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Descartes' Laws of MotionAuthor(s): Richard J. BlackwellReviewed work(s):Source: Isis, Vol. 57, No. 2 (Summer, 1966), pp. 220-234Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/227961 .

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Descartes' Laws of Motion

By Richard J. Blackwell *

D1 ESCARTES IS OFTEN said to be the first to have formulated a clear statement of the principle of inertia. As early as 1633 in a work

entitled Le monde Descartes had worked out a theory of motion which ascribes to each body of the universe the power either to remain at rest or to continue in motion in a straight line. However, this treatise was withheld from publication since it advocated a heliocentric astronomy at the height of the Galileo controversy. Le monde was finally published posthumously in 1664. Despite this postponement, Descartes' views on motion continued to develop and received their first presentation to the public with the appearance of his Principles of Philosophy in 1644. This work sidestepped the

controversy over Copernicanism with a thinly veiled appeal to the relativity of motion which apparently satisfied Descartes' hesitancy to become involved in the great debate between the astronomers and the theologians of his day. At any rate the Principles of Philosophy presents the same three laws of nature 1 which had first been worked out in Le monde, along with a much fuller explanation of their metaphysical background and physical conse-

quences. The claim that Descartes was the first to formulate the principle of inertia

is based on the first two of his laws of nature, which read as follows: 2

The First Law of Nature: Each thing, insofar as in it lies, always perseveres in the same state, and when once moved, always continues to move.

The Second Law of Nature: Every motion in itself is rectilinear, and therefore things which are moved circularly always tend to recede from the center of the circle which they describe.

Now compare this with Newton's first law of motion, which is the classic statement of the principle of inertia:

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.3

* Saint Louis University. 2 Principia philosophiae (Amsterdam, 1644), 1The main point of difference is that the II, 37, 39.

order in which the last two laws are presented 3 Isaac Newton, Mathematical Principles of is reversed in the two works. In this paper Natural Philosophy, Motte translation revised we will follow the sequence given in the Prin- by F. Cajori (Berkeley: Univ. California Press, ciples of Philosophy since this is the more 1962), Vol 1 p. 13 logical ordering of the three laws.

Isis, 1966, VOL. 57, 2, No. 188. 220



DESCARTES' LAWS OF MOTION

Both of these statements of scientific law describe basically the same state of physical affairs. A body at rest will stay at rest in the absence of an external influence; a body in motion will continue always in motion in a

straight line in the absence of an external influence. Let us call this the

descriptive meaning of the principle of inertia. At this level Descartes has a legitimate claim to priority in the formulation of the principle of inertia. But the meaning of a scientific law includes much more than a description of the physical state of affairs. Also involved is an understanding of why the designated physical state of affairs obtains. Let us call this latter factor the theoretical meaning of a scientific law, referring to the entire conceptual apparatus which explains the physical event under discussion. Now it is

quite conceivable that two versions of a scientific law may agree at the level of descriptive meaning but not at the level of theoretical meaning. In this case the two versions are not really " the same law." For example, Kepler and Newton would agree in regard to the actual paths described by the

planets around the sun, but not in regard to why this is the case. When

Kepler's laws were subsumed within Newtonian physics, they acquired a new theoretical meaning.4

The purpose of this paper is to determine the theoretical meaning of Descartes' version of the principle of inertia. Does this differ from the theoretical meaning of Newton's first law? To answer this question it will be necessary to investigate the conceptual framework within which Descartes sets his discussion of the problem. Fortunately Descartes is rather explicit on this issue. He carefully worked out the metaphysical preliminaries which he thought were needed as a base for his laws of nature. He also developed a complicated theory of collision (his seven rules of impact), which illustrates the use of his laws at the concrete level. From this data we should be able to establish the theoretical meaning of his version of the principle of inertia.

Our task is complicated, however, by the fact that his rules of impact are almost completely erroneous. (This is perhaps already an indication that the theoretical meaning of his inertial principle is considerably different from Newton's.) As Alexandre Koyre remarked, the obvious falsity of the Cartesian rules of impact has led the history of science to reject them out of hand without sufficient examination of their systematic role within Cartesian physics.5 It will not be our purpose here to assess the empirical validity of Descartes' rules of impact.6 Our approach rather will be to assume that they are strictly logical consequences of his more basic three laws of nature. If these logical connections can be worked out, then the factors standing behind the rules of impact provide a tool for evaluation of the theoretical meaning of Descartes' version of the principle of inertia. In short, by granting the logical consistency of the laws of nature and the

4 For a convenient summary of the Keplerian Hermann, 1939), Vol. 3, p. 176. theory of celestial motion and its differences 6 For Tannery's comparison of the Cartesian from the Newtonian view, see J. L. E. Dreyer, and Newtonian analyses of the seven cases A History of Astronomy from Thales to Kepler involved, see (Euvres de Descartes, eds. Charles (New York: Dover, 1953), pp. 393-400. Adam and Paul Tannery (1st ed. Paris: Cerf,

5 Alexandre Koyre, Etudes Galileennes (Paris: 1897-1910), Vol. 9, pp. 327-330.

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RICHARD J. BLACKWELL

rules of impact as Descartes presents them, the overall conceptual framework which establishes their theoretical meaning should emerge. With this in mind, then, let us turn to Descartes' first two laws of nature.

THE BACKGROUND OF THE FIRST TWO LAWS OF NATURE

The Cartesian physical universe is generated out of two principles: matter and motion. Once these two factors are given by the divine act of creation, nothing more is needed for the evolution of the endless variety and differ- entiation on display before us in the universe. But there is a very unequal distribution of labor between these two principles. The sole contribution of matter is to provide the spatial extendedness of physical bodies. This is the consequence of Descartes' identification of the essence of material substance with extension.7 Considered in itself, matter is nothing more than three-dimensional extension, or spatial volume. All of the other properties which common sense attributes to material bodies are really due to the motions of the particles of matter among themselves and in relation to man's

organs of sense perception. Furthermore, the only type of motion involved here is local motion, defined as the translation of a particle of matter from one set of contiguous neighbors to another.8 Thus spatial extension and local motion generate the universe and constitute its proper explanatory principles. This is indeed a mechanistic philosophy in its fullest sense. Within this mechanism local motion is asked to bear a terrifically heavy burden: everything in the universe other than extension is due to local motion. It remains to be seen whether Descartes' theory of motion can hold up under this strain.

Granting these two principles, how do they function in the generation of the universe? What governs their interrelations so that a variable yet stable and intelligible universe is produced? Koyre has located the key issue here when he points out that " The supreme law of the Cartesian universe is the law of persistence."

9 Considered in itself, each body of the universe is endowed with the power to maintain its own extension and motion. If left to itself, each body will continue always to persist in its given state of extension and motion. Why is this so? What justifies this law of persistence? If we could address these questions to Descartes, he would undoubtedly tell us that they must be answered at the metaphysical, not the physical, level. At the first moment of creation God brought into being a certain amount of matter and a certain amount of motion and rest. But the divine creation was not over and done with at that first moment; rather, the universe is

continually created at each moment of its history. Divine conservation is an ongoing process as long as the universe is to exist. If we are not to deny the stability and immutability of God in his relation to the physical universe, we must admit that He always conserves the same amount of matter and motion which He originally created. Descartes never tires of emphasizing this point, and he makes it abundantly clear that the immutability of God

7 Principia philosophiae, II, 4. 8 Ibid., 25. 9 Koyre, op. cit., Vol. 3, p. 159.

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is the main justification for his first two laws of nature.10 Persistence in extension and persistence either in rest or in motion in a straight line is a divine guarantee to each body in the universe. If these bodies are to inter- act with each other, as they must, especially in a plenum universe, the law of persistence must still be observed at the overall level in the conservation of matter and motion.

This is a curious development in the history of science. Conservation

principles will be destined to play a major role in the formulation and

subsequent history of Newtonian physics. And here at the foundation of Cartesian physics we also find an essential role played by the notion of conservation. The theoretical differences, however, are of major propor- tions. If it is legitimate to say that Descartes' emphasis on conservation has contributed to the origins of modern physics, it must be noted that for Descartes the word "conservation" refers more to something present in God (i.e., immutability) than to anything properly inherent in the structure of the material universe. In short, the conservation of matter and motion is fundamentally a metaphysical law in Cartesian thought. However, its physical counterpart has proven to be a very fertile concept in the history of physics. Scientific concepts have a habit of developing in very devious ways, and Descartes' contribution to the origin of the conservation laws is perhaps not as negligible as it might at first seem.

Within this background the theoretical meaning of Descartes' first two laws of nature emerges rather clearly. The first law states that if a body is at rest, it will forever stay at rest unless disturbed by an external influence; a body in motion will forever continue to move unless disturbed by an external influence. But the reason for this persistence is not located in the structure of either matter or motion. If these two were the only elements to be considered here, the law of persistence quite possibly might not obtain in the Cartesian universe. The first law does obtain because of the immutability of God in His conservation of the same quantity of matter and motion in the universe. The persistence involved here is first and foremost the persistence of God.

The second law of nature states in effect that a moving body will forever continue to move in a straight line unless disturbed by an external influence. The reason for this is once again the divine immutability rather than the structure of matter or motion. Let us permit Descartes to speak for himself:

God conserves motion precisely as it is at the moment of time in which He conserves it and independently of the motion which chanced to occur a little bit earlier. And although no motion occurs in an instant, nevertheless it is clear that whatever is moved, in the individual instants which can be designated while it is moving, is determined to continue its motion on further in a straight line but never in any kind of a curved line.11

Descartes illustrates the above with an example of a stone being twirled in a sling. At each point of its motion the stone would fly off on a tangent

lo Principia philosophiae, II, 37, 39.

223

il Ibid., 39.




to the circle if it became freed from the sling. So far, so good. But the reason for this is that God conserves its motion independently of any previous motion, and as Descartes says, ". . . none of this curvature can be said to remain in it...." What this really means is that the divine immutability would be imperiled if the stone at any given point of its motion could continue on in any path other than a straight line. Any such irregu- larity would signal a change in the divine act of conservation. Also involved here is an atomistic view of time and a commitment that a straight line is more perfect than a curved line. But Descartes' main concern is to argue that the immutability of God guarantees that a moving body will continue its motion in a straight line.

THE THIRD LAW OF NATURE AND ITS COROLLARIES

The first two laws of nature deal with only one body. Descartes has told us that each body perseveres in its state of motion or of rest and that a moving body tends to continue its motion in a straight line. The situation must now be expanded to consider the interactions of a plurality of bodies, each one of which observes the first two laws. The problem at hand is to examine the phenomena of collision; this is the task of Descartes' third law of nature and its corollaries. The simplest case of collision involves only two bodies, each of which is assumed to be perfectly elastic, or as Descartes puts it, perfectly hard.

Like the first two laws, this final law of nature is based on an appeal to the immutability of God. The same quantity of motion must be constantly maintained in the universe if the unchangeability of God is to be affirmed. However, if the same bodies always possessed the same quantity of motion which they had received from God at creation, the universe would be totally regular and without novelty. But the various bodies of the universe, and especially of a plenum universe, constantly collide with each other and impart motion to each other. The principle of the conservation of motion is, however, perfectly well preserved if a balanced communication of motion occurs between the various parts of the universe. The conditions under which this mutual sharing of the available motion occurs are ex- pressed in the third law:

The third law of nature is this. If a moved body collides with another, then if it has less force to continue in a straight line than the other body has to resist it, it will be deflected in the opposite direction and, retaining its own motion, will lose only the direction of its motion. If it has a greater force, then it will move the other body along with itself and will give as much of its motion to that other body as it loses.12

The basic principles standing behind Descartes' rules of impact are contained in this law. We are told that each body possesses a force of perse- verence in its state of motion or of rest and that each body possesses a force of resistance to the motions of other bodies. Within any given body these

12 Ibid., 40.

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two forces are identical. To determine the result of a collision between two bodies, one need only to calculate the force of persistence in the first body, the force of resistance in the second body, and then compare these two forces. The larger force will overcome the smaller one, and the result of the collision is determined accordingly.

How are these forces to be calculated? What determines the quantity of resistance in the second body? Examination of Descartes' rules of impact reveals that two factors are involved here: namely, size and velocity.13 The larger a body is, the greater is its persistence in motion or in rest, and the greater is its resistance to another body. The faster a body moves, the greater is its persistence in motion, and the greater is its resistance to another body. Since in any given body the force of its persistence in its state of motion or rest is equal to its force of resistance to another body, only one calculation need be made for each of the two bodies involved in a collision. If a body is in motion, its persistence in that state is measured by its size and its velocity. Let us call this its quantity of motion.14 If a body is at rest, its persistence in that state is measured by its size and the velocity of another body of equal size. Descartes explicitly says that " a body at rest gives more resistance to a larger velocity than to a smaller one in proportion to the excess of the one velocity over the other." 15 Since a body at rest can adjust its resistance to any exterior velocity in a body of equal size, its persistence in rest is reduced to a function of its size. Let us call this its quantity of rest.16

This latter concept strikes a discordant note to the modern ear. Since the time of Newton we have become accustomed to think in this context of the concept of mass, or as Newton puts it, the " force of inactivity " (vis inertiae), to understand why a body at rest resists being put in motion. As we shall see more clearly, however, Newtonian mass is a quite foreign concept to Cartesian physics; this will have major repercussions on the precise meaning of Descartes' version of the principle of inertia. Nevertheless, Descartes is well aware that a body at rest tends to stay at rest and resists being put in motion. If such bodies are to enter meaningfully into his account of collision, he must have some way of calculating their resistance to motion. Hence he uses the notion which we have called the quantity of rest. This

13 In the Latin this is celeritas, in the French vitesse. It should be noted, however, that for Descartes velocity is a scalar and not a vector

quantity. He is aware of the directional

aspect of motion and tries to take account of it in his laws of impact. But he does not look

upon direction as a component of velocity considered as a vector. As a result, his law of the conservation of motion demands that the same scalar quantity of motion be preserved but not the same vector quantity. This is one of the sources of difficulty in his rules of

impact. Throughout this paper the term

"velocity" will be used to refer to the scalar

quantity of motion only. 14 P. Mouy (Le d,eveloppement de la

physique Cartesienne [Paris: Vrin, 1934], pp.

20 ff.) takes Descartes' "quantity of motion" to mean momentum (mv). This in effect attributes to Descartes the concept of mass. But this seems contrary to both his definition of matter as extension and the details of his rules of impact. For Descartes the quantity of motion is volume times velocity, not mass times velocity. Descartes is not a Newtonian on this point.

15 Principia philosophiae, II, 49. 16 For a very helpful analysis of the meaning

of Descartes' notion of rest and its systematic role in Cartesian physics, see M. Gueroult, " Metaphysique et physique de la force chez Descartes et chez Malebranche," Revue de metaphysique et de morale, 1954, 59:1-37, 113- 134; Koyre, op. cit., Vol. 3, p. 180.

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quantity is designated as a function of size or spatial volume in a physics whose only principles are extension and motion. The same type of New- tonian criticism might be directed toward the Cartesian concept of the quantity of motion; Descartes looks upon this as a function of velocity and size or spatial volume, not velocity and mass. Scientific concepts develop slowly and painfully and not without blind alleys. Be that as it may, the

quantities of persistence and resistance which enter into Descartes' rules of impact are determined by the various sizes and velocities of the bodies involved.

With this in mind, the specific content of Descartes' third law can now be put in focus. If the persistence of the first body is overcome by the resistance of the second body, then the first body will have the direction, but not the quantity, of its motion changed. From this it follows that the

quantity of motion or of rest in the second body remains unchanged and unaffected by the collision. In his rules of impact Descartes draws out the consequences of this first part of his third law with complete, logical rigor. The factual errors of the rules of impact are rooted largely in the first part of the third law. The second part of the law states that if the persistence of the first body overcomes the resistance of the second body, then both the direction and the velocity of the second body will be changed in such a proportion as to observe the principle of the conservation of the quantity of motion. Descartes claims that this third law contains all the principles that are needed to determine the outcome of various cases of collision.17 All that is required is the actual calculation of the forces involved in any given case. Once this is done the result of the collision can be deduced. In Le monde Descartes leaves this to the ingenuity of his reader.l8 However, in the Principles of Philosophy he fortunately works out the rules covering seven basic cases. I say " fortunately " because it is not at all clear how these seven rules are deduced from the three laws of nature; further specifications, discoverable from his presentation of the seven rules, are needed.

Before turning to the rules of impact themselves, let us take note of a major difficulty which is implicit in the third law and which we have tried to bring out in our discussion of this law. The two parts of the law describe what Descartes thinks happens when the force of the first body is either larger or smaller than the force of the second body. But in a collision of two bodies, which one should be designated as the first body and which one the second? If the two bodies involved are B and C, should we say that B collides with C or that C collides with B? The answer, it seems, is both. But on this basis the first and the second parts of the third law are incon- sistent. According to the first part the smaller force of persistence in B is overcome by the larger force of resistance in C, and as a result B is deflected with the same veocity and C's velocity is unaffected. But according to the second part of the law the larger force of persistence in C overcomes the smaller force of resistance in B, and as a result B is deflected with a greater

17 Principia philosophiae, II, 40. (New York: Scribner's, 1927), pp. 328-329. is See Descartes Selections, ed. R. M. Eaton (Le monde, Ch. 7.)

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velocity and C's velocity is proportionately decreased. This seems to be a flat contradiction. And indeed it is, if we accept the relativity of motion, as Descartes himself does in many places. If we are to give Descartes all benefit of the doubt here, we must conclude that the relativity of motion does not play a systematic role in Cartesian physics. Rather, his appeal to the relativity of motion seems to have a polemical purpose enabling him to reconcile his heliocentric theory of astronomy with the prevailing attacks

against Copernicanism. If this be the case, then his third law of nature is not contradictory since it assigns a privileged status to one of the bodies in a collision as being active and the other as being passive. This will emerge more clearly as we examine his seven rules of impact in detail.

Since Descartes' presentation of his rules of impact is not available in

English and since we wish to examine them individually in relation to his third law of nature, let us provide here a full translation of the texts.

First rule:

if two bodies B and C are completely equal and are moved with equal velocity, B from right to left and C from left to right, then when they collide, they are reflected and afterward continue to be moved, B toward the right and C toward the left, without losing any part of their velocities.

Second rule:

if B is slightly larger than C, and the other conditions above still hold, then only C is reflected and both bodies are moved toward the left with the same velocity.

Third rule:

if they are equal in size, but B is moved slightly faster than C, then not only do they both continue to be moved toward the left but also B transmits to C part of its velocity by which it exceeds C. Thus, if B originally possessed six degrees of velocity and C only four, then after the collision they both tend toward the left with five degrees of velocity.

Fourth rule:

if C is completely at rest and is slightly larger than B, then no matter how fast B is moved toward C, it will never move C but will be repelled by C in the opposite direction. For a body at rest gives more resistance to a larger velocity than to a smaller one in proportion to the excess of the one velocity over the other. Therefore there is always a greater force in C to resist than in B to impel.

Fifth rule:

if C is at rest and is smaller than B, then no matter how slowly B is moved toward C, it will move C along with itself by transferring part of its motion to C so that they are both moved with equal velocity. If B is twice as large as C, it transfers a third of its motion to C because a third part of the motion moves the body C as fast as the two remaining parts move the body B which is twice as large. And thus, after B has collided with C, B is moved one third

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slower than it was before, that is, it requires the same time to be moved through a space of two feet as it previously required to be moved through a space of three feet. In the same way if B were three times larger than C, it would transfer a fourth part of its motion to C, etc.

Sixth rule:

if C is at rest and is exactly equal to B, which is moved toward C, then C is partially impelled by B and partially repels B in the opposite direction. Thus, if B moves toward C with four degrees of velocity, it transfers one degree to C and is reflected in the opposite direction with the remaining three degrees.

Seventh rule:

let B and C be moved in the same direction with C moving more slowly and B following C with a greater velocity so that they collide. Further let C be greater than B, but the excess of velocity in B is greater than the excess of magnitude in C. Then B will transfer as much of its motion to C so that they are both moved afterward with equal velocity and in the same direction. On the other hand, if the excess of velocity in B is less than the excess of magnitude in C, then B is reflected in the opposite direction and retains all of its motion. These excesses are computed as follows. If C is twice as large as B but B is not moved twice as fast as C, then B does not impel C but is reflected in the opposite direction. But if B is moved more than twice as fast as C, then B impels C. For example, if C has only two degrees of velocity and B has five, then C acquires two degrees from B which, when transferred into C, become only one degree since C is twice as large as B. And thus the two bodies B and C are each moved afterward with three degrees of velocity. And other cases must be evaluated in the same way. These things need no proof because they are clear in themselves.19

AN EVALUATION OF THE SEVEN RULES OF IMPACT

Descartes maintains that his rules of impact are implicitly contained in his third law of nature.20 The two parts of this law tell us what to expect when the force of persistence in the first body in a collision is either larger or smaller than the force of resistance in the second body. All we need to do in any given case is to calculate and to compare these forces. The general guidelines are that a larger size overcomes a smaller one and a faster velocity overcomes a slower one.

Further specifications are needed - for an obvious reason. The possible combinations of size, motion, rest, direction of motion, and velocity are rather large, and it is not clear at first sight how these various possibilities conform to the third law. In presenting his rules of impact, Descartes has used certain assumptions which are only implicit in the text. To assist

19 Principia philosophiae, II, 46-52. We pre- additions. We will consider the more im- sent here a translation of the original Latin portant additions in our evaluation of his text. The later French version, which was ap- theory of collision in the next section. proved by Descartes, includes a number of 20 Ibid., 40.

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us in understanding his rules of impact, therefore, let us formulate the following theorems:

I. Theorems regarding size:

a) Two colliding bodies of equal size mutually reflect each other.

b) A larger body upon colliding with a smaller body perseveres unchanged in its state of motion; that is, it maintains the same velocity and direction, and it imposes its direction on the smaller body.

c) A smaller body upon colliding with a larger body changes its state of motion; that is, its velocity is maintained but its direction is reversed.

II. Theorems regarding velocity: a) Two bodies of equal velocity retain their original velicities after

a collision.

b) A faster body upon colliding with a slower body imposes its direction and part of its velocity upon the slower body.

It should be noted that these theorems are not explicitly formulated by Descartes. However, they are implicitly contained in his rules of impact, and we have stated them here explicitly to clarify the logical relations between these rules and the third law of nature. It will also be noted that these theorems, like the third law itself, do not take account of the relativity of motion, but assign a privileged status to one body in a collision as active and the other as passive. In our evaluations below we will consider B to be the body which is active (i.e., persisting) and C the body which is passive (i.e., resisting). Also, it is not yet clear as to what happens when the bodies involved in a collision differ in both size and velocity. Nevertheless, with these theorems and the two parts of Descartes' third law in mind, we can now evaluate his rules of impact.

Rule 1. This is the special case in which both the sizes and the velocities of each body are exactly equal. Therefore the force of persistence in each body is exactly equal to the force of resistance in the other body. Theorems Ia and IIa apply. Because, of the former, the bodies mutually reflect each other; because of the latter, each body retains its original velocity. Here, as in all these rules of impact. Descartes assumes that he is dealing with a perfectly elastic collision, or as he puts it, each body is perfectly hard.21

As a corollary to this rule, one is tempted to add the case in which the unequal sizes of the two bodies are exactly compensated for by proportionately unequal velocities. For if the relative sizes and velocities are properly proportioned, the force of persistence in one body would be equal to the force of resistance in the other body. Although Descartes does not discuss this case, it seems to belong here as a variation of the first rule.

21 The French version repeatedly emphasizes ences from any other bodies. This is an that only two bodies are involved. The bodies obvious qualification and applies to all seven are considered in abstraction from all influ- rules.

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Rule 2. B differs from C only in being larger than C. Therefore the persistence of B overcomes the resistance of C. Theorems Ib and IIa apply. B continues on unchanged and imposes its direction on C. Since the original velocities were equal, each body retains its original velocity after the collision. The amount of B's excess of size over C is immaterial; any difference however small is sufficient to produce the effect.

However, Descartes' presentation of this rule is ambiguous. When he says, " both bodies are moved toward the left with the same velocity," what does he mean by the expression " same velocity "? Two interpretations are available here. Either (a) each body retains its original velocity, or (b) the velocities of both bodies are equal after the collision, but differ from their original velocities. If the latter interpretation is accepted, the law of the conservation of motion is violated. Instead of considering velocity a vector quantity including a directional component, Descartes postulates that the quantity of motion is a function of size and velocity, as we explained earlier. By velocity he means the scalar quantity speed. Therefore, if there is any change of velocity or speed in rule two, the conservation of motion is not observed. As a result interpretation (a) seems preferable.

Rule 3. B differs from C only in being faster than C. Therefore the persistence of B overcomes the resistance of C. Theorem IIb applies. B imposes its direction and part of its motion upon C. The law of the conservation of motion is evoked in the calculations, for the sum of the two velocities after the collision is the same as it was before the collision.

It is not clear why Theorem Ia does not apply here. In the other two cases in which the colliding bodies are of equal size (Rules 1 and 6) mutual reflection occurs. For some reason or other Descartes apparently feels that the difference in velocity is the only relevant factor in Rule 3. This is perhaps reflected in his remark that only a slight excess of velocity in B is sufficient to produce the effect predicted by Rule 3.

Rule 4. B is in motion; C is at rest. B is smaller than C. The persistence of B is overcome by the resistance of C. Theorem Ic applies. B is reflected in the opposite direction with the same velocity and C remains at rest. The velocity of B must remain unchanged in accordance with the law of the conservation of motion.

Descartes' concept of the quantity of rest is operative in this rule. The limit of resistance offered by C at rest is a function of its size and the velocity of any other body of equal size. Since B is smaller than C, it can never overcome the resistance of C no matter how fast it moves. In short, C, which is at rest, can adjust its resistance to any smaller body of any velocity, as Descartes explicitly says here. As a result the comparative sizes of the two bodies are the only really relevant factors in Rule 4.22 However, if B were larger than C, the resistance of C would be overcome. And if B were exactly

22 The French version of Rule 4 adds a long note giving examples of how the force of indication of the ability of a body at rest to resistance in a body at rest is always larger adjust its resistance to any velocity in a smaller than the force of persistence in a smaller body body. The reason given in the French version having any velocity whatsoever. This is a clear is precisely the larger size of the body at rest.

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equal to C, C would reach the limit of its resistance. These latter two

possibilities are the subject of the next two rules. Rule 5. B is in motion; C is at rest. B is larger than C. The persistence

of B overcomes the resistance of C. Theorem Ib applies. The larger body B continues moving in the same direction and imposes its direction on the motion of the smaller body C. But Theorem IIb also applies. The faster

body imposes its direction and part of its motion on the slower body. According to the law of the conservation of motion the amount of motion

acquired by C is exactly equal to the amount of motion lost by B. As a result they are both moved in the same direction with the same velocity. In the calculations presented in the latter part of Rule 5 we have a clear indication that Descartes measures the quantity of motion as a function of size and velocity conjointly.

Rule 6. B is in motion; C is at rest. B and C are exactly equal in size. In this case the limit of resistance in a body at rest is reached. The force of persistence in one body seems to be equal to the force of resistance in the other. Theorem Ia applies. Since B and C are equal in size, they mutually reflect each other. One might also argue that Theorem IIb applies. The faster body communicates its direction and part of its motion to a slower

body (taking a body at rest as being " slower " than a body in motion). But the calculation presented at the end of Rule 6 is not clear.23 Why does the

moving body communicate less than half of its motion to the body at rest? Is Descartes implicitly assuming here that a body at rest is in a " more natural state" than a body in motion, and therefore that rest offers more resistance than motion? The reason is not apparent. But at any rate the law of the conservation of motion is carefully observed.

Rule 7. B moves faster than C in the same direction. Therefore Theorem IIb applies. The faster body communicates part of its motion to the slower

body, all else being equal. B is smaller than C. Therefore Theorem Ic

applies. B is reflected in the opposite direction with the same velocity, all else being equal. But it is not true that all else is equal. The bodies differ in both size and velocity. Descartes tells us we must compare the sizes and velocities to determine which takes precedence. If the excess of B's velocity is greater than the excess of C's size, then Theorem IIb governs the result. On the other hand, if the excess of B's velocity is smaller than the excess of C's size, then Theorem Ic governs the result.

The computations given in the latter half of Rule 7 are of no help in

indicating precisely how these comparisons are to be made. The measurement of both size and velocity will result in a scalar quantity for Descartes.

23 The French version of Rule 6 offers an the result is that these two effects are equally explanation of this calculation. If B were to divided. Hence B retains three degrees of move C without rebounding, it would transfer motion and communicates one degree to C. half of its motion, i.e., two degrees, to C. If This explanation is not very convincing. It B were to rebound without moving C, it would seems to assume that consideration of size retain all of its motion, i.e., four degrees. Now takes precedence over consideration of relative since the bodies are of equal size, neither of motion and rest, but the reason for this is the above possibilities takes precedence, and not clear.

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But what are the units of measurement? How is the measurement of size (in terms of three-dimensional volume) to be compared with the measurement of velocity (in terms of distance per unit of time)? Descartes is very disappointing on this score. Even though he was very well equipped to approach the problem of motion quantitatively, he seems to prefer to think in quali- tative terms throughout his entire presentation of the rules of impact. Nevertheless, his computations here do reemphasize two key points: (a) the

understanding of the quantity of motion as a function of size and velocity; and (b) the sanctity of the law of the conservation of motion.

Our attempt to reconstruct Descartes' theory of colliding bodies has been

only partially successful.24 Many questions remain unanswered. How would the theorems we have introduced apply to other cases not examined by Descartes? Does Descartes have an adequate apparatus for a complete and consistent theory of impact? Precisely how are the theorems introduced related to his laws of nature? Assuming that these theorems are true to Cartesian thought, how would Descartes explain their origin? Would he consider them as empirically grounded in some way or as deductions from more general principles? How are the required calculations to be made in an exact quantitative manner? The answers to these questions are not clear. The texts of Descartes are too vague and too brief to clarify these points. Nevertheless our analysis has explicated a number of concepts very basic to the Cartesian theory of motion and the understanding of his first two laws of nature. Chief among these are the concepts of the quantity of motion and the law of conservation.

CONCLUSION

Successful scientific theories do not just suddenly appear overnight. They are almost always the product of a laborious evolution which, when viewed in midstream, appears confused, misdirected, sometimes purposeless. The

history of science is strewn with valiant efforts which proved to be failures or only partial successes. The Cartesian theory of motion is a case in point.

24 A different and quite interesting attempt to reconstruct the Cartesian theory of collision is to be in D. Dubarle, O. P., " Remarques sur les regles du choc chez Descartes," in Cartesio (Milan: Vita et Pensiero, 1937), pp. 325-334. Dubarle emphasizes the fact that Descartes' Principia lacks an explanation of how the seven rules of impact are related to the third law of nature. Some kind of supplementation is needed. For this Dubarle turns to Descartes' letter to Clerselier, 17 February 1645 (Adam and Tannery [eds.], (Euvres de Descartes, Vol. 4, pp. 183-188). The key point in this letter is the principle of minimum change, i.e., when two bodies of unequal state collide, the result is determined by the requirement that the least possible change of state occur. Dubarle concludes that this principle clarifies Rules

4, 5, and 6 but is inadequate to explain the other rules.

Mouy (op. cit., pp. 22-23) suggests the following classification of the rules of impact. Group I: cases in which the inequalities of the two bodies regarding their respective sizes and states of velocity or rest compensate for each other (Rules 1, 4, and 7, part 2). Group II: cases in which the inequalities are very clear and do not compensate for each other (Rules 2, 3, 5, and 7, part 1). Group III: the intermediate cases which need to be resolved by the method of averaging (Rules 6 and 7, part 3). These classifications are not very clear either in themselves or in relation to the third law of nature. Moreover they are not of much help in indicating the result predicted by each of the seven rules.

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Although defective in many ways, it played an essential role in the evolution of Newtonian science, which was destined to make its appearance half a century later. We can more easily assess the successes and failures of Descartes' theory of motion when it is viewed from this vantage point of Newtonian mechanics.

The Cartesian rules of impact are notoriously erroneous; their defects arise no doubt from many sources. Our analysis of these rules and their presuppositions has highlighted three main deficiencies. First Descartes does not take serious systematic account of the relativity of motion. This is surprising and disappointing. We know that he was well aware of the relativity of motion since he used it to avoid controversy over Coperni- canism. In his astronomy he can say both that the earth is at rest in its vortex and also that it revolves around the sun in a larger vortex.25 But when he comes to the problem of two bodies mutually colliding, he con- siders only one body as persisting and the other as resisting. If he had considered the latter as mutually relative in the two bodies, his rules of impact would have taken a very different form which would have been perhaps much closer to the empirical facts.

Secondly, in his evaluation of the forces of persistence and resistance, he fastens on the geometrical size of a body as its key factor. This is not surprising in the Cartesian system. In a philosophy which designates extension as the essence of material substance, matter in itself can possess no properties other than size or three-dimensional volume. The Newtonian concept of mass would have been incomprehensible to Descartes. Or perhaps we should say that the term " mass," designating inertial resistance to change, would have been identified by Descartes with spatial volume and nothing more.

Thirdly, Descartes chooses to think qualitatively rather than quantitatively in working out his rules of impact. This is disappointing in a man who was very well equipped as a mathematician. In presenting his rules of impact, at least, he seems to look upon mathematics as a model of deductive order to be emulated rather than as the proper tool to deal with the intelli- gibilities involved. If he had adopted units of measurement and carried out exact calculations, he might have been able to clarify some of his concepts and avoid some incoherence.

Nevertheless, there are some very fruitful concepts involved in the Car- tesian analysis. High on this list is his concern with the quantity of motion - a concept, however, not easy to define. Which properties of a moving body should be included here? Shape, size, weight, density, distance traveled, time, direction, the power of the moving cause - all of these factors and perhaps others also might seem at first sight to be legitimate candidates for consideration. Descartes' selection of size and velocity helped considerably in bringing the meaning of this important concept into focus. One is reminded of the second definition at the beginning of Newton's Principia where the quantity of motion is defined as velocity times mass. This concept

25 Principia philosophiae, III, 25-30.

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of momentum is the key for which Descartes was searching and which would have unlocked for him the mysteries of the theory of collision. Yet he is so

very far away. Mass is not extension, and extension is the only inherent property of matter considered in itself for Descartes.

Also of major importance is Descartes' emphasis on the law of the conservation of motion. The fruitfulness of conservation principles was about to blossom in the history of physics, but without the concept of mass, Descartes is forced to identify this conservation ultimately with the immutability of God. If there were no God, the total quantity of motion may well not be conserved in the Cartesian universe. It is a long way from Descartes' conservation of motion to Newton's conservation of momentum, and once

again the gap between these two universes is bridged by the concept of mass. However, Descartes' concern with the problem of the conservation of motion undoubtedly played a major role in defining the dimensions of this problem.

The meaning of Descartes' version of the principle of inertia should now be quite clear. At the descriptive level his first two laws of nature designate the same state of physical affairs as Newton's first law of motion, but the theoretical differences are immense. A body perseveres in its state of motion or rest for Descartes because this is demanded by the immutability of God. The same holds true for Newton because of the body's mass. In the first case the reason is external, in the second case internal, to the material world. At the level of theoretical meaning these are not the same laws. One is

tempted to speculate as to what Cartesian physics would have turned out to be if Descartes had grasped the concept of mass. But this is asking too much. Matter is extension, and any change in this basic Cartesian commitment would modify his entire physics.

In its specific teachings, Cartesian physics was destined to be repudiated almost totally in the later history of science. Yet, paradoxically enough, it is almost impossible to imagine what that later history would have been without it.

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