moons were revolving around the earth, in the same way as it shall be in the system of Jupiter or Saturn : the periodic times (by an inductive argument) obey the laws of the planets found by Kepler, and therefore the centripetal forces of these shall be inversely as the squares of the distances from the centre of the earth, by Prop. I. of this Book. And if the smallest of these shall be infinitely small, and it may nearly touch the peaks of the highest mountains: the centripetal force of this by which it may be held in orbit, will be almost equal to the weights of bodies on the peaks of those mountains (by the preceding computation), and it may be effected that the same small moon, if it may be deprived of all the motion by which it goes on in orbit, with the deficiency of the centrifugal force by which it may remain in orbit, may fall to the earth, and that with the same velocity by which weights fall from the peaks of those mountains, on account of the equality of the forces by which they fall. And if that force by which that smallest moonlet fell, were different from gravity, and if that moonlet should have the customary force of gravity on the earth of bodies on the peaks of mountains : the same moonlet from each taken together would fall with twice the speed. Whereby since each of the forces, both these of the weights of bodies, and these of the moonlets, with regard to the centre of the earth, and between themselves shall be similar and equal, the same (by rules I. and II.) will have the same cause. And therefore that force, by which the moon may be kept in its orbit, that itself will be as we are accustomed to call the weight : and especially that the moonlet at the peak of a mountain may neither be without weight, nor that it will fall with twice the velocity that heavy bodies are accustomed to fall.

PROPOSITION V. THEOREM V. The circumjovial planets gravitate towards Jupiter, the circumsaturn planets towards Saturn, and the circumsolar planets towards the sun, and by the force of their gravity they are always drawn from rectilinear motion, and held in curved orbits. For the revolutions of the circumjovial planets around Jupiter, the circumsaturnal planets around Saturn, and of Mercury and Venus and of the remainder around the sun are



phenomena of the same kind as with the revolution of moon around the earth ; and therefore (by rule II.) depend on causes of the same kind : especially since it may be shown that the forces, on which these revolutions depend, may be with respect to the centre of Jupiter, Saturn and the sun, and by receding from Jupiter, Saturn or the Sun they may decrease in the same ratio and by the law, by which the force of gravity decreases in receding from the earth. Corol. 1. The force of gravity therefore is given towards all the planets. For no one can doubt that Venus, Mercury and the others are bodies of the same kind as Jupiter and Saturn. And since the attraction of all shall be mutual by the third law of motion, Jupiter towards all its satellites, Saturn towards its satellites, and the earth towards the moon, and the sun will gravitate towards all the primary planets. Corol. 2. The gravity, with regard to any planet, is inversely as the square of the distance of the places from its centre. Corol. 3. All the planets have a weight between each other, by Corol.1. and 2. And hence Jupiter and Saturn by attracting each other near conjunction, noticeably perturb each others motion, the sun disturbs lunar motions, the sun and the moon disturb our sea, as will be explained in the following.

Scholium. Until now we have called that force, by which celestial bodies may be retained in their orbits, the centripetal force. Now the same is agreed to be the force of gravity, and therefore we will now hereafter call this the force of gravity. For the cause of that centripetal force, by which the moon may be held in orbit, must be extended to all the planets by rules I, II. & IV.

PROPOSITION VI. THEOREM VI. All bodies are attracted by gravity to the individual planets, and the weights of these for whatever planet, for equal distances from the centre of the planet, are proportional to the quantity of matter in the individual planets. Others have observed for a long time now that [equal] descent of all weights on the earth (at any rate with the very small unequal retardation removed which arises from the air) happen in equal times ; and certainly one can note the most accurate equality of the times in pendulums. I have tested the matter with gold, silver, lead, glass, common salt, wood, water, and with wheat. I have prepared two equal small round wooden boxes. One I filled with wood, and likewise I suspended a weight of gold at the other centre of oscillation (as nearly as I could). The boxes constituted the pendulums, hanging from equal eleven feet strings, in order that the weight, the shape and the air resistance generally were equal; and with equal oscillations, placed next to each other, they were going and coming together for a long time. Hence the amount of matter in the gold (by Corol. I. and 6. Prop. XXIV. Book II.) was to the amount of matter in the wood, as the action of the motive force in all the gold to the same action in all the wood ; that is, as the weight to the weight. And thus for the others. From these experiments clearly in bodies of the same weight the difference of the materials to be taken would have [an effect] to be



less than a thousandth part of the whole matter. Truly now there is no doubt that the nature of gravity on the planets is the same as on the earth. For these terrestrial bodies raised as far as the orbit of the moon, and released together with the moon deprived of all motion, may fall likewise as on the earth; and now by what has been shown before it is certain that they describe equal spaces with the moon, and thus so that they [i.e. the quantities of matter in the bodies] shall be to the quantity of matter in the moon, as their weights shall be to the weight of the moon itself [at the radius of the moon's orbit]. Again because the satellites of Jupiter are revolving in times which are in the three on two ratio of the distances from the centre of Jupiter, the accelerative weights of these will be towards Jupiter inversely as the square of the distances from the centre of Jupiter ; and therefore at equal distances from Jupiter, the accelerative weights of these become equal. Hence by falling in equal times from the equal heights they describe equal distances ; thence so that it shall be as with weights here on our earth. And the circumsolar planets by the same argument, released from equal distances from the sun, describe equal distances in their descent towards the sun in equal times. Moreover the forces, by which unequal bodies may be equally accelerated, are as the bodies; that is, the weights are as the quantities of matter in the bodies. Again is apparent from the especially regular motion of the satellites that the proportional weights of Jupiter and of its satellites towards the sun are as the quantities of matter of these ; by Corol 3. Prop. LXV. Book I. For, if some of these should be attracted more towards the sun than the rest, for their quantity of matter : the motions of satellites (by Corol. 2. Prop. LXV. Book I.) would be perturbed from the inequality of the attraction. If, with equal distances from the sun, some satellite should be heavier towards the sun by its quantity of matter, than Jupiter by its quantity of matter, in some given ratio, for example d to e: the distances between the centre of the sun and the centre of the orbit of the satellite, always should be greater than the distance of the centre of the sun and the centre of Jupiter approximately in the three on two ratio ; as found in a certain initial calculation. And if a satellite should weigh less towards the sun in that ratio d to e, the distance of the centre of the orbit of the satellite from the sun would be less than the distance of the centre of Jupiter from the sun in the square root ratio approximately. And thus if in the distances from the sun equalities, the accelerative gravity of some satellite towards the sun should be greater or less than the accelerative weight of Jupiter towards the sun, by only a one thousandth part of the whole force of gravity ; the distance of the centre of the orbit of the satellite from the sun would become greater or less than the distance from Jupiter by a 1

2000th part of the whole distance, that is,

by a fifth part of the extreme distance of the satellite from the centre of Jupiter : which indeed would be quite detectable for an exocentric orbit. But the orbits of the satellites of Jupiter are concentric, and therefore the gravitating acceleration of Jupiter and of the satellites towards the sun are equal to each other. And by the same argument the weight of Saturn and of its attendants towards the sun, at equal distances from the sun, are as the quantities of matter within themselves : and the weights of the moon and of the earth towards the sun are either zero, or accurately proportional to these masses. But they dos have some masses by Corol. 1. and 3. Prop. V. For indeed the weights of parts of each of the individual planets to some other planet are between themselves as the matter in the individual parts. For if some parts should exert a greater force of gravity, others less, than for the quantity of the matter; the whole



planet, for the kind of part with which it should abound the most, would gravitate more or less than for the quantity of the whole matter. But it does not matter if the parts refer to external or internal parts. If for example earthly bodies, which are now about us, were supposed raised to the orbit of the moon, and brought together with the body of the moon : if the weights of these were to the weights of the external parts of the moon as the quantities of matter in the same, truly in a greater or smaller ratio to the weights of the internal parts, the same would be in a greater or smaller ratio to the weight of the whole moon: contrary than what has been shown above. Corol.1. Hence the weights of bodies do not depend on the forms and textures of these. For if they were able to be varied with the shape, they could be greater or less, according to the kind of shape, in an equal amount of matter, completely contrary to experience. Corol. 2. Generally bodies, which are near the earth, are heavy on the earth ; and all weights, which are equally distant from the centre of the earth, are as the quantities of matter in the same. This is a characteristic of all that it is possible to establish by experiment, and therefore has been confirmed generally by rule III. If the ether or some other either may be freed from gravity, or for a quantity of its matter that may experience gravity less : because that (by the reasoning of Aristotle, de Cartes, and of others) may not differ from other bodies except in the shape of the matter, likewise it may be possible by a change of shape gradually to be transformed into a body of the same condition as these, but which may gravitate as greatly as possible, and in turn especially heavy bodies, by gradually adopting the form of these, gradually would be able to lose their weight. And hence the weights depend on the shapes of bodies, and could be varied according to their form, contrary to what has been proven in the above corollary. Corol. 3. All spaces are not equally occupied. For if all spaces were to be equally filled, the specific gravity of the fluid by which the region of the air would be filled up, on account of the maximum density of matter, would concede nothing to the specific gravity of mercury, gold, or to any other body of the greatest density ; and therefore neither would it be possible for a body of gold nor of any other kind of body to descend in air. For bodies do not descent in fluids, unless they shall be of greater specific gravity. Because if the quantity of matter in a given space may be diminished by rarefaction, why may it not be possible to diminish it indefinitely? [This is an attempt to explain the differentiation of matter by density in a body held together by gravity; the densest in the centre, and the least dense in the outer regions.] Corol. 4. If all the particles of all solid bodies should be of the same density, nor able to be rarefied without pores, a vacuum is given. I say that bodies are to be of the same density, of which the forces of inertia are as their magnitudes [i.e. masses]. [In this case, if no differentiation is possible as the density is constant, then the body ends abruptly on the outside in a vacuum.]



Corol. 5. The force of gravity is of a different kind from the magnetic force. For magnetic attraction is not as matter attraction. Some bodies are attracted more, others less, most are not attracted. And the magnetic force in one and the same body is able to be retained and to be lost , and sometimes greater by far than the force of gravity, and in receding from a magnet it does not decrease in the ratio of the inverse square, but almost as the inverse, as far as can be ascertained from certain gross observations.

PROPOSITION VII. THEORM VII. Gravitation happens in bodies universally, and that is to be proportional to the quantity of matter in the individual bodies. All planets attract each other mutually as we have approved previously, and so that the gravity in each one separately can be considered to be reciprocally as the square of the distance of the places from the centre of the planet. And hence it is a consequence (by Prop. LXIX. Book 1. and its corollaries) that gravity be in the same proportion in all materials. Again since the gravitational forces of all the parts of some planet A shall be present on some planet B, and the gravity of its parts shall be to the whole gravity, as the matter of its parts to the whole, and for every action there shall be an equal reaction (by the third law of motion); planet B in turn will attract planet A by gravity in all its parts, and by its gravity acting on each and every part to its total gravity, as the matter of the parts to the matter of the whole. Q. E. D. Corol. 1. Therefore the whole gravity on a planet arises and is prepared from the gravity of all its individual parts. We have an example of this affair from magnetic and electric attractions. For the whole attraction arises from the attractions of the individual parts. This matter is understood in gravitation, by considering many smaller planets to run together into one sphere and to compose a larger planet. For the whole force must arise from the forces of the composing parts. If this may be objected to because all bodies, which are around us, must mutually attract each other by gravity, since yet gravity of this kind may by no means be experienced : I respond that the gravity in these bodies, since it shall be to the gravity of the whole earth as these bodies are to the whole earth [i.e. in terms of mass], is by far smaller than that which it may be possible to experience. Corol.2. Gravitation in the individual particles of bodies is reciprocally as the square of the distance of the places between the particles. This is apparent from Corol 3. Prop. LXXIV. Book I.

PROPOSITION VIII. THEOREM VIII. If the matter of two spheres mutually attracting each other shall be homogeneous on all sides in all directions, which are equally distant from the centres: the weight of the spheres of the one to the other shall be reciprocally as the square of the distance between the centres. After I had found that the gravitation towards a whole planet arose from and was composed from the gravitation of the parts ; and the gravitation between the parts to be



reciprocally proportional to the distances between the parts : I was doubting whether that reciprocal square proportion was obtained accurately with the whole force composed from the many forces, or whether indeed it was as an approximation. For it may come about that the proportion, which may be obtained well enough at greater distances, might be perceptibly in error near the surface of a planet on account of the unequal distances of the particles and on account of dissimilar situations. Yet truly, by Prop. LXXV and Prop. LXXVI. of Book I and their corollaries, I understood the truth of the proposition upon which the proposition here is acting. Corol. 1. Hence the weights of the bodies in different planets are able to be found and to be compared. For the weights of equal bodies revolving in circles around planets are (by Corol. 2. Prop. IV. Book I.) as the diameters of the circles directly and inversely as the squares of the periodic times ; and the weights at the surfaces of the planets, or at any other distances from the centre, are greater or less (by this proposition) inversely in the square ratio of the distances. Thus from the periodic times of Venus around the sun of 224 and 3

416 hours, of the outermost circumjovial satellite around Jupiter of 16 days and 82516 hours, of the Huygens satellite around Saturn of 15 days and 2

322 hours, and of the moon around the earth of 27 days, 7 hours and 43 min., by gathering together the mean distance of Venus from the sun and with the maximum heliocentric elongation of the outer circumjovial satellite from the centre of Jupiter 8'. 16"; with the mean distance of the Huygens satellite from the centre of Saturn 3'. 4"; and of the moon from the centre of the earth 10'. 33'' ; by entering upon a calculation [see L & J note following below] I found that of equal bodies and at equal distances from the centre of the sun, Jupiter, Saturn, and the earth , that their attractions towards the centre of the sun, Jupiter, Saturn, and the earth shall be as 1 1 1

1067 3021 1692821 and , , , respectively [Cohen in his text unfortunately includes a comma, or equivalent to a decimal point in Newton's work, in his translation at this point, which is evidently incorrect.], and with the distances increased or diminished in the square ratio: the weights of equal bodies towards the sun, Jupiter, Saturn and the earth at the distances 10000, 997, 791, and 109 from the centres of these, and thus at the surfaces of these, will be as 10000, 943, 529, and 435 respectively. However great the weights of bodies shall be on the surface of the moon will be talked about in the following. [Abridged note 68, L & J, p. 35, Book 3. All these may be returned by algebraic proofs. Let S be the centre of the sun, V the centre of Venus, P the centre of a primary planet, L a satellite at its maximum heliocentric elongation that the angle LSP measures, for which the angle SLP is right. The periodic time of Venus may be called t ; the periodic time of the satellite L about the primary planet P may be called θ . The distance SP, however great it may be, shall be called z; the ratio SP

SV which is given by the Phenomena IV may be expressed by the ratio a to b, and hence there will be



( ) and SP z a bzSV b aSV= = = ; and with the radius present the sine of the maximum heliocentric

elongation of the satellite L, or the sine of the angle LSP may be called e; Hence in the right-angled triangle SLP,

( )2

z PLsin sin LSP eπ =

= ; hence the side PL ez= .

Because the force of the sun on Venus and the force of the sun on the satellite are by Cor. 2 of Prop. IV, Book I, as the distance of Venus and the distance of the satellite from the centre of the sun and of the primary planet divided by the squares of the periodic times, either as 2 2to bz ez

at θ, or if the force

of the sun on Venus is taken as 1, the force of the primary planet on the satellite will be as

2

2aetbθ

.

But the force of the primary planet on the satellite at the distance PL, is to the force by which it may itself act if it were at just as great a distance from the sun as Venus, inversely as the square of the distances, therefore there becomes

( )2 2 2 2 3 3 2

2 2 2 2 2 2 3 21 to , or 1 to as to a a e aet a e t

e z b z b b bθ θ× ; and again it will

be found that the force of the sun on Venus is to the force of the primary P planet on the satellite, if it may be at just as great a distance by that amount that Venus is distant from the sun

3 3 2

3 2as 1 to a e tb θ

× . This result

follows for any positions of the circular orbits of the bodies, and hence is true for all positions; the quantities in the ratios are known from observation, and hence the force of the sun will be to the force of the planet

3 3 2

3 2as 1 to a e tb θ

× . Thus, for example, let P be the

centre of the planet Jupiter, and L the outer satellite, then b is to a as 72333 to 520096, while ( )8 16e sin ' "= , on inserting the ratio of the periodic times in seconds, and in those days using logarithms, the force of the attraction towards the centre of Jupiter is 1

1067 of the attraction towards the sun. Subsequent to Newton, astronomers have found more accurate values for the various distances and angles. Cohen gives a table comparing the present accepted values and Newton's values, in his Introduction. p.225.] Corol. 2. Also the quantities of matter in the individual planets becomes known. For the quantities of matter in the planets are as the forces of these at equal distances from their centres; that is, on the sun, on Jupiter, Saturn, and the earth, they are as

1 1 11067 3021 1692821, , ,& respectively. If the parallax of the sun may be taken as greater than 10",

and less than 30"', the quantity of matter on the earth must be increased or diminished in the cubed ratio. Corol. 3. The densities of the planets become known also. For the weights of equal and homogeneous bodies on homogeneous spheres are at the surfaces of the spheres as the



diameters of the spheres, by Prop. LXXII. Book I, and thus the densities of heterogeneous spheres are as these weights applied to the diameters of the spheres. But indeed the diameters of the sun, Jupiter, Saturn and of the earth in turn will be as 10000, 997, 791, and 109, and the weights at the surfaces of the same will be as 10000, 943, 529 and 435 respectively, and therefore the densities are as 100, 1

294 , 67 and 400. The density of the earth which arises from this calculation does not depend on the parallax of the sun, but is determined from the parallax of the moon, and this therefore is defined correctly. Therefore the sun is a little denser than Jupiter, and Jupiter than Saturn, and the earth is four times denser than the sun. For the sun is rarefied by its remarkable heat. Truly the moon is denser than the earth, as will become apparent in the following. Corol. 4. Therefore the smaller planets are denser, with all other things being equal. Thus the force of gravity approaches more to equality on the surfaces of these. And also the planets are denser, with all else being equal, which are closer to the sun ; as Jupiter with Saturn, and the earth with Jupiter. Certainly planets were to be gathered together at different distances from the sun so that as it were by a change in the density they might enjoy the heat from the sun more or less. Our water, if the earth were located in the orbit of Saturn, would be frozen, if in the orbit of Mercury it would depart at once into vapours. For the light of the sun, to which the heat is proportional, is seven times denser in the orbit of Mercury than with us : and with a thermometer I have found that with a seven-fold increase in the heat of the summer sun, water boils off. Truly there is no doubt why the matter of Mercury should not be accustomed to the heat, and therefore shall be denser than ours ; since all denser matter requires greater heat to take part in natural processes.

PROPOSITION IX. THEOREM IX. Gravity on going downwards from the surfaces of planets decreased approximately as the distance from the centre. If the matter of a planet were uniform as far as density is concerned, this proposition would be obtained accurately: by Prop. LXXIII. Book I. Therefore there is as great an error, as can arise from the inequality of the density.

PROPOSITION X. THEOREM X. The motions of the planets in the heavens is able to be conserved for a very long time. In the Scholium of Proposition XL, Book II, it has been shown that a sphere of frozen water, by moving freely in our air and by describing the length of its radius, may lose a

14586

th part of its motion by the resistance of the air. Moreover the same proportion will be obtained approximately in spheres with any size and speed. Now truly I deduce thus that sphere of our earth is denser than if the whole were made from water. If this sphere were entirely aqueous, in which some parts were rarer than water, on account of the smaller specific gravity these would rise and float on top. And by that reason the earthly sphere would arise covered by water on all sides, if some parts were rarer than water, and thus all the water flowing away might meet together in the opposite direction. And the account of our earth surrounded by great seas is the same. This, if it were not denser, would emerge



from the seas, and by its part for the degree of lightness would stand out from the water, with seas flowing together in the opposite region. By the same argument the sunspots are lighter than the luminous matter of the sun on which they float. And in the formation of some kinds of planets, in which time all the heavier matter would seek the centre from the mass of water. From which, since with common ground uppermost, it shall be as if twice as dense as water, and a little further down in mines it may be found to be as much as three or four or even five times heavier: it is plausible that the abundance of all the matter in the earth shall be as if five times or greater than six times greater than if the whole were made from water; especially since it was shown before that the earth is to be as if four times denser than Jupiter. Whereby if Jupiter shall be a little denser than water, here in an interval of thirty days, in which it will describe a length of 459 of its radii, it may lose almost a tenth part of its motion in a medium of the same density as the density of our air. Truly since the resistance of the mediums may be reduced in the ratio of the weight and density, thus as water, which is 3

513 lighter than quicksilver, resists less in the same ratio ; and the air, which is in 860 parts lighter than water, resists less in the same ratio : if it [i.e. a body of the constitution of Jupiter] should ascend to the heavens where the weight of the medium, in which the planets are moving, is diminished indefinitely, the resistance will almost cease. At any rate we have shown in the scholium to Prop. XXII. Book II, that if it were to ascend to a height of 200 miles above the earth, the air there would be rarer than at the surface of the earth in the ratio 30 to 0,0000000000003998, or around 75000000000000 to 1. And hence the star of Jove by revolving in a medium of the same density with that upper air, in a time of 1000000 years, would not lose a ten hundred thousandth part of its motion to the resistance of the medium. Certainly in spaces closer to the earth, nothing is found that creates resistance other than exhalations and vapours. With these from a hollow glass cylinder with the air must diligently exhausted weight within the glass fall freely and without any sensible resistance ; a weight of gold itself and the finest feather dropped together fall with the same velocity, and in their case describing a height of four, six or eight feet reach the bottom at the same time, as has been ascertained by experiment. And therefore if it may ascend into the heavens devoid of air and exhalations, the planets and comets will be moving without any sensible resistance through that space for an exceedingly long time.

HYPOTHESIS I. The centre of the system of the world is at rest.

This has been conceded by everyone, while some may contend that the earth, others the sun to be the centre of the system at rest. We may see which thence may follow.



PROPOSITION XI. THEOREM XI. The common centre of gravity of the earth, the sun, and of all the planets is at rest. For that centre (by Corol. IV of the laws) either may be at rest or may be progressing uniformly in a straight direction. But with that centre always progressing, the centre of the world also will be moving, contrary to the hypothesis.

PROPOSITION XII. THEOREM XII. The sun is disturbed by the continual motion, but at no time does it recede far from the common centre of gravity of all the planets. For since (by Corol. 2. Prop. VIII.) the matter in the sun shall be to the matter in Jupiter as 1067 to 1, and the distance of Jupiter from the sun shall be to the radius of the sun in a little greater ratio ; the common centre of gravity of Jupiter and the sun falls on a point a little above the surface of the sun. By the same argument since the matter in the sun shall be to the matter in Saturn as 3021 as 1, and the distance of Saturn from the sun shall be as the radius of the sun in a slightly smaller ratio : the common centre of gravity of Saturn and the sun lies a little below the surface of the sun. And by following in the footsteps of the same calculation, if the earth and all the planets were put in place at one side of the sun, the common centre of gravity of all scarcely would be different by the diameter of the whole sun from the centre of the sun. In all the other cases the distance of the centres is always less. And therefore since that centre of gravity will always be at rest, the sun will constantly be moving by the variation of the position of the planets in all directions, but at no time will it recede far from the centre. [Thus, in this proposition, Newton gives us essentially a criterion for investigating the possibility of planetary systems circulating around nearby stars ; and in the same manner, gives a sign to other intelligent beings on planets elsewhere, if they exist, that the sun itself has such a planetary system.] Corol. Hence the common centre of gravity of the earth, the sun and of all the planets is obtained for the centre of the world. For since the earth, the sun and the planets all gravitate amongst themselves mutually, and therefore, by its force of gravity, following the laws of motion will always be disturbed : it is evident that from the mobility of these, for the centre of the world to be at rest, they are unable to be consider at rest. If that body should be located in the centre in which all the bodies are attracted especially (as is the common opinion) that must be conceded to be the privilege of the sun. But since the sun may be moving, a point at rest must be selected, from which the centre of the sun departs minimally, and from which likewise besides it will depart less, only if the sun were denser and greater, so that it would be moving less.



PROPOSITION XIII. THEOREM XIII. The planets are moving in ellipses having the centre of the sun as one focus, and with the radii drawn to that centre describing areas proportional to the times. [See, e.g. Smart, Celestial Mechanics, the first few chapters, for a clear modern exposition of Kepler's Laws in terms of the law of gravitation.] Above we have argued about these motions from phenomenae. Now with the principles of the motion known, we may deduced the motions from basic principles. Because the weights of the planets towards the sun are inversely as the squares of the distances from the centre of the sun ; if the sun were at rest and the remaining planets did not act on each other mutually, the orbits of these would be ellipses, having the sun at a common focal point, and the areas would be described in proportional times (by Prop. I. and Xl. and Corol. I. Prop. XIII. Book I.) ; but the actions of the planets among themselves are very small (so that they may be disregarded) and the motion of the planets in ellipses around the moving sun is less perturbing (by Prop. LXVI. Book I.) than if the motion were being undertaken as around the sun at rest. Certainly the action of Jupiter on Saturn cannot be dismissed entirely. For the gravity towards Jupiter is to the gravity towards the sun (with equal distances) as 1 to 1067;

[i.e. 11067

Sat Jup .

Sat sun

M . MM . M

×× = ] and thus with Jupiter and Saturn in conjunction, because the

distance of Saturn from Jupiter is as the distance of Saturn from the sun almost as 4 to 9 the gravity of Saturn towards Jupiter to the gravity of Saturn towards the sun will be as 81

to 16 1067× or as 1 to 211 approximately. [i.e. ( )( )

811 11067 16 211

1681

Sat Jup .SatJup

Satsun Sat sun

M . M /accacc M . M /

×

×= = × ≈ ]

And hence the perturbation of the orbit of Saturn arises from the individual conjunctions of this planet with Jupiter thus are perceptible so that astronomers are in difficulties with the same. On account of the variation in position of the planets in these conjunctions, the eccentricity of this now may be increased, now decreased, now the aphelion may be moved forwards and then perhaps it is drawn backwards, and the mean motion may be accelerated by the forces and then retarded. Yet the error in all the motion of this around the sun arising from such a force (except in the mean motion) can be almost avoided by putting in place the lower focus of its orbit at the common centre of gravity of Jupiter and the sun (by Prop. LXVII, Book I.) and therefore when it is a maximum, the error scarcely exceeds two minutes. And the maximum error in the mean motion scarcely exceeds two minutes per annum. But in the conjunction of Jupiter & Saturn the accelerating gravity of the sun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards the sun are almost as 16, 81 and 16 81 3021

25× × or156069, and therefore the difference of the forces of

gravity of the sun towards Saturn and of Jupiter towards Saturn is to the force of gravity of Jupiter towards the sun as 65 to 156609 or 1 to 2409. [Because at the conjunction of Jupiter and Saturn, the distance of Saturn from the sun, Saturn from Jupiter, and Jupiter from the sun, are to each other as 9, 4, and 5,



approximately; we already have that ( )( )

16 1 16 811067 81 1067 1681

Sat Jup .Sat / Jup

Sat / sun Sat sun

M M /F /F /M M /

×××

= = = ; while

( )( )

25 3021 811067 2581

sun Jup .Jup/ sun

Sat / sun Sat sun

M M /FF M M /

× ×××

= = , and 3021 81 1067 16 3021 161067 25 81 25

Jup/ sun Jup / sunSat / sun

Sat / sun Sat / Jup Sat / Jup

F FFF F F

× × ××× = = × = .

Or,

1 1 13021 81 1067 3021 16 1067 25

1 1 181 3021 3021 16 25

: : : : : :

or : : : : : :

Sat / sun Sat / Jup. Jup/ sun

sun/ Sat Jup / Sat Jup/ sun

F F F

acc acc acc

× × × ×

× ×

i.e. 30211 1

81 16 25 and, , , by Cor. 1, Prop. VIII, that is, as 16, 81 and 16 81 302125 or 156609× × .

Again, the difference of the accelerations of the sun on Saturn and of Jupiter on Saturn to

the acceleration of Jupiter on the sun is as 1 1

3021 81 3021 161 3021 81 1625 25

12409

65sun/ Sat Jup/ Sat

Jup/ sun

a aa

× ×× ×

− −= = = ; thus

the effect of Saturn on Jupiter is much less than the effect of the sun on Jupiter (from an L & J. note). ] But the difference of this proportionality has the maximum effectiveness of Saturn in perturbing the motion of Jupiter, and therefore the perturbation of the orbit of Jupiter is far less than that of Saturn. The perturbations of the rest of the orbits are besides a great deal smaller except that the orbit of the earth is perceptibly disturbed by the moon. The common centre of gravity of the earth and the moon, travels around the sun – situated at the focus, in an ellipse, and a radius drawn to the sun will describe areas proportional to the time, truly the earth is revolving around this common centre in its monthly motion.

PROPOSITION XIV. THEOREM XIV. The aphelions and nodes of orbits are at rest.

The aphelions are at rest by Prop. XI. Book I, and so that both the planes of the orbits, by Prop. I of the same book, and the nodes are at rest in the planes at rest. But yet from the revolution of the planets and from the actions of comets between themselves other inequalities may arise, but which here they are to be ignored on account of their insignificance. Corol. 1. Also the fixed stars are at rest, therefore so that they maintain the given positions at the nodes and aphelions. Corol. 2. And thus since nothing shall be evident of the parallax of these, arising from the annual motion of the earth, the forces of these, on account of the immense distance of the bodies, will give rise to no noticeable effect in the region of our system. Perhaps the fixed stars dispersed equally in all the parts of the heavens mutually destroy the forces from mutual attractions, by Prop. LXX. Book I.



Scholium.

Since the nearer planets of the sun (evidently Mercury, Venus, the earth, and Mars) on account of the smallness of the pairs of bodies acting in turn between themselves : the aphelions and nodes of these are at rest, unless perhaps they may be disturbed by the forces of Jupiter, Saturn and further bodies. And thence it can be deduced from the theory of gravitation, that the aphelia of these are moving a little in consequence with respect of the fixed stars, and that in the three on two proportion of the distances of the planets from the sun. So that if the aphelion of Mars should make 33' , 20" in a hundred years with respect of the fixed stars as a consequence; the aphelions of the earth, of Venus, and of Mercury in a hundred years would make 17', 40", 10', 53", and 4', 16" respectively. And these motions, on account of the smallness, are ignored in this proposition.

PROPOSITION XV. PROBLEM I. To find the principal diameters of the orbits.

These are to be taken in the two on three ratio of the periodic times, by Prop. XV, Book I, then one by one increased in the ratio of the sum of the masses of the sun and of each planet revolving to the first of the two mean proportionals between that sum and the sun, by Prop. LX, Book I. [See Smart, Celestial Mechanics, p.15]

PROPOSITIO XVI. PROBLEMA II. To find the eccentricities and aphelions of the orbits.

The problem has been done in Prop. XVIII. Book I.

PROPOSITION XVII. THEOREM XV. The daily motions of the planets is uniform, and the libration of the moon arises from its daily motion. It is apparent by the first law of motion and Corol. 22. Prop. LXVI. Book I that Jupiter certainly is revolving with respect to the fixed stars in 9 hours and 56 minutes, Mars in 24 hours and 39 minutes, Venus in around 23 hours, the earth in 23 hours 56 minutes, the sun in 1

225 and the moon in 27 days 7 hours 43 minutes. It is evident that these are found from the phenomena [i.e. experimental data in modern jargon]. Spots in the body of the sun return at the same place on the solar disc in around 1

227 days, with respect to the earth ; and thus with respect to the fixed stars the sun is rotating in around 1

225 days. Truly because there is the monthly revolution of the moon about its axis : the same face of this will always look at the more distant focus of its orbit, as nearly as possible, and therefore according to the situation of that focus will hence deviate thence from the earth. This is the libration of the moon in longitude: For the libration in latitude has arisen from the latitude of the moon and the inclination of its axis to the plane of the ecliptic. N. Mercator has explained this theory of the libration of the moon more fully in letters from me, published in his Astronomy at the start of the year 1676.



[Institutionum Astronomicarum libri duo, p.286-7; see note on p.16 of Book I of Newton's Letters; these letters have been lost, but Newton's help had been fully acknowledged in Mercator's work.] The outer satellite of Saturn may be seen to be revolving in a similar manner about its axis, with its same face always looking towards Saturn. For by revolving around Saturn, as often as it arrives at the eastern part of its orbit, it appears most decayed [in brightness] and the fullness seen to cease: as which can arise through certain spots on that part of the body which then is turned towards the earth, as Cassini noted. Also, the outer satellite of Jupiter may be seen to be to be revolving about its axis in a like motion, therefore so that it may have a spot on the part of its body turned away from Jupiter as in the body of Jupiter is discerned whenever the satellite passes between Jupiter and our eyes.

PROPOSITION XVIII. THEOREM XVI.

The axes of the diameters of the planets which are drawn normally to the same axes are the minor axes. Planets must conform to a spherical shape with all the daily circular motion removed, on account of the equal weight of the parts on all sides. Through that circular motion it comes about that parts next to the equatorial axis will try to rise, receding from the axis. And thus the matter, if it shall be fluid, rising by itself will increase the diameter of the equator, truly the axis at the poles will be diminished by its falling. Thus the axis of Jupiter (in common agreement from observations) it taken to be shorter between the poles than from East to West. By the same argument, unless our earth were a little higher at the equator than at the poles, the seas would subside at the poles, and by ascending adjoining the equator, there they would flood everything. [Again, need it be said, we have a present instance of Newton's insight : the melting of glaciers and polar ice caps due to global warming is and will continue to increase the equatorial bulge of the earth, leading to permanent flooding of equatorial regions, increased tidal activity, etc. ; in the same manner, the sea levels in the northern polar regions may remain unchanged or actually fall.]

PROPOSITION XIX. PROBLEM III. To find the proportion of the axis of a planet to the same perpendicular diameter. Our countryman Norwood around the year 1635 by measuring the distance between London and York as 905751 London feet, and by observing the difference of the latitude to be 2 0 , 28' deduced the measure of one degree to be 367196 London feet, that is, 57300 Paris toises. Picart by measuring the arc of 10 , 22' , 55" in the meridian between Amiens and Malvoisine, found the arc of 10 to be 57060 Paris hexafeet [or toises]. Cassini the elder measured the distance along the meridian from the town of Colliore in Roussilon to the observatory in Paris; and his son had added the distance from the observatory to the tower of the town Dunkirk. The total distance was 1

2486156 toises and the difference of the latitudes of the towns Collioure and Dunkirk was 80 , 31', 1

611 ". From which the arc of 10



produced 57061 toises. And from these measurements a circuit around the earth is deduced to be 123249600 Paris feet, and its radius 19615800 Paris feet, from the hypothesis that the earth shall be a sphere. In the latitude of Paris a heavy body by falling in a time of one second will describe 15 Paris feet, 1 inch and 7

91 lines as above, that is, 792173 lines. The weight of the body is

diminished by the weight of the surrounding air. We may suppose the weight lost to be one eleven thousandth part of the total weight, and that heavy body by falling in a vacuum describes a height of 2174 lines in one second. [recall that 1 line is 1

12 part of an inch.] A body revolving uniformly in a circle at a distance of 19615800 feet from the centre, in single sidereal days of 23 hours, 56' minutes, and 4" seconds will describe in a time of one second an arc of 1433,46 feet, the versed sine of which is 0,052656, or of 7,54064 lines. And thus the force, by which weights descend at the latitude of Paris, is to the centrifugal force of bodies at the equator arising from the daily motion of the earth, as 2174 to 7,54064. The centrifugal force of bodies at the equator of the earth is to the centrifugal force, by which bodies tend to move directly from the earth at the latitude of Paris of 480 , 50' , 10", in the square ratio of the radius to the sine of the complement of its latitude, that is, as 7,54064 to 3,267. This force may be added to the force by which the bodies descend at that latitude of Paris, and a body at that latitude by falling with that total force, in a time of one second describes 2177,267 lines, or 15 feet 1 inch and 5,267 lines. And the total force of gravity at that latitude will be to the centrifugal force of bodies at the equator of the earth as 2177,167 to 7,54064 or 289 to 1. From which if APBQ may designate the figure of the earth now no longer spherical but generated by the revolution of an ellipse around the minor axis PQ, and ACQqca shall be a channel full of water, from the pole Qq to the centre Cc, and thence going to the equator Aa : the weight of water in the legs of the channel ACca, is to the weight of water in the other leg QCcq as 289 to 288, because the centrifugal force arising from the circular motion will sustain one part of the 289 parts and may be taken from the weight of the parts, and the weight 288 in the other leg will sustain the rest. [The alternate diagram for a prolate spheroid rather than the true oblate spheroid has been drawn in all three editions of the Principia, with PQ taken as the bulge; I have changed this to the oblate shape, as Chandrasekhar has done, but the orientation is now unusual, as the polar axis PQ is now horizontal ; this is the usual view of someone standing at the equator and looking North along CQ.] Again (from Proposition XCI. Corol. 2. Book I.) by entering upon a computation, [which is quite long and is presented by Chandrasakher in Ch. 20, p. 281 of his work on Newton,] I find that if the earth should be constructed from uniform matter, and it may be deprived of all motion, and its axis PQ to be to the diameter AB as 100 to 101; the gravity at the location Q on the earth becomes to the gravity at the same location Q in a sphere with centre C with the radius PC or QC described, as 126 to 125. And by the same argument the weight at the position A on the spheroid, by convoluting the ellipse APBQ described about the axis AB, is to the weight at the same place A on the sphere described with centre C and



with the radius AC, as 125 to 126. But the weight at the position A on the earth is the mean proportional between the weights on the said spheroid and sphere : so that therefore the sphere, by diminishing the diameter PQ in the ratio 101 to 100, is turned into the figure of the earth ; and this figure with a third diameter which is perpendicular to the two diameters AB and PQ, being diminished in the same ratio, is turned into the said spheroid ; and the gravity at A, in each case, is diminished in approximately the same ratio. Therefore the gravity at A on the sphere with centre C and described with the radius AC is to the gravity at A on the earth as 126 to 1

2125 , and the gravity at the position Q on the sphere with centre C and described with the radius QC, is to the gravity at the location A on the sphere described with centre C and with radius QC, in the ratio of the diameters (by Prop. LXXII. Book I.) that is, as 100 to101. Now these three ratios may be taken together, 126 to 125, 126 to 1

2I25 , and 100 to 101: and the gravity at the location Q on the earth becomes to the gravity at the position A on the earth, as

12126 116 100 to 125 125 101× × × × , or as 501 to 500.

Now since (by Corol. 3, Prop. XCI, Book I.) the weight in either leg of the canal ACca or QCcq shall be as the distances of the places from the centre of the earth ; if those legs may be separated into transverse and equidistant surfaces proportional to the total parts, the weights of the individual parts in the leg ACca will be to the weights of just as many parts in the other leg, as the magnitudes and accelerating weights jointly ; that is, as 101 to 100 and 500 to 501, that is, as 505 to 501. And thus if the centrifugal force of each part in the leg ACca arising from the daily motion were to the weight of each part as 4 to 505, so that therefore from the weight of each part, by dividing into 505 parts, four parts may be subtracted; the weights will remain equal in each leg, and therefore the fluid might remain in equilibrium. Truly the centrifugal force of each part is to the weight of the same as 1 to 289, that is, the centrifugal force must be equal to the 4

505 part of the weight is only the 1289 .

part. And therefore I say, following the golden rule of proportions, that if the centrifugal force were made 4

505 as the height of the water in the leg ACca may exceed the height of water in the leg QCcq by the hundredth part of the total height: the centrifugal force 1

289 may be made as the excess of the height in the leg ACca shall be of the height in the other leg QCcq by only the 2

229 part. Therefore the diameter of the earth along the equator to the diameter through the poles is as 230 to 229. And thus since the mean radius of the earth, close to the measurement of Picart, shall be 19615800 Paris feet, or of 3923,16 miles (on putting that a mile shall be a measure of 5000 feet), the earth will be higher at the equator than at the poles by an excess of 85472 feet, or of 1

1017 miles. And its altitude at the equator will be around 19658600 feet, and at the poles 19573000 feet. If a planet shall be greater or smaller than the earth with its density remaining and the daily periodic time of revolution, the proportion of the centrifugal force to gravity will remain, and therefore the proportion of the diameter between the poles to the diameter along the equator will remain also. Or if the daily motion may be accelerated or retarded in some ratio, the centrifugal force will be augmented or decreased in that ratio squared, and therefore the difference of the diameters will be increased or diminished in the same ratio approximately. And if the density of the planet may be increased or decreased in some ratio, the drawing force of gravity also will be increased or decreased in the same ratio, and the difference of the diameters in turn will be diminished in the ratio of the of



the increased gravity or diminished in the ratio of the diminished gravity. From which since the earth with respect to the fixed stars may be revolving in 23hours and 56 minutes, but Jupiter in 9 hours and 56 minutes and the squares of the times shall be as 29 to 5, and the densities of the revolving bodies as 400 to 1

294 : the difference of the diameters of Jupiter will be to the smaller diameter as 1

2

29 400 15 22994× × to 1, or 1 to 1

39 approximately.

Therefore the diameter of Jupiter drawn from the east to the west , to the diameter between the poles is as 1

310 to 139 approximately. From which since the major diameter

of this shall be 37", the minor diameter lying between the poles will be 33", 25'''. From the erratic [refraction of the] light there may be added around 3", and the apparent diameters of this planet emerge 40'' and 36" : which are to each other approximately as

1 16 611 to 10 . This thus may be itself had from the hypothesis that the body of Jupiter shall

be of uniform density. But if its body shall be denser towards the plane of the equator rather than the poles, it diameters may be in turn as 12 to 11, 13 to 12, or perhaps as 14 to 13. And indeed Cassini observed in the year 1691, that the diameter of Jupiter stretched out from east to west surpassed the other by about a fifteenth part. Moreover our countryman Pound with a telescope of 123 feet long and with the best micrometer, measured the diameters of Jupiter in the year 1719, as follows.

Times. Diam. max. Diam. min. Diameters ratio. days hours. part. part. Jan. 28 6 13,40 12,28 as 12 to 11 Mar. 6 7 13,12 12,20 3 3

4 413 12 Mar. 9 7 13,12 12,08 2 2

3 312 11 Apr. 9 9 12,32 11,48 1 1

2 212 11 Therefore the theory has agreed with the phenomena. For planets are warmed more in the light of the sun towards their equators, and therefore they are heated there a little more than towards the poles. Even though the force of gravity be diminished at the equator by the daily rotation of our earth, and therefore there the earth rises higher than at the poles (if its matter shall be of uniform density), it will be apparent from experiments with pendulums which are reviewed in the following proposition. [See Chandrasekhar Ch. 20 for a modern exposition of this proposition and related work.]

PROPOS1TION XX. PROBLEM IV.



To find and to compare amongst themselves, the weights of bodies in different regions of the earth. Because the weights of the unequal legs of the channels of water ACQqca are equal; and the weights of the parts proportional to the whole legs, and similarly situated to the whole, are in turn as the weights of the whole, thus they are also equal to each other ; the weights of the equal and similarly situated parts are inversely as the legs, that is, inversely as 230 to 229. And likewise the ratio of any homogeneous and similarly situated bodies in the legs of the channels. The weights of these are reciprocally as the legs, that is, reciprocally as the distances of the bodies from the centre of the earth. Thus if bodies may be considered in the upper parts of the channels, or on the surface of the earth ; the weights of these in turn will be inversely as the distances of these from the centre. And the weights by the same argument, in any other regions over the whole surface of the earth, are inversely as the distances of the places from the centre; and therefore are given in proportion, from the hypothesis that the earth shall be a spheroid. From such thence a theorem arises, that the increment of the weight in going from the equator to the pole, shall be nearly as the versed sine of twice the latitude, or which is the same thing, as the sine square of the right latitude. And the arc of the degrees of latitude may be increased in around the same ratio in the meridian. Therefore since the latitude of Paris shall be 480, 50', that of places on the equator 00 0, 00', and that of places at the poles 900, and the squares of the versed sines shall be as 11334, 00000 and 20000, with the radius present 10000, and the weight at the pole shall be to the weight at the equator as 230 to 229, and the excess of the weight at the pole to the weight at the equator shall be as 1 to 229: therefore the excess of the weight at the latitude of Paris to the weight at the equator shall be as 11334

200001 to 229 or 5667 to 2290000,× . And therefore the total weighs in these places will be in turn as 2295667 to 2290000. Whereby since the lengths of isochronous pendulums shall be as the forces of gravity, and at the latitude of Paris the length of a pendulum oscillating in individual seconds shall be three Paris feet and

128 lines, or rather on account of the weight of the air 5

98 : the length of the pendulum at the equator will be overcome by the length of the Parisian pendulum by an excess of 1,087 parts of a line. And by a similar calculation the following table can be constructed.

General latitude of the place.

Length of the pendulum.

Measure of 10 along a meridian.

Degrees Ft. Lin. Toises. 0 3 7,468 56637 5 3 7,482 56642 10 3 7,526 56659 15 3 7,596 56687 20 3 7,692 56724 25 3 7,812 56769 30 3 7,948 56823 35 3 8,099 56882 40 3 8,261 56945 1 3 8,294 56958



2 3 8,327 56971 3 3 8,361 56984 4 3 8,394 56997 45 3 8,428 57010 6 3 8,461 57022 7 3 8,494 57035 8 3 8,528 57048 9 3 8,561 57061 50 3 8,594 57074 55 3 8,756 57137 60 3 8,907 57196 65 3 9,044 57250 70 3 9,162 57295 75 3 9,258 57332 80 3 9,329 57360 85 3 9,371 57377 90 3 9,387 57382

Moreover it is agreed by this table that the inequalities of degrees shall be so small that in it may be taken as a sphere in the geographical shape of the earth: especially if the earth were a little denser towards the equatorial plane than towards the poles. Now truly some astronomers sent to far away regions to make astronomical observations, observe that oscillating clocks may move slower near the equator than in our regions. And indeed Richer observed this in the year 1672 on the island of Cayenne. For while he observed the transit of fixed stars through the meridian in the month of August, he found his clock to be moving slower than for the mean motion of the sun, with a difference present of 2'.28" for individual days. Thereupon by arranging so that a simple pendulum might oscillate in single seconds as measured by a reliable clock, he noted the length of the simple pendulum, and this he did often each day for ten months. Then on returning to France he compared the length of this pendulum with the length of the Parisian pendulum (which was of three Parisian feet, and 3

58 ) and he found it to be shorter, with the difference being 1

41 lines. Afterwards our countryman Halley sailing to the island of St. Helens in the year 1677, found the oscillations of his clock there to be moving slower than in London, but he did not measure the difference. Truly he rendered the pendulum shorter by 1

8 part of an inch, or by 1

21 lines. And in order to do this, since the length of the screw at the end of the pendulum was insufficient, he interposed a wooden ring between the sheath of the screw and the pendulum bob. Then in the year 1682, Varin and Des Hayes found the length of the pendulum oscillating in a time of one second at the royal observatory in Paris to be 3ft., 5

98 lin. And on the island of Gorée by the same method they found the length of the isochronous pendulum to be 3 ft., 5

96 lin., with a difference of the lengths present of 2 lin. And in the same year sailing to the islands of Guadeloupe and Martinique, the found the length of the isochronous pendulum on these islands to be 3ft., 1

26 lin.



After this Couplet the younger in the month of July of the year 1697, thus setting his pendulum clock to the mean motion of the sun at the royal observatory in Paris, so that in a long enough time the clock was in agreement with the motion of the sun. Then sailing to Lisbon he found that late in the month of November the clock would go slower than before, with a time difference of 2', 13" in 24 hours. And in the month of March following sailing to Paraiba he found there his clock to go slower than in Paris, with a difference present of 4', 12" in 24 hours. And it confirms that the pendulum swinging in one second should be shorter in Lisbon by 1

22 lin. and in Paraiba by 233 lines than in Paris. He would

have corrected the differences better to be 513 91 and 2 . For these differences [of the

lengths] correspond to the differences of the times 2', 13", and 4', 12". There is less trust in gross observations of this kind. In the next years (1699 and 1700) des Hayes sailing again to America, determined that on the islands of Cayenne and Granada the length of a pendulum swinging in a time of one second, should be a little less than 3 ft., 1

26 lin., and that on the island of St. Christophers that length should be 3 ft., 3

46 lin., and that on the island of St. Dominica the same should be 3 ft., 7 lin. In the year 1704 Father Feuillée found at Portobello in America that the length of a pendulum oscillating in seconds was 3 Parisian ft., and only 7

125 lin, that is, almost a line shorter than at Paris, but from an erring observation. For then sailing to the island of Martinique, he found the length of the isochronous pendulum to be 3 Parisian ft., and 10

125 lin. But the latitude of Paraiba is 60, 38' to the South, and that of Portobello 90, 33' to the North, and the latitudes of the islands of Cayenne, Gorée, Guadeloupe, Martinique, Granada, & St. Christophors [St. Kitts] and St. Dominica are respectively 40, 55', 140, 40', 140, 00', 140, 44', 120, 6', 170, 19', and 190, 48' to the North. And the excess of the longitude of the Parisian pendulum over the lengths of the isochronous pendulums at these latitudes are a little greater than for the table of lengths of pendulums calculated above. And therefore the earth is a little higher at the equator than for the above calculation, and denser at the centre than in the mines near the surface, except perhaps the heat in the torrid zones may have increased the lengths of the pendulums a little. Certainly Picart observed that iron rods, which in winter time were one foot in length frozen solid, heated in the fire emerged one foot with the quarter of a line. Then de la Hire observed that an iron rod which in the same manner in winter time was six foot long, when it was exposed to the summer sun emerged six feet in length with the two thirds part of a line. In the former case the heat was greater than in the latter case, truly in this latter case it was greater than the heat of the external parts of the human body. For metals become exceedingly hot in the summer sun. But the rod of a pendulum in the pendulum clock at no time is accustomed to be exposed to the heat of the summer sun, nor at any time to take in heat equal to the heat of the external surface of the human body. And therefore the rod of the pendulum in the clock three feet in length, indeed will be a little longer in summer time than in winter time, but scarcely by exceeding a quarter of a line. Therefore the total difference of the lengths of the pendulums which are isochronous in diverse regions, cannot be attributed to different amounts of heat. But nor is this difference to be attributed to errors of the astronomers sent from France. For whatever of



their observations were not in perfect agreement among themselves, yet the errors are small and thus may be ignored. [One may note that at this time the science of statistics was virtually unknown.] And they all agree, that isochronous pendulums are shorter at the equator than at the royal observatory in Paris, with a difference not less than 1

41 , not greater than 2

32 . Through the observations made by Richer in Cayenne, the difference was 1

41 . From these made by des Hayes that corrected difference produced 121 lines or

341 lines. From these measurements of others made with less accuracy the same result was

produced as around 2 lines. And this discrepancy was able to arise partially from errors in observations, partially from the differences of the internal parts of the earth and the height of mountains, and partially from the different heats [that we would now call temperatures] in the air. An iron rod three feet long, in winter time in England, is shorter than in summer time, I believe, by the sixth part of a line. On account of the heat at the equator this quantity [ 1

6 th of a line] may be taken away from the difference of the lines 1

41 observed by Richer, and 1

121 line will remain which now agrees properly with that deduced before by theory 8710001 .

Moreover the observations made by Richer in Cayenne, that he repeated in the individual weeks through ten months, and the lengths of the pendulum in the iron rod noted with the length of that similarly noted when he returned to France. Which diligence and caution seems to be lacking in the observations of others. If it is required to put trust in the observations of this, the earth is higher at the equator than at the poles in excess of around 17 miles as predicted by the above theory.

PROPOSITION XXI. THEOREM XVII. The equinoctial points regress, and the axis of the earth by nutating in the individual yearly revolution, inclines twice towards the ecliptic and twice to the first position. It is apparent by Corol. 20. Prop. LXVI. Book I. But that motion of nutation must be very small, and scarcely or not perceptible at all.


Book III Section II. Translated and Annotated by Ian Bruce. Page 777

Translator's Introduction.

For an elementary understanding of lunar theory, if such a thing is possible, a good place to start is the Wikipedia article of the same name, and related topics, followed perhaps for a brief account of the history of lunar theory, by reading if available J. F. Scott's note at the start of Vol. IV of Newton's Letters, and progressing to a text such as Godfray : An Elementary Treatise on the Lunar Theory (1853), the Appendix of which contains a nice non-technical summary of early developments, and then to the more compendious volume of Brown (1895) of the same name, etc. The Lectures on the Lunar Theory by J.C. Adams are also of interest, a little of which is quoted here below. All of these except the Letters can be downloaded from the Open Library website. Most books on classical mechanics including Whittaker, Synge & Griffith, do not mention the moon at all (!), an indication of the difficulty of the task, although Goldstein's work is an exception, and one can get some idea of the immense labour that individuals such as Hill and Delauney put into this study, since Newton's time, before the dawn of the computer age, whereby we may gain numerical prediction, but not necessarily physical understanding. A modern and highly relevant mathematical commentary is found in Chadrasekhar's delightful book, Newton's Principia for the common reader, Ch.13. In addition, the lecture notes of Prof. Richard Fitzpatrick on Newton's Dynamics, Ch. 14 are of special interest, where the problem is given a modern setting, and some of the inequalities resolved to some extent. Newton used the moon as a template for his theory of gravitation, in which endeavour he was highly successful, but he could not complete the task : indeed, John Couch Adams in the introduction to his lunar lectures, tells us that: ' Newton's Principia did not profess to be and was not intended to be a complete exposition of the Lunar Theory. It was fragmentary; its object was to shew that the more prominent irregularities admitted of explanation on his newly discovered theory of universal gravitation. He explained the Variation completely, and traced its effect in Radius Vector as well as in Longitude; and he also saw clearly that the change in eccentricity and motion of the apse that constitute the Evection could be explained on his principles, but he did not give the investigation in the Principia, even to the extent to which he had actually carried it. The approximations are more difficult in this case than that of the Variations, and require to be carried further in order to furnish results of the same accuracy as had already been obtained by Horrox from observation. He was more successful in dealing with the motion of the node and the law of the inclination. He shewed that when the Sun and Node were in conjunction, then for nearly a month the moon moved in a plane very approximately, and that the inclination of the orbit then reached a maximum, namely, 50 17' about; but as the Sun moved away from the Node the latter also began to move, attaining its greatest value when the separation was a quadrant, and that at this instant, the inclination was 50 very nearly. He also assigned the law of intermediate positions. The fact that the there was no motion when the Sun was at the Node, that is, in the plane of the moon's orbit, confirmed his theory that these inequalities were due to the Sun's action. When we spoke of Newton's results as being fragmentary and incomplete, let it not be understood that he gave only a very rude approximation to the truth. His results are far more accurate than those arrived at in elementary works of present day on the subject (i.e. towards the end of the 19th Century).'



It seems appropriate to insert here some definitions of astronomical terms and to quote relevant formulas for the various inequalities described by Newton. For this we follow Chandrasakher mainly: the diagram and the lettering used is that of Newton.

Some Descriptive Terms Celestial sphere : an imaginary spherical shell of very large radius concentric with the earth and rotating with the earth, onto which all heavenly bodies are projected. Plane of the ecliptic: the plane of the earth's orbit around the sun, which changes slowly in time. Eclipses of the sun or moon can occur only in this plane, and hence the name. Syzygy : when the sun S, the moon P (or a planet) & earth T lie on, or almost on, the same straight line, whether in conjunction or opposition. Quadrature : when the moon (or a superior planet) lies in the first or last quarter position so that PT is perpendicular to ST. Apogee: point in the moon's orbit furthest from the earth. Perigee: point in the moon's orbit nearest to the earth. Nodes : the two points were the moon's orbit cuts the plane of the earth's orbit or ecliptic, one rising and the other falling. Vernal equinox : a reference point on the earth's orbit, when the sun rises in the first point of Aries, at which the tilt of the earth is at 90 degrees to the plane of the ecliptic, and the sun is directly overhead at the equator. Synodic month : the time for a complete orbit of the moon around the sun measured by a line drawn from the earth to the sun. This is taken from a long time average to be 29.530589 days.

Main Phenomena Evection (Ptolemy) : changes in the apparent size of the moon over monthly periods of time, measured by its subtended angle. Elliptic Inequality, or, Equation of the Centre (Tycho.): changes in the rate of motion of the moon's centre or longitude as it rotates around the earth. Variation (Tycho.): additional speeding up of the moon as it approaches full moon or new moon, and slowing down as it approaches quadrature. Annual Equation : Gradual expansion of the moon's orbit as the earth gets closer to the sun at perihelion in January, and contraction as it goes through aphelion in July, due to the sun's perturbation. Two effects that arise from the interaction with the sun are the precessions of the perigee and of the nodes. The perigee precesses in the same sense as the moon rotates about the earth, completing a complete cycle in 8.85 years, while the right ascending node precesses in a retrograde manner, and completes a cycle in 18.6 years.



The following notes relate mainly to Prop. XXV and Prop. XXVI. Initially we assume the moon's orbit is almost circular around the earth as centre, and lies in the plane of the ecliptic.

1. The perturbing function TL : The standard unit of acceleration, being that experienced by the earth in the Sun's gravity, is given by 2

SGMST

, and 22SGM

STN ST= , where N is the angular frequency of the

earth about the sun; n is similarly defined as the mean angular frequency of the moon around the earth : and 2

2TGMPT

n PT k= = .

In the above diagram, we may note that SL denotes the force of the sun on the moon at P, while ST denotes the force of the sun on the earth at T; from which we note that the force TL must be added to ST to give SL and hence in the position shown, TL is the extra perturbing force due to the sun acting on the moon, all else being equal; on the other side of the quadrature line CD, the force acting on the moon is less by TL than the force acting on the earth; these forces thus distort the circular orbit of the moon. In the above diagram, ST PT>> , and it will be shown later that in this case 2KL PK , where K is the average distance of the moon from the sun in its orbit round the earth, and the figure PLMT is assumed to be a parallelogram. The extra force TL can then be resolved into components TE and LE parallel and perpendicular to TP : the component of the perturbing force LE, or 3 3

23 3 3 2PK TK PKTP TPF PK sin PT cos PT sin cos PT sinψ ψ ψ ψ ψ×

⊥ = = = = = acting perpendicular to the radius TP; while the original component PT TE+ acts along TP; or the parallel perturbing force

( ) ( ) ( ) ( )12

2 2 23 3 1 3cos 1 3cos 1 1 3 2PK PKPT PTTE F PK PT PT PT PT PT cos .ψ ψ ψ⎞⎛= = − = − = − = − = +⎜ ⎟

⎝ ⎠Note finally that these lengths are to be multiplied by N2 to give absolute forces as above, and the force 23

2 2F N .PT sin ψ⊥ = − , while ( )12

2 1 3 2F N .PT cos .ψ= + The perturbing forces (or accelerations towards the sun) acting at the syzygies, when

0 or ψ ψ π= = are both parallel to the axis AB, and given by ( ) 22 SF syzy. N .r= ; in the position A, the moon P is pulled by an acceleration greater by this amount then the earth,



while in the position B, in the above diagram, the force at B has the same value, but in this case, the earth is now accelerated more than the moon, by the same amount. The perturbing forces acting at the quadratures C and D are given by

312 2 or ψ π ψ π= = , when the forces along CD are now both ( ) 2

QF quad. N .r= − . 2. The max. and min .centripetal acceleration: The initial circular orbit satisfies 2

2TGMPT

n PT= ; this centripetal force becomes at the

syzygies, ( ) ( ) ( ) ( )2

2 22 2 2 2 22 1 2 2T

S

GM NS S S Sr n

F F syzy. n N r n r n N r= − = − = − = − ,

where n is the corresponding angular frequency of the moon, assumed to rotate in a circle about the earth as centre. At quadrature, the centripetal force becomes

( ) ( ) ( ) ( )2

2 22 2 2 2 21T

Q

GM NQ Q Q Qr n

F F quad. n N r n r n N r= − = + = + = +

Thus,

( ) ( ) ( )2 2 2 22 2 1S SF n N r n N AT x= − = − − and ( ) ( ) ( )2 2 2 2 1Q QF n N r n N AT x= + = + + ,

where x is the departure from the initial radius. These accelerations can also be written in the form :

( ) ( ) ( ) ( )2 222 2 2 2

112 1 2 1 2TGM k

S S S xx PTF n r N r m m

−−= − = − = − and similarly,

( ) ( ) ( ) ( )2 222 2 2 2

111 1TGM k

Q Q Q xx PTF n r N r m m

++= + = + = + ; results established by

Chandrasekhar in his notation. Thus, the orbit is a prolate ellipse, and reverting to a general point on the orbit, the value of the radius becomes ( )1 2r AT x cos ψ= − .]

3. Variation of the areas swept out: Recall that 232 2F N .PT sin ψ⊥ = − , then the torque

exerted on the moon is given by 2 232 2dH

dt F .PT N .PT sin ψ⊥= = − , which we can equate to a quantity proportional to the rate of change of the area swept out by the moon, relating to Kepler's Second Law. For the moon in a circular orbit, d

dtn ψ= , where n is the average time of the rotation of the moon for a synodic month, that is, following the phases of the moon. Thus,

22 2 23 32 22 2N

ndH N .PT sin .dt .PT sin .dψ ψ ψ= − = − ; recall that ψ is the angle between the earth-moon and the earth-sun radii, and on these being referred to the earth-moon ascending nodal angles of these radii, and v v' (or U) respectively, we have

( ) ( )1 1v'vv v' v v mψ = − = − = − , where v'

v can also be expressed by the ratio Nnm = ,

where N is the angular frequency of the earth around the sun, and n the synodic angular frequency of the moon around the earth. Thus,

( ) ( )2 2 2 2 2 2 23 32 22 2 1 3 2 1dv

ndH N .PT sin .dt m n .PT sin v m . m n.PT sin v m .dvψ= − = − − = − −

this can be integrated to give : 2 23

4 1constant 2m nmH .PT cos .ψ−= + To turn this into a

manageable form, recall that for an almost circular orbit, we have



2 2av.H nr nPT H= = = , and hence ( )22 3

4 11 2dv mav.dt mH PT H cos .ψ−= = + As m is very

small, in fact, 1000178718m , then the variation in the area between syzygies and quadrature

is given approximately by 2 23 3

4 1 42 233

44 1

1 1 111311102311

msyz . m

mquad .m

H mH m

−

−

+ +−−

= = .

4. Finding the other quantities. Since the radial velocity is very small, the velocity V of the moon in orbit can be approximated by 2

tV PT nPTπ= = . Clearly this is related to the approximate rate at

which the area A is swept out 12

2dA dvdt dtPT .PT PT nπ= = , and we have

( )21 1 12 2 2

2 34 11 2dA dv m

av.dt dt mH PT H cos ψ−= = = + , while the velocity V is found from

( )2 2

2

22 34 11 2av .H m

mPTV cos ψ−= + .

Thus at the syzygies and at quadrature, we have ( ) ( )2 2

222 3

2 111av .H m

S mx PTV −−

+ and

( ) ( )2 2

222 3

2 111av .H m

Q mx PTV −+

− . Other quantities of interest can be found in the relevant

chapters of Chandrasekhar. As was his custom, Newton gives word summaries only in this initial proposition, for the quite extensive mathematical investigations that he had carried out in relation to the various inequalities of lunar motion, which he sketches out briefly in subsequent propositions, and which are treated here hopefully in more detail, with the help of Chadrasekhar's and the work of others.

PROPOSITION XXII. THEOREM XVIII. All lunar motions, and all the inequalities of motions, follow from the principles put in

place. The major planets, while they are carried around the sun, meanwhile are able to have other smaller planets conveyed revolving around them, and these smaller ones must appear by Prop. LXV. Book I to revolve in ellipses having foci at the centres of the major planets. But the motions of these are disturbed in various ways by the action of the sun, and from these they will bring about the inequalities which are observed in our moon. Certainly this is moving faster at the syzygies than at the quadratures (by Corol. 2, 3, 4, & 5, id.), and with a radius drawn to the earth will describe a greater area in the time, and have an orbit less curved, and thus may approach closer to the earth, unless perhaps it may be prevented by an eccentric motion. [Note that Newton's use of the term eccentricity referring to ellipses is not that of more modern times; he meant presumably merely the curvature of the approximately elliptic orbit at the point in question. The word of course has a long history in astronomy, dating



from Hipparcus, where it referred to eccentric circular orbits and the use of epicycles in describing inequalities in the motion of the moon and planets.] For the eccentricity is a maximum (by Corol. 9. id.) where the apogee of the moon is situated at the syzygies, and the minimum where likewise it stands at the quadrature positions ; and thence the moon is faster at the perigee and closer to us , but slower at the apogee, and [in this case] further from us at the syzygies than when at the quadratures. In addition the apogee progresses forwards, and the nodes go backwards, but in unequal motions. And indeed the apogee is progressing faster at its syzygies, regressing slower at its quadratures (per corol. 7. & 8, id.), and as a consequence it is carried forwards annually by the excess of the progression over the regression. But the nodes (by Corol. 2, id.) are at rest in their quadratures and regressing the most quickly at the syzygies. [We have inserted the correction made by Chandrasekhar on p. 422; by transposing the words quadratures and syzygies, thus correcting a mistake made in all three editions of the Principia & in all the translations.] And also the latitude of the moon is greater at its quadratures (by corol.10, id.) then in its syzygies: and the mean motion is slower at the perihelion of the earth (by corol. 6. id.) than at its aphelion. And these are the most significant inequalities observed by astronomers. Also there are certain other inequalities not observed in earlier times by astronomers, by which the lunar motions thus may be disturbed, so that they are unable to be reduced by a law to some certain rule so far. For the velocities or the hourly motions of the moon at the apogee and nodes, and the equations of the same, so that both the difference between the maximum eccentricity at the syzygies and the minimum at the quadratures, and which inequality is called the variation, may be increased or diminished annually (by Corol. 14, id.) in the cubic ratio of the apparent diameter of the sun. And therefore the variation is increased or diminished in the square ratio of the time between the quadratures approximately (by Corol. 1 & 2, Lem. X. & Corol. 16, idem) but this inequality is usually referred to in astronomical calculations as part of the lunar prosthaphaeresis [i.e. the central equation of lunar theory], and combined with that.

PROPOSITION XXIII. PROBLEM V. To derive the inequalities of the satellites of Jupiter and Saturn from lunar motions.

From the motions of our moon, the analogous motions of the moons or satellites of Jupiter thus may be derived. The mean motion of the nodes of the outermost of Jupiter's satellites [Callisto], is to the mean motion of the nodes of our moon, in a ratio composed from the square ratio of the periodic time of the earth around the sun to the periodic time of Jupiter around the sun, and in the simple ratio of the periodic time of the satellite around Jupiter to the periodic time of the moon around the earth (by Corol. 16. Prop. LXVI. Section XI. Book I.) and thus in 100 years the nodes are advanced by 8 gr. 24'.

[Following Chandrasakher, we have 2

2 216 69126 3211 86

J CE

EE J

TT .T .T .

ΔΩΔΩ

= × = × ; from the known

average value for the moon, 0 0=19 286 per year, we have 8 . 22'E C.ΔΩ ΔΩ = for Callisto for 100 years, in close agreement with what is observed.]



The mean motions of the nodes of the inner satellites are to the motion of this outer satellite, as the periodic times of these inner satellites to the periodic time of this outer satellite (by the same corollary) and thence are given. Moreover the motion of the apogee of each satellite consequently is to the antecedent motion of its nodes [i.e. carried backwards], as the motion of the apogees of our moon to the antecedent motion of its nodes (by the same Corol.) and thence is given. Yet the motion of the perigees thus found must be diminished in the ratio 5 to 9 or approximately as 1 to 2, by a reason that cannot be explained here. The equations of the greatest nodes and of the apogee of each are almost to the greatest equations and of the apogee of the moon respectively, as the motion of the nodes and of the apogees of the satellites in the time of one revolution of the first equations, to the motion of the nodes and of the lunar apogee in the time of one revolution of the latter equations. The variation of the satellite seen from Jupiter, is to the variation of the moon, as in turn are the whole motion of the nodes in the times in which the satellites and the moon are rotating around the sun, by the same corollary ; and thus in the outer satellite does not exceed 5". 12"'.

PROPOSITION XXIV. THEOREM XIX. The flow and ebb of the sea arise from the actions of the sun and the moon.

[We may note that for this proposition, Chandrasakher has produced an annotated version on pp. 403–411 of his work on the Principia, setting out where he agrees and where he disagrees with what Newton has stated, from his modern knowledge concerning tides; the 3 hour rule introduced by Newton for the rising of the tide after the sun or moon has passed its meridian is thus discarded, as are other rather vague assertions, while the main body of the argument is retained.] The sea must be raised up twice in each day both by the sun as well as by the moon, and it is apparent by Corol. 19. & 20. Prop. LXVI. Book I, that the maximum height of the water in deep and open seas, follows the influence of the luminous bodies at the meridian of a place, in an interval of less than six hours, as occurs in the seas of the Atlantic and of Ethiopia on being drawn eastwards between France and the Cape of Good Hope, and as in the seas of the Pacific along the coasts of Chile and Peru : on all the shores of which the tide comes in for around two, three or four hours, except where the motion propagated from the ocean depths may be retarded by shallow places as much as five, six, or seven hours or more. The number of hours I count from the influence of each luminous body at the meridian of the place, both below the horizon as well as above, and by the hours said of the moon I understand the twenty four parts of the time in which the moon by its apparent daily motion is returned to the place of the meridian. The force of the sun or of the moon required for the maximum elevation in the sea by the luminous body itself is at the meridian of the location. But the force from that remains acting on the sea for some time, and the influence may be increased by the new force, then the sea may rise to a new height, which happens in a space of one of two hours but more often in a space of around three hours on a shore, or even more if the sea shall be shallow. The two motions, which the two luminous bodies excite, cannot be discerned separately, as they bring about a certain mixed motion. In the conjunction or opposition of the luminous bodies the effect of these will be added together, and the maximum flow and



ebb will be put in place. At the quadratures, the sun may raise the water where the moon depresses it, or it may depress the water where the moon may raise it ; and from the difference of the effects the smallest of all the tides may arise. And because, by experience, it may be seen that the moon has a greater effect than the sun, the maximum height of the water occurs around the third lunar hour. Outside the conjunctions and the quadratures, the maximum tide which arises only from the force of the moon must always be in the third hour of the moon, and by the sun alone in the third hour of the sun, with the sum of the forces it is incident at some intermediate time that is close to the third hour of the moon ; and thus in the transition of the moon from the syzygies to the quadratures, where the third hour of the sun precedes the third hour of the moon, the maximum height of the water will also precede the third hour of the moon, and that in a maximum interval a little past the eight part of the lunar time ; and with equal intervals the maximum tide will follow the third lunar hour in the transition of the moon from quadrature to conjunction [or syzygies]. These are thus in the open sea. For at the estuaries of rivers the greatest flow with all else being equal reach a peak later. But the effect of the luminous bodies depends on their distances from the earth. Indeed at the smallest distances the greater are the effects of these, at greater distances the lesser, and that in the cubic ratio of the apparent diameters. Therefore the sun in winter time, present at the perigee, produces a greater effect, and it comes about that the tide at the syzygies shall be greater, and at the quadratures less (with all else being equal) than in the summer time ; and the moon in perigee in individual months will disturb the tide more than before or after the fifteenth day, when it is moving into apogee [i.e. the half-cycle of the moon's orbit nearest the perigee ]. From which two causes it happens that at successive syzygies the tides may not follow each other entirely as two high tides. Also the effect of each luminous body depends on the declination of that, or the distance from the equator. For if a luminous body may be put in place at a pole, that may attract the individual parts of the water constantly, without an increase or decrease in the action, and thus it will generate no reciprocation of the motion. Therefore the luminaries by receding from the equator towards the pole, gradually lose their effect, and therefore they will raise smaller tides in the solstical syzygies than in the equinoxal syzygies. But they will raise greater tides in the solstical quadratures than in the equinoxal quadratures ; because now there the effect of the moon situated at the equator certainly exceeds the effect of the sun. Therefore the maximum tides arise in the syzygies and the minima at the quadratures of the luminaries, around the equinoxal times of each. And the maximum tide in the syzygies is always accompanied by the minimum in the quadrature, as is known from experience. But during the minimum distance of the sun from the earth, as in winter rather than in summer, it arises that the maximum and minimum tides often precede rather than follow the vernal equinox, and often follow rather than precede the autumnal equinox.



Also the effects of the luminous bodies depends on the latitude of the places. The figure ApEP designates the earth completely overwhelmed with deep water ; C is centre; P, p the poles; AE the equator ; F some place above the equator; Ff parallel to the place ; Dd parallel to that corresponding to the other side of the equator; L the place that the moon will occupy three hours before ; H the place on the earth directly under that ; h the place opposite to this; K, k places thus at 90 degrees distance; CH, Ch the maximum heights of the sea measured from the centre of the earth ; and CK, Ck the minimum heights : and if with the axes Hh, Kk an ellipse is described, then by revolving about the major axis of this ellipse Hh the spheroid HPKhpk may be described; this will designate the figure of the sea approximately, and CF, Cf, CD, Cd will be the heights of the sea at the places F, f, D, d. Truly indeed if in the aforemade ellipse by revolving some point N it may describe the circle NM, cutting the parallel Ff, Dd at some points R, T, and the equator AE in S; CN will be the height of the sea at all the places R, S, T, situated on this circle. Hence in the daily revolution of some place F, the influx will be a maximum at F, three hours after the influence of the moon at the meridian above the horizon ; later the maximum outflow will be at Q three hours after the setting of the moon; then the maximum influx will be at f three hours after the influence of the moon at the meridian below the horizon ; the final maximum outflow will be at Q three hours after the rising of the moon ; and the last influx at f will be less than the first influx at F. For the whole sea may be separated into two completely hemispherical seas, one in the hemisphere KHk inclined to the north, the other towards the south in the opposite hemisphere Khk; which can be called the northern and southern floods. These seas always come in turn at the meridians of individual places mutually opposed to each other, between lunar intervals of twelve hours. And since the northern regions partake more in the northern seas, and the southern regions in the southern seas, thence the tides arise alternately greater and lesser, in the individual places beyond the equator, in which the luminous bodies rise and set. But the greater tide, with the moon declined at the vertical of the place, will happen at around three hours after the influence of the moon at the meridian above the horizon, and with the declination of the moon changed it becomes the lesser tide. And the greatest difference of the maximum flux arises at the time of the solstices ; especially if the ascending node of the moon is passing through the principle point of Aries. Thus it is found from experience, that in winter time the morning tide surpasses the evening tide, and in summer the evening tide surpasses the morning tide, indeed at Plymouth by a height of as much as one foot, at Bristol truly by a height of fifteen inches : from the observations of Colepress and Sturmy. But the motions described up to this point are changed a little by that returning force of the waters, by which the tides of the sea, even with the cessation of the luminous bodies, may persevere for some time. This persistence of the impressed motion diminishes the difference of the alternate tides ; and the tide just after the syzygies is returned the most, and that diminished the most just after the quadratures. From which it shall be that the alternate tides at Plymouth and Bristol shall not be more different in turn than one foot or fifteen inches ; and so that the maxima of all the tides in the same ports shall not be the first from the syzygies, but the third. Also all the motions are retarded by passing through channels, thus so that the maximum of all the tides, in certain straits and estuaries of rivers, shall be even as the fourth or fifth day after the syzygies.



Again it can come about that the tide will be propagated from the ocean through different channels to the same port, and it may pass faster by one channel than by another : in which case the same tide, divided into two or more arriving successively, may consist of new motions of different kinds. We may consider two equal tides to arrive at the same port from different locations, the foremost of which precedes the other by a space of six hours, and may happen in the third hour after the influence of the moon at the meridian of the port. If the moon were turning at the equator, then by its own influence, equal flows would arrive every six hours, which being equal to the mutual reflux the same flows shall be equal, and thus in the space of that day the tides will effect that the still water remain at rest. If the moon then should decline from the equator, major and minor tides in turn are produced in the ocean, as has been said ; and then two major and two minor inflows are propagated in turn to this port. But the two major influxes compound the water to the highest level in the middle between each, the major and the minor influx act so that the water rises to the average height in the middle of these, and between the two minor influxes the water ascends to the minimum height in the middle of these. Thus in the space of twenty four hours, the water is accustomed to arrive not twice, but only once at the maximum height and once at the minimum ; and the maximum height, if the moon declines towards the pole above the horizon of the place, falls either in six or thirty hours after the influence of the moon at the meridian, and by changing the lunar declination it will become a deflux [i.e. an ebb-tide]. An example of all of which has been revealed from the observations of Halley's sailors at the port of Batsham in the kingdom of Tunquin, at the northern latitude of 200. 50'. [This channel is now the Qiongzhou Strait, a region where unusual tides are found, separating the island of Hainan from the P.R. of China, and connecting the South China Sea to the Gulf of Beibu.] There during the day following the transit of the moon through the equator, the water remains unmoved, then with the moon declining to the north the tides begins to flow and ebb, not twice, as in other ports, but only once in individual days ; and the tide rises on the setting of the moon, the maximum deflux on the moon rising. Here the tides increase with the declination of the moon, as far as the seventh or eighth day, then through another seven days falling by the same amounts, by which before it had risen ; and finally stops changing with the lunar declination [again passing through the equator], and soon it changes into deflux. For indeed at once the deflux starts at the setting of the moon, and the influx at its rising, then again it changes with the declination of the moon [until the whole process repeats itself]. The approach to this port is apparent from two narrow channels, the one from the Chinese Sea between the continent and the island of Leuconia [Hainan Island], the other from the Indian Sea between the continent and the island of Borneo. But the tide in the space of twelve hours arrives from the Indian Sea, and in the space of six hours from the Chinese Sea arrives through that narrow channel, and thus arising in the third and ninth hours of the moon, motions of this kind add together ; lest there shall be a condition imposed by other seas, I leave that determination from the observations of neighbouring shores. [For an early paper, see Whewell. Phil. Trans. 1833, p. 224.] Up to this point I have given an account of the motions of the moon and of the seas. Now it is fitting to add a little on the quantity of the motions.



PROPOSITION XXV. PROBLEM VI.

To find the forces of the sun perturbing the motion of the moon. Let S designate the sun, T the earth, P the moon, and CADB the orbit of the moon. On SP there is taken SK equal to ST [i.e. the mean Sun-Earth distance]; and let SL to SK be in the square ratio SK to SP,

[i.e. the forces and SL SK vary inversely with the distance from the sun : 2

2

SL SKSPSK

= ;]

and LM is acting parallel to PT itself ; [i.e. resolve LS into components so that LS LM MS= + ; ] and if the gravitational acceleration of the earth on the sun may be put in place by the distance ST or SK, SL will be the gravitational acceleration of the moon on the sun. [Note that due to the inverse square law, the line SL is hence longer than the line ST, while the forces of attraction on the moon and the earth due to the sun act along SL and

ST, while the ratio of the accelerations of these is2

2

SP STSLST

= .]

This is composed from the parts SM and LM, [on re-arranging the above forces per unit mass : SL LM SM= + ] of which LM and the part TM of SM perturb the motion of the moon ; as established in Book I, Prop. LXVI., and in its corollaries. [Thus, the earth and the moon form a sub-system, falling with almost the same centripetal acceleration aTS towards the sun, while extra accelerations exist between these two bodies, so that LS LT TSa a a= + , where LT LM MTa a a= + ] Certainly the earth and the moon are revolving about a common centre of gravity, also the motion of the earth will be perturbed around that centre by similar forces [by the third law of motion] ; moreover both the sums of the forces as well as of the motions as referred to the moon [instead of the earth], and the sums of the forces designated by the analogous lines TM and ML themselves. The force ML in its mean magnitude is the centripetal force, by which the moon may rotate at a distance PT in its orbit around the earth at rest, [i.e. with the sun's perturbation acting independently in the quadrature position, or 0MTa = ; or (according to a note of L. & J. on p. 2 of vol. IV) : on account of the great distance of



the sun from the earth, the figure PTML is a parallelogram, and thus ML is approximately equal to PT, and hence the force ML will be to the force by which the sun acts at the point T, as PT to SK or ST, but any central forces are between themselves as the radii of the circles which are described by these, and inversely as the square of the periodic times of these, and hence this force by which the sun acts at the point T is to the force by which the moon is retained in its orbit (supposing that to rotate about the earth at rest) as PT directly, and as the square of the periodic time of the moon around the earth to the square of the periodic time of the earth around the sun; hence from the composition of the ratios, the force ML is to the force holding the moon in its orbit, as the square of the periodic time of the moon to the square of the periodic time of the earth around the sun; that is, in the square ratio of ..... ], as the square ratio of the periodic times of the moon around the earth and of the earth around the sun (by Corol. 17. Prop. LXVI. Book I.) that is, in the square ratio of 27days, 7 hours, and 43 minutes to 365 days, 6 hours and 9 minutes, that is, as 1000 to 178725, or 1 to 29

40178 .

[i.e. recall that ( )2PT PT TST tST

∝ × , where ST ST= has a scaling factor of 1 in the diagram,

but other lengths such as PT have to be scaled as shown to become forces ; we can write

this ratio of the centripetal forces in modern terms as ( )( )

22

221000

178725PT t

ST T

F PT PTSTF ST

π

π

× ×××

= = .]

But we have found in Proposition IV that, if the earth and the moon may be revolving about a common centre of gravity, the mean distance between these will be in turn as 1

260 mean earth radii approximately. And the force by which the moon can rotate around the earth at rest, to the distance PT of 1

260 earth radii, is to the force, by which it can revolve in the same time at a distance of 60 radii, as 1

260 to 60; and this force to the force of gravity with us on the earth as 1 to 60 60× approximately. And thus the average force ML compared to the force of gravity on the earth is as 291

2 401 60 to 60 60 60 178× × × × , or 1 to

638092,6. [i.e. the acceleration ML at the moon's orbit can be taken as 638092 6g

, .] Thence

truly, and from the known proportion of the lines TM, ML, the force TM is also given : and these are the sun's forces disturbing the motion of the moon. Q.E.I.



PROPOSITION XXVI. PROBLEM VII.

To find the hourly increment to the area that the moon will describe in a circular orbit, by a radius drawn to the earth.

We have said the area that will be described by a radius drawn from the moon to the earth, to be proportional to the time, unless perhaps the motion of the moon may be disturbed by the action of the sun. Here we propose to investigate the inequality of the moments, or of the hourly increments. In order that the computation may be carried out more easily, we may consider the orbit of the moon to be circular, and we may neglect all the inequalities, with the sun excepted, by which this orbit is disturbed. On account of the truly huge distance of the sun, we may also consider the lines SP, ST to be parallel to each other. With this agreed on, the force LM will always be reduced to its average value

TP, and so that the force TM is reduced to its average value 3PK. [For the distance ST, with the sun S imagined very far to the left in the diagram, is on average given by the distances SP PK+ , and SL is given by SP PL+ ; while the magnitudes of the forces and SP SL vary inversely as their associated distances squared

as above : 2

2

SL SKSPSK

= or 3 3

2 2SK STSP SP

SL = = . Hence if we consider the constant distance ST to

be given by ST SP PK= + , then

( )( ) ( )3 3 2 2 2 2 22ST SP ST SP ST ST .SP SP PK SP .SP.PK PK ST .SP SP− = − + + = + + + +

( )2 2 3 2 3 2 23 2SP .PK .SP.PK PK LS.SP SP SP LS SP SP .PL+ + = − = − = ; or, on

dividing, ( )2

223 3

3 1 .PK PK.SP SP

PL PK + + ; then, since PK is much less than SP, we

have 3PL PK , as was required to be shown.] These forces (by Corol. 2 of the laws ) make the force TL; and this force, if the perpendicular LE may be sent to the radius TP, is resolved into the forces TE, EL, of which TE, by always acting along the radius TP, neither accelerates nor retards the description of the area TPC made by that radius TP ; and EL by acting along the perpendicular, does accelerate or retard the area described, at the same times as it



accelerates or retards the moon. That acceleration of the moon, in the transition of that from the quadrature C to the conjunction A, made in individual moments of time, is as the accelerating force EL, that is, as 3PK TK

TP× . [i.e., from the similar triangles PEL and PKT.]

The time may be established from the mean motion of the moon, or (because it returns almost the same thing) by the angle CTP, or also by the arc CP. [L. & J. note 110 a : Thus, the perturbation produced by the sun is extremely small, and thus the arc or angle may be taken in proportion to the time.] The normal CG may be erected equal to CT. And with the arc of the quadrant AC divided into innumerable equal parts Pp, &c. by which just as many equal parts of the time shall be put in place, and by drawing pk perpendicular to CT, TG may be crossed by KP and kP produced to F and f ; and FK will be equal to TK, and = PpKk

PK Tp , that is, in a given ratio

[L. & J. note 110 b : this proportion follows from a most noteworthy property of the circle, for if from the point p the line increment pq may be drawn normal to PK, that will be parallel and equal to the line Kk, and the differential triangle Ppq will be formed similar to the triangle PKT, for since the angles pPK and KPT together make a right angle, and equally the angles KPT and PTK, the angles pPK and PTK are equal, from which = PpKk

PK Tp .],

and thus FK Kk× or the area FKkf, will be as [the applied tangential force] 3PK TKTP× , that

is, as EL; and by adding the increments, the whole area GCKF will be as the sum of all the forces EL impressed on the moon in the total time CP, and thus also as the velocity generated by this sum, that is, as the acceleration of the described area CTP, or the increment of the moment. The force by which the moon can revolve about the earth at rest at the distance TP, in its periodic time CADB of 27 days, 7 hours, and 43 minutes, effects that a body, by falling in the time CT, describes the distance 1

2 CT , and likewise it acquires a velocity equal to the velocity, by which the moon is moving in its orbit, which is apparent by Corol. 9. Prop. IV, Book I. But since the perpendicular Kd sent to TP shall be the third part of EL, and the half of TP itself or of ML at the octant, the force EL at the octant , where it is a maximum, will exceed the force ML in the ratio 3 to 2, and thus it will be to that force, by which the moon will be able to revolve about the quiescent earth in its periodic time, as 100 to 2 1

3 217872× or to 11915, and in the time [corresponding to] CT it must be able to generate a velocity which will be the 100

22915 part of the lunar velocity, but in the time CPA it will generate a greater velocity in the ratio CA to CT or TP. The maximum force EL in the octant may be established by the area FK Kk× equal to the rectangular area 1

2 TP Pp× . And the velocity, that the maximum force can generate in some time CP, will be to that velocity that all the smaller forces EL can generate in the same time, as the rectangle 1

2 TP CP× to the area KCGF : but in the total time CPA, the velocities arising will be in turn as the rectangle 1

2 TP CA× and the triangle TCG, or as the arc of the quadrant CA and the radius TP. And thus (by Prop. IX, Book V, Euclid's



Elements.) the latter velocity, generated in the total time, will be the 10022915 part, of the

moon's velocity. To this velocity of the moon, which is analogous to the average moment [i.e. increment] of the area, there may be added and taken away half the other velocity ; and if the average moment may be expressed by the number 11915, the sum 11915+50 or 11965 will show the maximum moment of the area at the syzygies A, and the difference 11915 – 50 or 11865 the minimum moment of the same at quadrature. Therefore the areas described in equal times at the syzygies and quadratures, are in turn as 11965 to 11865. To the minimum moment 11865 there may be added the moment, which shall be to the difference of the moments 100 as the trapezium FKCG to the triangle TCG (or which is the same thing, as the square of the sine PK to the square of the radius TP, that is, as Pd to TP), and the sum will show the moment of the area, when the moon is at some intermediate place P. Thus all these may themselves be had, from the hypothesis that the sun and the earth are at rest, and the moon is revolving in the synodic time of 27 days, 7 hours, and 43 min. But since the synodic period of the moon shall truly be 29 days, 12 hours and 44 min., the increments of the moments must be increased in the ratio of the times, that is, in the ratio 1080853 to1000000. With this agreed upon, the total increment, that was the 100

11915 part of the average moment, now will become the 10011023 part of this. And

thus the moment of the area of the moon at the quadrature will be to the moment of this at the syzygies as 11023–50 to 11023+50, or as 10973 to 11073; and to the moment of this, when the moon is moving through some intermediate place P, as 10973 to 10973+Pd, clearly with TP arising equal to 100. [See Chandrasakher, p. 239 onwards for a modern interpretation of the perturbing function (a) and the variation of the 'constant of areas' (c)] Therefore the area, that the moon will describe with the radius drawn to the earth in individual instants of time, is as an approximation as the sum of the numbers 219,46 and the versed sine of double the distance of the moon from the nearest quadrature, in the circle the radius of which is one. Thus these themselves are found when the variation at the octant is of average magnitude. But if the variation there shall be greater or less, that versed sine has to be increased or diminished in the same ratio.

PROPOSITION XXVII. PROBLEM VIII.

From the hourly motion, to find the distance of the moon from the earth. The area, that will be described by a radius drawn from the moon to the earth in individual instants of time, is as the hourly motion of the moon and the square of the distance of the moon from the earth jointly ; and therefore the distance of the moon from the earth is in a ratio composed from the square root of the area directly, and inversely as the ratio of the square root of the hourly motion. Q.E.D. Corol. 1. Hence the apparent diameter of the moon is given: which certainly shall be inversely as its distance from the earth. Astronomers can test whether this rule agrees correctly with the phenomena. Corol. 2. Hence also it is possible for the orbit of the moon to be defined more accurately than before from these phenomena.



PROPOSITION XXVIII. PROBLEM IX.

To find the diameters of the orbit in which the moon can move without eccentricity.

[Here initially we give a modern view on solving this problem, following from the results expressed in Prop. XXVI : ( )23

4 11 2mav. mH H cos .ψ−= +

] The curvature of the trajectory that will be described by a moving body, if it may be drawn along the perpendicular of that trajectory, is as the attraction directly and inversely as the square of the velocity. I consider the curvatures of lines to be between the final ratio of the sine or tangent of the angle of contact pertaining to equal radii, when these radii are reduced indefinitely. But the attraction of the moon towards the earth at the syzygies is the excess of its gravity 2PK over the force of the sun (see fig. Prop.25.), by which the gravitating acceleration of the moon towards the sun exceeds the gravitating acceleration of the earth towards the sun, or may be increased by that amount. But in quadrature that attraction is the sum of the gravities of the moon towards the earth and of the force of the sun KT, by which the moon is drawn towards the earth. And these attractions, if 2

AT CT+ is called N [the average radius], shall be as

2 2178725 2000 178725 1000 and +CT N AT NAT CT× ×− approximately; or as

2 2 2 2178725 2000 and 178725 1000N CT AT CT N AT CT AT× − × × + × . For if the accelerative gravity of the moon towards the earth may be expressed by the number 178725, then the average force ML, which is PT or TK in quadrature, and is drawn towards the earth, will be 1000, then the average force TM in syzygies will be 3000; from which, if the average force ML may be removed there will remain 2000 by which the moon is drawn from the earth, formerly called 2PK. But the velocity of the moon in syzygies A and B is to its velocity in the quadratures C and D, as CT to AT and the moment of the area which the radius drawn from the moon to the earth will describe at the syzygies to the same moments conjointly, i.e. as 11073 CT to 10973 AT. This inverse ratio is taken squared and the first ratio taken once directly, and the curvatures of the orbit of the moon in syzygies to the same curvature in quadrature becomes as

2 2 4120406729 178725 120406729 2000AT CT N AT CT× × × − × × to

2 2 4122611329 1178725 122611329 l000AT CT N CT AT× × × + × × ,



i.e. as

3 32151969 24081 to 2191371 N 12261AT CT N AT AT CT CT .× × − × × + Because the figure of the moon's orbit is unknown, we may assume this in turn to be the ellipse DBCA, in the centre of which the earth T may be located, and the major axis of which is between the quadratures DC, and the minor axis AB may be placed between the syzygies . But since the plane of this ellipse is revolving in an angular motion about the earth, and the we may consider the curved trajectory of this that is completely free from any angular motion: the figure will be required to be considered, which the moon will describe in that ellipse by revolving in that plane, that is the figure Cpa, the individual points p of which may be found by taking some point P on the ellipse, that may represent the position of the moon, and by drawing Tp equal to TP, from that law that the angle PTp shall be equal to the apparent motion of the sun completed from the time of the quadrature C ; or (what returns almost the same) that the angle CTp shall be to the angle CTP as the synodic time of revolution of the moon to the periodic time of revolution or 29d. 12h. 43', to 27d. 7h. 43' . Therefore the angle CTa may be taken in the same ratio to the right angle CTA, and the longitude Ta shall be equal to the longitude TA; and a will be the inner apse and C the outer apse of this orbit Cpa. Because I find from the required ratio to be entered, that the difference between the curvature of the orbit Cpa at the vertex a, and the curvature of the circle described with centre T and with the radius TA, shall be to the difference between the curvature of the ellipse at the vertex A and the curvature of the same circle, in the square ratio of the angle CTP to the angle CTp; and because the curvature of the ellipse at A shall be to the curvature of that circle, in the square ratio TA to TC; and the curvature of that circle to the curvature of the circle with centre T described with radius TC, as TC to TA ; but the curvature of this to the ellipse curvature at C, in the square ratio TA to TC; and the difference between the ellipse curvature at the vertex C and the newest curvature of the circle , to the difference between the curvature of the figure Tpa at the vertex C and the same curvature of the circle, in the square ratio of the angle CTp to the angle CTP. Which ratios indeed are easily deduced from the sines of the angle of contact and the difference of the angles. But with these deduced between themselves, the curvature of the figure Cpa at a to the curvature of this at C can be produced, as

16824 16824200000 200000

3 2 3 2 to AT CT AT CT AT CT+ × + × . Where the number 16824200000 may designate

the difference of the squares of the angle CTP and CTp applied to the square of the smaller angle CTP, or (which amounts to the same) the difference of the squares of the times 27d. 7h. 43', and 29d. 12h. 44' 1.9, applied to the square of the time 27d. 7h. 43'. Therefore since a may designate the syzygies of the moon, and C the quadrature of this, the proportion now found must be the same as the proportion of the curvature of the



orbit of the moon in syzygies to the same curvature at quadrature, as we have found above. Hence so that the proportion CT to AT may be found, I multiply the extremes and means between themselves. And the terms arising to the applied term AT CT× , becomes

4 3 2

2 3 2 3 4

2062 79 215I969 368676

36342 362047 2191371 4051 4 0

, CT N CT N AT CT

AT CT N AT CT N AT , AT .

− × + × ×

+ × − × × + × + =

Here for the half sum of the terms AT and CT , N I write 1, and by putting x for the semi difference of the same, there shall be 1 and 1CT x, AT x= + = − : with which written into the equation, and with the equation produced resolved, there will be obtained x equal to 0,00719, and thence the radius CT shall be 1,00719, and the radius AT 0,99281, which numbers are approximately as 1 1

24 2470 and 69 . Therefore the distance of the moon from the earth at syzygies is to the distance of this in quadrature as 69 to 70 (clearly with the consideration of the eccentricity disregarded), or with rounded numbers as 69 to 70.

PROPOSITION XXIX. PROBLEM X. To find the variation of the moon.

This inequality arises partially from the elliptic form of the lunar orbit, partially from the inequality of the moments of the area, that the moon describes by the radius drawn to the earth. If the moon P is moving in the ellipse DBCA around the earth at rest at the centre of the ellipse, and with the radius TP drawn to the earth describing an area CTP proportional to the time ; moreover the maximum radius CT of the ellipse shall be to the minimum radius TA as 70 to 69: the tangent of the angle CTP to the tangent of the mean motion at the quadrature C shall be computed, as the radius of the ellipse TA to the same radius TC or as 69 to 70. But the description of the area CTP, in the progression of the moon from quadrature to syzygies, to be accelerated in that ratio, so that the moment of this in the syzygies of the moon shall be to the moment of this in quadrature as 1073 to 10973, and so that the excess of the moment at some intermediate place P over the moment in quadrature shall be as the square of the sine of the angle CTP. As that comes about well enough, if the tangent of the angle CTP may be diminished in the square root ration of the number 10973 to the number 11073, that is, in the ratio of the number 68,6877 to the number 69, and with which agreed on, the tangent of the angle CTP now will be to the tangent of the mean motion as 68,6877 to 70, and the angle CTP in the octants, where the mean motion is 450, will be 440. 27'. 28", which taken from the angle of the mean motion 450 leaves the maximum variation 32' .32". These thus themselves are found if the moon, by going from the quadrature to the syzygy, may describe an angle CTA of almost 900. Truly on account



of the motion of the earth, by which the sun as a consequence by moving, has been apparently carried across, the moon, before it overtakes the sun, will describe the angle CTA greater than a right angle in the ratio of the synodic time of revolution of the moon to the periodic time of revolution, that is, in the ratio 29d. 12h. 44' to 27d. 7h. 43'. And with this agreed on, all the angle about the centre T may be enlarged in the same ratio, and the maximum variation which otherwise shall be 32'. 32", now increased in the same ratio shall be 35'. 10''. This is its magnitude at the average distance of the sun from the earth, with the differences ignored which may arise from the curvature of the great orbit [of the earth] and from the greater attraction of the sun on the sickle-shaped moon at the new moon rather than at the gibbous and full moon [phases]. At the other distances of the sun from the earth, the maximum variation is in a ratio that is composed from the square of the ratio of the synodic time of the moon (at a given time of the year) directly, and in the inverse cubic ratio of the distance from the sun. And thus at the apogee of the sun, the maximum

variation is 33'. 14", and at its perigee 37'.11", but only if the eccentricity of the sun shall be to the transverse radius of the great orbit as 15

1616 to 1000. Up to the present we have investigated the variation in an orbit without eccentricity, in which the moon certainly in its octants is always at its average distance from the earth. If the moon because of its eccentricity, may be more or less distant from the earth than if located in this orbit, the variation can be a little more or a little less than produced here by this rule : but I leave the excess or deficiency requiring to be determined through the phenomena being observed by astronomers.

PROPOSITION XXX. PROBLEM XI. To find the hourly motion of the moon's nodes in a circular orbit.

[The Latin word horarius means 'related to the hours or seasons' or perhaps 'scheduled', as applied to a timetable.] Let S represent the sun, T the earth, P the moon, NPn the orbit of the moon, NPn the projection [lit. footprint] of the orbit in the plane of the ecliptic; N, n the nodes, nTNm the nodal line produced indefinitely ; PI, PK the perpendiculars sent to the lines ST, Qq; Pp the perpendicular sent to the plane of the ecliptic; A, B the syzygies of the moon in the plane of the ecliptic ; AZ the perpendicular sent to the nodal line Nn; Q, q the quadratures



of the moon in the plane of the ecliptic, & pK meeting the perpendicular to the line of Qq. The force of the sun perturbing the motion of the moon is twofold (by Prop. XXV.), the one proportional to the line LM in the diagram of that proposition, the other proportional to the line MT. And the moon is attracted to the earth by the first force, by the latter towards the sun along a right line drawn parallel to ST from the earth to the sun. The first force LM acts along the plane of the lunar orbit, and therefore changes nothing in the situation of the plane. This force therefore can be ignored. The latter force MT by which the plane of the orbit of the moon may be perturbed, may be expressed by the force 3PK or 3IT. And this force (by Prop. XXV.) is to the force by which the moon may be revolving uniformly almost in a circle in its periodic time about the earth at rest, as 31T to the radius of the circle multiplied by the number178,725, or as IT to the radius multiplied by 59,575. For what remains in this calculation, and from that which follows generally, I consider all the lines drawn from the moon to the sun as parallel to the line which is drawn from the earth to the sun, therefore so that by diminishing all the effect in some cases almost as much as it may be increased in others ; and we seek the mean motion of the nodes, with minute details of this kind ignored, which may exceedingly impair the calculation. Now let PM designate the arc, that the moon will describe in a given time taken as minimal, and ML the small line the half of which the moon, with the before mentioned force 3IT impressed, may describe in the same time. PL and MP may be joined, and these may be produced to m and l, where they cut the plane of the ecliptic ; and the perpendicular PH may be sent to Tm. And because the right line ML is parallel to the plane of the ecliptic, and thus which cannot cross with the right line ml lying in that plane, and yet these lines lie in the common plane LMPml ; these right lines will be parallel and therefore the triangles LMP and lmP are similar. Now since MPm shall be in the plane of the orbit, in which the moon will be moving at the place P, it meets the line Nn drawn through the nodes N, n of this orbit at the point m. And because the force by which half of the line LM is generated, if likewise the whole force were impressed once at the place P, it would generate that whole line ; and puts into effect that the moon should move in an arc, of which the chord must be LP, and thus it should carry the moon from the plane MPmT into the plane LPIT; the motion of the angle of the nodes generated by that force, will be equal to the angle mTl. But ml is to mP as ML to MP, and thus since MP shall be given on account of the time given, mt is as the rectangle ML mP× , that is, as the rectangle IT mP× . And the angle mTl, but only if the angle Tml shall be right, is as ml

Tm ,

and therefore as IT PmTm× , that is (on account of the proportionals Tm and mP, TP and PH)

as IT PHTP× , and thus on account of TP given, as IT PH× . Because if the angle Tml, or

STN shall be oblique, the angle mTl now becomes less, in the ratio of the sine of the angle STN to the radius, or AZ to AT. Therefore the velocity of the nodes is as IT PH AZ× × , or as contained by the product of the sines of the three angles TP1, PTN and STN. If these angles may be right, with the nodes in quadrature and with the moon present in syzygies, then the small line ml will go off to infinity, and the angle mTl will emerge equal to the angle mPl . But in this case, the angle mPl is to the angle PTM, that the moon will describe in the same time by its apparent motion around the earth, as 1 to 59,575. For the angle mPl is equal to the angle LPM, that is, to the angle of deflection of the moon



from a straight course, that only the aforementioned force of the sun 3IT, may generate in a given time, if the gravity of the moon should cease ; and the angle PTM is equal to the angle of deflection of the moon from a straight line course, by that force, by which the moon may be retained in its orbit, that could be generated in the same time, if the force

31T of the sun should then cease. And these forces, as we have said above, are in turn as 1 to 59,515. Therefore since the mean hourly motion of the moon with respect to the fixed stars shall be 1

2iv32 56 27 12' . " . ''' . , the hourly motion of the node in this case will be

iv v33 10 33 12'' . '" . . . Moreover in other cases this hourly motion will be to iv v33 10 33 12'' . '" . . as the multiple of the three sines of the angles TPI, PTN, and STN (or

of the distances of the moon from quadrature, of the moon from the node, and of the node from the sun) to the cube of the radius. And just as often as the sign of some angle changes from positive to negative, or from that in which a negative will be changed into a positive, the motion of regression is changed into progression, and progression is changed into regression. From which it shall arise that the nodes shall be progressing as often as the moon shall be moving between one quadrature and the node of the nearest quadrature. In other cases it will be regressing, and by the excess of the regression over the progression are carried forwards each month in advance. Corol.1. Hence if from a given minimal arc PM with the ends P and M there may be sent the perpendiculars PK, M k to the line Qq joining the quadratures, and the same may be produced cutting the nodal line Nn at D and d; the hourly motion of the nodes will be as the area MPDd and the square of the line AZ jointly. For indeed PK, PH and AZ become the three aforesaid sines. Clearly PK is the sine of the distance of the moon from the quadrature, PH the sine of the distance of the moon from the node, and AZ the sine of the distance of the node from the sun : and the velocity of the node will be as the product PK PH AZ× × . But PT is to PK as PM to Kk, and thus on account of the given PT and PM , there is Kk proportional to PK itself. Also AT is to PD as AZ to PH, and therefore PH is proportional to the rectangular PD AZ× , and with the adjoining ratios, PK PH× it as the product and Kk PD AZ , PK PH AZ× × × × as Kk PD AZqu× × ; that is , as the area PDdM and AZ 2 jointly. Q. E.D.



Corol. 2. At some given position of the nodes, the hourly mean motion is half the hourly motion at the syzygies of the moon, and therefore is to iv v16 35 16 . 36" . "' . . as the square of the sine of the distance of the nodes from the syzygies to the square of the radius, or as AZ2 to AT2. For if the moon may be going around the semicircle QAq with a uniform motion, the sum of all the areas PDdM, in which time the moon goes from Q to M, will be the area QMdE which is terminated by the tangent to the circle QE ; and in which time the moon reaches the point n, that sum will be the total area EQAn that the line PD will describe, then with the moon going from n to q, the line PD falls outside the circle, and the area nqe described terminated by the tangent qe to the circle ; which, because the nodes at first were regressing, now indeed because they are progressing, the area must be taken from the first area, and since it shall equal the area QEN, there will remain the semicircle NQAn. Therefore since the sum of all the areas PDdM, in which time the moon will describe the semicircle, is the area of the semicircle ; and the whole sum in which time the moon will describe the circle is the whole area of the circle. But the area PDdM, when the moon is turning in the syzygies, is the rectangle under the arc PM and the radius PT; and the sum of all the areas equal to this area, in which time the moon will describe the circle, is the rectangle under the whole circumference and the radius of the circle ; and this rectangle, since it shall be equal to two circles, is more than twice the first rectangle. Hence the nodes, that by continuing with a uniform velocity as they have at the lunar syzygies, describe a distance twice as great as they actually describe ; and therefore the average motion by which, if it were continued uniformly, they would be able to describe a distance between themselves by the unequal motion they actually complete, is half the motion that they have in the moon's syzygies. From which since the maximum hourly motion, if the nodes are passing through quadrature, shall be

iv v33 10 33 . 12" . "' . , and the mean hourly motion in this case will be iv v16 35 16 36" . ''' . . . And since the hourly motion of the nodes always shall be as AZ2 and the area PDdM jointly, and therefore the hourly motion of the nodes at the syzygies of the moon shall be as AZ2 and the area PDdM jointly, that is (on account of the area PDdM described at the syzygies) as AZ2 also the mean motion will be as AZ2, and thus this motion, when the nodes are moving beyond the quadratures, will be to iv v16 35 16 36" . "' . . as AZ2 to AT2. Q. E. D.



PROPOSITION XXXI. PROBLEM XII.

To find the hourly motion of the nodes of the moon in an elliptic orbit. Let Qpmaq designate the ellipse described with the major axis Qq, and with the minor axes ab, QAqB the circumscribed circle, T the earth in the common centre of each, S the sun, p the moon moving in the ellipse, and pm the arc that will be described in a given particular element of time, Nn the line joining the nodes N and n, pK and mk perpendiculars sent to the axis Qq and hence produced from there until they cross the circle at P and M, and the line of the nodes at D and d. And if the moon, with the radius drawn to the earth, describes an area proportional to the time, the hourly motion of the nodes will be as the area pDdm and AZ2 jointly. For if PF is a tangent to the circle at P, and produced crosses TN in F, and pf is a tangent at p and produced meets the same line TN at f, moreover these tangents meet the axis TQ at Y; and if ML may designate the distance that the moon revolving in a circle, while meanwhile it describes the arc PM, urged and impelled by the aforementioned force 3IT, or can describe 3PK by a transverse motion, and ml may designate the distance that the moon revolving in the ellipse in the same time can describe, also urged by the force 3IT or 3PK ; and LP and lp produced cross the plane of the ecliptic at G and g; and FG and fg are joined, of which FG produced cuts pf, pg and TQ in c, e and R respectively, and fg produced cuts TQ in r. Because the force 31T or 3PK in the circle is to the force 3IT or 3pK in the ellipse, as PK to pK, or AT to aT; the distance ML generated by the first force will be to the distance ml generated by the second force, as PK to pK, that is, on account of the similar figures PYKp and FYRc, as FR to cR. But ML is to FG (on account of the similar triangles PLM, PGF) as PL to PG, that is (on account of the parallel lines Lk, PK, GR) as pl to pe, that is to say, (on account of the similar triangles ( plm, cpe) as lm to ce ; and inversely as LM is to 1m, or FR to cR, thus FG is to ce. And therefore if fg shall be to ce as fY to cY, that is, as fr to cR (that is, as sr to FR and FR to cR jointly, that is, as fT to

FI and FG to ce jointly) because the ratio FG to ce with each taken away leaves the ratios fg to FG and fT to FT, there becomes fg to FG as fT to FT; and thus the angles, which



FG and fg subtend at the earth T, are equal to each other. But these angles (by that which we have put in place in the preceding proposition) are the motions of the nodes, by which in the time the moon travels in the arc of the circle PM, it travels in the arc of the ellipse pm : and therefore the motions of the nodes in the circle and in the ellipse are equal to each other. These thus are found, but only if fg shall be to ce as fY to cY, that is, if fg shall be eqaual to ce fY

cY× . Truly on account of the similar triangles fgp, cep, there is fg to

ce as fp to cp; and thus fg is equal to ce fpcp× ; and therefore the angle, that fg actually

subtends, is to the first angle, that FG subtends, that is, the motion of the nodes in the ellipse to the motion of the nodes in the circle, as this line fg or to the first value of fg or ce fY

cY× , that is, as to fp cY fY cp× × , or fp to fT and cY ad cp, that is to say, if ph crosses

FP at h parallel to TN itself, so that Fh to FY and FY to FP; that is, as Fh to FP or Dp to DP, and thus as the area Dpmd to the area DPMd. And therefore, since (by Corol. I. Prop. XXX.) the latter area and AZ2 jointly shall be proportional to the hourly motion of the nodes in the circle, the former area and AZ2 jointly shall be proportional to the hourly motion of the nodes in the ellipse. Q.E.D. Corol. Whereby since, in the given position of the nodes, the sum of all the areas pDdm, in which time the moon goes through from quadrature to some place m, shall be the area mpqQEd, which is terminated by the tangent line of the ellipse QE ; and the sum of all those areas, in a whole revolution, shall be the area of the whole ellipse: the mean motion of the nodes in the ellipse will be to the mean motion of the nodes in the circle, as the [area of the] ellipse to the [area of the] circle; that is, as Ta to TA, or 69 to 70. And therefore, since (by Corol. 2. Prop. XXX.) the mean hourly motion of the nodes in the circle shall be to iv v16 35 16 36" . ''' . . .as AZ2 to AT2, if the angle iv v16 21 3 30" . "' . . . may be taken to the angle iv v16 35 16 36" . ''' . . .as 69 to 70, the mean hourly motion of the nodes in the ellipse will be to iv v16 21 3 30" . ''' . . . as AZ2 to AT2; that is, as the square of the sine of the distance of the node from the sun to the square of the radius. For the remainder the moon, with the radius drawn to the earth, will describe areas faster at the syzygies than at the quadratures, and by that account the time is diminished at the syzygies, and augmented at the quadratures ; and together with the time the motion of the nodes is augmented or diminished. But the moment of the area at the quadrature of the moon to the moment of this in syzygies was as 10973 to 11073, and therefore the mean moment at the octants is to the excess in syzygies, and the defect at quadrature, as half the sum of the numbers 11023 to half the difference 50. From which since the time of the moon in equal individual particular orbits shall be inversely as its velocity, the mean time in the octants to the excess of the time at quadrature, and the defect in syzygies, arising from this effect, will be as 11023 to 50 approximately. Moreover in going from quadrature to syzygies, I find that the excess of the moments of the area in individual places, above the minimum moment at the quadratures, shall be as the square of the sine of the distance of the moon from the quadrature approximately ; and therefore the difference between the moments at some place and the average moment at the quadrant, is as the difference between the square of the sine of the distance of the moon from the



square of the sine of 45 degrees, or half the square of the radius ; and the increment of the time in the individual places between the octants and the quadratures, and the decrement of this between the octants and the syzygies, is in the same ratio. But the motion of the nodes, in which time the moon traverses equal individual parts of the orbit, may be accelerated or retarded in the square ratio of the time. For this motion, while the moon passes through PM (with all else equal) is as ML, and ML is in the square ratio of the time. Whereby the motion of the nodes at syzygies, performed in that time in which the moon traversed a small part of the orbit, is diminished in the square ratio of the number 11073 to the number 11023 ; and the decrement to the motion of the rest as 100 to 10973, to the true total motion as 100 to 11073 approximately. But the decrement in places between the octants and the syzygies, and the increment in places between the octants and quadrature, is approximately to this decrement, as the ratio of the whole motion in these places to the whole motion at the syzygies, and the difference between the square of the sine of the distance of the moon from the quadrature and half the square of the radius to half the square of the radius taken jointly. From which, if the nodes are present in quadrature, and two places hence may be taken equally distant from the octant on each side, and two others may be taken, one equally distant from the syzygies and the other from the quadrature, then, from the decrements of the motion in the two places down between the syzygy and the octant, the increments of the motions in the two remaining places may be taken away, which are between the octant and quadrature ; the remaining decrement will be equal to the decrement in the syzygies : the reason for that will be easily seen on entering into a computation. And therefore the mean decrement, that must be taken from the mean motion of the nodes, is the fourth part of the decrement in the syzygy. The whole hourly motion of the nodes in syzygies, when the moon was supposed to describe equal areas by the radius drawn to the earth, was iv32 42 7'' . ."' . .And the decrement of the motion of the nodes, the moon in which time now moving faster will describe the same distance, we have said to be to this motion as 100 to 11073 ; and thus that decrement is

iv v17 43 11''' . , the fourth part of which iv v4 25 48''' . . , of the mean hourly motion, taken from i v16 21 3 30v" . ''' . . found above, leaves iv v16 16 37 42'' . ''' . . . , the correction of the mean hourly motion. If the nodes may be moving beyond the quadratures, and hence two places may be considered equally distant from the syzygies ; the sum of the motions of the nodes, when the moon is moving through these places, will be to the sum of the motions, when the moon may be present in the same places and the nodes may be at the quadratures, as AZ2 to AT2. And the decrements of the motions arising, from the causes now discussed, will be in turn as the motions themselves, and thus the remaining motions will be in turn as AZ2 to AT2 and the mean motions as the remaining motions. And thus the correct mean hourly motion, in some given situation of the nodes, is to v16 16''' 37 42iv'' . . . .as AZ2 to AT2

, that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius.



PROPOSITION XXXII. PROBLEM XIII.

To find the mean motion of the nodes of the moon. The mean annual motion is the sum of all the mean horary motions during the year. Consider the node to be moving through N, and with an individual hour completed to be drawn back to its first position, so that with no obstacle to its proper motion, it may maintain a given situation with regard to the fixed stars. Meanwhile truly the sun S, by the motion of the earth, has moved from the node, and to complete uniformly its annual course. Moreover let Aa be that minimum given arc described in the minimum given time, by the direction of the right line TS always drawn to the sun and the circle NAn: and the mean hourly motion (by what has just been shown) will be to AZ2, that is (on account of the proportionals AZ, ZT) as the rectangle under AZ and ZY, that is, as the area AZTa. And the sum of all the mean hourly motions from the beginning, is as the sum of all the areas aTZA, that is, as the area NAZ. But the maximum AZTa is equal to the rectangle under the arc Aa and the radius of the circle ; and therefore the sum of all the rectangles in the whole circle to the sum of just as many maximas, as the whole area of the circle to the rectangle under the whole circumference and the radius, that is, as 1 to 2. Moreover the hourly motion, corresponding to the maximum area, was iv vl6 16 37 42'' . ''' . . . And this motion, in the whole sidereal year of 365 days. 6 hours. 9 min. shall be

039 38 7 50. ' . " . ''' . And thus the half of this gr19 49 3 55. ' . " . "' . is the mean motion of the

nodes corresponding to the whole circle. And the motion of the nodes, in which time the sun goes from N to A, is to gr19 49 3 55. ' . " . ''' . as the area NAZ to the whole circle. These results themselves thus are obtained from the hypothesis, that the node of the individual hour is redrawn into its former position, thus so that the sun by completing a whole year may return to the same node from which in the beginning it had set out. Truly by the motion of the node it shall arise that the sun shall return to the node quicker, and by computation now is in a shorter time. Since the sun in a whole year completes 3600 and the node by the maximum motion completes 039 38 7 50. ' . " . ''' in the same time, or 39,6355 degrees ; and the mean motion of the node at some place N shall be to the mean motion of its quadrature, as AZ2 to AT2: the motion of the sun to the motion of the node at N, shall be as 360 AT2 to 39,6355 AZ2 ; that is, as 9,0827646 AT2 to AZ2. From which if the circumference of the whole circle NAn may be divided into equal small parts Aa, the



time in which the sun travels through the element Aa, if the circle were at rest, will be to the time in which it traverses the same element, if the circle together with the nodes were revolving around the centre T, reciprocally as 9,0827646 AT2 to 9,0877646AT2 +AZ2. For the time is reciprocally as the velocity in which an element is traversed, and this velocity is the sum of the velocities of the sun and of the node. Therefore if the time, in which the sun without the motion of the node traversed the arc NA, may be expressed by the sector NTA, and the element of the time in which it traversed that element of the arc Aa, may be expressed by the element of the sector ATa; and (with the perpendicular aY sent to Nn) if dZ may be taken on AZ , its length shall be as the rectangle dZ by ZY to the element of the section ATa as AZ2 to 9,0827646AT2 +AZ2, that is, as dZ shall be to 1

2 AZ as AT2 to 9,0827646AT2 +AZ2, the rectangle eZ into ZY will designate the decrement of the time arising from the motion of the node, in the total time in which the arc Aa is traversed. And if the point d touch the curve NdGn, the curved area NdZ will be the total decrement, in which time the whole arc NA is traversed; and therefore the excess of the sector NAT above the area NdZ will be that total time. And because the motion of the node in a smaller time is smaller in the ratio of the time, also the area AaYZ must be diminished in the same ratio. Because that arises if eZ may be taken in the length AZ, which shall be to the length AZ as AZ2 to 2 29 0827646, AT AZ+ . Thus indeed the rectangle eZ in ZY will be to the area AZYa as the decrement of the time, in which the arc Aa is traversed, to the total time in which it may be traversed, if the node may be at rest : and therefore that rectangle will correspond to the decrement of the motion of the node. And if the point e may touch the curve NeFn, the total area NeZ, which is the sum of all the decrements, will correspond to the total decrement, in which time the arc AN is traversed ; and the area remaining NAe will correspond to the remaining motion, which is the true motion of the node, in which time the whole arc NA is traversed by the motion of the sun and node conjointly. Now truly the area of the semicircle is to the area of the figure NeFn, sought by the method of infinite series, as 793 to 60 approximately. But the motion which corresponds to the whole circle was 019 49 3 55. ' . " . "' , and therefore the motion, which corresponds to duplicate figure NeFn, is gr1 29 58 2. ' . '' . "' . Which taken from the first motion leaves 018 19 5 53. ' . " . "' , the total motion of this with respect to the fixed stars between conjunctions with the sun ; and this motion taken from the annual motion of the sun of 3600, leaves 0341 40 54 7. ' . ' . "' , the motion of the sun between the same conjunctions. But this motion is to the annual motion 3600. So that the motion of the nodes is now found, 018 19 5 53. ' . " . "' . to the motion of the year itself, which therefore will be 019 18 1 23. ' . " . ''' . This is the mean motion of the nodes in a sidereal year. The same from astronomical tables is 019 21 21 50. ' . " . ''' . The small difference is a one three hundredth part of the total motion, and may be seen to arise from the lunar eccentricity and from the inclination to the plane of the ecliptic. The motion of the nodes is accelerated by the eccentricity of the orbit, and by this in turn the inclination may be slowed down a little, and at the same time the velocity is reduced.



PROPOSITION XXXIII. PROBLEM XIV. To find the true motion of the lunar nodes.

In the time which is as the area NTA-NdZ, (in the fig. preced.) the motion is as this area NAe, and thence is given. Truly on account of the excessive difficulty of the calculation, it is better to keep to the following construction of the problem. With the centre C, with some radius CD, the circle BEFD may be described. DC is produced to A, so that AB to AC shall be as the mean motion to half the true mean motion, when the nodes are in

quadrature, that is, as 0 019 18 1 23 to 19 49 3 55. ' . " . ''' . . ' . " . ''' , and thus BC to AC as the difference of the motions 00 31 2 32. ' . " . ."' , to the latter motion 019 49 3 55. ' . " . ''' , that is, as 1 to 3

1038 ; then through the point D the indefinite line Gg is drawn, which is a tangent to the circle at D; and if the angle BCE or BCF may be taken equal to twice the distance of the sun from the place of the node, by the mean motion found; and AE or AF may be acting cutting the perpendicular DG in G; and the angle may be taken which shall be to total motion of the node between its syzygies (that is, to 09 11 3. ' . " .) as the tangent DG to the circumference of the whole circle BED; and this angle (for which the angle DAG can be used) may be added to the motion of the nodes when the nodes pass from quadrature to syzygies, and may be subtracted from the same mean motion when they pass from syzygies to quadratures; the true motion of these will be had. For the true motion thus found agrees approximately with the true motion which arises by putting in place the time by the area NTA-NdZ, and the motion of the node by the area NAe; so that the matter considered carefully will agree with the computations put in place. This is the half yearly equation of the motion of the nodes. And there is a monthly equation, but which is hardly necessary for finding the latitude of the moon. For since the variation of the inclination of the moon's orbit to the plane of the ecliptic shall be liable to a two-fold inequality, the one half-yearly, but the other monthly ; the monthly inequalities of this and the monthly equation of the nodes thus mutually modify and correct each other, so that both can be ignored in determining the latitude of the moon. Corol. From this and from the preceding proposition it is evident that the nodes are at rest at their syzygies, but in the quadratures they are regressing by the hourly motion

iv16 19 26" . ''' . . And because the equation of motion in the octants shall be 01 30. ' . Which are rightly in order with all celestial phenomena.



Scholium.

J. Machin, Prof. of Astronomy at Gresham College, and Henry Pemberton M. D. have separately discovered the motion of the nodes by another method. Certain mention of this method has been made elsewhere. And each paper that I have seen, contained two propositions, and they agreed between each other in each case. Truly Machin's paper is here adjoined, since it came first into my hand.

' Concerning the motion of the moon's nodes.

PROPOSITION I. The mean motion of the sun from a node, is defined by the mean geometrical proportion, between the mean motion of the sun, and that mean motion by which the sun recedes most rapidly from the node in quadrature.

T shall be the place where the earth is, Nn the line of the lunar nodes at some given time, KTM a right angle drawn to this, TA a right line revolving around the centre with that angular velocity by which the sun and the nodes recede from each other in turn, thus so that the angle between the line Nn at rest and the revolving line TA, always becomes equal to the distance of the places of the sun and of the node. Now if some right line TK may be divided into the parts TS and SK which shall be as the motion of the mean hourly sun to the motion of the mean hourly node at quadrature, and the right line TH may be put as the mean proportional between the part TS and the whole TK, this right line between the rest of the proportions will be the mean motion of the sun from the node. For the circle NKnM with centre T and radius TK may be described, and with the same centre and radius TH and TN the ellipse NHn is described, and in the time in which the sun recedes from the node through the arc Na, if the right line Tba is drawn, the area of the sector NTa expresses the sum of the motions of the sun and of the node in the same time. Therefore by the afore mentioned law let the arc aA be the element the right line Tba will describe revolving in the given element of time, and that minimal sector TAa will be as the sum of the velocities by which the sun and the node are carried separately in that time. But the



velocity of the sun is almost constant, just as the small inequality of this will lead to scarcely any variation in the mean motion of the nodes. The other part of this sum, clearly the mean value of the velocity of the node, may be increased by the departure from the syzygies in the square ratio of the sine of its distance from the sun ; by the Corollary to Prop. 31, Book 3 Princip. and since it is a maximum at the quadrature to the sun at K, it will be obtained in the same ratio to the velocity of the sun which SK has to TS, that is as (the difference of the squares from TK and TH or) the rectangle KHM to the square TH. But the ellipse NBH divides the sector ATa which expresses the sum of these two velocities, into the two parts ABba and BTb proportional to these velocities. For BT may be produced to the circle at β , and from the point B the perpendicular BG may be sent to the major axis, which each produced cross the circle at the points F and f, and because the distance ABba it to the sector TBb as the rectangle ABβ to the square BT (for that rectangle is equal to the difference of the squares from TA and TB on account of the right line Aβ cut equally and unequally at T and B.) Therefore this ratio, when the distance ABba is a maximum at K, will be the same as the ratio of the rectangle KHM to the square HT, but the maximum mean velocity of the nodes was in this ratio to the velocity of the sun. Therefore in the quadratures the sector ATa is divided into parts proportional to the velocities. And because the rectangle KHM is to the square HT as FBI to the square BG and the rectangle ABβ is equal to the rectangle FBf. Therefore the little area ABba when it is a maximum to the sector TBb of the remaining, as the rectangle ABβ to the square BG. But the ratio of these small areas always was as the rectangle ABβ to the square BT ; and therefore the element of the area ABba at the place A is less than the similar element in quadrature, in the square ratio BG to BT that is in the square ratio of the sine of the distance of the sun from the node. And therefore the sum of all the elements of area ABba surely the interval ABN will be as the motion of the node in the time in which the sun is moving away from the node through the arc NA. And the distance of the remaining, surely the elliptic sector NTB will be as the mean motion of the sun in the same time. And therefore because annual mean motion of the node, that is which shall be in the time in which has completed its period, the mean motion of the node from the sun will be to the mean motion of the sun itself, as the area of the circle to the elliptic area, that is as the right line TK to the mean right line TH evidently the proportional between TK and TS ; or what amounts to the same, as the mean proportional TH to the right line TS.

PROPOSITION II. From the given mean motion of the nodes, to find the true motion.

The angle A be the distance of the sun from the mean location of the node, or the mean motion of the sun from the node. Then if the angle B is taken, its tangent shall be to the tangent of the angle A as TH to TK, that is, in the square root ratio of the mean hourly motion of the sun with the node moving in quadrature ; the angle B will be the same distance of the sun from the true position of the node. For FT may be joined and from the demonstration of the ratio in the above proposition, the angle FTN will be the distance of



the sun from the mean place of the node, but the true distance of the angle ATN from the node, and the tangents of these angles are to each other as TK to TH. Corol. Hence the angle FTA is the equation of the lunar nodes, and the sine of this angle when it is a maximum in the octants, is to the radius as KH to TK TH+ . But the sine of this equation at some other place A is to the maximum sine, as the sine of the sum of the angles FTN ATN+ to the radius : that is almost as the sine of twice the distance of the sun from the mean place of the node (clearly 2.ETN) to the radius.

Scholium.

If the mean hourly motion of the nodes in quadrature shall be iv v16 16 3 42" . "' . . .that is, 39 38 7 50. ' . '' . '''° in the whole sidereal year, TH will be to TK in the square root ratio of the number 9,0827646 to the number 10,0827646, that is, as 18,6524761 to 19,6524761. And therefore TH shall be to HK as 18,6514761 to 1, that is, as the motion of the sun in one sidereal year to the mean motion of the nodes 2

3019 18 1 23. ' . '' . ''' .

But if the mean motion of the nodes of the moon in the 20 Julian years shall be 0386 50 15. ' . " . as thus from observations it is deduced in the theory of the moon used : the

mean motion of the nodes in a sidereal year shall be 019 20 31 58. ' . '' . ''' . And TH will be to HK as 3600 to 019 20 31 58. ' . '' . ''' , that is, as 18,61214 to 1, from which the mean hourly motion of the nodes in quadrature becomes iv16 18 48'' . ''' . . And the maximum equation of the nodes in the octants will be 01 29 57. ' . " . '

PROPOSITION XXXIV. PROBLEMA XV. To find the variation of the hourly inclination of the moon's orbit to the plane of the

ecliptic. Let A and a designate the syzygies; Q and q the quadratures; N and n the nodes; P the location of the moon in its orbit ; p the projection of that place into the plane of the ecliptic, and mTl the instantaneous motion of the nodes as above. And if the perpendicular PG may be sent to the line Tm, pG may be joined, and with that then produced then it may cross Tl in g, and also may be joined PG: the angle for the inclination to the orbit of the moon to the plane of the ecliptic will be PGp, when the moon may be passing through P ; and the angle Pgp will be the inclination of the same after a completed instant of time; and thus the angle GPg is the instantaneous variation of the inclination. But this angle GPg is to the angle GTg as TG to PG and Pp to PG jointly. And therefore if for the instant of time an hour may be substituted; since the angle GTg (by Prop. XXX.) shall be to the angle iv33 10 33" . "' . . as 3 to IT PG AZ AT× × , the angle GPg (or the variation of the hourly inclination) shall be to the angle iv33 10 33" . "' . .as

3 to PpPGIT AZ TG AT× × × Q. E.I.



Thus these may be had themselves from the hypothesis that the moon is gyrating

uniformly in its orbit. Because if that orbit is elliptical, the mean motion of the nodes may be diminished in the ratio of the minor axis to the major axis ; as has been explained above. And the variation of the inclination may be diminished in the same ratio also. Corol. 1. If the perpendicular TF may be erected to Nn, and pM shall be the hourly motion of the moon in the plane of the ecliptic ; and the perpendiculars pK, Mk may be sent to QT and each produced crossing TF in H and h: IT will be to AT as Kk to Mp, and TG to Hp as TZ to AT, and thus IT TG× equals Kk Hp TZ

Mp× × , that is, equal to the area HpMh

taken by the ratio TZMp : and therefore the hourly variation of the inclination

iv33 10 33" . "' . .as HpMh multiplied by 3 to PpTZMp PGAZ AT× × .

Corol. 2. And thus if the earth and the nodes of the individual completed hours may be withdrawn from the new places of these, and always reintroduced at once again into the first place, so that their position, through the period of a whole month, may remain in place ; the whole variation of the inclination in the time of a month of its inclinations will be to iv33 10 33" . "' . . as the sum of all the areas HpMh, described in the revolution of the point p, and taken together with the appropriate signs + and – , multiplied into

3 to PpPGAZ TZ Mp AT× × × , that is to say, as the whole circle QAqa multiplied into

3 to PpPGAZ TZ Mp AT× × × that is, as the circumference QAqa multiplied into

2 to 2PpPGAZ TZ Mp AT× × × .

Corol. 3. Hence in the given situation of the nodes, the mean hourly variation, from which that monthly variation can be generated continually each month, is to



iv33 10 33" . "' . .as 2 to 2PpPGAZ TZ AT× × , or as 1

2 to 4AZ TZ

ATPp PG AT×× × , that is (since Pp

shall be to PG as the sine of the aforesaid inclination to the radius, and 12

shall be to 4AZ TZAT AT× as the sine of double the angle ATn to the radius quadrupled) as the

sine of the same inclination multiplied into the sine of double the distance of the nodes from the sun, to four times the square of the radius. Corol. 4. Because the variation of the hourly inclination, when the nodes are turning in

the quadratures, is (by this proposition) to the angle iv33 10 33" . "' . .as 3 to Pp

PGIT AZ TG AT× × × , that is, as 12

to 2PpIT TGPGAT AT× × ; that is to say, as the sine of

twice the distance of the moon from the quadrature multiplied into PpPG to the radius

doubled: the sum of all the hourly variations, in which time the moon in this situation of the nodes passes from quadrature to syzygies (that is, the space of 1

6177 hours,) will be to

the sum of just as many angles iv33 10 33" . "' . , or 5878", as the sum of all the sines of twice the distances of the moon from quadrature multiplied into Pp

PG to the sum of just as

many diameters ; that is, as the diameter multiplied by PpPG to the circumference; that is,

if the inclination shall be 05 1. ' , as 874100007 to 22× , or 278 to 10000. And hence the whole

variation, from the sum of all the hourly variations in the aforementioned time put together, is 163", or 2'. 43".

PROPOSITION XXXV. PROBLEM XVI. To find the inclination of the moon to the ecliptic arising at a given time.

Let AD be the sine of the maximum inclination, AB the sine of the minimum inclination ; BD is bisected in C, and the circle BGD may be described with centre C, and with the radius BC. On AC, CE may be taken in that ratio to EB that EB has to 2BA: and if in a given time the angle AEG may be put in place equal to twice the distance of the nodes from the quadratures, and to AD there may be sent the perpendicular GH: AH will be the sine of the inclination sought. For GE2 is equal to



2 2 2

2 2 2

2

2

2 2 2 2

GH HE BHD HE HBD

HE BH HBD BE BH BE

BE EC BH EC AB EC BH EC AH .

+ = + = +

− = + − ×

= + × = × + × = ×

And thus since 2EC is given, GE2 is as AH. Now AEg designates twice the distance of the nodes from the quadratures after some given instant of time completed, and the arc Gg on account of the given angle GEg will be as the distance GE. But Hh is to Gg as GH to GC and therefore Hh is as the productGH Gg× , or GH GE× ; that is, as

2 or GH GHGE GEGE AH× × , that is, as AH and the sine of the angle AEG jointly. Therefore if

AH in some case shall be the sine of the inclination, that will be increased in the same increments with the sine of the inclination, by Corol. 3 of the above proposition, and therefore it will always remain equal to that sine. But AH, when the point G falls on either the point B or D, is equal to the sine of this, and therefore always remains equal to the same. Q. E. D. I have supposed in this demonstration that the angle BEG, which is twice the distance of the nodes from quadrature, is increased uniformly. For it would be superfluous on this occasion to consider all the small inequalities. Now consider the angle BEG to be right, and in this case Gg is to be the increase of the hours of twice the distance of the nodes and of the sun in turn ; and the hourly variation of the inclination in the same case (by Corol. 3 of the most recent Prop.) will be to iv33 10 33" . "' . as the product of the sine of the inclination AH and the sine of the right angle BEG, which is twice the distance of the nodes from the sun, to four times the square of the radius ; i.e. as the sine of the mean inclination AH to four times the radius ; that is (since that mean inclination shall be as

12

05 8. ' as its sine 896 to the fourfold radius 40000, or as 224 to 10000. Moreover the whole variation, corresponding to the difference of the sines BD, is to that hourly variation as the diameter BD to the arc Gg; i.e., as the diameter BD to the semi circumference BGD and the time of the hours 7

102079 , in which a node goes from quadrature to syzygies, to one hour jointly ; i.e., as 7 to 11 and 7

102079 to 1. Whereby if

all the ratios may be joined together, the total variation BD becomes to iv33 10 33" . "' . as 7

10224 7 2079× × to 110000, i.e., as 29645 to 1000, and thence that variation BD will be produced 1

216 23' . " . This is the maximum variation of the inclination as long as the position of the moon in its orbit may not be considered. For the inclination, if the nodes may be turning in syzygies, are not changed from the variation of the moon. But if the nodes are present in quadrature, the inclination is less when the moon turns in syzygies, as when that may be in quadrature, by the excess 2'. 43"; as we have indicated in corollary four of the above proposition. And the whole mean variation BD with the moon in quadrature, diminished by half of its excess 1

21 21' . " , shall become 15'. 2", but in the syzygies increased by this amount to become 17'. 45". Therefore if the moon may be in place in syzygies, the whole variation of the nodes from quadrature to syzygies will be 17'. 45": and thus if the



inclination, when the nodes are present in syzygies, shall be 05 17 20. ' . " ; the same, when the nodes are in quadrature, and the moon in syzygies, will be 04 59 35. ' . '' . And this is thus confirmed from observations. If now that inclination of the orbit may be desired, when the moon is turning in syzygies and the nodes anywhere ; AB becomes to AD as the sine of 40. 59'. 35" to the sine of 50. 17'. 20", and the angle AEG may be taken equal to twice the distance of the nodes from quadrature ; and AH will be the sine of the inclination sought. For the inclination of this orbit is equal to the inclination of the orbit , when the moon shall be set at 900 from the nodes, [as just found]. In other locations of the moon the monthly inequalities, that are allowed in the variation of the inclination, may be compensated in the calculation of the latitude of the moon, and in some manner may be corrected by the monthly inequality of the motion of the nodes (as we have said above), and thus it can be ignored in the calculation of the latitude.

Scholium. I wished to show from these computations of the moon's motions, how the causes of the lunar motions could be calculated from the theory of gravity. In addition by the same theory I found that the annul equation of the mean motion of the moon arose from the various dilations of the moon's orbit by the force of the sun, as in Corol. 6. Prop. LXVI. Book 1. This force is greater at the perigee of the sun, and dilates the moon's orbit ; at its apogee it is less, and it allows that orbit to be contracted. In a dilated orbit the moon is revolving slower, in the contracted orbit faster ; and the annual equation, by which this inequality may be balanced, is nothing at the apogee and perigee of the sun, at the mean distance of the sun from the earth it rises to around 11'. 50'", at other places it is proportional to the equation of the centre of the sun ; and it is added to the mean motion of the moon when the earth goes from the aphelion to its perihelion, and it is subtracted in the opposite part of the orbit. On assuming the radius of the orbit to be of size 1000 and the eccentricity of the earth to be 7

816 , this equation, when it is a maximum, will produce 11'. 49", by the theory of gravity. But the eccentricity of the earth may be seen to be a little greater, and with increased eccentricity this equation must be increased in the same ratio. The eccentricity may be 11

1216 , and the maximum equation will become 11'. 51". I have also found that at the perihelion of the sun, because of the greater force of the sun, the apogee and nodes of the moon move faster than at its aphelion, and that inversely in the cubic ratio of the distance from the earth. And thence the annual equations of these motions arise proportional to the equation of the sun's centre. But the motion of the sun is in the inverse square ratio of the distance from the sun to the earth, and the maximum equation of the centre, that these inequalities may generate, is 01 . 56' 20". agreeing with the predicated eccentricity of the sun 11

1216 . But if the sun's motion were inversely in the cubic ratio of the distance, this inequality would generate the maximum equation 02 .54'. 30'' . And therefore the greatest equations, that the inequalities of the motions of the apogee, and of the nodes of the moon can generate, are to 02 . 54 30' . " as the mean diurnal motion of the apogee and the mean diurnal motions of the nodes of the moon are to the mean diurnal motion of the sun. From which the greatest equation of the



mean motion of the apogee becomes 19'.43", and the maximum mean motion of the nodes 9'. 24". Truly it is added to the first equation and subtracted from the second, when the earth travels from it perihelion to its aphelion : and it shall be the contrary on the opposite side of the orbit. It will be agreed also by the theory of gravity that the action of the sun on the moon shall be a little greater, when the transverse diameter of the moon's orbit passes through the sun, than when the same is at a right angle to the line joining the earth and the sun : and therefore the orbit of the moon is a little greater in the first case than in the second. And hence another equation of the mean motion of the moon arises, depending on the position of the moon's apogee relative to the sun, which indeed is a maximum when the apogee of the moon is in the same octant as the sun ; and nothing when that arrives at the quadrature or syzygies : and it is added to the mean motion in the transition of the lunar apogee from the sun's quadrature to syzygies, and taken away in the transition of the apogee from the syzygy to the quadrature. This equation, that I will call half-yearly, when it is a maximum in the octants of the apogee, increases to around 3'. 45", as many times as I can gather from the phenomena. This is the quantity of this in the mean distance of the earth from the sun. Truly it may be increased or diminished in the inverse cubic ration of the distance of the sun, and thus at the maximum distance of the sun it is 3'. 34'" and at the minimum 3'.56" approximately: truly when the apogee of the moon is placed beyond the octants, it emerges smaller ; and is to the maximum equation, as the sine of twice the distance of the moon's apogee from the nearest syzygy or quadrature to the radius. By the same theory of gravity the action of the sun on the moon it is a little greater when the right line drawn through the nodes of the moon pass through the sun, than when that line is at right angles with the right line joining the earth and the sun. And thence arises another equation of the mean motion of the moon, that I will call the second half-yearly, and which is a maximum when the nodes are turning in the octants of the sun, and which vanishes when they are in syzygy or in quadrature, and in other positions of the nodes proportional to the sine of twice the distance of each node from the nearest syzygy or quadrature: truly is added to the mean motion of the moon, if the sun is moving away ahead of its nearest node, subtracted if it is moving away behind its nearest node, and in the octants, when it is a maximum, it rises to 47" in the mean distance of the sun from the earth, as I have deduced from the theory of gravity. At other distances of the sun this maximum equation at the octants of the nodes is inversely as the cube of the distance of the sun from the earth, and thus at the perigee of the sun it rises to around 49" , and to around 45" at the apogee. And by the same theory of gravity the moon's apogee is progressing at a maximum rate when it is either in conjunction with the sun or in opposition to the same, and recedes when it goes in quadrature with the sun. And the eccentricity shall be a maximum in the former case and a minimum in the latter, by Corol. 7, 8 and 9. of Prop. LXVI. Book I. And these inequalities by the same corollaries are very great, and they generate the principle equation of the apogees, that I have called half-yearly. And the maximum of this equation is around 012 .18' , as much as I have been able to gather from observations. Our Horrox first put in place that the moon revolved in an ellipse around the earth situated at its lower focus. Halley located the centre of the ellipse in an epicycle, the centre of which is revolving uniformly around the earth. And from the motion in the epicycle the



inequalities now discussed arise in the progression and recession of the apogees as well as the amount of the eccentricity. It is supposed that the distance between the moon and the earth be divided into 100000, and T refers to the earth and TC the mean eccentricity of the of the moon from 5505 parts. TC may be produced to B, so that CB shall be the sine of the equation of the maximum half-yearly inequality 012 .18' to the radius TC, and the circle BDA described with centre C, radius CB will be that epicycle in which the centre of the orbit of the moon is located and is revolving following the order of the letters BDA. The angle BCD is taken equal to twice the annual argument, or to twice the true distance of the place of the sun from the lunar apogee corrected once, and CTD will be the half-yearly first correction of the lunar apogee and TD the eccentricity of it orbit tending towards the apogee, corrected in the second place. Moreover, having the mean motion, the eccentricity, and the apogee of the moon, as well as with the major axis of the orbit of 200000 parts; and from these there is extracted the place in the orbit and its distance from

the earth, and that by well-known methods. At the perihelion of the earth, on account of the greater force of the sun, the centre of the lunar orbit will move faster around the centre C than at the aphelion, and that in the inverse cubic ratio of distance of the earth from the sun. On account of the equation of the centre of the sun included in the annual argument, the centre of the moon's orbit will be moving faster in the epicycle BDA in the inverse square ratio of the distance of the earth from the sun. So that also at this time it will be moving faster in the simple inverse ratio of the distance ; from the centre D of the orbit there may be drawn the right line DE towards the lunar apogee, or parallel to the right line TC , and the angle EDF may be taken equal to the excess of the aforementioned angular argument over the distance of the lunar apogee from the following perigee of the sun; or what is the same, the angle CDF may be taken equal to the complement of the true anomaly of the sun to 3600. And DF shall be in the ratio to DC as twice the eccentricity of the great orbit to the mean distance of the sun from the earth, and the mean diurnal motion of the sun from the lunar apogee to the mean diurnal motion of the sun from the appropriate apogee jointly, that is, as 7

833 to 1000 and 52'. 27". 16'" to 59'. 8". 10'" jointly, or as 3 to 100. And consider the centre of the moon to be located at the point F, and revolving on an epicycle whose centre is D, and the radius DF, and meanwhile the point D is progressing on the circumference of the circle DABD. For by this account, the velocity by which the centre of the moon will be moving along some curve described about the centre C, will be inversely as the cube of the distance of the sun from the earth approximately, as required. The computation of this motion is difficult, but it may be rendered easier by the following approximation. If the mean distance of the moon from the earth shall be of 100000 parts, and the eccentricity TC shall be of 550 parts as above, the right line CB or



CD will be found of 341172 parts, and the right line DF of 1

535 parts. And this right line DF at the distance TC, subtends an angle at the earth, which the translation from the centre of the orbit from some place D to some place F, produced in the motion of its centre : and twice the same right line 2DF in a parallel position to the line which joins the earth and the upper focus of the lunar orbit, will subtend the same angle at the earth, that certainly that translation generates in the motion of the focus, and at the distance of the moon from the earth, it will subtend an angle that the same translation will generate in the motion of the moon, and which therefore can be said to be the second equation of the centre. And this equation, at the mean distance of the earth from the sun, is as the sine of the angle, which that right line DF contains approximately with the right line drawn from the point F to the moon, and when it is a maximum, 2'. 25" emerges. But the angle that the right line DF makes with the right line drawn from the point F to the moon taken together, is found either by subtracting the angle EDF from the mean anomaly of the moon, or by adding the distance of the moon from the sun to the distance of the lunar apogee from the apogee of the sun. And as the radius is to the sine of the angle thus found, thus 2'. 25", is required to be added to the second equation of the centre, if that sum shall be less than a semicircle, and to be subtracted if greater. Thus its longitude will be found in the syzygies of the luminous bodies themselves. Since the atmosphere of the earth refracts the light of the sun as far as a height of 35 or 40 miles, and by refracting may scatter the same into the shadow of the earth, and on being scattered the light in the confines of the shadow dilates the shadow : to the diameter of the shadow, which is produced by parallax, I add 1 minute or 1

31 minutes in eclipses of the moon. Truly lunar theory must be examined through the phenomena and firmly established first in syzygies, then in the quadratures, and finally in the octants. And with this need in mind, I have observed rather precisely the average movements of the moon and of the sun at the time of the meridian, in the royal observatory in Greenwich, and for the last day in December of the year 1700 in the old style , it will not be an inconvenience to use the following : truly the mean motion of the sun 020 .43 40' . ' , and of its apogee

07 . 44 30' . " , and the mean motion of the moon 015 . 21' 00". , and of its apogee 08 20' 00". , and of the ascending node 027 .24' 20". ; and the difference of the

meridians of this observatory and of the royal observatory in Paris to be h0 .9' 20''. but the mean motion of the moon, and of the apogees of this have not yet been found accurately enough.


Book III Section III. Translated and Annotated by Ian Bruce. Page 843

PROPOSITION XXXVI. PROBLEM XVII.

To find the force of the sun required to move the waters of the sea. The force [per unit mass] ML or PT of the sun required to perturb the lunar motion acting at the lunar quadratures, was to the force of gravity with us on earth (by Prop. XXV. of this Book;), as 1 to 638092,6. And the force TM LM− or 2PK at the lunar

syzygies is twice as great. [see the notes at the start of the previous section, from which the results presented here follow; see also Chandrasekhar, p. 412.] But these forces, if one descends to the surface of the earth, are diminished in the ratio of the distances from the centre of the earth, that is, in the ratio 1

260 to 1; and thus the first force on the surface of the earth is to the force of gravity as 1 to 38604600. It is by this force that the sea is lowered in places, which stand apart at 90 degrees from the sun. The sea is raised both under the sun and in the region opposite the sun by the other force, which is greater by twice as much,. The sum of the forces is to the force of gravity as 1 to 12868200. And because the same force produces the same motions, that either lowers the water in regions which are set at 90 degrees to the sun, or raises the same in regions under the sun and in regions opposite to the sun, thus sum will be the total force of the sun required to set the seas in motion ; and the same effect will be had, if the whole sea may be raised in regions under the sun and in regions opposite the sun, but in regions which are set at 90 degrees to the sun it gives rise to no effect. This is the force of the sun required to set the sea in motion at some given place, both when the sun is situated overhead as well as at its mean distance from the earth. In other places the force required to raise the sea from the position of the sun is directly as the versed sine of twice the altitude of the sun above the horizon of the place, and inversely as the cube of the distance of the sun from the earth. Corol. Since the centrifugal force of the parts of the earth arises from the diurnal motion of the earth, which is to the force of gravity as 1 to 289, it has the effect that the height of the water at the equator exceeds the height of this at the poles by an amount of 85472 Parisian feet, as in Prop. XIX above ; the force of the sun by which we are driven, since it shall be to the force of gravity as 1 to 12868200, and thus to that force as 289 to 12868200 or 1 to 44527, effects that the height of water in regions under the sun and opposite to the sun shall surpass the height of this in places, which are set at 90 degrees to the sun, by a measure of one Parisian foot and 1

3011 inches. For this measure is to the measure of 85472 feet as 1 to 44527.



PROPOSITION XXXVII. PROBLEM XVIII.

To find the force of the moon required to move the waters of the sea. The force of the moon required to move the sea can be deduced from the proportion of its force to the force of the sun, and this proportion is to be deduced from the proportion of the motions of the seas, which arise from these forces. [At the time of writing, the mass of the moon was not known precisely: Newton could hardly go to the moon and measure the acceleration of gravity there, and as there was no small satellite on hand on which measurements could be made, so he adopted this rather imprecise method, the only one available to him at the time.] Before the mouth of the river Avon at the third milestone below Bristol, at spring time and in autumn the whole rise of the water at the conjunction and opposition of the moon, according to the observation of Samuel Sturmy, is around 45 feet, but in the quadratures it is only 25 feet. The first height arises from the sum of the forces, and the last from the difference of the same. Therefore let S and L be the forces of the sun and moon situated above the equator and at their mean distances from the earth, and to L S L S+ − will be as 45 to 25, or as 9 to 5. [These tide-producing forces, due to the sun and moon individually as viewed from the earth, act on a unit mass at some place in question; this mass is situated a whole earth-radius from the centre of the mean earth-sun distance and the mean earth-moon distances respectively at the meridian, by which the force is increased by the inverse square law above the average value at the centre of the earth ; each force in turn sweeping through 1800 , from acting initially along one tangent to the sea through the normal at the meridian and finally acting along the tangent in the opposite direction, in a time of 12 hours, due to the earth's diurnal rotation, while the sun or moon can be considered as almost at rest. These actions are simultaneously performed, by the forces of the sun and the moon acting through the earth on the water on the far side of the earth, and now diminished by the inverse square law, giving rise on the whole to corresponding tidal bulges on opposite sides of the earth. This sum or difference of the forces arises from the simple addition of the vectors L and S at the various places : at the new moon syzygies these forces are simply added L S+ , allowing 6 hours for the earth to rotate into the tangential position of the forces; the full moon case requiring both forces to act in the opposite direction, this being the case for the sun as the force is now less than the average at the centre of the earth by the same amount S; An elementary derivation of part of this argument is given, e.g. in Ohanian, Gravitation and Spacetime, p.26. At the quadrature positions, the tangential moon's force and that of the sun after a quarter rotation of the earth, are considered to act in opposite directions, giving the magnitude of the force to be now L S− in this simplified scheme. See the notes at the start of the last section. For a concise modern analysis, you may consult p.355 onwards of Volume 7 of the Encyclopaedic Dictionary of Physics, where the whole matter of tide-generating forces is addressed from the point of view of potentials, introduced some hundred years later by Laplace. This is one section of such a work that does not get dated! Thus, the potential



energy of a unit mass due to the moon, situated on the earth's surface at a point P a distance d from the centre of the moon, is given by

mGM / d , where Mm is the mass of the moon, G the constant of universal gravitation; a negative sign may be attached to give the gradient forces the correct sign for the direction in which they act. Hence, if R is the distance between the centres of the earth and moon, and e the radius of the earth making an angle ϕ to R, then

2 2 2 2d e R eRcosϕ= + − ; in which case the potential becomes

( )221 2

nm m

ne e

RR

GM GM em nR RR cos n

GM / d P cosϕ

ϕ+ −

= = ∑ ,

where the coefficients ( )nP cosϕ are the Legendre polynomials defined by

( ) ( )

( ) ( ) ( ) ( )1 12 2

0 1

2 32 3

1

3 1 5 3 etc

P cos ;P cos cos ;

P cos cos ;P cos cos cos ; .

ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ

= =

= − = −

Now, the mean value of eR is 0.01659, and due to its small magnitude, only a few terms

need be considered; on differentiation of the potential function to obtain the forces, the first term is constant and disappears; the second term can be shown to give rise to a constant force along the line of the centres, and hence only the derivative of the third term involving P2 need be considered in most cases to cause relative motion. Since 2

EGM

eg = ,

where ME is the mass of the earth and e the mean radius of the earth , then the potential

generating the lunar tides has the form ( ) ( )232

32 10 03 and m

E

Me eM Re

V gV cos Vϕ= − = ; from

which we see that the tangential component 0 2dVed gV sinϕ ϕ= − is the only large

component, generating tides over large distances on the earth's surface, since the vertical component ( )2

02 1dVde gV cos ϕ= − is negligible in comparison with the local acceleration

of gravity g, since V0 has a mean value of 85 6 10. −× . From the historical viewpoint, we may note that Newton worked with forces in his analysis, a hard task, while Laplace on introducing the idea of potential, was able to perform a simpler analysis; but of course Laplace could not have seen the easier method without Newton's great labours.] In the port of Plymouth the tides of the sea from the observations of Samuel Colepress rise around 16 feet from the average, and at spring time and autumn the height of the tide in syzygies can surpass the height at quadrature by 7 or 8 feet. If the maximum difference of these heights shall be 9 feet, then to L S L S+ − will be as 1 1

2 220 to 11 or 41 to 23.



Which proportion agrees well enough with the previous. On account of the magnitude of the tide at the port of Bristol, it may appear that the observations of Sturmy to be more reliable, and thus we will use the proportion 9 to 5, as it is a little more certain. For the remainder, on account of the reciprocal motion of the waters, the maximum tides do not occur at the syzygies of the luminary bodies themselves, but are the third tides after the syzygies as it has been said, either nearest to the third passage of the moon through the meridian of the place, or rather, (as it has been noted by Sturmy) they are the third tides to arrive after the day of the new moon or the full moon, or within an [additional] interval of [half of] 12 hours of the new or full moon, and thus are incident a little more or less than the 43rd hour from the new or full moon. [Such situations may arise when the moon acts as a forcing oscillator on a body of water in a confining basin, which builds up a resonance, as it were.] Truly these arrive at this port at around the seventh hour from the approach of the moon at the place of the meridian ; and thus the tides follow the passage of the moon through the meridian very closely, when the moon either stands apart from the sun, or is in opposition by 18 or 19 degrees as a consequence. [In fact, the tidal forces at the time act along the tangent rather than normally, as Chandrasakher points out; also, this is a place where Mott translates Newton's octodecim vel novemdecim as 18 or 19, which would seem to be correct – though this is not the usual way of writing these numbers, and which is translated by Madame du Chatelet as 80 or 90.] The summer and winter tides are especially vigorous, not in the solstices themselves, but when the sun stands at around a twentieth part of the whole circuit, or around 36 or 37 degrees. And similarly the maximum tide of the sea arises from the approach of the moon to the meridian of the place, when the moon stands away from the sun by a tenth part of the whole motion from tide to tide. That distance shall be of around

1218 degrees. And the force of the sun at this distance of the moon from syzygies and

quadratures, will have a smaller value towards increasing or decreasing the motion of the seas arising from the force of the moon, than at the syzygies and the quadratures, in the ratio of the radius to the sine of twice the complement of the angle 1

218 , i.e. of 370, that is, in the ratio 10000000 to 7986355. And thus in the above analogy 0,7986355 S must be written for S. And thus the force of the moon in quadrature, on account of the declination of the moon from the equator, must be diminished. For the moon in quadrature, or rather at

12018 after quadrature, is present at a declination of around 220.13'. And the force required

to move the sea is diminished by the decline from the equator in the square ratio approximately as the square of the complement of the sine of the declination. [Note from L & S. p. 110, De Mund. Syst.: Let TBD be the plane of the equator, T the centre of the earth, and the moon shall be at L, the angle LBD will measure the declination from the equator, or on account of the very small angle TLB, that declination will be approximately equal to the angle LTD, the cosine of which angle is TF, by taking TL for the radius. Now the force which pulls on the water from the centre T, at the location B of



the equator, when the moon is present in the equatorial plane at D, is to force that pulls the same water directly from the centre, when the moon is at L, as TL to TF, that is, as the radius to the sine of the compliment of the declination LDT, distinct from the centripetal force of the water towards T. But with that centripetal force increased, the other force drawing the water from the centre is diminished in the same ratio; whereby, on compounding the effects, the force of the moon at the location D, is to the force of this at L, as the square of the whole sine TL, to the square of the complement of the sine, TF, of the declination of the moon LTD.] And thus the force of the moon in these quadratures is only 0,857327 L. Therefore there is

0 7986355 to 0 857032 7 L 0 79863558 as 9 to 5L , S , . ,+ − . Besides the diameters of the orbit, in which the moon must be moving without eccentricity [as in the Horrox sense], are in turn as 69 to 70; and therefore the distance of the moon at syzygies is to its distance at the quadratures as 69 ad 70, with all else being equal. And the distances of this at 1

2018 from syzygies, when the maximum tide is

generated, and at 12018 from quadratures, when the minimum tide is generated, are to the

mean distance of this as 69,098747 and 69,897345 to 1269 . But the lunar forces requiring

to move the sea are inversely in the cubic ratio of the distances, and thus the forces at the maximum and minimum of these distances are to the force at the mean distance as 0,9830427 and 1,017522 to 1. From which there arises 1 017522 0 7986355, L , S+ to 0 9830427 0 8570327 0 7986355, , L , S× − as 9 to 5. And S to L is as 1 to 4,4815. And thus since the force of the sun shall be to the force of the gravity 1 to 12868200, the force of the moon shall be to the force of gravity as 1 ad 12871400. [Because this value of S to L adopted is even further than the original estimate of 3.5 from the now accepted value of 2.34, Chandrasekhar wisely decides not to investigate the following Corollaries.] Corol. 1. Since the water disturbed by the force of the sun rises to a height of

1301ft 11 ''., , by the force of the moon the same will rise to a height of 5

228ft.,7 '' and the force with each acting shall be to a height of 1

210 ft., and when the moon is in perigee to a height of 1

212 ft.and more, especially when the tide is helped by the winds blowing. Moreover so much force suffices in abundance for all the excited motions of the sea, and corresponds properly to the quantity of the motions. For in seas which extend out widely from east to west, as in the Pacific, and in the parts of the Atlantic & Ethiopic [i.e. Indian] Oceans outside the tropics, the water is accustomed to be raised to a height of 6, 9, 12, or 15 feet. But in the Pacific Ocean, because it is deeper and extends wider, the tides are said to be greater than in the Atlantic and Indian Oceans. And indeed so that the tide shall be full, the width of the sea from east to west cannot be less than 900. In the Indian Ocean the rise of the water within the tropics is less than in the temperate zone, on account of the narrow seas between the African and the southern part of America. The water is unable to rise in the middle of the sea unless at it may fall on each eastern and western shore at the same time: since yet with our narrow seas it must fall alternately in turn on



these shores. For that reason the flow and ebb for islands which are farthest from the shores, is usually very small. In certain ports, where water is trying to flow in and out with great force through the shallow places, alternately filling and emptying the bays, the flow and ebb must be customarily greater, as at Plymouth & at Chepstowe bridge in England ; at mount St. Michael and at Avranches in Normandy ; at Cambaia and Pegu in the East Indies. In these places the sea, by approaching and receding with great speed, now inundates the shore and again now leaves it dry for many miles. Neither the impetus of the inflow or of the return of the first can be broken, as the water is raised or lowered by 30, 40, or 50 feet and more. And equal is the account of long narrow and shallow seas, as the Magellanic Straits, and of that by which England is surrounded. The tide in ports of this kind and in narrow channels is augmented by the impetuosity of the flow and ebb above the normal. Truly, the magnitude of the tide corresponds to the forces of the sun and moon for shores which descend to the depths of the abyss and which are seen to be open, and where water without precipitation can flow and ebb by rising and falling without an impetuous motion. Corol. 2. Since the force of the moon requiring to move the seas shall be to the force of gravity as 1 to 2871400, it is evident that that force shall be much less than those considered in any experiments with pendulums, or in statics or hydrostatics. This force is able to produce a sensible effect only in the tides in the sea. Corol. 3. Because the force of the moon required to move the sea compared to the force of the sun is as 4,4815 to 1 [recall that this is an overestimate roughly by a factor of 2, so that the following values are not numerically correct and the conclusion is wrong], and these forces (by corol. 14 Prop. LXVI, Book I.) are as the densities of the bodies of the moon and of the sun and the cubes of the apparent diameters conjointly ; the density of the moon will be to the density of the sun as 4,4815 to 1 directly, and the cube of the diameter of the moon to the cube of the diameter of the sun inversely : that is (since the mean apparent diameters of the moon and sun shall be 1

231 16'' and 32 12' . , ' . " ) as 4891 to 1000. But the [mean] density of the sun was to the density of the earth as 1000 to 4000; and therefore the density of the moon is to the density of the earth as 4891 to 4000 or 11 to 9. Therefore the body of the moon is more dense that our earth. Corol. 4. And since the true diameter of the moon from astronomical observations shall be to the true diameter of the earth as 100 to 365 the mass of the moon to the mass of the earth shall be as 1 to 39,788. Corol. 5. And the gravitational acceleration on the surface of the moon will be as if three times smaller than the acceleration of gravity on the surface of the earth. Corol.6. And the distance of the centre of the moon from the centre of the earth will be to the distance of the centre of the moon from the common centre of gravity of the earth and the moon, as 40,788 to 39,788.



Corol. 7. And the mean distance of the centre of the moon from the centre of the earth in the lunar octants will be approximately 2

560 of the maximum earth radii. For the maximum radius of the earth became 19658600 Parisian feet, and the mean distance of the centres of the earth and the moon from 2

560 constant radii [these are called diameters in the original] of this kind, is equal to 1187379440 feet. And this distance (by the above corollary) is to the distance of the centre of the moon from the common centre of gravity of the earth and the moon, as 40,788 to 39,788: and thus the latter distance is 158268534 feet. And since the moon is revolving with respect to the fixed stars, in 27 days, 7 hours, an 4

943 minutes; the versed sine of the angle, that the moon will describe in a time of one minute, is 12752341 parts, for a radius of 1000,000000,000000 parts. And as the radius is to this versed sine, thus as 1158268534 feet is to 14,7706353 feet. Therefore the moon, by that force by which it is held in orbit, by falling towards the earth, describes a distance of 14,7706353 feet in a time of one minute. And by increasing this force in the ratio

29 2940 40178 to 177 , the total force of gravity of the moon in orbit will be had, by the corol.

of Prop. III. And by this force the moon by falling for a time of one minute describes a distance of 14,8538067 feet. And to the sixtieth part of the distance of the moon from the centre of the earth, that is to the distance of 197896573 feet from the centre of the earth, a heavy body in a time of one minute will also describe a distance of 14,853067. And thus at the distance of 19615800 feet, which is the mean radius of the earth, a weight by falling describes 15,11175 feet, or 15 feet, 1 inch, and lines 1

114 [a line was the twelfth part of an inch]. This will be the descent of bodies at 450 latitude. And by the table described in the preceding Prop. XX, the descent will be a little greater arising at the latitude of Paris by an excess of as much as the 2

3 parts of a line. Therefore a weight through this computation by falling in a vacuum at the latitude of Paris, describes around 15ft,, 1inch, and 25

334 lines in a time of one second. And if the weight may be diminished by taking away the centrifugal force, which arises from the diurnal motion of the earth at that latitude, then weights by falling there describe15 feet,1 inch, and 1

21 lines in a time of one second. And a weight had been shown above in Prop. IV and XIX to be falling with this speed at the latitude of Paris. Corol.8. The mean distance between the centres of the earth and of the moon at syzygies is 60 of the earth's maximum radius, with around a 30th part of the radius taken away. And in quadrature the mean distance of the moon from the same centres is 5

660 earth radii. For these two distances are to the mean distance of the moon in the octants as 69 and 70 to 1

269 by Prop. XXVIII. Corol.9. The mean distance between the centres of the earth and the moon in the lunar syzygies is 1

1060 radii of the mean earth radius. And in the lunar quadratures the mean distance of the same centres is 61 mean radii of the earth, with the 30th part of a radius taken away.



Corol.10. In the lunar syzygies, its mean horizontal parallax at the latitudes of 0 0 0 0 0 0 00 30 38 45 52 60 90, , , , , , , is

57 20 57 16", 57 14 57 12 57 10 57 8" 57 4' . ", ' . ' . ", ' . ,", ' . ", ' . , ' . " respective. [This quantity of almost 10 can be considered as the difference in the angle the moon subtends against the fixed stars, by viewing the moon vertically at one place on the earth, and horizontally at another.] In these computations I have not considered the magnetic attraction of the earth, as its quantity is very little and it has been ignored. Indeed if whenever this attraction will be able to be investigated, then all the measurements of degrees at the meridian, the lengths of isochronous pendulums at different places, the laws of motion of the seas, the parallax of the moon, the different apparent diameters of the sun and moon, will have to be determined more carefully from the phenomena: then one would be permitted to repeat all these calculations more accurately.

PROPOSITION XXXVIII. PROBLEM XIX. To find the figure of the body of the moon.

If the lunar body were fluid just like our seas, the force of the earth required to raise that fluid into parts both nearest and furthest from the earth, would be to the force by which our sea is raised in parts both under the moon and opposite the moon, as the acceleration of gravity of the moon on the earth is to the gravitational acceleration of the earth on the moon, and the diameter of the moon to the diameter of the earth conjointly : that is, as 39,788 to 1 and 100 to 365, or as 1081 to 100. From which since our sea is raised to 3

58 feet by the force of the moon, the lunar fluid by the force of the earth must be raised to 93 feet. And for that reason the figure of the moon must be a spheroid, of which the maximum diameter produced passes through the centre of the earth, and exceeds the perpendiculars by an excess of 186 feet. Therefore the moon affects such a figure, and the same must endure from the beginning. Q.E.I. Corol. Hence indeed it shall be for this reason that the same face of the moon is always turned towards the earth. For the moon cannot be at rest in any other situation, but by oscillating always returns to that situation. Yet the oscillations, on account of the smallness of the forces of agitation, have to be the slowest, acting over a long period of time : and thus so that that face, which must always look towards the earth, can look towards the other focus of the moon's orbit (on account of the reason referred to in Prop. XVII), and cannot at once be turned to look away from the earth.



LEMMA I.

If APEp may designate an earth of uniform density, with centre C and with the poles P, p and with the equator delineated by the line AE; and furthermore from the centre C and with the [smallest] radius CP, the sphere Pape is understood to be described ; moreover QR shall be a plane, to which the right line drawn from the centre of the sun to the centre of the earth stands normally; and the individual particles of all the exterior part of the earth PapAPepE, only those particles which have been described beyond the sphere will try to recede thence from the plane QR, and the attempt of each particle to recede shall be as its distance from the plane. [This Lemma starts the calculation of the torque exerted on the earth by the sun due to the equatorial bulge of the earth, as the earth rotates about an axis at an angle of 23.50 to the normal to the plane of the ecliptic. The forces acting away from the plane QR towards the sun arise from the inverse square law applied at the slightly shorter distance than average at the centre of the earth, while those in the opposite direction away from the sun arise from the slightly smaller than average inverse square forces. Such forces are proportional to their distance from the plane QR, and exert a torque along an axis formed by the intersection of the plane of the equator with the plane QR. There is an uncertainty about this proposition, as the particles within the inner sphere are not supposed to contribute to the bulge by having an unbalanced force acting on them] I say in the first place, that the total force and effectiveness [i.e. the torque] in making the earth rotate around its centre [i.e. precess], of all the particles that have been placed in the plane of the equator AE, and which are arranged regularly around the globe in the form of a ring, will be to the whole force exerted by an equivalent number of particles placed at the point A of the equator, which shall be maximally distant from the plane QR, and which constitute a similar force and effectiveness for the circular motion [i.e. precession] of the earth moving around its centre, in the ratio one to two. And this circular motion will be carried out around the axis lying in the common section of the equator and the plane QR. [The diagram below is actually in three dimensions : and represents the earth with its axis Pp tilted to the plane of the ecliptic, which is itself normal to the plane QR, and both planes can be imagined to extend out of the plane of the diagram (note that Newton calls QR a plane and not a line); the line AE is then the outermost part of the permanent semicircular equatorial bulge as viewed normally, and the lines AH, FG, etc. are the perpendicular distances of points on this bulge to the plane QR of the average distance to the sun, passing through the centre of the earth C. Thus, the axis normal to the plane of the diagram passing through C is the one about which the torques act (according to the right-hand thumb rule, with the thumb pointing out of the plane of the page), due to the equal and opposite attractions of the sun on the equatorial bulges in front of and behind the earth w.r.t. the sun, leading to an unbalanced torque acting on the earth, which thus precesses slowly about its angular momentum vector direction Pp. The component of the torque described normal to Pp, performs this task ; and which is proportional to the whole



torque acting, because of the constant angle. The first Lemma shows that the torque acting on all the points on the bulge at the equator added together is half the torque required to produce the same motion with all the equatorial mass of the bulge placed at A, directly under the sun, at the maximum separation from the plane QR ; note that the mass of a line or curve can be taken as proportional to its length, as detailed in Lemma III below. We may visualise the arch starting at C out of the plane, passing through A and going behind the plane of the diagram. Note that we should use phrases such as moment of inertia and angular momentum with due caution, as these notions came later from Euler's work on mechanics. This then is Newton's excursion into the realm of extended bodies acted on by torques, rather than particles being acted on by forces. It is then perhaps inappropriate to consider these lemmas from the point of view of vector calculus, as Chandrasekhar does. The approach adopted by the old faithful Le Seur and Jacquier is probably more useful.]

For with centre K and diameter IL the semicircle INLK is described. It may be understood that the semi circumference INL has been divided into innumerable equal parts, and from each of the individual parts N the sine NM may be sent to the diameter IL. And the sum of the squares from all the sines NM will be equal to the sum of the squares from all the [co]sines KM, and both sums will be equal to the sum of the squares from the whole radius KN; and thus the sum of the squares from all NM is half as large as the sum of the squares from the whole radius KN. [Thus, from a more modern viewpoint, the moment of inertia of the semicircular arch through K normal to the diagram is twice as great as the moment of inertia about either diameter IL or the vertical one through K in the plane of the diagram. Physically, the torque required to generate the same angular speed is twice as great in the first case than in the following two equal cases.] Now the perimeter of the circle AE may be divided into just as many equal parts, and from any of these F a perpendicular FG is sent to the plane QR, just as the perpendicular AH is sent from the point A. And the force, by which a small particle F tries to recede from the plane QR, will be by hypothesis as that perpendicular FG, and this force multiplied by the distance CG will be the effectiveness [i.e. the word Newton uses here for torque] of the small particle F in turning the earth about its centre [i.e. about an axis normal to the plane of the diagram, passing through C]. And thus the effectiveness of the particle at the place F, will be to the effectiveness of the particle at the place A, as

to FG GC AH HC× × , that is, as 2 2 to FC AC ;



[For the ratio to FG GC AH HC× × or FG GC FC FC

AH HC AC AC×× = × by the similar triangles AHC

and FGC. Thus the torques are as the equivalent moments of inertia. For, the sum of all the terms 2 2 2 2FC .dm KM .dm AC d cosσ θ θ= = , and the summations give

( ) 12

2 2 2 22

M M.AC ACi

iFC . M AC d cos ACπΔ θ θ= = ×∑ ∫ , where AC KN= , and where

AC. M MΔ = ; while ( )2 2MAC

iAC . M ACΔ = ×∑ , thus giving the required ratio. Thus, the

torque produced by the annulus is half the torque exerted by the whole mass placed at A required to produce the same motion, where both the force and distance are a maximum. ] and therefore the total effectiveness of all the particles in the positions of F will be to the effectiveness of an equivalent number of particles at the place A, as the sum of all FC2 to the sum of just as many AC2, that is (as now shown) as one to two. Q. E. D. And because the particles exert forces by receding perpendicularly from the plane QR, and that equally from each side of the plane: the same will cause the circumference of the equatorial circle to turn, and with that holding fast to the earth, around both the axis lying in that plane QR as well as that in the plane of the equator.

LEMMA II. With these in place: in the second place I say that the force and the effectiveness of all the individual particles situated on both sides outside the globe, for making the earth rotate [i.e. precess] around the same axis, shall be as the total force of just as many particles set out uniformly on the equator of the circle AE in the manner of a ring, to set the earth moving in a similar circular motion, as two to five. [Thus, if we sum over the whole bulge, it will exert a torque equal to 2

5th of the torque

provided by a uniform ring with the same number of particles in the bulge over the equator at the maximum distance.]



Let IK be some smaller circle parallel to the equator AE, and L, l shall be any two equal particles placed on this circle beyond the globe Pape. And if perpendiculars LM, lm are sent to the plane QR, that has been drawn perpendicularly to a ray from the sun: the total forces, by which these particles flee from the plane QR, will be proportional to these perpendiculars LM and lm. But the right line Ll [lying under the arc Ll] shall be parallel to the plane Pape, and is bisected by the same at X and through the point X there is drawn Nn, which shall be parallel to the plane QR and crosses the perpendiculars LM, lm at N and n, and the perpendicular XY may be sent to the plane QR. And the opposite forces of the particles L and l, rotating in opposite directions to the earth, are as the forces [here and subsequently, Newton uses the word force, but clearly as the quantities are forces times perpendicular distances from the vertical axis through C, these are moments or torques]

and LM MC lm mC× × , that is, as LN MC NM MC× + × and ln mC nm mC× − × , or and LN MC NM MC LN mC NM mC× + × × − × : and the difference of these

LN Mm NM MC mC× − × + , is the force [i.e. resultant torque] of both the particles taken together required to rotate the earth [i.e. precess]. The positive part of this difference : or 2LN Mm LN NX× × is to force of two particles of the same magnitude put in place at A , 2 22 as to AH HC, LX AC× . And the negative part : NM MC mC× + or 2XY CY× is to the force of the same two particles put in place at A, 2 22 as to AH HC, CX AC× . And hence the difference of the parts, that is, the force of the two particles L & l taken together at the place A required to turn the earth is to the force of the same two equal particles situated in the same place turning the earth in the same manner, as

2 2 2 to LX CX AC− . [By part of Lemma I.] But if the circumference IK of the circle may be divided into innumerable equal parts L, all the terms LX2 will be to a comparable number of terms IX2 as 1 to 2 (by applying the same argument as in Lemma I.) and as a consequence to a like number particles in AC2, as IX2 to 2AC2; and the equivalent number of particles in CX2 to just as many in AC2 is as 2CX2 to 2AC2 [i.e. this ratio is maintained, and written as shown for later convenience in subtracting.].



[Thus, ( ) ( )2 2

2 212= =

i ii ii i

i

m LX m LX

IX m M IX

∑ ∑

∑ ×, and

( ) 22

2 22=

i ii

i

m LXIX

AC m AC

∑

∑; while

2 2

2 222

CX CXAC AC

= ; hence the total

torques acting at A are as 2 2 22 to 2IX CX AC− .] Whereby the forces of all the particles taken together in the course of the circle IK are to the forces taken together of just as many particles at the place A, as

2 2 22 to 2IX CX AC− : and therefore (by Lem. 1.) to the forces taken together of an equivalent number of the particles on the circumference of the circle AE, as

2 2 22 to IX CX AC− . [Thus, a circuit of the bulge has been replaced by a mass at A, and then this mass is distributed about the equatorial ring AE, and as the distance of the masses is now variable to the plane QR, the torque is decreased to half its maximum value, according to Lemma I.] Now truly if the diameter of the sphere Pp may be divided into innumerable equal parts, in which there are present just as many circles IK; the matter in the perimeter of each circle IK will be as IX2 : [Recall that all the matter in the ring IK was put at this position above, where it exerted the maximum torque for that ring; and which matter is proportional to the length of the IX, so that the quantity of motion is taken as IX 2 .], and thus the force of that matter requiring to rotate the earth will be as IX 2 into 2 22IX CX− . But the force of the same matter, if it shall be present in the circumference of the circle AE, shall be as IX 2 into AC2. And therefore the force [torque] of all the particles of all the matter, outside the globe present in the perimeters of all the circles [constituting the bulge], is to the force of just as many particles present in the perimeter of the great circle AE, as all IX 2 by 2 22IX CX− to just as many IX2 by AC2, that is, as all

2 2 2 2 by 3AC CX AC CX− − to just as many 2 2 2 into AC CX AC− , [For from the centre C, to the point I, the right line CI is supposed to be drawn, and there will be 2 2 2IX CI CX= − : but CI AC= , whereby 2 2 2IX AC CX= − , and hence the total number of particles by their appropriate torques,

( ) ( ) ( )2 2 2 2 2 2 22 3IX IX CX AC CX AC CX× − = − × − .]

that is, as the whole of 4 2 2 44 3AC AC CX CX− × + to just as many 4 2 2AC AC CX− × , that is, as the total fluent quantity, the fluxion of which is 4 2 2 4 3AC AC CX CX− × + , to the total fluent quantity , the fluxion of which is 4 2 2AC AC CX− × ; and hence by the method of fluxions, as 34

3 54 2 3 5AC CX AC CX CX× − × + to

13

4 2 3AC CX AC CX× − × , that is, if the total Cp or AC may be written for CX, as 4 2

15 35 5 to AC AC , that is, as two is to five. Q.E.D.



[The quantities 4 2 2 44 3AC AC CX CX− × + and 4 2 2AC AC CX− × , taken multiplied together by the fluxion of the line CX, and with the fluents taken, will be the fluent of the first quantity 34

3 54 2 3 5AC CX AC CX CX× − × + ; but the fluent of the second quantity

becomes 13

4 2 3AC CX AC CX× − × , and from that the total effectiveness may be obtained, for CX there is written Cp or AC, the first fluent to the second fluent will be as

4 215 3

5 5 to AC AC , giving the ratio quoted. (A note from L. & S.). Basically, the torque required to rotate the extended shell around earth is 2

5th of the torque required to rotate a

uniform ring of the same mass about the earth with the same angular acceleration.] [Chandrasekhar establishes a result similar to this by modern methods, but he appears to have solved a different problem, involving the moment of inertia of the whole earth and solid discs. The results as stated here are also shown by Cohen.]

LEMMA III.

With the same in place: I say in the third place that the motion of the whole earth about the axis already described, composed from the sum of all the motions of the particles, will be to the motion of the aforesaid ring around the same axis in a ratio, which is composed from the ratio of the matter in the earth to the matter in the ring, and in the ratio of three times the square of the arc of the quadrant of some circle to twice the square of the diameter ; that is, in the ratio of the matter in the ring to the matter in the earth, and of number 925275 to the number 1000000. [This ratio amounts to

2332π ]

For the motion of a cylinder revolving about its fixed axis is to the motion of the inscribed sphere and likewise rotating, as any four equal squares are to three circles inscribed in these squares ; and the motion of the cylinder to the motion of a very thin ring, going around with the sphere and the cylinder at their common point of contact, as twice the matter in the cylinder to three times the matter in the ring ; and the continued uniform motion of this ring around the axis of the cylinder is to its uniform motion around its own, made in the same periodic time, as the circumference of the circle to twice the diameter. [L. & J. Note 126: Lemma III is demonstrated. By rotating the semicircle AFB, and the circumscribed rectangle AEDB of the same, a sphere and a circumscribed cylinder may be described. Let the radius 1CB = , the periphery of the circle described in this ratio

( )or 2n, ,π= the abscissa CP x= , the ordinate PM y= , some part of this



PR v= , rR dv= ; the periphery of the circle described with the radius PR is 2π v; the circular ring from the revolution of the line increment 2rR vdvπ= ; the velocity of the point R v= ; (taking the angular velocity as 1 for convenience and setting the density as 1, so that the mass and volume are given by the same variable ; thus the angular momentum considered – here called the motion, for a given increment is the volume times the radius) the motion of the aforementioned ring 22 v dvπ= , the motion of the whole circle described (i.e. disc with unit thickness) with the radius PR 2

33vπ= ; the motion of the whole disc with

unit thickness described with the radius PM 23

3yπ= ; the motion of the unit disc described with the radius PN 2

3π= , and the motion of the whole cylinder 4

3π= .

Let Pp dx= ; the motion of the solid ring described by the revolution of the figure

PMmp is equal to ( )322 2 2

3 3 33 2 31y dx dx x sin d sinπ π π θ θ θ= × − = ×− × , on setting x cosθ= .

From which the motion of the solid figure of revolution of figure described CFMB is equal to 22

3 84sin d .π πθ θ =∫ Therefore the motion of the half cylinder to the motion of

the hemisphere is as 2

23 8 to or 16 to 3π π π ; that is, as some four equal squares 24 2 16× =

to three circles with 21π × ; and so for the whole cylinder and sphere. The most tenuous matter of the ring going around touching the sphere and the cylinder adjacent to the common point F shall be at the height m, and the velocity shall be as CF, or as 1; and thus the motion m= , and thus the motion (or angular momentum) of the cylinder to the motion of this ring is 4

3 to mπ , or as 4π to 3m, that is, as twice the matter in the cylinder (taken as its volume) to three times the matter in the ring; for the base of the cylinder is the circle 21π × and the diameter height 2AF = , and thus the cylinder

2π= . The matter of the aforesaid ring shall be 2 2a . π (where a2 is the line density), and thus the motion around the axis of the cylinder itself 2 2a . π= . Now likewise the ring may be revolving around closer to the axis that the diameter AB may be showing, and the particles of the matter of the arc corresponding to the infinitesimal Mm, will be 2a Mm× and the motion of this 2 2a y Mm a dx× = , on account of the proportion

( )( )( )

1CM i.e.arc MmmH or dx PM i.e. y= . Whereby the motion of the part FM, of the ring is 2a x , and by

making 1x = , the motion of the quadrant of the ring 2a= ; and the motion of the whole ring nearer the axis is 24a= . Therefore the motion of the ring about the axis of the cylinder to its motion about the nearer axis is as 2 22 to 4a . aπ , or as 2π to 4 ; that is, as the circumference of the circle 2π to twice the diameter 4. On account of which we have the ratios : the motion of the cylinder to the motion of the sphere is as 16 to 3 π ; the motion of the ring around the axis of the cylinder is to the motion of the cylinder as 4

3 to m π ;



and the motion of the ring around the nearer axis is to its motion around the axis of the cylinder as 4 to 2 .π Whereby, by the composition of the ratios and from the equation, the motion of the sphere

around the closer axis is to the motion of the ring as ( )32 to 64 .mπ But24 123

3 16864 8=m m

π ππ ×× , and

43π is the quantity of matter in the earth of radius 1; m, the quantity of matter in the

ring;212

16π is the sum of three squares from the arc of the circle AFB, and 8 is the sum of

two squares from the diameter AB. Whereby the motion of the whole earth around the axis now described, composed from the motions of all the particles, will be to the motion of the aforesaid ring around the same axis, in the ratio that is composed from the ratio of the matter of the earth to the matter in the ring, and from the ratio of three squares from the arc of the quadrant of some circle to two squares from the diameter, that is, in the ratio of the matter to the matter and the number 925275 to the number 1000000 or

2332π , with

the ratio of the diameter to the periphery taken approximately as 1 to 3.141 approximately. Q.e.d.]

HYPOTHESIS II. If the aforesaid ring, with all of the remaining earth removed, alone may be carried in orbit around the sun in its annual motion, and meanwhile revolving in a diurnal motion around its axis inclined to the plane of the ecliptic at an angle of 1

223 degrees : likewise the motion of the equinoctial points shall be the same, whether the annulus shall be fluid or constructed from rigid and firm matter.

PROPOSITION XXXIX. PROBLEM XX. To find the precession of the equinoxes.

The mean hourly motion of the lunar nodes in a circular orbit, when the nodes are in quadrature, was iv v16 35 16 36" . "' . . , and the mean hourly motion of the nodes in such an orbit is half of this iv v8 17 38 18" . "' . . (on account of the reasons explained above); and in a sidereal year in total it shall be gr20 11 46. ' . " . Therefore because in the preceding the lunar nodes in such an orbit put such an annual amount in place gr20 11 46. ' . " ; and if there were several moons, the motion of the nodes of each (by Corol. 16. Prop. LXVI. Book I.) would become as the periodic times ; if a moon were revolving next to the surface of the earth in the space of a sidereal day, the annual motion of the nodes would be to gr20 11 46. ' . " as the sidereal day of 23. 56' hours to the periodic time of the moon of 27 days, 7 hours, 43minutes ; that is, as 1436 to 39343. And the ratio of the nodes of a ring of moons around the earth is the same ; whether these moons do not affect each other, or they merge together and form a continuous ring, or in fact that ring may be rendered rigid and inflexible. Therefore we may consider that this ring as a quantity of matter that shall be equal to all the earth PapAPepE which is above the globe Pape; (See the figure for Lemma II ) and



because such a globe is to the earth above as aC2 to 2 2AC aC− , that is (since the smaller radius of the earth PC or aC shall be to the greater radius AC as 229 to 230) as 52441 to 459; if this ring should surround the earth around the equator and each were revolved around the diameter of the ring, the motion of the ring would be to the motion of the interior globe (by Lem. III of this part) as 459 to 51441 and 1000000 to 925175 conjointly, that is, as 4590 to 485223 ; and thus the motion of the ring would be to the sum of the motions of the ring and the globe, as 4590 to 489813. From which if the ring were attached to the globe, and its motion, by which nodes of this or the equinoctial points were regressing, since it may share with the globe: the motion which will remain in the ring will be to the first motion, as 4590 to 489813; and therefore the motion of all the equinoctial points will be diminished in the same ratio. Therefore the annual equinoctial motion of all the points of the body composed from the ring and the globe will be to the motion gr20 11 46. ' . " , as 1436 to 39343 and 4590 to 489813 conjointly, that is, as 100 to 292369. But the forces by which the lunar nodes (as I have set out above) and thus by which the equinoctial points of the ring are regressing (that is the forces 3IT in the figure to Prop. 30) are by the individual particles as the distances of the particles from the plane QR, and by these forces these particles flee from that plane; and therefore (by Lem. II.) if the matter of the ring may be spread out over the whole globe in the manner of the figure PapAPepE above that part of the earth put in place, the force and the total turning effect of all the particles rotating around the equator of the earth in some manner, and thus to the movement of the equinoctial points, will emerge smaller than in that first ratio of 2 to 5. And thus the annual regression of the equinoxes now will be to gr20 11 46. ' . " as 10 to 73092 : and hence it becomes iv9 56 50" . ''' . . Further this motion on account of the inclination of the plane of the equator to the plane of the ecliptic is to be diminished, and that in the ratio of the sine 91706 (which is the complement of the sine of 1

223 degrees) to the radius 100000. On which account this

motion now becomes iv9 7 20" . ''' . . This is the annual precession of the equinoxes arising from the force of the sun. Moreover the force of the moon requiring to move the sea was to the force of the sun as 4,4815 to 1 approximately. And the force of the moon required to move the equinoxes is in the same proportion to the force of the sun. And thence the annual precession of the equinoxes by the force of the moon arises : iv40 52 52" . ''' . , and thus the total annual precession arising from both forces will be iv50 00 12" . ''' . . And this motion agrees with the phenomena. For the precession of the equinoxes from astronomical observations is annually a little more or less than fifty minutes. If the height of the earth at the equator should exceed that at the poles, by more than

1617 miles, its matter will be rarer at the circumference than at the centre : and the

precession of the equinoxes on account of that height will be increased, on account of the rareness it must be diminished. Now we have described the system of the sun, of the earth, moon and planets; it remains that something may be added concerning comets.


Book III Section IV. Translated & Annotated by Ian Bruce. Page 869

LEMMA IV. Comets occupy places beyond the moon, moving in the region of the planets.

Just as the lack of diurnal parallax raises comets beyond sub lunar regions, so one is convinced from the annual parallax that they fall in the regions of the planets. For comets, which are progressing along the order of the signs [of the Zodiac], are all at the end of their appearance, either going more & more slowly or backwards, if the earth is between them & the sun ; but just as much faster if the earth turns in opposition. And on the contrary, those which go against the order of the signs are equally faster at the end of their appearance, if the earth is present between them & the sun ; & equally they go slower or backwards, if the earth is situated on the other side [of its orbit] . This is in maximum agreement with the motion of the earth : in the variation of its position, in the same way as it shall be with the planets, which for the motion of the earth are either in agreement, or in opposition, in the one case seen to be progressing slower, in the other quicker. If the earth travels in the same direction as the comet, & in the angular motion about the sun it goes a little faster, so that a right line drawn through the earth & the comet always meet at places beyond the comet, the comet seen from the earth on account of its slower motion appears to be moving backwards ; but if the earth were carried forwards slower [than the comet], the motion of the comet (with the motion of the earth subtracted) shall be at least slower. But if the earth proceeds in the opposite direction, the comet will appear a great deal faster. Moreover either from the acceleration or retardation or from the backwards motion, the distance of the comet can be deduced in this manner. Let QA, QB, QC be three observations of the longitudes of the comet from the initial motion, & let QF be the final longitude observed, when the comet ceases to be

seen. The right line ABC is drawn, the parts of which AB, BC lying between QA & QB, QB & QC , shall be in turn as the times between the first three observations. AC is produced to G so that AG shall be to AB as the time between the first observation & the final to the time between the first observation & the second, & QG may be joined. And if the comet may be moving uniformly in a straight line, & the earth is either at rest or also may be progressing along a line with a uniform motion; the angle QG will be the final longitude of the comet observed in the final time of observation. Therefore the angle FQG, which is the difference of the longitudes, arises from the inequality of the motion of



the comet & the earth. But this angle, if the earth & the comet are moving in opposite directions, is added to the angle QG & thus the apparent motion of the comet is returned faster : but if the comet goes in the same direction as the earth, the same is

subtracted, & the motion of the comet is returned either slower, or perhaps backwards ; as I have just explained. Therefore here the angle arises particularly from the motion of the earth, & therefore consequently can be taken for the parallax of the comet, clearly with some increment or decrement of this ignored – because an inequality in the motion of the comet in its own orbit may arise. Truly the distance of the comet thus can be deduced from this parallax. Let S designate the sun, acT the great orbit, a the position of the earth at the first observation, c the place of the earth at the third observation, T the place of the earth at the final observation, & T a right line drawn towards the first star in Aries. The angle

TV is taken equal to the angle QF, that is, equal to the longitude of the comet when the earth is present at T. Join ac, & that is produced to g, so that ag shall be to ac as AG to AC, & g will be the position that the earth will reach in the time of the final observation, by continuing uniformly in a straight line. And thus if gr is drawn parallel to Tr itself, & the angle gV is taken equal to the QG, this will be the angle gV equal to the longitude of the comet seen from the place g ; & the angle TVg will be the parallax, which arises from the translation of the earth from the place g to the place T: & hence V will be the position of the comet in the plane of the ecliptic. Moreover this place V is accustomed to be within the orbit of Jupiter. Likewise the paths of comets is deduced from the curvature. These bodies go almost in large circles while they travel with their greatest speed ; but at the end of their course, when that part of the apparent motion which arises from parallax, has a greater proportion to the whole apparent motion, they are accustomed to be deflected from these circles, & just as often as the earth is moving in one direction, to go off in the opposite direction. This deflection arises mainly from the parallax, because that corresponds to the motion of the earth ; & the amount of this is signified, by my calculation, to have deduced the disappearance of comets well enough far beyond Jupiter. From which it follows that in the perigees & perihelions, when they are present closer, often fall within the orbit of Mars & of the inner planets.



Also the nearness of comets is confirmed by the light of the heads. For the light of a heavenly body illuminated by the sun, & going off into distant regions, is diminished in splendour in the square of the distance : clearly in the square ratio on account of the increase of distance from the sun, & in another square ratio on account of the diminution of the apparent diameter. From which if the apparent diameter & the quantity of light of the comet is given, the distance of the comet will be given, by saying that the distance will be to the distance of a planet, in the ratio of the diameter [of the comet] to the diameter of the planet directly & inversely in the inverse square root ratio of the light [of the comet] to the light of the planet. Thus if the smallest diameter of the hairs of the comet of the year 1682, observed by Flamsted with an optical tube 16 feet long & measured by a micrometer, was equal to 2 0' . " ; but the nucleus or the star in the middle of the head was occupying scarcely the tenth part of this width, & thus the width was only 11" or 12". Truly with the light & clarity of the head it exceeded the head of the comet of the year 1680, & was emulating a star of the first or second order magnitude. We may consider Saturn & its ring to be as if four times brighter : & because the light of the ring will be in the same manner equal to the light of the globe in the centre, & the apparent diameter of the globe shall be as if 21", & thus the light of the globe & of the ring jointly will be equal to the light of a globe, the diameter of which shall be 30": the distance of the comet to the distance of Saturn will be as 1 to 4 inversely, & as 12" to 30" directly, that is, as 24 to 30 or 4 to 5. Again the comet of the year 1665 in the month of April, as Helvelius is the authority, by its clarity surpassed almost all the fixed stars, even Saturn, on account of the vividness of its colour for a long while. Certainly this comet was brighter than that other one, which had appeared at the end of the preceding year, & might be compared with the stars of the first magnitude. The width of the hairs was around 6', but the nucleus compared with the planets with the aid of a telescope clearly was less than Jupiter, & now smaller than the intermediate body of Saturn, & at other times judged to be equal. Again since the diameter of the hairs of the comet rarely exceeded 8' or 12', truly the diameter of the nucleus, or of the central star shall be around a tenth or perhaps a fifteenth part of the diameter of the hairs, it is apparent that these stars & most of the same kind are of the same apparent magnitudes as the planets. From which since the light of these can often be compared with the light of Saturn, & which may exceed that a little ; it is evident, that all comets taken together at the perihelion are either below Saturn, or not much more. Therefore, those who relegate comets to the region of the fixed stars err completely: by which account certainly comets would be illuminated no more by our sun, than the planets, which are here, can be illuminated by the fixed stars. We have discussed these things without considering the obscurity of comets through all that most dense & abundant smoke, by which the head is surrounded, as if it were always shining through a dense cloud. For as much as the body is rendered obscure by this smoke, so it is necessary that it must approach much closer to the sun, so that the abundance of light reflected from them emulates the planets. Thence it appears to be the case that comets fall far below Saturn's globe, just as we have shown from parallax. The same indeed as may be confirmed fully from the tails; these arise either from reflection from the smoke scattered through the ether, or from the light of the head. In the first case it is to be lessened with the distance of the comets, lest the smoke from the head always arises & by expanding may be propagated with an incredible speed through an



exceedingly great distance, in the latter case all the light for both the tails as well as the hairs are to be attributed to the nucleus of the head. [It is of course the solar wind that is responsible for the tail.] Therefore if we may consider all this light to be gathered together & to be encircled by the disc of the nucleus, certainly that nucleus now, whenever it sends off a great & most brilliant tail, will exceed the splendour of Jupiter itself many times. Therefore with an apparent smaller diameter emitting more light, the comet must be much more illuminated by the sun, & thus much closer to the sun. Furthermore, when their head is hidden under the sun, & with great smoking tails sometimes emitted in the image of tree trunks on fire, by the same argument such comets must be deduced to be within the orbit of Venus. For all that light if it be supposed to be gathered together in a star, I may say that it may surpass Venus itself & lest whenever several Venuses are in conjunction. Finally the same may be deduced from the increasing light of the heads of comets receding from the earth towards the sun, & in the decrease in their recession from the sun. For since the comet at the end of the year 1665 (from the observations of Hevelius) from when it began to be seen, always maintained its apparent motion, & thus went beyond perigee ; truly the splendour of the head likewise was increasing from day to day, until it ceased to be evident on account of being hidden by the rays of the sun. The comet of the year 1683 (from the observations of the same Hevelius) at the end of the month of July, when first it was evident, was moving most slowly, completing around 40 or 45 minutes of arc in its orbit. From that time its diurnal motion always increased as far as Sept. 4. when it became around five degrees. Therefore in this whole time the comet was approaching the earth. Which was also deduced from the diameter of the head measured with a micrometer : evidently as Hevelius found on Aug. 6. to be as much as 6'. 5" with the hair included, but on Sept. 2 to be 9'. 7''. Therefore the head appeared less long than at the end of the motion, but yet initially in the vicinity of the sun it was more visible in length than around the end, as the same Hevelius reports. Hence in this whole time, the light decreased on account of its receding from the sun, not withstanding its approach to the earth. The comet of the year 1618 about the middle of the month of December, & that one of the year 1680 around the end of the same month, were moving with great speed, & thus soon they were in perigees. The true maximum splendour of the heads was seen almost two weeks before, but only when they emerged from the rays of the sun, & the maximum splendour of the tails a little before, in the near vicinity of the sun. The head of the first comet, beside the observations Cysat on December 1, was seen to be greater than stars of the first order magnitude, & on December 16. (now present in perigees) with a small magnitude, with the splendour or clarity of the light a great deal reduced. On Jan. 7, with the head uncertain, Kepler finished his observations. On the 12th day of December Flamsteed observed & noted the head of the latter comet at a separation of 9 degrees from the sun ; it was conceded to be a star of scarcely the third order magnitude. On December the 15th & 17th is appeared the same as a star of the third order magnitude, certainly diminished compared to the splendour of the clouds of the setting sun. December 26th , the day of the greatest motion, with that present near the perigee, it dropped to the mouth of Pegasis, a star of the third magnitude. Jan. 3rd it appeared as a star of the fourth magnitude, Jan. 9th as a star of the fifth, Jan.13th it disappeared on account of the brilliance of the rising moon. Jan. 25. scarcely being equal to a star of the seventh magnitude.



Thence if the times are taken equally from the perigee, then the heads put in more distant regions at these times [from the earth], on account of the equal distances from the earth, ought to shine equally with maximum brilliance; but it is of maximum brightness in the direction of the sun, & nearly disappearing from the other side of the perigee. Therefore from the great difference of the light in each situation, it must be concluded that the maximum light comes from the comet & the sun nearby in the first situation. For the light of comets is accustomed to be regulated, & the maxima to appear when the heads are moving the fastest, & thus they are at the perigees; unless perhaps that is greater in the vicinity of the sun. Corol. 1. Therefore comets shine by the light of the sun reflected from them. Corol. 2. Also from what has been said it is understood why comets frequent the region of the sun so much. If they may be discerned in regions far beyond Saturn, often they must appear on opposite sides of the sun. Indeed they must be closer to the earth, which is present in these parts; & the sun interposed obscures the rest. Truly by running through the history of comets, I have found that four or five times more have been detected in the hemisphere towards the sun, than in the opposite hemisphere, besides, without doubt a few others, that the light of the sun has concealed. Without doubt in the descent to our regions neither do they emit tails, nor thus are they illuminated by the sun, as required so that they may be found in the first place by the naked eye, as they shall be closer to Jupiter itself. But of the space described in the small interval around the sun, the greater length is situated on the side of the earth, which looks at the sun ; & in that greater part comets are usually illuminated more by the sun, as they are very much closer. Corol. 3. Hence also it is evident, that the heavens are free from resistance. For comets have followed oblique paths sometimes contrary to the course of the planets, are moving most freely in all cases, & their motion, even considered for a long time contrary to the course of the planets. I am mistaken if they shall not be bodies of the same kind as planets, & may be moving in orbit, returning perpetually. For as writers wish to regard comets as some kind of meteors, the argument following from the constantly changing heads, may be seen to be without foundation. The heads of comets are surrounded by huge atmospheres ; & the atmospheres below must be denser. From which it is from these clouds, not from the bodies of comets themselves, that these changes are seen. Thus the earth if it may be viewed from the planets, without doubt brilliant from the light of its clouds, & the firm body beneath the clouds almost hidden. Thus belts have been formed by the clouds of the Jovian planet, in which the positions may change among themselves, & the firm body of Jupiter is discerned with difficulty by these clouds. And the bodies of comets must be buried under both much more deep & thicker atmospheres.



PROPOSITION XL. THEOREM XX.

Comets move in conic sections having foci in the centre of the sun, & with rays drawn to the sun describe areas proportional to the times. This is apparent by Corol. 1. Prop. X11I. Book I, together with Prop. V11I, X11, & X11I. of Book 11I. Corol. 1. Hence if comets return in orbit; the orbits will be ellipses, & the periodic times will be to the periodic times of the planets in the three on two ratio of the principle axes. And thus comets are present for the most part beyond the planets, & by that account describing orbits with greater axes, revolving slower. So that if the axis of the orbit of a comet shall be four times greater than the orbit of Saturn, the time of revolution of the comet shall be to the time of revolution of Saturn, that is, to 30 years, as 4 4 (or 8) to 1, & thus it will be of 240 years. Corol 2. But the orbits will be thus so close to parabolas, that parabolas may be taken in turn without sensible error. Corol. 3. And therefore (by Corol. 7, Prop. XVI, Book 1.) the velocities of all comets, will be to the velocity of any planet revolving in a circle around the sun, in the square root ratio of twice the distance of the planet from the centre of the sun to the distance of the comet from the centre of the sun approximately. We may consider the radius of the great orbit, or the maximum radius of the ellipse in which the earth is revolving to be of 100000000 parts: & the earth from its daily motion describes on average 1720212 parts, & in its hourly motion 1

271675 parts. And thus a comet at the same mean distance from the sun, with that velocity which shall be to the velocity of the earth as 2 to 1, describes in its daily motion 2432747 parts, & in the hourly motion 1

2101364 parts. But in the greater or lesser distances, the daily as well as the hourly motion will be inversely in the square root ratio of the distances, & thus is given. [Chandrasekhar points out the formula on page 480 from which this constant is derived, correct to 6 places, which is now commonly called the Gaussian constant of gravitation.] Corol. 4. From which if the major latus rectum of the parabola shall be four times the radius of the great orbit, & the square of that radius is put to be of 100000000 : the area that will be described by the radius drawn from the comet to the sun in single days will be of 1

21216373 parts, & the area of the individual hours will be of 3450682 parts. But the

major latus rectum may be either less or greater in some ratio, the diurnal area & hourly areas will be greater or less inversely in the square root of that ratio.



LEMMA V.

To find the curved line of the parabola generated, that will pass through any number of given points.

[This is Newton's theory of interpolation. An excellent article on this can be found by Duncan C. Fraser, Newton & Interpolation, in the commemorative volume published in 1927 for the Mathematical Gazette by Bell & Co. This has subsequently been set out by Chandrasekhar in his book, Newton's Principia for the Common Reader, p.481 (Oxford).] Let these be the points A, B, C, D, E, F, etc, and from the same send some number of perpendiculars AH, BI, CK, DL, EM, FN to some right line in the given position HN. Case. 1. If the intervals HI, IK, KL of the points H, I, K, L, M, N shall be equal, gather together the first differences b, 2b, 3b, 4b, 5b, of the perpendiculars AH, BI, CK, and the second differences c, 2c, 3c, 4c ; the third d, 2d, 3d ; that is, so that thus there shall be

2 3 4 5AH BI b, BI CK b,CK DL b, DL EM b, EM FN b− = − = − = + = − + = ; then 2b b c− = , thus it may go on to the last difference, which is f.

Then erect some perpendicular RS, which shall be the applied ordinate to the curve sought : in order that the length of this may be found, put the intervals HI, IK, KL, LM, etc equal to one, & call

1 1 1 12 3 4 5AH a, HS p , p IS q, q SK r, r SL s, s SM t= − = ×− = ×+ = ×+ = ×+ = ;

but it is clear that by proceeding as far as to the second last perpendicular ME, & by putting negative signs in front of the terms HS, IS, which lie on the one side of the point S towards A, & with a positive sign for the terms SK, SL, etc which lie towards the other direction of the point S. And with the signs observed properly, there will be RS a bp cq dr es ft= + + + + + , Case 2. However if the intervals HI, IK, etc of the points H, I, K, L, etc shall be unequal, take the first differences b, 2b, 3b, 4b, 5b etc to be the difference of the perpendiculars AH, BI, CK, etc , divided by the intervals of the perpendiculars ; the



second differences c, 2c, 3c, 4c, etc to be the previous differences divided by the interval between every two intervals; the third d, 2d, 3d, etc those divided by the interval between every three intervals; the fourth e, 2e, the third divided by every four intervals, & thus henceforth; that is, so that there shall be thus : 2 2 3 3 4, 2 , 3 , then 2 3BI CK CK DL b b b b b bAH BI

HI IK KL HK IL KMb b b & c , c , c ,&c.− − − − −−= = = = = =

then again, 2 2 32c c c cHL IMd , d ,− −= =

With the differences found call

AH a, HS p, p IS q, q SK r, r SL s, s SM t= − = ×− = ×+ = ×+ = ×+ = ; clearly by proceeding as far as the penultimum perpendicular ME, & the applied ordinate [i.e. the y coordinate] RS a bp cq dr es ft= + + + + + , Corol. Hence the areas of all curves can be found approximately. For if from some points of any curve are found of which the quadrature is required, & a parabola is understood to be drawn through the same points : the area of this same parabola will be approximately equal to the quadrature sought. Moreover the quadrature of the parabola can always be found geometrically by very well–known methods.

LEMMA VI. By observing some number of places of a comet to find its place at some given intermediate time. In fig. preceding figure, HI, IK, KL, LM may designate the times between observations, HA, IB, KC, LD, ME are five observations of the longitude of the comet, HS the given time between the first observation & the longitude sought. And if the regular curve ABCDE is understood to be drawn through the points A, B, C, D, E ; then by the above lemma, its applied ordinate RS can be found, & RS is the longitude sought. By the same method from five latitudes observed the latitude at a given time can be found. If the differences of observed longitudes shall be small, for example only of 4 or 5 degrees ; three or four observations will suffice to find a new longitude & latitude. But if the differences shall be greater, for example of 10 or 20 degrees, five observations must be used.



LEMMA V11. To draw a right line BC through a given point P, the parts of which PB, PC have a given ratio in turn, with the two given positions of the abscissa AB, AC. From that point P to one of the lines AB some line PD is drawn, & the same is produced towards the other right line AC as far as to E, so that PE shall be to PD in that given ratio; EC shall be drawn parallel to AD itself ; & if CPB, is drawn, PC will be to PB as PE to PD. Q.E.F.

LEMMA V11I. Let ABC be a parabola having the focus S. The chord AC, bisected in I, cuts the segment ABCI, of which the diameter shall be Iμ and the vertex μ . On Iμ produced there is taken Oμ equal to half of Iμ . OS is joined, & that is produced to ξ , so that Sξ shall be equal to 2SO. And if the comet B is moving in the arc CBA, & Bξ is drawn cutting AC in E: I say that the point E will cut a segment AE from the chord AC, approximately proportional to the time. [From the construction, 1 1 1

2 3 3 and O I IO, SO Oμ μ ξ= = = .]

For EO is joined cutting the parabolic arc ABC in Y & Xμ may be drawn, which is a tangent to the same arc at the vertexμ , & crossing EO in X; & therefore the curvilinear area AEX Aμ will be to the curvilinear area AEY Aμ as AE to AC. And therefore since the triangle ASE shall be in the same ratio to the triangle ASC as AE to AC, the total area ASEX Aμ will be to the total area ASCY Aμ as AE to AC. But since, Oξ shall be to SO as 3 to 1, & EO to XO shall be in the same ratio, SX will be parallel to EB : & therefore if BX is joined, the triangle SEB will be equal to the triangle XEB. From which if the triangle EXB is added to the area ASEX Aμ , & from the sum the triangle SEB is taken, there will remain the area ASBX Aμ equal to the area ASEX Aμ , & thus to the area ASCY Aμ as AE



to AC. But the area ASBY Aμ is approximately equal to the area ASBX Aμ , & this area ASBY Aμ is to the area ASCY Aμ , as the time of the arc described AE to the time of the whole arc described AC. And thus AE is to AC in the approximate ratio of the times. Q. E. D. [Chandrasekhar had trouble with this proof on account of the now obscure theorem, (the basis of which L & S attribute to Archimedes : de Parabola , see Heath, The Works of Archimedes, Quadrature of the Parabola, Prop. XXIV & prior to this, available in Dover), & the poor quality of the diagram, which he has taken the trouble to have re–drawn, & he has finally tracked down a more modern proof in Salmon's Treatise on Conic Sections, p.372 in the 6th edition. I have added some colours to the diagram, which is almost completely illegible in the original Principia, & which is hard to see even in Cohen's redrawn version, though of course the idea behind it is readily understood, when everything else is tidied up. Thus, in Newton's day, the works of Archimedes were generally well-known, so he felt disinclined to elaborate on a result that presumably was obvious to him. However, here is the note produced by L. & J., showing that

: :ASBY A ASCY A AE ACμ μ = : Since the chord AC has been bisected in I, the semi-segment A I ICμ μ= . Likewise because Xμ is a tangent to the parabola at μ , Xμ will be parallel to the chord AC (by Lem. IV, de Conics, Apoll., Book I.), & hence the triangle OIE is similar to the triangle O Xμ , & thus, on account of 3IO Oμ= × , 9

89 and trap.IOE OX IOE I XEΔ Δμ Δ μ= × = × . Therefore the triangle 3

2:IAO IAΔ Δ μ = , on omitting lines in the diagram to avoid confusion, since likewise A shall be a vertex of each triangle, & the base OI shall be 32 Iμ× ; truly 3

4 semi-segment A I A IΔ μ μ= × , from Prop. XXIV of Archimedes quoted above. Whereby 9

8 semi-segment AOI A IΔ μ= × , that is, in the ratio composed from 3 32 4× ,

& hence 98: semi-segment : trapezium AOI A I IOE XEIΔ μ Δ μ= = , just as in turn we have

the trapezium : semi-segment :XIE A I IE AIμ μ = , & hence on compounding, since the curvilinear area half-segment trapA XE A I . XEIμ μ μ= + , then the curvilinear area : semi-segment :A XE A I AE AIμ μ = , & hence the curvilinear area : total segment :A XE A C AE AC.μ μ = ] Corol. When the point B falls upon the vertexμ of the parabola, AE is to AC in the accurate ratio of the times.

Scholium. If μξ is joined cutting AC in δ , & on that there is taken nξ , which shall be to Bμ as 27MI to 16Mμ : with Bn drawn cutting the chord AC in the ratio of the times more accurately than before. But the point n may be placed beyond the point ξ , if the point B is more distant from the principle vertex of the parabola than the point μ ; & closer, if the point B is at a lesser distance from the same vertex.



LEMMA IX.

The right lines 4 AI ICSI , M , and the length μμ μ × are equal to each other.

For 4Sμ is the latus rectum of the parabola pertaining to the vertex μ .

LEMMA X. If Sμ is produced to N & P, so that Nμ shall be the third part of Iμ , & SP shall be

to SN as SN to Sμ . A comet, in which time it will describe the arc A Cμ , if it may be progressing in that time with the velocity that it has at the altitude equal to SP, describes a length equal to the chord AC. For if the comet with the velocity that it has at μ , may be progressing in the same time uniformly in a right line, which will be a tangent to the parabola at μ ; the area, that it will describe by a radius drawn to the point S, shall be equal to the area of the parabola ASCμ . & thus the product formed from the length described by the tangent & the length Sμ shall be to the product formed from the lengths AC & SM, as the area ASCμ to the triangle ASC; that is, as SN to SM. Whereby AC is to the length of the tangent described, as Sμ to SN. But since the velocity of the comet at the altitude SP [i.e. distance from the sun at S] shall be (by Corol. 6, Prop. XVI, Book I.) to its velocity at the altitude Sμ inversely in the square root ratio of SP to Sμ , that is, in the ratio Sμ to SN ; the length described with this velocity in the same time will be to the length described in the tangent, as Sμ to SN . Therefore AC & the length described with this new velocity, since they shall be in the same ratio to the length described by the tangent, are equal to each other. Q.E.D. Corol. Therefore a comet with that velocity, that it has at the altitude 2

3S Iμ μ+ , will describe the chord AC in approximately the same time.



LEMMA XI.

If a comet deprived of all its motion may be dropped from the altitude SN or 23S Iμ μ+ , so shat it falls into the sun, & it is continued always to be acted on by force,

by which it was acted on from the beginning; the same in half the time, in which it described the arc AC in its orbit, will describe a distance equal to the length Iμ in its descent. For the comet, in the time during which it describes the arc of the parabola AC, will describe the chord AC with that velocity which it did from the altitude SP (by the most recent lemma), & thus (by corol. 7, Prop. XVI, Book I.) in the same time in a circle, its diameter would be SP, & by the force of gravity by its revolving, would describe an arc, the length of which would be to the parabolic chord of the arc AC, in the square root ratio of 1 to 2. And therefore with that weight, which it had at the altitude SP to the sun, by falling from that altitude into the sun, describe in half of that time (by corol. 9, Prop. IV, Book I.) a distance equal to the square of half the chord applied to the square of the altitude SP, that is, the distance

2

4AISP . From which since the weight of the comet towards

the sun at the altitude SN shall be to its weight towards the sun at the altitude SP, as SP to Sμ : the comet with the weight that is has at the altitude SN, by falling towards the sun in

the same time, describes the interval 2

4AISμ , that is, a distance equal to the length

or I Mμ μ . Q.E.D.

PROPOSITION XLI. PROBLEM XXI. To determine the trajectory of a comet moving in a parabola from three given observations. This most difficult problem has been attacked for a long time in many ways, & I have set out certain problems in Book I, which are concerned with its solution. Afterwards I have thought out the following simpler solution.



Three observations are selected with equal time intervals from the approximate intervals one after the other. Moreover let that time interval of the time, when the comet is moving the slowest, be a little greater than the other, thus so that it is apparent that the difference of the times shall be to the sum of the times, as the sum of the times to around 600 days ; or so that the point E (in the figure of Lemma VIII.) may fall nearly on the point M, & thence it wanders towards I rather than towards A. If such observations shall not be on hand, the place of a new comet can be found by Lemma VI. Let S designate the sun, T , t , τ three places of the earth in its great orbit ; TA, tB,

Cτ three observations of the longitudes of the comet ; V the time passing between the observations of the first & the second, W the time passing between the second & the third ; X the longitude, that the comet in that whole time may be able to describe, with that velocity that it has at the mean earth to sun position, & which (by Corol. 3, Prop. XL, Book 11I) is required to be found, & tV the perpendicular to the chord Tτ . In the middle longitude observed tB , some point B is taken as the location of the comet in the plane of the ecliptic, & thence the line BE is drawn towards the sun S, which shall be to the sagittam tV, as the product 2SB St× to the cube of the hypotenuse of the right angled triangle, the sides of this are SB & the tangent of the latitude of the comet in the second observation for the radius tB. And the right line AEC is drawn through the point E (by Lemma V11 of this section), the parts of which AE, EC terminated by the lines TA &

Cτ , shall be in turn as the times V & W: & both A & C will be locations of the comet in the plane of the ecliptic approximately in the first & third observations, but only if B shall be the place correctly assumed in the second observation. To AC bisected at I erect the perpendicular Ii. Through the point B draw the hidden line Bi parallel to AC itself. Join the hidden line Si cutting AC in λ , & completer the

parallelogram iI pλ . Take Iσ equal to 3Iλ , & through the sun S draw the hidden line σξ equal to 3 3S iσ λ+ . And now with the letters A, E, C, & I deleted ; from the point B



towards the point ξ draw a new hidden line BE, which shall be to the first BE in the square ratio of the distance BS to the quantity 1

3S iμ λ+ . And through the point E again draw the right line AEC from the same rule as before, that is, so that the parts of this AE & EC shall be in turn, as the times between the observations V & W; both A & C will be more accurate places of the comet. The perpendiculars AM, CN, IO are erected to AC bisected at I, of which AM & CN shall be the tangents of the latitudes in the first & third observations to the radii TA & TC ; MN is joined cutting IO in O ; the rectangle iIλμ is put in place as before. In IA produced ID is taken equal to 2

3S iμ λ+ . Then on MN , MP is taken towards N, which shall be to the above longitude found X, in the square root ratio of the mean distance of the earth from the sun (or the radius of the great orbit) to the distance OD. If the point P falls on the point N; A, B & C will be three places of the comet, through which it must describe its orbit in the plane of the ecliptic. But if the point P is not incident in the point N; then on the right line AC there is taken CG equal to NP, thus so that the points G & P may lie in the same direction as the right line NC. By the same method, by which the points E, A, C, & G have been found from the assumed point B, with whatever other assumed points b & G, new points e, a, c, g, &

, , ,ε α χ γ may be found. Then if through G, g, γ the circumference of the circle Ggγ may be drawn, cutting the right line Cτ in Z: Z will be the position of the comet in the plane of the ecliptic. And if on AC, ac, αχ , the right lines AF ,af ,αϕ are taken respectively equal to CG,cg ,χγ , & through the points F , f ,ϕ the circumference of the circle Ffϕ is drawn, cutting the right line AT in X ; X will be another position of the comet in the plane of the ecliptic. To the points X & Z the tangents of the latitudes of the comet may be erected to the radii TX & TZ; & two places of the comet will be found in its proper orbit. Finally, (by Prop. XIX, Book I.) with the focus S, a parabola is described through two places, & this will be the trajectory of the comet. Q. E. I. The demonstration of this construction follows from the lemmas : certainly since the right line AC may be cut at E in the ratio of the times, by Lemma V11, as required by Lemma V11I ; & BE by Lemma XI would be the part of the right line BS or Bξ in the plane of the ecliptic, put in place between the arc ABC & the chord AEC ; & MP (by Corol. Lem. X.) would be the length of the chord of the arc, that the comet in its proper orbit would describe between its first & third observations, & thus it should be equal to MN, but only if B were a true place of the comet in the plane of the ecliptic. The remainder of the points B, b, β are not taken as you wish, for it is agreed to choose them near each other. If the angle AQt may be known roughly, in which projection of the orbit described in the plane of the ecliptic cuts the line tB ; in that angle it will be necessary to draw the hidden line AC, which shall be to 4

3 Tτ in the square root ratio of SQ to St. And with the right line SEB acting, the part of which EB is equal to the length Vt, the point B will be determined that it was allowed to use in the first place. Then with the right line AC deleted & with the following preceding construction drawn again, & with the above length MP found ; on tB the point b is taken, by that rule, so that if TA, TC shall mutually cut each other in T, the distance Tb shall be to the distance TB, in the ratio



composed from the ratio MP & MN & the ratio of the square root of SB to Sb. And by the same method the third point β can be found but only if there is a desire to repeat the operation a third time. Moreover by this method the two operations at most will suffice. For if the distance Bb becomes very small, after the points F, f & G, g have been found the right lines Ff & Gg drawn will cut TA & Cτ in the points sought X & Z.

Example.

The comet of the year 1680 may be proposed. The motion of this has been observed by Flamsted , & from the observations the following table shows the computation, & with the observations from the same corrected by Halley.

Apparent time

in hours. True time in hours.

Longitude of the sun.

Longitude of the comet.

Northern Latitude of the comet.

1680.Dec. 12 4.46' 4.46'. 0'' 10.51'. 23'' 60.32'.30'' 80. 28'.0'' 21 1

26 32. 6.36.59 11. 6. 44 5. 8 .12 21. 42.13 24 6.12 6.17.52 14. 9. 26 18.49. 23 25. 23. 5 26 5.14 5.20.44 16. 9. 22 28.24. 13 27. 0.52 29 7.55 8. 3. 2 19.19. 43 13.10. 41 28. 9.58 30 8.02 8.10.26 20. 21. 9 17.38. 20 26. 15. 7

I681.Jan. 5 5.51 6. 1.38 26. 22.18 8.48. 53 24. 11.56 9 6.49 7. 0.53 0. 29. 2 18.44. 4 23. 43.52

10 5.54 6. 6.10 1. 27.43 20.40. 50 16. 42.18 13 6.56 7. 8.55 4. 33.20 25.59. 48 16. 4. 1 25 7.44 7.58.42 16. 45.36 9.35. 0 17. 56.30 30 8.07 8.21.53 21. 49.58 13.19. 51 16. 42.18

Feb. 2 6.20 6.34.51 24. 46.59 15.13. 53 16. 4. 1 5 6.50 7. 4.41 27. 49.51 16.59. 6 15. 27. 3

Longitude of sun. Longitude of comet. Latitude of Comet

1M ,E 1 2 3E a E ,E= 1 1 2M ,E ,E a,E 3E 1 2M ,E ,E 3E 1680.Dec. 12 10.53'. 2'' 10.51'.23'' 60.33'.0''

1M , E

6.31.211 2

E aE

60.32'.30'' 80. 26'.0'' 80. 28'.0''

21 11.8.10 [E1] 11.8.10 1

3 [M] 11. 6.44 5. 8 .12 5. 8 .12 21. 45.13 21. 42.13

24 14.10.49 14. 9.26 18.49. 10 18.49. 23 25. 23.24 25. 23. 5 26 16.10.38 16. 9.22 28.24. 6 28.24. 13 27. 0. 57 27. 0.52 29 19.20.56 19.19.43 13.11. 45 13.10. 41 28. 10.05 28. 9.58 30 20.22.20 20.21. 9 17.37. 5 17.38. 20 28. 11.12 28. 11.53

1682.Jan. 5 26.23.19 26.22.18 8.49. 10 8.48. 53 26. 15. 26 26. 15. 7

9 0.29.54 0.29. 4 18.43. 18 18.44. 4 24. 12.42 24. 11.56 10 1.28.34 1.27.43 20.40. 57 20.40. 50 23. 44. 0 23. 43.52 13 4.34. 6 4.33.20 25.59. 34 25.59. 48 22. 17. 36 22. 17. 28 25 16.45.58 16.45.36 9.55. 48

1M , E 9.35. 0 17. 56. 54 17. 56.30

9.55. 481 2

E aE

30 21.50. 9 21.49.58 13.19. 36 13.19. 51 16. 40. 57 16. 42.18 Feb. 2 24.47. 4 24.46.59 15.13. 48 15.13. 53 16. 2. 2 16. 4. 1

5 27.49.51 27.49.51 16.59. 52 16.59. 6 15. 27. 23 15. 27. 3



To these observations add several made myself :

Apparent time. Longitude of comet. Northern latitude of comet 1682.Feb. 25 8h.30' 260.18'.35'' 120. 46'.46''

27 8. 15 27. 4 .30 12. 36. 12 Mar. 1 11. 0 27. 52. 42 12. 23. 40

2 8. 0 28. 12. 48 12. 19. 38 5 11.30 29. 18. 0 12. 3. 16 7 9.30 0. 4. 0 11. 57. 0 9 8.30 0. 43. 4 11. 45. 52

I have completed these observations with a seven foot telescope, & that I have carried out with a micrometer with threads located at the focus of the telescope : with which instruments we have determined the positions of the fixed stars amongst themselves & the position of the comet to these fixed stars.

Longitude of comet Latitude of comet

1M ,E 2E 3E 1M ,E 2E 3E 1682.Feb. 25 260.19'. 22'' 260.18'.17'' 260.18'.35'' 120.46 7

8 120.46' 78 120. 46'.46''

27 27. 4. 28 27. 4.24 27. 4.30 12. 36 12. 36 15 12. 36.12

Mar. 1 27.53. 8 27.53. 6 27.52.42 12.24 37 12. 24 6

7 12. 23.40 2 28.12.29 28.12.27 28.12.48 12. 19 1

2 12. 20 12. 19.38 5 29.20.51 29.20.51 29.18. 0 12. 19 2

3 12. 3 12 12. 3.16

7 0. 4. 0 11. 57. 0 9 0. 43. 2 0. 43. 4 0. 43. 4 11. 44 3

5 11. 45 78 11. 45.52

A shall describe the star of the fourth magnitude in the left heel of Perseus ( by Bayer ο ) B the following star of the third magnitude in the left foot ( Bayer ζ ) & C a star of the sixth magnitude ( Bayer n ) in the heel of the same foot, & D, E, F, G, H, I, K, L, M, N, O, Z, , , ,α β γ δ other lesser stars in the same foot. And p, P, Q, R, S, T, V, X shall be the places of the comet in the observations described above : & with the distance present AB of 7

1280 , & there was : 5 5 6 71 2 4 1 11

4 6 12 11 3 7 2 12 127 5 5 71 2 1 4

6 12 12 3 9 4 7 85

12

of the parts 52 , 58 , 57 82 23 29 57 49 27 52 36 , 53 38 , 43 31 29, 23, 36 18 50 , 46

AC BC AD , BD ,CD , AE , CE , DE , AIBI , CI DI ,AK BK , CK , FK FB FC ,AH ,DHBN , 5 51

3 12 7 31 45 31 ;CN , BL ,NLHO was to HI as 7 to 6 & produced passed between the stars D & E, thus so that the distance of the star D from this right line was 1

6 CD . LM was to LN as 2 to 9, & produced passed through the star H. From these the positions of the fixed stars were determined among themselves. Finally our countryman Pound in turn observed the positions of these fixed stars between themselves, & produced the longitudes & latitudes of these in the following table.



Fixed Stars Longitude. Latitude North. Fixed Stars Longitude. Latitude North. A 26. 41. 50 12. 8.36 L 29.33.34 12. 7.48 B 28 .40. 23 11.17.54 M 29.18.54 12. 7.20 C 27.58.30 12.40.25 N 28.48.29 12.31.91 E 26.27. 17 12.52. 7 Z 29.44.48 11.57.13 F 28.28.37 11.52.22 a. 29.52. 3 11.55.48 G 26.56 . 8 12. 4.58 β 11 0. 8.23 11.48.56

H 27.11 .45 12. 2. 1 γ 0.40.10 11.55.18 I 27.25. 2. 11.53.11 δ 1. 3.20 11. 30.42 K 27.42. 7 11.53.26

Now I have observed the positions of the comet to these fixed stars as follows. On Friday Feb. 25. (old calendar) at 8 30. p.m. the distance of the comet present at p was a little less from the star E than 3

13 AE, & greater than 15 AE, & thus approximately equal to

314 AE ; & the angle ApE was somewhat obtuse, but almost right. Certainly if a perpendicular is sent from A to pE, the distance of the comet from that perpendicular was 15 pE.

In the same night at the hour of 9 30. p.m., the distance of the comet present at P from the star E was greater than 1

2

14 AE, & less than 1

4

15 AE, & thus equal to 7

8

14 AE, or

approximately 839 AE . But with a perpendicular sent from the star A to the right line PE ,

the distance of the comet was 45 PE.

On Sunday, Feb. 27 at 8 15. p.m., the distance of the comet present at Q from the star O was equal to the distance of the stars O & H, & the right line QO produced passed between the stars K & B. The position of this right line on account of intervening clouds could not be defined more accurately. On Tuesday, March 1 at 11 p.m., the comet was present at R, accurately placed between the stars K & C , & the part CR of the right line CRK a little greater than 2

3 CK, & a little less than 1 1

3 8CK CR+ , & thus equal to 161 13 16 45 or CK CR CK+ .



On Wednesday, March 2, at 8 p.m. , the distance of the comet present at S was approximately 4

9 FC from the star C. The distance of the star F from the right line CS produced was 1

24 SC; & the distance of the star B from the same right line, was five times greater than the distance from the star F. Likewise the right line NS produced passed between the stars H & I, being five or six times closer to the star H than to the star I. On Saturday, March 5, at 11 30. p.m., with the comet present at T, & with the right line MT equal to 1

2 ML, & with the right line LT produced passing between B & F four or five times closer to F than B, taking from BF a fifth or sixth part of that towards F. And MT produced passed beyond the distance BF in the direction of the star B, being four times closer to the star B than the star F. Star M was exceedingly close which scarcely could be seen with the telescope, & L a star perhaps of magnitude greater than eight. On Monday, March 7, at 9 30. p.m., with the comet present at V, with the right line Vα produced, it passes between B & F, bearing off from BF towards F 1

10 by BF, & it was to the right line Vβ as 5 to 4. And the separation of the comet from the right line αβ was 1

2 Vβ . On Wednesday, March 9, at 8 30. p.m., with the comet present at X, the right line Xγ was equal to 1

4 γδ , & a perpendicular sent from the star δ to the right line Xγ was 25 γδ . On the same night at midnight, the comet was present at Y, the right line Yγ was equal to 1

3 γδ , or a little less, perhaps 516 γδ , and a perpendicular sent from the star δ to the right

line Yγ was equal to around 1 16 7 or γδ γδ . But the comet on account of the vicinity of the

horizon was scarcely able to be seen, nor its place to be defined as distinctly as with the preceding. And from the observations of this kind by constructing figures & doing computations I derived the longitudes & latitudes of the comet, & our countryman Pound from the correct locations of the fixed stars corrected the places of the comet, & the corrected places are given above. I used a micrometer constructed very poorly, but still the errors in the latitude & longitude scarcely exceeded one minute (as far as they arose from our observations). Moreover the comet (according to our observations) at the end of its motion began to be noticeably deflected towards the North, from the parallel that it had held at the end of February. Now towards determining the orbit of the comet, I have selected three from the observations just described, which Flamsted had determined on Dec. 21, Jan. 5, & Jan. 25. From these I have found St of 9842,1 parts & Vt of 455 parts, such that 10000 is the radius of the great orbit. Then according to the first operation on assuming tB to be of 5657 parts, I have found 9747SB = , BE at first in turn 412, 9503Sμ = , 413iλ = : BE in the second case becomes 421, while [ '=' signs have been introduced for convenience,]

10186 8528 4 8450 8475 25OD ,X , MP ,MN ,NP= = = = = . From which according to the second operation I have deduced the distance 5640tb = . And by this operation I have found at last that the distances 4775 and 11322TX Zτ= = . From which by defining the orbit, I have found the descending node in 10. 53' & the ascending node in 10. 53'; the inclination of its plane to the plane of the ecliptic to be 610. 20' 1

3 ; its vertex (or the perihelion of the comet) to stand at 80. 38' from the node, & to be in 270. 43' with the



southern latitude 70.34'; & its latus rectum to be 236,8, & the area described in single days by the radius drawn to the sun to be 93585, on putting 100000000 for the square of the radius of the great orbit ; truly the comet was progressing along the signs of the zodiac, & on the 8th day of December at 4 minutes past midday was in the vertex of its orbit or perihelion. All these things I have determined graphically from a scale of equal parts & the chords of the angles taken from a table of natural sines ; by constructing a large enough diagram, in which as it were the radius of the great orbit of 10000 parts was equal to 1

316 English inches. Comet to sun

distance . Long. deduced. Lat. deduced. Long.obs. Lat.obs. Differ.

Long. Differ. Lat.

Dec.12 2792 60. 32' 12

08 18'. 13

06 31'. 80.26' +1' 127'−

29 8403 2313 13. 28. 0 3

413 11. 11228 10. +2 1

1210− Feb. 5 16669 17. 0 2

315 29. 7816 59. 2

515 27. +0 142+

Mar. 5 21737 3429 19. 12. 4 6

729 20. 1212 3. –1 1

2+

Finally so that is could be agreed that the comet truly was moving in such an orbit found, I deduced partially through arithmetical operations & partially graphically the locations of the comet in its orbit according to the required times of observation : as can be seen in the following table. Certainly after our countryman Dr. Halley had determined the orbit more accurately by arithmetical calculations than one could do graphically by described lines ; indeed the position of the nodes was retained in (Cancer) & (Capricornus) as 10. 53', & the inclination of the plane of the orbit to the ecliptic 610. 53', & so that the time of the perihelion of the comet was on the 8th day of December at 4 minutes past noon, truly the distance of the perihelion from the ascending node of the comet measured in the orbit of the comet was found to be 90.20' , & the latus rectum or the parabola was of 2430 parts, with the mean distance of the earth from the sun being 100000 parts. And from these given, with an accurate arithmetical calculation put in place, the places of the comet computed at the times of observation were as follows.

True time Comet sun distance.

Long.computed Lat.computed Differ. Long.

Differ. Lat.

Dec. 12d 4h.46' 28028 60.29'.25'' 80.26'.0'' Bor. –3'.5'' – 2'. 0'' 21. 6.37 61076 5. 6.30 21.43.20 –1.42 + 1. 7 24. 6.18 70008 18.48.20 25.22.40 –1. 3 – 0.25 26. 5.21 75576 28.22.45 27. 1.36 –1.28 + 0.44 29. 8. 3 l4021 13.12.40 28.10.10 +1.59 + 0.12 30. 8.10 84021 17.40. 5 28.10.20 +1.45 – 0.33 Jan. 5. 6. 1 1

2 101440 8.49.49 26.15.15 +0.56 + 0. 8

9. 7. 0 110959 18.44.36 24.12.54 +0.32 + 0. 8 10. 6. 6 113162 20.41.02 23.44.10 +0.10 +0.18 13. 7. 9 120000 26. 0.21 22.17.30 +0·33 +0.25 25. 7.59 145370 9.33.40 17.57.55 –1.20 + 1.25 30. 8.22 155303 13.17.41 16.42. 7 –2.10 –0.11 Feb. 2. 6.35 160951 15.11.11 16. 4.15 –2.42 +0.14 5. 7. 4 1

2 166686 16.58.25 15.29.13 –0.41 +2.10

25. 8.41 202570 26.15.46 12.48. 0 –2·49 +1.14 Mar. 5.11.39 216205 29.18.35 12. 5. 40 + 0·35 + 2.24



Also this comet appeared in the preceding month of November in Coburg in Saxony, & was observed by Mr. Gottfried Kirch in the days of the month I have shown, on the 4th , the 6th & on the 11th , in the old style ; & from its position to nearby fixed stars observed first with a two foot long telescope & then more accurately with a ten foot telescope, & the difference of the longitudes of Coburg & London of 110 & with the places of the fixed stars observed by our countryman Pound, our Halley has determined the locations of the comet as follows. On November 3rd at I7h.2', at the apparent time in London, comet was in 290.51' with latitude North 10.17'.45". On November 5th at 15h.58', comet was in (Virgo) 30. 23' with latitude North 10.6'. On November 10th at I6h.31', the comet was equally distant from the stars of Leo σ and τ according to Bayer ; indeed it did not touch the right line joining these, but was a little away from that. In Flamsteed's catalogue of stars σ then had the position 140.15' with almost the latitude North 10.41', τ truly 170.39' 1

4 , with latitude North 00.34'. And the mean point between these stars was 150.39' 1

4 with latitude North 0gr. 33 12 . The distance

of the comet from that right line shall be around 10' or 12', & the difference of the longitudes of the comet & the mean of these points was 7', & the difference of the latitudes around 7' 1

2 . And thence comet was in 150.32' with latitude North around 26'. The first observation from the position of the comet to some small fixed stars was clearly accurate enough. The second also was accurate enough. In the third, which was less accurate, the error was under 6 or 7 minutes, & scarcely greater. Truly the longitude of the comet in the first observation, which was more accurate than the others, computed in the predicted parabola, was in (Leo) 290.30'.22'' with a latitude of 10. 25'.7" North, & its distance from the sun 115546. Again Halley, by observing that a conspicuous comet had appeared four times with an interval of 575 years, evidently in the month of September after the death of Julius Caesar ; in the year of Christ 531 under the consulate of Lampadius & Orestes ; in the year of Christ 1106 in the month of February, & during the end of the year 1680, & that with a long & conspicuous tail (except that during the death of Caesar, the tail was less apparent on account of the inconvenient position of the earth:) sought an elliptic orbit the major axis of which would be of 1382957 parts, with a mean distance of the earth from the sun being 10000 parts: certainly in which orbit the comet would be able to revolve in 575 years. And on putting the ascending node in at 20. 2'; the inclination of the plane of the orbit to the plane of the ecliptic 610 . 6'.48"; the perihelion of the comet in this plane

(Sagittarius) 220. 44'. 25'';the equal time of the perihelion December 7d. 23h. 9' ; the distance of the perihelion from the ascending node in the plane of the ecliptic 90. 17'. 35"; & the conjugate axis 184081,2: I have computed the motion of the comet in this elliptic orbit. Moreover its places in this orbit both deduced from observations as well as by computation are shown in the following table.



True time Long. observed Lat. N. obs. Long.computed. Lat.computed. Differ.

Long. Differ. Lat.

Nov. 3.16.47 290.51'. 0'' 10.17'. 45'' 290.51'. 22'' 10.17'. 32'' + 0'.22'' – 0'.13'' 5.15.37 3.23. 0 1. 6. 0 3.24. 32 1. 6. 9 + 1.32 + 0. 9 10.16.18 15.32. 0 0.27. 0 15.33. 2 0.25. 7 + 1. 2 – 1.53 16.17. 0 [no readings] [no readings] 8.16.45 0.52. 7 A 18.21.34 " " 18.52.15 1.26.54 20.17. 0 " " 28.10.36 1.53.35 23.17. 5 " " 13.22.42 2.29. 0 Dec. 12d 4h.46' 60.32'.30'' 8.28. 0 6.32.20 8.29. 6 B – 1.10 + 1. 6 21. 6.37 5. 8.11 21.42.13 5. 6.14 21.44.42 – 1.58 + 2.29 24. 6.18 18.49.23 25.23. 5 18.47.30 25.23.35 – 1.53 + 0.30 26. 5.21 28.24.13 27. 0.52 28.21.42 27. 2. 1 – 2.31 + 1. 9 29. 8. 3 13.10.41 28. 9.58 13.11.14 28.10.38 + 0.33 + 0.40 30. 8.10 17.38.20 28.11.53 17.38.27 28.11.38 + 0. 7 – 0.16 Jan. 5. 6. 1 1

2 8.48.53 26.15. 7 8.48.51 26.14.57 – 0. 2 – 0.10 9. 7. 1 18.44. 4 24.11.56 18.43.51 24.12.17 – 0.13 + 0.21 10. 6. 6 20.40.50 23.43.32 20.40.23 23.43.25 + 0.10 – 0. 7 13. 7. 9 25.59.48 22.17.28 26. 0. 8 22.16.32 + 0·20 – 0.56 25. 7.59 9.35. 0 17.56.30 9.34.11 17.56. 6 – 0.49 – 0.24 30. 8.22 13.19.51 16.42.18 13.18.28 16.40. 5 – 1.23 – 2.13 Feb. 2. 6.35 15.13.53 16. 4. 1 15.11.59 16. 2. 7 – 1.54 – 1.54 5. 7. 4 1

2 16.50. 6 15.27. 3 16.59.17 15. 27. 0 + 0.11 + 2.10 25. 8.41 26.18.35 12.46.46 26.16.59 12.45. 22 – 1·36 – 1.24 Mar. 1.11.10 27.52.42 12.23.40 27.51.47 12.22.28 – 0·55 – 1.12 5.11.39 29.18. 0 12. 3.16 29.20.11 12. 2.50 + 2.11 – 0·26 9. 8.38 0.43. 4 11.45.52 0.42.43 11.45.35 – 0·21 – 0·17

The observations of this comet from the beginning to the end are not in less agreement with the motion of the comet in the orbit now described, since the motions of the planets are accustomed to agree with their theories, & by agreeing they prove that the comet was one & the same, which appeared in this whole time, & its orbit was correctly designated here. In the preceding table we have set aside the observations from the 16th , 18th , 20th & 23rd days of November as less accurate. For the comet was also observed in these times. Certainly Ponteo & associates, on November 17th old time, at the hour of 6 in the morning in Rome, that is, at 5 hours & 10 minutes London time, by threads pointing towards fixed stars, had observed the comet in (Libra) 80.30' with the southern latitude 00. 40'. These observations can be found in a tract that Ponteo published about this comet. Cellio, who was present there & who sent his observations to Mr. Cassini in a letter, saw the same comet at the same hour in 80.30' with a southern latitude of 00. 30'. At the same hour Gallet of Avenion (that is, at the hour of 5. 41 a.m. London time) saw the comet in 80. without latitude; but the comet by the theory now was in 80. 16'. 45" with the southern latitude of 00.53'. 7", On November 18th in the morning at 6. 30 a.m. Rome time (that is, at 5.40 a.m. London time) Pontio saw the comet in 130. 30' with the southern latitude 10. 20'. Cellio observed it in 130.30', with the southern latitude 10. 00'. But Gallet at the early morning hour of 5h.30' in Avenion saw the comet in 13gr. 00', with the southern latitude 10. 00'.



And R.P. Ango in the university of La Fleche in France, at five o'clock in the morning (that is, at 5h. 9' London time) saw the comet half-way between two small stars, of which one is the middle star of three in line in the southern hand of Virgo, Bayer's ψ , & the other is the end of the wing, Bayer's θ . Thence the comet then was in 120 .46' with the latitude 50' South. In the same day in Boston in New England at a latitude of 42 1

20, in the

hour of five in the morning, (that is at the London time of 9h.44' a.m.) the comet was seen near 140, with the southern latitude 10. 30', as I have been informed by the most illustrious Halley. On Nov. 19. at the morning hour of 4 1

2 in Cambridge, the comet (observed by a certain young man) was separated from Spica in by about 20 towards the North-West [There is some confusion here : Madame du Chatelet considers spica to be an ear – as in ear of corn or wheat, or a tuft of hair, in a constellation, while Cohen considers Spica to be the star of this name, which he now writes with a capital S, belonging to Virgo; Motte makes the best of both worlds, & sometimes calls it by one name, then by the other.] Or it was in 19gr.23'. 47" with the southern latitude 20. 1'. 59". In the same day at the hour of 5 in the morning in Boston in New England, the comet was distant from Spica in by one degree, a difference of the latitudes present of 40'. On the same day on the island of Jamaica, the comet was distant from Spica by an interval of around one degree. On the same day Mr. Arthur Storer at the river Patuxent, near Hunting Creek in Maryland, on the border of Virginia at the latitude 38 1

20, at the hour of five in the morning (that is, at 10h London

time) saw the comet above the star Spica in & almost coincident with Spica, with the separation between the same around 1

40. And from these observations collated among

themselves I deduced that at the hour 9h. 44' of London time that the comet was in 180. 50' with a latitude of around 10 .25'. Moreover the comet by theory now was in 180. 52'. 15" with a latitude of 10. 26'. 54'' South. Nov 20. Dr. Montanari, professor of astronomy at Padua, at 6 o'clock in the morning in Venice (that is, at 5h.10' London time) saw the comet in 180 with the latitude 10.30' South. On the same day in Boston, the comet separation from Spica was 40 longitude to the East, & thus it was about 230. 24' in . Nov 21. Ponteo & his companions on the morning at 7 1

4 hours observed the comet in 270. 50' with a southern latitude of 10. 16', Cellius in 280. Ango at 5 o'clock in the

morning in at 270. 45", Montanari in at 270.51'. On the same day on the island of Jamaica saw the comet near the principle star of Scorpio, & that had around the same latitude as Spica in Virgo, that is, 20. 2'. On the same day at 5 o'clock in the morning, at Balasore in India, (that is at the hour of the preceding night in London of 11h. 20') took the distance of the comet from the ear in as 70. 35' to the East. In the right line between the ear & the scales, & thus it was present in 260. 58' with a southern latitude of around 10.11' ; & after 5h & 40' (which corresponds to the hour of around 5 a.m. in London) it was in 280. 10' with a southern latitude of 10. 16'. By the theory the comet now truly was in 280.12' .36", with a southern latitude of 10. 53'. 35". Nov. 22. The comet was seen by Montanari in (Scorpio) 20. 33', but in Boston in New-England it appeared in around 30 , with almost the same latitude as before, that is, 10. 30'. In the same day at 5 o'clock in the morning at Balasore the comet was observed in

10. 50'; & thus at the hour of five in the morning in London the comet was in around



30. 5'. In the same day at 6 12 hours in the morning in London our countryman Hook saw

the comet in at around 30. 30', & that in a right line which passed through the [corn] ear of Virgo [i.e. Spica] & the heart of Leo, not exactly however, but in a line a little deflected to the North. Montanari likewise noted that the line from the comet drawn through the ear, on this day & in the following days passed through the southern side of the heart of Leo, with a very small interval interposed between the heart of the lion & this line. A right line passing through the heart of Leo & the ear of Virgo, cut the ecliptic in 30. 46'; at an angle 20. 51'. And if the comet were located on this line in 30 its latitude would have been 20. 26'. But since the comet from the agreement of Hook & Montanari, to some extent was separated from this line a little towards the North, its latitude was a little less. On the 20th day, from the observation of Montanari, its latitude was almost equal to the latitude of the ear in , & that was around l0. 30', & by the agreement of Hook, Montanari, & Father Alga it was always increasing, & now significantly greater than 10. 30'. Now between these two limits found: 20. 26' & 10. 30', the magnitude of the average latitude was around 10.58'. The comet's tail, from the agreement of Hook & Montanari, was directed towards the ear in , falling a little from that star, nearly to the south according to Hook, & nearly to the north, according to Montanari ; & thus that decline was scarcely sensible, & the tail to be present almost parallel to the equator, was being deflected a little by the opposition of the sun towards the North. Nov. 23. old time, at the hour of 5 in the morning in Noreberg (that is at 4 1

2 hours in London) D. Zimmerman saw the comet in 80. 8', with a latitude of 20. 31' South, evidently with its separation taken from the fixed stars. Nov. 24. Before sunrise the comet was seen by Montanari in 120. 52', towards the northern side of the line drawn through the heart of Leo & the ear of Virgo, & thus it had a latitude a little less than 20. 38'. This latitude, as we have said, from the observations of Montanari, Ango & Hook, was always increasing ; & now thus it was a little greater than 180.58' & with a mean magnitude, without noticeable error, to be put in place at 20.18'. Ponteo & Galtetius now wished to decease the latitude, both Cellio & an observer in New-England retained the same magnitude, evidently one degree or one & a half degrees. The observations of Ponteo & Cellio were coarser, especially as those were taken by means of azimuths & altitudes, & as those of Gallet ; those are better which are taken through the positions of the comet relative to the fixed stars by Montanari, Hook, Ango & by the observer in New-England, & some that Ponteo & Cellio have made. On the same day at 5 o'clock in the morning the comet was seen in Balasore in 110. 45'; & thus at the fifth hour of the early morning in London it was in at around l30. Truly by the theory the comet now was in 130. 22'. 42". Nov. 25. Before sunrise Montanari observed the comet in 17 3

40 approximately. And

Cellius observed at the same time that the comet was in a right line drawn between the bright star in the right thigh of Virgo & the southern scale of Libra, & this right line cut the path of the comet in 180. 36'. Truly by the theory the comet was now approximately in 18 1

30.

Therefore these observations agree with the theory just as they agree amongst themselves, & by agreeing they prove it to be one & the same comet, which appeared in



the whole time from the fourth day of November as far as to the ninth of March. The trajectory of this comet cut the plane of the ecliptic twice, & therefore was not a straight line. It did not cut the ecliptic in opposite directions of the heavens, but at the end of Virgo & at the beginning of Capricorn, through an interval of around 98 degrees; & thus the course of the comet was deflected greatly from a great circle. For in the month of November its course dropped by at least three degrees below the plane of the ecliptic to the South, & after the month of December it moved away from the ecliptic to the North by 290, & the two parts of the orbit, the one with the comet approached the sun & the other with it receded from the sun, appearing to have an angle between each other of more than 300, as Montanari had observed. This comet went through nine signs, clearly from the last degree of Leo as far as to the first point of Gemini, besides the sign of Leo, through which it had gone before it began to be seen; & no other theory is extant that by which the comet traverses so much of the heavens with a regular motion. Its motion was greatly uneven. For around the 20th day of November it described around 50 each day ; then with the motion retarded between November 26th & December 12th, evidently in a time of 15 & a half days, it described only 400; truly again with an accelerated motion it

describes almost 50 per individual day, before the motion again began to be retarded. And the theory, which corresponded properly to so much of the unequal motion through the great part of the sky, & which observed the same laws as the theory of the planets, as well as agreeing accurately with accurate astronomical observations, cannot be otherwise than true. The rest of the trajectory that the comet described, & indeed the tail that it projected at individual places, has been shown traced out in the adjoining diagram in the plane of the trajectory : where ABC denotes the trajectory of the comet, D the sun, DE the axis of the trajectory, DF the line of the nodes, GH intersection of the sphere of the great orbit with the plane of the trajectory, I, the position of the comet on Nov. 4 th of the year 1680, K the place of the same on Nov. 11th , L the place on Nov. 19th, M the place on Dec. 12th, N the place on Dec. 21st, O the place on Dec. 29th, P the place on Jan. 5th, the following Q the place on Jan. 25th, R the place on Feb. 5th, S the place on Feb. 25th, T the place on Mar. 5th, & V the place on Mar. 9th. Now the following observations are used in determining the tail: Nov. 4th & 6th. The tail had not yet appeared.



Nov. 11th. A tail now appeared of half a degree, not visible except with a telescope ten feet long. Nov. 17th. The tail appeared to Ponteo more than 15 degrees long. Nov. 18th. The tail was 300 long, & was seen in New-England to be directly away from the sun, & and was extended as far as to the planet [Mars] , which then was in 90. 54'. Nov. 19 th. In Mary-land , the tail was seen to be 150 or 200 long. Dec. 10th. The tail (from Flamsteed's observation) was passing through the middle of the interval between the tail of the serpent of Ophiuchus & the star δ in the south wing of Aquilae, & stopped close to the stars A, ,bω in Bayer's tables. Therefore its end was in 19 1

2gr, with a northern latitude of around 34 1

40.

Dec. 11th. The tail increased to as far as the head of Sagittarius (Bayer ,α β ) ending in 260. 43', with a latitude of 380. 34' North.

Dec. 12th. The tail was passing through the middle of Sagittarius, not being extended any longer, stopping in 40, with a latitude North of around 42 1

20. These are understood

to be concerned with the clearer part of the tail. For in the more obscure light, perhaps in the heavens in a more clear weather, the tail on Dec.12 th, at the hour of 5. 40' in Rome (observed by Pontio) rose to 100 above the rump of the swan ; & from this star its edge stopped at 45' to the North-West. But the tail was 30 wider in these days towards the upper extremity, and thus its middle was 20. 15' distant from that star towards the South, and the upper end was in (Pisces) 220, with the latitude of 6l0 North. And hence the length of the tail was around 700. Dec. 21st. The same increased almost to the chair of Cassiopeia, being equally distant from β and Schedir [the brightest star in this constellation], and its distance from each of these two stars was equal to the distance between themselves, and thus stopping in 240, with the latitude of 47 1

20.

Dec. 29 th. The tail was touching Scheat placed on the left, and the interval of the two stars in the northern foot of Andromeda was filled up completely, and the length of the tail was 540 ; and thus was defined in 190, with the latitude 350. Jan. 5 th. The tail touched the starπ in Andromeda's breast on its right side, and the star μ in the girdle of this at the left side ; and (on a par with our observations) the length was 400; but it was a curve and with the convex side seen towards the South. And on one side it made an angle of 40 with a circle passing through the sun and the head of the comet; but near to the other boundary is was inclined to that circle at an angle of 100 or 110 and the chord of the tail contained an angle of 80 with the circle. Jan. 13th. With enough light the tail appeared to be finished sensibly between Alamech and Algol, and with a more tenuous light it was defined to the region of the star χ in the side of Perseus. The distance of the end of the tail from the circle joining the sun and the comet was 30.50', and the inclination of the chord of the tail to that circle was 8 1

20.

Jan. 25 th and 26 th. The tail shone with a rather feeble light to a length of 60 or 70 ; and with a night or two following when the sky was very clear, with the light most tenuous and barely visible, it reached a length of 12 degrees and a little beyond. But its axis was directed accurately to the bright star in the eastern shoulder of Auriga, and thus fell from opposing the sun towards the North at an angle of 100. And then on Feb. 10 th with the naked eye I viewed the tail two degrees long. For the aforesaid light was more tenuous



and not apparent through a glass. But Ponteo on Feb. 7 th wrote that he had seen the tail at a length of 120. Feb. 25 th and after that the comet appeared without a tail. On examining the orbit just described above, and on considering the other phenomena of this comet, it will be agreed without difficulty, that the bodies of comets are solid, compact, fixed [in shape] and durable in the image of the bodies of planets. For if they were nothing other than vapours or exhalations of the earth, sun and planets, here a comet in its passing in the vicinity of the sun ought to be dissipated at once. For the heat of the sun is as the intensity of the rays, that is, reciprocally as the square of the distances of the places from the sun. And thus since the distance of the comet from the centre of the sun on December 8th, when it was passing through the perihelion, was to the distance of the earth from the centre of the sun almost as 6 to 1000, the heat of the sun at the comet at this time was to the heat of our summer sun as 1000000 to 36, or 28000 to 1. But the heat of boiling water is around three times greater than the heat that the dry earth receives on the summer sun, as I have found out : and the heat of incandescent iron (if I conjecture rightly) will be around three or four times greater than the heat of the boiling water; and thus the heat, that dry earth present on the comet passing through the perihelion, is able to experience from the rays of the sun, is around 2000 times more than the heat of the glowing iron. But with so much heat, vapours and exhalations and all volatile matter ought to be at once consumed and dissipated. Therefore the comet in its perihelion receives an immense amount of heat from the sun, and that heat can be conserved for a very long time. For an incandescent iron globe one inch wide present in air, can scarcely sent off all its heat in the space of an hour. Moreover a greater globe will conserve the heat in the ratio of the diameters, because the surface area (according to the measure of this cooled by the surrounding air) is less in that ratio to the quantity of its included warm matter. And thus a glowing iron globe equal to this earth, that is, more or less 40000000 feet wide, or in just as many days, and thus in 50000 years, will scarcely be cooled. Yet I suspect that the duration of the heat, on account of hidden causes, will be increased in a smaller ratio than that of the diameter: and I would choose to have the true ratio found by experiment. Again it is to be observed that the comet in the month of December, but only when it had been heated by the sun, sent off a much longer and splendid tail than in the month of November before, when it had not yet reached the perihelion. And generally, all the greatest and most brilliant tails arise suddenly from comets after their passage through the region of the sun. Therefore the heating of the comet leads to the magnitude of the tail : and thence I seem to deduce that the tail shall be nothing other than the most tenuous vapour, that the head or nucleus of the comet emits by its heat. Otherwise there are three opinions concerning the nature of the tails of comets ; either to be the light of the sun passing through the transparent heads of comets, or to arise from the refraction of light in passing from the head of the comet to the earth, or finally the cloud to be either vapour from the head of the comet arising and passing away in directions opposite to the sun. The first opinion is that of those who have not yet imbued the science of optical matters. For the light of the sun cannot be discerned in a darkened chamber, unless the light is certainly reflected from the dust and smoke of particles always flying about in the



air : and thus in air filled with thicker smoke it is more brilliant, and strikes the senses more strongly; in clearer air it is more tenuous and scarcely perceived: but in the heavens nothing is able to be reflected without reflecting matter being present. The light certainly cannot be discerned that is in a heavenly body, except that from thence it is reflected into our eyes. For vision can only be from rays which impinge on the eyes. Therefore matter of some kind is required reflecting in the tail region, lest the whole heavens be illuminated uniformly in a brilliant light from the sun. The second opinion is overwhelmed by many difficulties. The tails are never variegated with colours : which yet are accustomed to be found whenever refraction occurs. The light of the fixed stars and of the planets transmitted distinctly to us demonstrates the medium of the heavens is not influenced by any refractive force. For as it is said that the fixed stars sometimes were seen as comets by the Egyptians, but as that happened most rarely, it can be ascribed to fortuitous refraction of the clouds. [It may be of course be that they were viewing a cosmic event such as a supernova.] The radiations and scintillation of the fixed stars [i.e. twinkling] also are accustomed to refractions both within the eyes as well as trembling of the air : which certainly vanish with the eye applied to a telescope. The rising of vapours in the air may cause a tremble, so that the rays are easily turned away in turn from the narrow space of the pupil, but by no means from the side of the wider glass in the aperture of the objective. Thus it is that such a scintillation arises in the first case, and ceases in the second: and the cessation in the second case demonstrates the regular transmission of light through the heavens without any sensible refraction. It has been said, incorrectly, that one cannot always see the tails of comets, because their light is not strong enough, as then the secondary rays do not have enough strength to affect the eyes, and it is for this reason that we cannot see the tails of fixed stars, as [likewise] they do not have enough strength to affect the eyes, and therefore the tails of fixed stars are not seen: but it is known that the light of fixed stars can be increased more than 100 times by means of telescopes, yet still tails are not discerned. The light of planets is more plentiful too, truly without tails: but often comets have the greatest tails, when the light of the head is feeble and very dull. Thus indeed the comet of the year 1680, in the month of December, in which time the head by its light was scarcely equal to a second order magnitude star, was sending out a magnificent tail as far as to 400, 500, 600 or 700 of longitude and beyond : after January 17th and the18th the head appeared as a star of only the 7 order magnitude, truly the tail by a certain weak light was enough to be seen 60 or 70 long, and with the most obscure light, scarcely made it possible to see, it was stretched out to as far as 120 or a little further: as has been said above. But on both February 9th and 10th when one had abandoned looking at the head with the naked eye, I considered the tail to be 20 in length with a telescope. Again if the tail was arising from the refraction of celestial matter, and by virtue of the heavens the figure as deflected in the opposite direction to the sun, that deflection must always be into the same region of heavens and always to be made in the same direction. And the comet of the year 1680 December 28, in the hour 8 1

2 p.m. London time, was present in 80.41', with a latitude North of 280. 6', with the sun present in 180.26'. And the comet of the year 1577, on December 29 was moving through 80. 41' with a latitude North of 280.40', with the sun present in 180.26' approximately. In each case the earth was present at the same place, and the comet appeared in the same part of the heavens : yet in



the first case the tail of the comet (from my own and from the observations of others) declined at an angle of 4 1

20 from the opposition of the sun towards the North ; truly in

second case (from the observations of Tycho) the declination was of 210 to the South. Therefore the refraction of the heavens is rejected, and it remains that the phenomena of tails is derived from some matter reflecting the light. The laws which the tails observe confirm that they arise from the heads of comets, and that they ascend into regions away from the sun. So that since the tails which the heads leave behind as they progress in these orbits, are in the planes of the orbits of comets passing the sun, these always deviate in directions opposite to the sun. Which appear to a spectator established in these planes to be in directions directly away from the sun ; moreover to an observer in these planes the deviation would appear to change little by little, and in days would become greater and greater. Because the deviation, with all else being equal, is less when the tail is more oblique to the orbit of the comet, as when the head of the comet approaches closer to the sun; especially if the angle of deviation of the tail is viewed near the head of the comet. Besides, since the non-deviating tails appear in straight lines, and moreover those deviating are curved. So that the curvature is greater where the deviation is greater, and more noticeable where the tail is longer, with all things being equal : for with shorter tails the curvature is scarcely noticed. In addition, the angle of deviation is less near the comet's head, and greater towards the other extremity , and thus as a consequence the convex side of the tail is turned towards the parts that have been carried away by its deviation, and which are in right lines from the sun drawn through the head of the comet to infinity. And because the tails which are more prolix and wider, and shine by light more vigorously, shall be a little more brilliant on the convex side and terminate less indistinctly than at the concave side. Therefore the phenomena of the tail depends on the motion of the head, but not in the region of the heavens in which the head is seen ; and therefore they do not come into being through refraction of the heavens, but arise from the material supplied by the head. And indeed just as in our air the top of any body on fire tries to become higher, and that either perpendicularly if the body is at rest, or obliquely if the body is moving to the side: thus in the heavens, where bodies gravitate towards the sun, smoke and vapours must rise from the sun (as has now been said) and reaches higher and straight up if the body is at rest, or obliquely, if the body by progressing always leaves behind places from which the higher parts of the vapour rise. And that obliquity will be less when the ascent of the vapour is faster : without doubt in the vicinity of the sun and near smoking bodies. But from the differences of the obliquity the column of vapour appears curved : and because the vapour in the preceding side of the column is a little more recent, thus also it likewise will be a little denser, and the light reflected on that account more abundant, and it will be terminated more distinctly. Concerning the sudden and uncertain agitations of the tails and with regard to the irregular figures of these, which some have described occasionally, here I add nothing; therefore as either from the changes in our air, and from the motions of clouds sometimes the parts of tails arise from obscurity ; or perhaps from parts of the milky way, which disregarded may be confused with the tails, and only parts of that are seen. But it is possible to understand from the rarity of our air, how vapours from which the atmospheres of comets are able to arise, which are sufficient to fill such immense spaces.



For the air adjacent to the surface of the earth occupies around 850 more parts of space than water of the same weight, and thus a cylindrical column of air 850 feet high is of the same weight as a column of water of the same width measuring a foot in height. Moreover a column of air rising to the top of the atmosphere equals by its weight a column of water around 33 feet high; and therefore if the lower part of the whole column of air 850 feet high is taken away, the part remaining above will equal by its weight a column of water to of height 32 feet. Thus truly (by a rule confirmed by many experiments, that the compression of air shall be as the weight of the incumbent atmosphere, and that the gravity shall be inversely as the square of the distance of places from the centre of the earth), by putting in place the computation according to the Corollary, Prop. XX11, Book II, I have found that the air, if it may rise from the surface of the earth to a height of one radius of the earth, shall be rarer than with us in a ratio greater by far than the distance greater than all of the distance between the orbit of Saturn and a globe of diameter one inch. And thus a globe of our air one inch wide, that with the rarity which it would have at a height of one radius of the earth, would fill all the planetary regions as far as to Saturn's sphere and more beyond. Hence since the air higher besides is rarefied indefinitely ; and the hair or atmosphere of a comet, by ascending from its centre, almost ten times higher shall be than the surface of the nucleus, then the tail may ascend higher than this altitude, and it must become the even more rarefied. And since on account of the great thickness of the atmosphere of the comet, and the great size of the gravitation of the bodies towards the sun, and the gravitation of the particles of the air and vapour between themselves, it can happen that the air in the celestial distances and in the tails of comets themselves thus may not be rarefied ; but yet the exceedingly small quantity of air and vapour suffices in abundance for all that phenomena of the tails, and is evident from this computation. For the signs of rarity of the tails is deduced from the stars by their translucence. With the brilliant light of the sun through the earth's atmosphere, with its thickness of a few miles, and the stars and the moon itself obscured and thoroughly extinguished : equally illuminated by the light of the sun, through the immense thickness of the tails, the stars are known to shine through without detriment and with minimum loss of clarity. Nor is the splendour of many tails usually greater than the air in our darkened chamber with the light from the sun in a space of one or two inches reflecting in the sunshine. The interval of time in which the vapour ascends from the head to the tail, can almost be known by drawing a line from the end of the tail to the sun, and by noting the place where that line cuts the trajectory. For the vapour at the end of the tail, if it ascends straight from the sun, starts to ascend from the head, at which time the head was at the point of the intersection. But the vapour does not ascend straight from the sun, for by retaining the motion of the comet, that it had before it had its ascent, and by adding the motion of the ascent to the motion of the same, it ascends obliquely. From which the solution of the problem will be more realistic, so that the straight line, which cuts the orbit, shall be parallel to the length of the tail, or rather (on account of the curved motion of the comet) as the same may diverge from the line of the tail. With this agreed upon I found that the vapour, which was at the end of the tail on January 25th , had began to rise from the head before December 11th , and thus in its whole ascent it took more than 45 days. But all that tail which appeared on December 10th , ascended in an interval of these two days, which had elapsed from the time of the perihelion of the comet. Therefore the



vapour from the beginning in the vicinity of the sun rose most quickly, and afterwards from the motion retarded by its own gravitation proceeded always to ascend more slowly ; and in ascending increased the length of the tail : but the tail, as soon as it appeared, was made nearly completely from vapour, which arose from the time of the perihelion ; nor did that first part vanish, the vapor which first ascended, and of which the end of the tail was composed, until on account of its excess distance even though being illuminated by the sun, it is ceased to be seen by our eyes. Also the tails of other comets, which are shorter and do not ascent with a rapid continual motion from the heads and soon vanish, but instead the tails are permanent columns of vapours and exhalations, propagated from the heads most slowly over many days with a motion which, by sharing that motion with the head that it had at the start, together with the head they are able to move through the heavens. And hence again it is gathered that the space of the heavens is free of resistance ; just as in which not only the solid bodies of the planets and of comets, but also the rarest vapours of the tails go forwards most freely, and conserve the most rapid velocities of their motion for a long time. Kepler ascribed the ascent of the tails from the atmospheres of the heads of comets, and their progression in directions away from the sun, to the action of the rays of light dragging the material of the tail along with it. And it is not absurd to reason that the finest vapours are able to be carried along in spaces free from all resistance by the action of the rays, and it cannot therefore be by any other reason, although dense vapours on being impeded are unable to be propelled sensibly by the rays of the sun in our regions. Another astronomer has considered that it is possible to be given both light as well as heavy particles, and the matter of the tails to be light, and by their lightness to ascend from the sun. But since the weight of terrestrial bodies shall be as the matter in the bodies, and thus as the quantity of matter remains the same [thus, particles that levitate do not exist], the weight cannot be made greater or less. I suspect that ascent to arise rather from the rarefaction of the matter in the tail. Smoke ascends in a furnace by the force of the air on which it floats. That air rarefied by heat ascends, on account of the diminution of the specific gravity, and the mixed-up smoke rises with it. Indeed, why may the tail of the comet not rise from the sun in the same way? For the sun's rays do not disturb the medium, through which they pass, except by reflection and refraction. The reflecting particles are heated by this action and in turn will heat the aether wind with which they are mixed up. It in turn is rarefied by that communicated heat, and from that rarefaction on account of the diminished specific gravity by which it is drawn towards the sun, it rises and takes the reflecting particles with it from which the tail is composed. The vapours which compose the tails of comets turn around the sun, and tend as a consequence to travel away from that star, which again contributes to their ascension, for the atmosphere of the sun and the material of the heavens either clearly remains at rest, or turns more slowly than the matter in the tails, and because it turns by that motion which it receives from the rotation of the sun. These are the causes for the ascent of the vapours which form the tails of comets when they are in the vicinity of the sun's atmosphere, or where their orbits are more curved, and where the comets are present within the denser and for that reason thicker part of the sun's atmosphere, and as a consequence the more heavy, and they will send out the longest tails. For the tails, which then begin to appear conserve their motion and yet gravitate



towards the sun, moving around the sun in ellipses in the manner of the heads, and by that motion they will always accompany the heads although they adhere most freely to these. For the gravity of these vapours acting towards the sun can no more make the tail longer by falling later, any more than the weight of the head can effect an increase in the length of the tail by falling towards the sun. Thus falling at the same time with the common [acceleration of ] gravity either into the sun, or being retarded in their ascent in the same manner ; and thus gravity hardly impedes the heads and tails of comets ; so that they easily take and afterward most freely observe some position of the head and of the tail in turn from the causes described, or from any others. Therefore the tails, which arise in the perihelions of comets, will go off with their heads into remote regions, and either thence after a long series of years will return to us with the same, or perhaps there little by little they vanish by rarefaction. For later in the descent of the heads to the sun new and very short tails must be propagated from the heads in a slow motion, and immediately in the perihelions of these comets, which descent as far as the atmosphere of the sun, are increased into an immensity. For the vapour will be perpetually rarefied and dilated in these free spaces. From which account it comes about that every tail shall be wider at the upper extremity than next to the head of the comet. But from that rarefaction the vapour perpetually spreads out wider and is scattered through the whole heavens, then little by little it is attracted to the planets by their gravity, and it seems likely to think that it may become mixed with their atmospheres. For just as the seas are required for the entire constitution of this earth, and so that from these by the heat of the sun enough vapours may be excited, which either begins to fall as rain from clouds, and all the earth may be watered and nourished for the growing of vegetables ; or to be condensed onto the freezing tops of mountains (as some conjecture with reason) running off in springs and rivers : thus comets may seem to be required for the preservation of the seas and of the moisture on the planets, and from the exhalations and condensed vapours of which, whatever is consumed by vegetation and decay from the condensed vapours and is changed into dry ground, will be continually supplied and replaced. For all vegetation generally grows by means of humidity, then that in the large part is return to dry ground by rotting, and slime always falls to the bottom of fluids that are putrefying. Hence the mass of the dry earth thence is increased, and the liquids, unless they are supposed to be increased from elsewhere, must always decrease, and at last cease to be. Again I suppose that the spirit, which is the smallest but the most subtle and the most excellent part of our air, and that is necessary to give life to all things, comes especially from comets. The atmospheres of comets are diminished in their descent towards the sun by the tails extending out, and (that certainly in the direction facing the sun) are rendered narrower : and in turn in the recession of these from the sun, when now extends less in tails, they may be made bigger ; but only if Hevelius noted these phenomena correctly. But they appear smallest, when the heads now heated by the sun and which send off the longest and brightest tails ; and the nuclei are surrounded by smoke perhaps thicker and darker in the lower parts of the atmosphere. For all the smoke generated by the great heat is usually thicker and darker. Thus the head of the comet, that we are discussing (i.e., the one of 1680), at equal distances from the sun and from the earth appeared more obscure after it perihelion rather than before. For in the month of December when it was accustomed to



be compared to a star of the third order magnitude, but in the month of November with stars of the first and second orders. And those who saw both, described the first as the greater comet. For on November 19th, to a certain young man of Cambridge, this comet with its light appearing dull and somewhat obtuse, was equal to the light of Spica Virgo, and was shining clearer than later. And to Montanari on November 20th in the old style, the comet appeared greater than a star of the first magnitude, with a tail of two degrees length. And from letters of Mr. Storer, that have come into our hands, he remarked that its head in the month of December was very small, when the maximum and most brilliant tail was sent off, and that it departed in grandeur from that who had seen the comet in the month of November, before the rising of the sun. It was conjectured that the reason for this was because in the head at the start was more plentiful, and it was being used up little by little. In the same way it seemed to be seen, that the heads of other comets, which had emitted great and most brilliant tails, had appeared somewhat obscure and very small. For in the year 1668 on March 5th in the new style, in the seventh hour in the evening, the R. P. Valentinus Estancius, viewing in Brazil, saw the comet almost horizontal at sunset on a wintry day, with a very small head and barely conspicuous, but truly with the tail above shining in such a manner, that standing on the shore an image of this could be easily discerned reflected from the sea. The image truly produced was that of a splendid beam 230 in length, going towards the south west, and almost parallel to the horizontal. But yet the splendour lasted only three days, suddenly perceptibly decreasing ; and meanwhile with the decrease in splendour with an increase in the magnitude of the tail. From which also in Portugal it was said to have occupied almost a quarter of the sky (that is, 450) from the west to the east with a most significant extension: for still the whole appeared, with the head always hidden in these regions below the horizon. From the increase of the tail and the decrease of the splendour it was evident that the head was receding from the sun, and it was nearest to it at the beginning of its appearance, in the manner of the comet of the year 1680. One reads in the Saxon Chronicle that a similar comet appeared in the year 1106, the star of which was small and obscure (like that in the year 1680) but the tail of which had a great brilliance, and which extended from that like a giant tree trunk towards the North-East, as Hevelius had also from the monk Simeon of Durham. Initially it appeared in the month of February, and then around the evening, at the setting of the winter sun. Then truly from the position of the tail it is gathered that the head was in the vicinity of the sun. Matthew of Paris said, that it was about a single cubit from the sun, from the third hour [more correctly six (the square brackets here are Newton's comments] until the ninth hour it was sending out a long ray from itself. Such also was the most outstanding light from that comet described by Aristotle in Book I. Meteor, 6. The head of which in the first day was not conspicuous, because that had set before the sun or perhaps appearing in the sun's rays, but in the following day so much of it was to be seen as possible. As it was able to leave the sun by a small distance, and soon set also. On account of the excessive brightness [evidently of the tail] the head did not appear covered in fire, but with time proceeding (according to Aristotle) with the tail now less ablaze, the head of the comet returned to its shape. And the splendour of the tail extended to a third part of the sky [i.e., to 600]. But it appeared in winter time [in the fourth year of the 101st Olympiad] and



ascending as high as the belt of Orion, when it vanished. With that comet of the year 1618, which appeared from the rays of the sun with the most splendid tail, was seen to be a star of the first magnitude or greater, but many larger comets appear, which have shorter tails. Of these, some are said to have been the equal of Jupiter, others of Venus, or even of the moon. We have said that comets are a kind of planets revolving in very eccentric orbits around the sun: And just as with the planets which are not accustomed to have small tails, which are revolving closer to and around the sun in smaller orbits, but thus also comets, which fall closer to the sun at their perihelions, it may be seen to be agreed upon for this reason, are to be very much smaller, lest they disturb the sun too much by their attraction. Now the transverse diameters of the orbits and the periodic times of the revolutions, remain to be determined from the gathering together of comets returning in the same orbits after long time intervals. Meanwhile the following proposition may shed some light on this business.

PROPOSITION XLLI. PROBLEM XX11. To correct the trajectory found of a comet.

[Chandrasekhar gives explanations of these operations from p.530 onwards.]

Operation 1. The position of the plane of the trajectory is assumed, by the above proposition found ; and three places are selected from the observations of the comet designated the most accurate, and which are as far apart from each other as possible ; and let A be the time between the first and the second, and B the time between the second and the third. Moreover it is agreed that the comet has turned through its perigee at one of these places, or at least not to be far away from the perigee. From these apparent positions, three true locations of the comet are found by trigonometrical operations in that assumed plane of the trajectory. Then with these locations found, around the centre of the sun or the focus, by arithmetical operations, put in place with the aid of Prop. XXI. Book I, a conic section is described: and the areas of this, terminated by the rays drawn from the sun to the places found shall be D and E; clearly D shall be the area between the first and second observations, and E area between the second and the third. And T shall be the whole time, in which the whole area D E+ it must describe with the velocity found by Prop. XVI. Book 1. Operation. 2. The longitude of the nodes of the plane of the trajectory is increased, with 20' or 30' added to that longitude, which are called P; and the inclination of that to the plane of the ecliptic is maintained. Then from the three aforementioned observed places of the comet, three true points may be found in this new place, as above : then also with the orbit passing through these places, and the two areas of the same described between the observations, which shall be d and e, so that the total time in which the whole area d e+ must be describe shall be t. Operation 3. The longitude of the nodes is used in the first operation, and the inclination of the plane of the trajectory is augmented to the plane of the ecliptic, with 20' or 30' added to that inclination, which is called Q. Then from the aforesaid observations with the three apparent places of the comet found in this new plane with the three true places, and with the orbit passing through these places, and so that the two areas of the



same described between the observations, which shall be &δ ε , and the whole time τ , in which the whole area δ ε+ must be described. Now let 1 1 1 1= = and =gC G dA D

B E e, , , γ δε= ; and let S be the true time between the first and

third observations ; and with the signs + and – properly observed the numbers m and n are sought from the observations, from this rule, that there shall be 2 2G C mG mg nG nγ− = − + − , and 2 2T S mT mt nT nτ− = − + − . And if in the first operation I designates the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of one or the node, I nQ+ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K mP+ the true longitude of the node. And finally if in the first operation, and in the second, and third, the quantities R, r and ρ designate the latera recta of the trajectory, and the quantities 1 1 1

L l, , λ the transverse widths [or diameters] of the same respectively : R mr mR n nRρ+ − + − will be the true latus rectum, and 1

L ml mL n nLλ+ − + − the true width of the trajectory that the comet will describe. But with the transverse diameter given also the periodic time of the comet is given. Q.E.I. Otherwise the periodic times of revolution of comets, and the transverse diameters of the orbits, cannot be determined with enough accuracy, except by collating the comets among themselves, which appear at different times. If more comets, after equal intervals of time, are found to describe the same orbit, it will be concluded that all these are one and the same comet, revolving in the same orbit. And then at last from the given times of revolution the transverse diameters will be given, and from the diameters the orbits of the ellipses will be determined. To this end therefore the trajectories of several comets are required to be computed, from the hypothesis that they shall be parabolas. For trajectories of this kind always agree approximately with the phenomena. That made clear, not only from the parabolic trajectory of the comet of the year 1680, as well as with the observations brought together above, but also from the particular features of that comet, which appeared in the years 1664 and 1665, and had been observed by Hevelius. This he had worked out from the observations of the longitude and latitude of this comet, but less accurately. From the same observations our countryman Halley computed the places of this comet anew, and then finally from such positions found determined the trajectory of the comet. Moreover he found its ascending node in 210.13'. 55", the inclination of the orbit to the plane of the ecliptic 210. 18'. 40'', the distance of the perihelion from the node in the orbit 490.17'. 30". The perihelion in 80.40'. 30" with the latitude south from the centre of the sun 160.1'.45". The comet in perihelion on November 24th at 11h. 52'. p.m. London time, or London, or 13h.8' Danzig [Gdansk], in the old style, and latus rectum of the parabola 410286, with the mean distance of the earth from the sun being at the distance 100000. Since the proper places of the comet in this orbit agree with the computation, it will be apparent from the following table computed by Halley.



Apparent time in Gdansk, old style.

Observed distances of comet. Positions observed. Positions computed in Orbit.

December.

3d.18h.29' 12 from heart of Leo 46gr.24'.20''

'' Spica Virginis 22.52.10Long. 7gr. 1'. 0''Lat. south. 21.39. 0

7gr.1' .29'' 21.38.50

4.18.1 12 '' heart of Leo 46. 2.45

'' Spica Virginis 23.52.40Long. 16.15. 0Lat. south. 22.24. 0

0.16. 5 22.24. 0

7.17.48 '' heart of Leo 44.48. 0 '' Spica Virginis 27.50.40

Long. 3. 6. 0Lat. south. 25.22. 0

3. 7. 33 25.21.40

17.14.43 '' heart of Leo 53.15.15 '' right arm of Orion 45.43.30

Long. 2.50. 0Lat. south. 49.25. 0

2.50. 0 49.25. 0

19.9.25 '' Procyone 35.13.15 '' Bright star jaw Cetus 52.56. 0

Long. 28.40.30Lat. south. 25.48. 0

28.43. 0 45.46. 0

20.9.53 12 '' Procyone 40.49. 0

'' Bright star jaw Cetus 40. 4. 0Long. 13. 3. 0Lat. south. 39.54. 0

3. 5. 0 39.53. 0

21.9.9 12 '' right arm of Orion 26.21.25

'' Bright star jaw Cetus 29.28. 0Long. 2.16. 0Lat. south. 33.41. 0

2.18.30 33.39.40

22.9. 0 '' right arm of Orion 29.47. 0 '' Bright star jaw Cetus 30.29.30

Long. 24.24. 0Lat. aust. 27.45. 0

24.27. 0 27.46. 0

26.7.58 '' Bright star Aries 23.20. 0 '' Aldebaran 26.44. 0

Long. 9. 0. 0Lat. south. 12.36. 0

9. 2.28 12.34.13

27.7.58 '' Bright star Aries 20.45. 0 '' Aldebaran 28.10. 0

Long. 7. 5.40Lat. south. 10.23. 0

7. 8.45 10.23.13

28.7.58 '' Bright star Aries 18.29. 0 '' Hyades 29.37. 0

Long. 5.24.45Lat. south. 8.22.50

5.27.52 8.23.37

31.6.45 '' Girdle Androm. 30.48.10 '' Hyades 32.53.30

Long. 2. 7.40Lat. south. 4.13. 0

2. 8.20 4.16.25

Jan. 1665 7.7. 37 1

2 '' Girdle Androm. 25.11. 0 '' Hyades 37.12.25

Long. 28.24.47Lat. bor. 0.54. 0

28.24. 0 0.53. 0

13.7. 0 '' Girdle Androm. 28. 7.10 '' Hyades 38.55.20

Long. 27. 6.54Lat. north. 3. 6.50

27. 6.39 3. 7.40

24.7. 29 '' Girdle Androm. 20.32.15 '' Hyades 40. 5. 0

Long. 26.29.15Lat. north. 5.25.50

26.28.50 5. 26. 0

Feb. 7.8. 37

Long. 27. 4.46Lat. north. 7. 3.29

27.24.55 7. 3.15

22.8.46 Long. 28.29.46Lat. north. 8.12.36

28.29.58 8.10.25

Mar. 1.8. 16

Long. 29.18.15Lat. north. 8.36.26

29.18.20 8.36.26

7.8.37 Long. 0. 2.48Lat. north. 8.56.30

0. 2.42 8.56.56

In the month of February at the start of the year 665, the first star of Aries, that in what follows I will call γ , was at 180. 30'. 15" with the latitude to the North 70. 8'. 58". The second star of Aeries was at 290. 17'.18" with the latitude to the North 80. 18'.16''. And a certain other star of the seventh magnitude, that I will call A, was at 280.14'.45" with the latitude to the North 80.28'. 33". Now the comet on Feb. 7d.7h. 30' Paris time (that is Feb. 7d. 8h. 37 Gdansk) old style, made a triangle with the stars γ and A right-angled at γ . And the distance of the comet from the starγ was equal to the distance of the stars γ and A, that is 10.19'. 46" on a great circle, and therefore that was 10. 20'.26'' in the parallel of the latitude of the starγ . Whereby if from the longitude of the star γ the longitude 10.20'. 26" were taken away, the longitude of the comet 270. 9'. 49'' will remain. Auzout from his observation put this comet in 270. 0' roughly. And from the diagram, from which Hook had traced out its motion, that now was in 260.59'. 24". From the mean ratio I have put the same in 270. 4'. 46". From the same observation



Auzout now put the latitude of the comet to be 70 and 4' or 5' towards the North. The same would be more correctly 70. 3'.29", clearly with the difference of the latitude of the comet and the starγ , being equal to the difference of the longitudes of the starsγ and A. Feb. 22d. 7h. 30' London, that is Feb. 22d. 8h. 46' Gdansk, the distance of the comet from the star A, according to the observation of Hook from his own delineated scheme, and according to the figure of Petit traced after the observations of Auzout, it was the fifth part of the distance between the star A and the first of Aries, or 15" 57''. And the distance of the comet from the line joining the star A and the first of Aries was the fourth part of the same fifth part, that is 4'. And thus the comet was in 280.29'. 46", with the latitude North 80. 12'. 36". March 1d. 7h. 0' London time, that is March 1d. 8h. 16' Gdansk, the comet was observed near the second star of Aries, with the distance between the same being to the distance between the first and the second of Aries, that is to 10. 33', as 4 to 45 according to Hook, or as 2 to 23 following Gottignies. From which the distance of the comet from the second star of Aries was 8'. 16" following Hook, or 8' 5" following Gottignies, or in the ratio of the mean 8'. 10". Now the comet according to Gottignies had just progressed beyond the second star of Aries around a distance of a quarter or a fifth part of the journey completed in a single day, that is, around 1'. 35" (with which Auzout agreed well enough) or a little less following Hook, thing of 1'. Whereby if to the first longitude of Aries there is added 1', and to its latitude 8'. 10", the longitude of the comet will be found 290. 18', and the latitude to the North 80. 36'. 26". March 7d. 7h.30' Paris time (that is March 7d.8h. 37' Gdansk) from the observations of Anzout the distance of the comet from the second star of Aries was equal to the distance of the second star of Aries from the star A, that is 52'.29''. And the difference of the longitudes of the comet and of the second star of Aries was 45' or 46', or in the mean ratio of 45'. 30". And thus the comet was in 00. 2'.48". From the diagram of observations of Auzout, that Petit constructed, Hevelius deduced the latitude of the comet to be 80.54'. But the engraver curved the path of the comet irregularly at the end of the motion, and Hevelius in the diagram of Auzout's observations by himself corrected the irregular curvature, and thus the latitude of the comet was made to be 80. 55'. 30''. And by correcting the irregularity a little more, the latitude emerged to be 80. 56', or 80. 57'. Here the comet was also seen on the 9th day of March, and then it must have been located in 00.18', with a latitude North of about 90.3' 1

2 . This comet appeared for three months, and to have described almost six signs, and in one day it completed almost 200. Its course was deflected a great deal from a great circle, curved in the North ; and its backwards motion at the end was made direct. And not standing in the way of such an unusual course, the theory agreed with the observations no less accurately from start to finish, as the theory of the planets is accustomed to agree with the observations of these, as will be apparent on inspecting the table. Yet around two minutes are required to be taken away at first, when the comet was the travelling the fastest; that which can be done by taking away 12' from the angle between the ascending node and the perihelion, or by putting that angle in place to be 490. 27'. 18". The annual parallax of each comet has been most noticeable (both of this and of the above), and thus the annual motion of the earth in a great circle.



Also the theory is confirmed by the motion of the comet, which appeared in the year l683. This was backwards in orbit, its plane with the plane of the ecliptic contained almost a right angle. The ascending node (from Halley's computation) was in 230.23'; the inclination of the orbit to the ecliptic 830.11'; the perihelion in 250. 29'. 30"; the distance of the perihelia from the sun, 56020, with the radius of the great orbit being of magnitude 100000, and with the time of the perihelion July 2d.3h.50'. Moreover the locations of the comet in this orbit computed by Halley, and collated with the places observed by Flamsted, are shown in the following table. 1683 Observed mean time.

Place of Sun. Comp. Long. of comet

Comp.Lat. North.

Long.obs. of comet

Lat. North Observ.

Differ. Long.

Differ. Lat.

d. h. gr. ' '' gr. ' '' gr. ' '' gr. ' '' gr. ' '' Jul. 13.12. 5' 1. 2.30 13. 5.42 29.28.13 13. 6.42 29.28.20 +1'.0'' +0'.7''

15.11.15 2.53·12 11.37. 4 29.34. 0 11.39.43 29.34.50 +1.55 +0.50 17.10.20 4.45.45 10. 7. 6 29.33.33 10. 8.40 29.34. 0 +1.34 +0.30 23.13.40 10·38.21 5.10.27 28.51.41 5.11.30 28.50.28 +1. 3 –1.14 25.14. 5 12.35.28 3.27.53 24.24.47 3.27. 0 28.23.40 –0.53 –1. 7 31. 9.42 18. 9.22 27.55. 3 26.22.52 27.54.24 26.22.25 –0.39 –0.27 31.14.55 18.21.53 27.41. 7 16.16.57 27.41. 8 26.14.50 +0. 1 –2. 7

Aug. 2.14.56 20.17.16 25.29.32 25.16.19 25.28.46 25.17.28 –0.46 +1.9 4.10.49 22. 2.50 23.18.20 24.10.49 23.16.55 24.12.19 –1.25 +1.30 6.10. 9 23.56.45 20.42.23 22.47. 5 20.40.32 22.49. 5 –1.51 +2. 0 9.10.26 26.50.52 16. 7.57 20. 6.37 16. 5.55 20. 6.10 –2. 2 –0.27 15.14. 1 2.47.13 3.30.48 11.37.33 3.26.18 11.32. 1 –4.30 –5.31

16.15.10 3.48. 2 0.43. 7 9.34.16 0.41.55 9.34.13 –1.12 –0. 3 18.15.44 5.45.33 24·52.53 5.11.15 24.49. 5 5. 9.11 –3.48 –2. 4

South. South. 22.14.44 9.35.49 11. 7.14 5.16.58 11. 7.12 5.16.58 –0. 2 –0. 3 23.15.52 10.36.48 7. 2.18 8.17. 9 7. 1.17 8.16.41 –1. 1 –0.28 26.16. 2 13·31.10 24.45.31 16.38. 0 24.44. 0 16.38.20 –1.31 +0.20

Also the theory is confirmed for the retrograde motion of the comet, which appeared in the year 1682. The ascending node of this (from the computation of Halley) was in 2l0.16'. 30''. The inclination of the orbit to the plane of the ecliptic 17gr. 6'. 0". The perihelion in 20. 52'. 50". The distance from the perihelion to the sun 58328, with the radius of the great orbit proving to be 100000. And the time equal to the perihelion September 4d.7h. 39'. Truly the places computed from the observations of Flamsted, and collated with the places computed by the theory, are shown in the following table.

1682. Observed mean time.

Position of the sun.

Long.Comp. of comet.

Comp. Northern Lat.

Obs. Long. of comet.

Observ. Northern Lat.

Differ. Long.

Differ. Lat.

d. h. gr. ' '' gr. ' '' gr. ' '' gr. ' '' gr. ' '' ' '' ' '' Aug. 19.16.38 7. 0. 7 18.14.28 25.50. 7 18.14.40 25.49.55 –0.12 +0.12 20.15.38 7.55.52 24.46.23 26.14.42 24.46.22 26.12.52 +0. 1 +1.50 21. 8.21 8.36.14 29.37.15 26.20. 3 29.38. 2 26.17.37 –0.47 +2.26 22. 8. 8 9.33.55 6.29.53 26. 8.42 6.30. 3 26. 7.12 –0.10 +1.30 29. 8.20 16.22.40 12.37.54 18.37.47 12.37·49 18.34. 5 +0. 5 +3.42 30. 7.45 17.19.41 15.36. 1 17.26.43 15.36.18 17.27.17 +0.43 –0.34 Sept. 1. 7.33 19.16. 9 20·30·53 15.13. 0 20·27· 4 15. 9.49 +3.49 +3.11 4. 7.22 22.11.28 25.42. 0 12.23.48 25.40.58 12.22. 0 +1. 2 +1.48 5. 7.32 23.10.29 27. 0.46 11.33. 8 26.59.24 11.33.51 +1.22 –0.43 8. 7.16 26. 5.58 29.58.44 9.26.46 29.58.45 9.26.43 –0. 1 +0. 3 9. 7.26 27. 5. 9 0.44.10 8.49.10 0.44. 4 8.48.25 +0. 6 +0.45



Also the theory for the retrograde motion of the comet is confirmed, which appeared in the year 1723. The ascending node of this (being computed by Mr. Bradley, Savilian professor of astronomy at Oxford) was in 140.16'. The inclination of the orbit to the plane of the ecliptic 490. 59'. The perihelion in 120. l5'. 20". The distance of the perihelion from the sun 998651, with the radius of the great orbit taken to be 1000000, and the corrected time of the perihelion to be September. 16d. 16h. 10'. Indeed the places calculated in this orbit by Bradley, and the places collated with the observations by our compatriots Dr. Pound and Dr. Halley are shown in the following table. 1723 Observed mean time.

Observed. Long. of comet.

Observ.Lat.North.

Comput. Long. of comet.

Comput. Lat. of comet..

Differ. Long.

Differ. Latit.:

Octob. 9d. 8h. 5' 70.22'.15'' 50. 2'. 0'' 70.21'.26'' 50. 2'.47'' +49'' – 47'' 10. 6 .21 6.41.12 7.44.13 6.41.42 7.43.18 – 50 + 55 12. 7.22 5.39.58 11.55. 0 5.40.19 11.54.55 – 21 + 5 14. 8.57 4.59.49 14.43.50 5. 0.37 14.44. 1 – 48 – 11 15. 6.35 4.47.41 15.40.51 4.47.45 15.40.55 – 4 – 4 21. 6.22 4. 2.32 19.41.49 4. 2.21 19.42. 3 + 11 – 14 22. 6.24 3.59. 2 20. 8.12 3.59.10 20. 8.17 – 8 – 5 24. 8. 2 3.55.29 20.55.18 3.55.11 20.55. 9 + 18 + 9 29. 8.56 3.56.17 22.20.27 3.56.42 22.20.10 – 25 + 17 30. 6.20 3.58. 9 22.32.28 3.58.17 22.32.12 – 8 + 16 Nov. 5. 5.53 4.16.30 23.38.33 4.16.23 23.38. 7 + 7 + 26 8. 7. 6 4.29.36 24. 4.30 4.29.34 24. 4.40 – 18 – 10 14. 6.20 5. 2.16 24.48.46 5. 2.51 24.48.16 – 35 + 30 20. 7.45 5.42.20 25.24.45 5.43.13 25.25.17 – 53 – 32 Dec. 7.6.45 8. 4.13 26.54.18 8. 3.55 26.53.42 + 18 + 36

From these examples it is made abundantly clear that the motion of comets set out by our theory are shown no less accurately, than the motion of the planets are accustomed to be shown by the same theories. And therefore the orbits of comets can be enumerated by this theory, and the periodic time of the comet revolving in some orbit finally becomes known, and then at last the transverse width of the elliptic orbits and the altitudes of the aphelions become known. The retrograde comet, which appeared in the years 1607, described an orbit, the ascending node of which (from Halley's computation) was in 120. 21' ; the inclination of the plane of the orbit to the plane of the ecliptic was 170. 2 ' ; the perihelion was in 20.l6'; and the distance of the perihelion from the sun was 58680, with the radius of the great circle taken to be 100000. And the comet was in perihelion in October 16d.3h.50'. This agreed approximately with the orbit of the comet which appeared in the year 1682. If these two were one and the same, this comet was revolving in a time of 75 years, and the major axis of its orbit will be to the major axis of the great orbit of the earth, as 3 75 75 to 1× , or around 1778 to 100. And because the distance of the aphelion of this comet from the sun, will be to the mean distance of the earth from the sun, almost as 35 to 1. With which understood, it would not be difficult to determine the elliptic orbit of this comet. And thus this will itself come about if the comet, in the interval of 75 years, may be returned hereafter in this orbit. The remaining comets are seen to be revolving with a greater time and to ascend higher. The remaining comets, on account of the great magnitude of their numbers, and the great distance of the aphelion from the sun, and the long delay at the aphelions, by gravity must disturb each other , and both the eccentricities and times of revolution at some times



are increased a little, at other times diminished. Hence it cannot be expected that the same comet will return accurately in the same orbit, and with the same periodic time. It will suffice if no greater changes come about, than may arise from the aforesaid causes. And hence the reason is provided, why comets are not restrained to the Zodiac in the manner of planets, but depart thence and by various motions be carried to all regions of the heavens. Evidently to that end, so that at their aphelions, where they are moving the slowest, as they shall be at great distances from each other so that their mutual attraction will not be experienced. It is by that reason why the comets which descend from the highest, and which as a consequence moving the most slowly at their aphelions, must return to the highest places. The comet which appeared in the year 1680, was at a shorter distance from the sun at its perihelion than a sixth part of the diameter of the suns ; and therefore because of the maximum velocity in that vicinity, and some density of the sun's atmosphere, it must have experienced some resistance, and be retarded by a small amount, and to approach closer to the sun : and by getting closer to the sun in the individual revolutions, finally it will fall into the body of the sun. But at the aphelions where it is moving the slowest, by the attraction of other comets it can be retarded, and suddenly fall into the sun [at its next approach]. Thus also the fixed stars, which exhaust themselves little by little from light and vapours, are able to renew themselves by the comets that fall into them, and rekindled by new fuel arise as new stars. Fixed stars are of this kind, which suddenly appear, and at the start with maximum brilliance, and subsequently vanish little by little. Such was the star that Cornelius Gemma barely saw on the quiet night of 8th of November 1572 in the chair of Cassiopeia, illuminating that part of the heavens ; but the following night (November 9th ) it was seen most brilliant among all the fixed stars, and with its light scarcely conceding to the light from Venus. Tycho Brahe saw this on the eleventh day of the same month when it was maximally brilliant ; and from that time decreasing little by little and in the space of the sixteen months it was observed to be vanishing. In the month of November, when it first appeared, it was equal to the light from Venus. In the month of December diminished a certain amount it was seen to equal the light of Jupiter. In the year 1573, in the month of January it was less than Jupiter and greater than Sirius, to which at the end of February and the beginning of March it emerged equal. In the month of April and in May it was a star of the second magnitude, in June, July and August it was equal to stars of the third magnitude, September, October and November to stars of the fourth magnitude, December and in the month of January of the year 1574, to stars of the fifth magnitude, and in the month of February it was seen equal to stars of the sixth magnitude, in the month of March it vanished from sight. In the beginning the colour was clear, white and very bright, afterwards yellow, and in the month of March in the year 1573 reddish in the image of Mars or of the star Aldebaran; but in May it took on a bluish- white appearance, such as we see in Saturn, which colour it maintained until the end, when it was made more obscure, yet always becoming more obscure. Such also was the star in the right foot of the Serpent, that students of Kepler saw to appear initially on the 30th day of September in the year 1604, in the old style, and by its light surpassed Jupiter, when in the preceding night it had been barely apparent. From that truly in a little time it decreased, and in the space of fifteen or sixteen months it had vanished from view. It was a new star of this kind which appeared so brilliant to the



fixed stars, in the time of Hipparchus, that persuaded him, it is said [as reported by Pliny], to observe the fixed stars, and to give them in a catalogue. But the fixed stars, which in turn appear and vanish, and which increase in brightness little by little, and by their light scarcely surpass any fixed stars of the third magnitude, are seen to be stars of another kind and are seen in turn revolving with one part bright and another part obscure. But vapours, which arise from the sun and from the fixed stars and from the tails of comets, can fall by their gravity into the atmospheres of planets and there to be condensed and converted into water and humid spirit, and in the end by a slow heat to be changed little by little into salts and sulphur, tinctures, slime and mud, clay and sand, stones and coral, and other terrestrial substances.

GENERALE SCHOLIUM . The hypothesis of vortices is beset with many difficulties. Since each and every planet by a ray drawn to the sun describes areas proportional to the times, the periodic times of the parts of the vortex must be in the square ratio of the distances from the sun. In order that the periodic times of the planets shall be in the three on two proportion of the distances from the sun, the periodic times of the parts of the vortices must be in the three on two proportion of the distances. In order that the smaller vortices around Saturn, Jupiter, and the other planets are able to be kept rotating and quietly swimming in the vortex of the sun, the periodic times of the parts of the vortex of the sun must be equal. The revolutions of the sun and planets about their axes, which must agree with the motions of the vortices, disagree with all these proportions. The motions of comets are the most regular, and they observe the same laws of motion as the planets, and are unable to be explained by vortices. Comets are carried in extremely eccentric motions into all parts of the heavens, which cannot happen, unless vortices are renounced. Projectiles, in our air, only experience the resistance of the air. In the most subtle air, as it becomes in a Boyle vacuum, the resistance stops, accordingly a fine feather and a gold solid fall together with the equal velocity in this vacuum. And equal is the account of celestial spaces, which are above the atmosphere of the earth. All bodies must be able to move most freely in these spaces ; and therefore planets and comets to be revolving perpetually in orbits, given in kind and position, following the laws set out above. Indeed they will persevere in their orbits by the laws of gravity, but the original situation of the orbits cannot be accounted for by these laws. The six principal planets are revolving around the sun in concentric circles, moving in the same direction, approximately in the same plane. Ten moons are revolving around the earth, Jupiter, and Saturn in concentric circles, moving in the same direction, in approximately the planes of the planets. And all these regular motions do not have their origins from the causes of mechanics ; if indeed the comets are in very eccentric orbits, and are sent freely into all parts of the heavens. By which motions comets generally cross the orbits of the planets quickly and easily, and into their aphelions where they are slower and they remain there for a long time ; since they are most distant from each other, so that they attract each other minimally. This most elegant structure of the sun, planets and comets could only arise from the planning and in the dominion of an intelligent and



powerful being. And if the fixed stars shall be the centres of similar systems, all these likewise constructed under the same plan in the dominion of One: especially since the light of the fixed stars shall be of the same nature as the light from the sun, and all the light is sent from all the systems into every system in turn. And so that the systems of fixed stars do not fall into each other by gravitation, here in turn the same immense distance has been put in place. This universal presence rules not as the soul of the world, but as the master of everything. And therefore he is usually called Παντοχράτωρ the master of all his dominion. For deus [god] is a relative word and is referring to servants: and the deity is the master of the gods, not of a special body, as they believe in which deus is the soul of the world, but only of servants. The Master is an eternal, infinite, absolutely perfect being: but some perfect being is not the master god without a dominion. For we may say my deus [god], your deus [god], the god of Israel, god of the gods, and master of the masters : but we cannot say my eternal one, your eternal one, the eternal one of Israel, the eternal one of the gods ; nor can we say my infinite one, your perfect one. These names do not have a relation to servants. The word deus actually signifies master [dominum]: but every master is not a god. [I have omitted Newton's note on the origin of this: see Cohen p.941 if interested.] The dominion of a spiritual being constitutes god; indeed the true god come from true gods, the greatest god from the greatest, an imaginary god from the imaginary. And from the true domination it follows that the true god is alive, intelligent and powerful ; and from his remaining perfections to be the greatest, or the peak of perfection. He is eternal, infinite, omnipotent, omnipresent, and omniscient, that is, enduring from infinity to infinity, and present from the infinite to the infinite : ruling all; and knowing everything, which can arise or not able to arise. Not eternity and infinite but eternal and infinite ; not duration or space, but enduring and present. He always, and is present everywhere, and by existing always and everywhere, constitutes duration and space. Since any small particle of space shall be always, and any indivisible instant everywhere, certainly the maker and master of all things cannot be missing at any time or place. Every perceiving soul who experiences in different times, with different senses, and by the motion of several organs, is still always the same indivisible person. There are these successive parts in the duration [of a person's experiences of his existence], and these parts coexisting in space : there is nothing [in ordinary space and time] that has any resemblance to that which constitutes the person that is the man, nor in his thinking principle [I have relied here on Madame du Chatelet's translation]; and there will be much less than in the thinking substance of god. Every man, whatever thing he is thinking about, is one and the same man during his life from all the individual sense organs. God is one and the same god always and everywhere. Omnipresence is not by virtue alone, but also by substance : for virtue without substance cannot exist. Everything is moved generally and containing the person himself, but not without the some action from other to be experienced by him. God suffers nothing from the motions of bodies : they experience no resistance from the omnipresence of god. It has been confessed that it is necessary that the supreme god must exist : and by the same necessity it is always and everywhere. From which also the whole being the same to him, all eyes, all ears, all mind, all arms, all the strength of feeling, of intelligence, and of action, but in a manner not human, still less than with a body, and in a manner completely unknown to us. As a blind man has no idea



of colours, thus we have no ideas of the ways in which the wisest god experiences and knows all things. Without any body and bodily shape, and thus unable to be seen, nor to be heard, touched, or worshipped under any kind of thing related to the body. We have ideas of his attributes, but that shall be of other substances unknown to us. We can see only the figures and colours of bodies, we hear only their sounds, we touch only their external surfaces, we smell only their odours, and we taste only their flavours: but as for the insides of substances, we know them not by any sense, nor by any reflection ; and we have far less idea about the substance of god. This we know only by his properties and attributes, and by the wisest and most optimal structures of things, and by their final causes, these we admire on account of their perfections ; moreover they are venerated and worshipped on account of his dominion. For we adore as servants, and god without a dominium, providence, final causes, is nothing else but fate and nature. From blind necessity metaphysics, which certainly is the same always and everywhere, no variation of things arises. The diversity which reigns over everything, whatever the times and the places , by necessity can come only from the ideas and will of an existing being. But it is said that god allegorically can see, hear, and that he can laugh, love, hate, desire, give, take, enjoy, be angry, fight, fabricate. For everything that one can say about god is taken from some comparison with human things; but these comparisons, although they are very imperfect, yet have some likeness. And thus concerning god, from which certainly from the difference of phenomena, pertain to natural philosophy. At this point I have established the phenomena of the heavens and of our seas through the force of gravity, but I have not assigned the cause of gravity. Certainly this force arises from some cause, which penetrates as far as to the centre of the sun and of the planets without diminution of strength; and which acts not only on the quantity of particles on the surface, on which it acts, (as they are the customary mechanical causes) but also on the quantity of the solid material ; and the action of this is extended thence over immense distances, always by decreasing in the inverse square of the distances. Gravity in the sun is composed from the gravity of the individual particles of the sun, and by receding from the sun it decreases accurately in the square ratio of the distances as far as to the orbit of Saturn, as that is evident from the quiet of the aphelion of the planets, and as far as to the final aphelions of comets, but only if these aphelions are at rest. Indeed I have not yet been able to deduce an account of these pleasing properties from the nature of the phenomena, and I devise no hypothesis. For whatever cannot be deduced from phenomena, it is required to call hypothesis ; and hypothesis, whether it be of some metaphysical, physical, occult, or mechanical qualities, have no place in experimental philosophy. In this philosophy the propositions are deduced from the phenomena, and rendered general by induction. Thus the impenetrabilities, mobilities, and the impetus of bodies and the laws of the motions and of gravity have become known. And it is enough that gravity actually exists, and acts according to the laws set forth by us, and it is sufficient to explain all the motions of heavenly bodies and those of our seas. Now this will be the place to add something on this most subtle kind of spirit that penetrates through all solid bodies, and which is hidden in their substance ; it is by this force and the action of this spirit that the particles of bodies attract each other mutually to the smallest distances, and are made to stick together ; and it is by this same means that electrified bodies are acted on at greater distances, both by repelling as well as attracting



small bodies in the vicinity ; and the light is emitted , reflected, refracted, inflected [i.e. internally reflected], and the bodies heated; all the sensations are excited, and the members of animals are moved according to its will, evidently by the vibrations of this spirit through the solid filaments of the nerves from the external sense organs to the brain and propagated to the muscles. But these are things that cannot be explained in a few words; nor do we have a sufficient supply of experiments, by which the laws of action of this spirit must be accurately determined and shown.

The End.

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Philosophiæ Naturalis Principia Mathematica