D. Apollonius

§ 1. Biographical Data

Apollonius ofPerga has always been rated with Euclid and Archimedes as one ofthe greatest mathematicians of antiquity. A good indication of the high esteem his work enjoyed is that Edmund Halley untertook the edition of his "Conic Sections ", the first four books of which survived in Greek and three of the remaining four in Arabic. l For Kepler it was a matter of course to consult Apollonius in his search for approximations to the oval curve he had found for the orbit of Mars.2

It is mainly from the introductions to the individual books of the Conic Sec­tions that the time of life of Apollonius can be determined within narrow limits to about 240 to 170 B.c. We also know that he lived for some time in Alexandria. 3

It is evident that Apollonius had a decisive influence on the development of Greek mathematical astronomy from Ptolemy's discussion in the Almagest of his theory of eccentric and epicyclic motion. It seems clear that these models for planetary motions were actually invented by Apollonius, thus opening the way for a rational astronomy in contrast to the speculative cosmogony of his predecessors.4 As we shall discuss presently, however, it is not known to what extent Apollonius himself tried to determine empirically the parameters of his models; nor do we know whether or not he had any knowledge of contemporary Babylonian astronomy.

No reference to Apollonius is made in the Almagest in connection with lunar theory. Surprisingly it is always the moon which is related to Apollonius in the few accidental references to his astronomical work found in other ancient sources. Hippolytus in his treatise "Against Heresies" (about A.D. 230) ascribes to him the estimate of 5000000 stades for the distance from the earth to the moon.s

A quotation in the "Library" of Photius (:::::;A.D. 835) from Ptolemaios Chennos (:::::; A.D. 190) tells us that Apollonius was nicknamed "e" because the shape of his letter is reminiscent of the moon about which he knew so much.6

I No printed edition of the Arabic text exists; Halley's edition (Oxford 1710) gives only a Latin translation. An epigram on the Conic Sections from the Byzantine period is found in the Greek Anthology (Loeb III, p.323, No. 578). 2 Kepler, Astronomia Nova (Werke III), Chaps. 59 and 60. 3 A careful discussion of these biographical data has been given by G.J. Toomer in the Dictionary of Scientific Biography I (1970), p.179f. 4 Cf. for this earlier phase below IV B 3, 4. 5 Apollonius, Opera II (ed. Heiberg1 p. 139 frgm. 60. Cf. also below pp. 650 and 655. 6 Opera II, p. 139 frgm. 61 or Phodus, ed. Henry, vol. III, p.66 (Collection Bude). The connection of the letter e with the moon probably originated in the coordination of the seven vowels of the Greek alphabet with the seven planets; cf. the restoration of P.Ryl. 63 in Neugebauer-Van Hoesen [1964], p.64, No. 131 and Dornseiff, Alph., p.43.

O. Neugebauer, A History of Ancient Mathematical Astronomy© Springer-Verlag Berlin Heidelberg 1975

I D 2. Eccenters and Epicycles 263

The most specific reference, however, is a much discussed passage of Vettius Valens (~A.D. 160) where he says 7 that he used for the computation of eclipses Hipparchus for the sun, Sudines, Kidenas, and Apollonius for the moon and for both types of eclipses, placing equinoxes and solstices at 8° of their signs. We have here factually correct references to Babylonian astronomy8 and related to it would be not only Hipparchus but also Apollonius.

Concerning" Apollonius", however, modern scholars usually follow Cumont, suggesting that Vettius Valens had Apollonius of Myndos in mind. He is perhaps contemporary with the mathematician, or a little earlier, calling himself a pupil of the Chaldeans.9 This relationship of the Myndian has been supported by a reference in a treatise of the "Anonymus of 379"10 specifically to Apollonius Myndius. This, however, does not exclude Babylonian connections for anyone else around 200 B.C. nor does it explain how Vettius Valens, writing more than three centuries later, could expect his readers to think of the Myndian when he only spoke about Hipparchus and Apollonius.

§ 2. Equivalence of Eccenters and Epicycles

One can well imagine that the discovery of the inequality of the seasons suggest­ed an eccentric position of the observer with respect to the uniformly revolving sun. Similarly the quite noticeable variations in the velocity of the moon can be readily explained by the assumption of an eccentric orbit. The planets, however, behave in a different manner: an oscillatory motion is superimposed on the mean progress along a circular orbit. Again one can consider it as a quite natural step to describe such a phenomenon as produced by a planet rotating in a small circular orbit which in turn is carried around us on another circular track. It is, however, a brillant geometric discovery to see that the solar and lunar motion can also be described as an epicyclic motion while the planetary phenomena can equally well be generated by an eccenter model.

The concept of eccentric motion is much older than Apollonius and is explicitly attested to by the remark that Polemarchus purposely ignored eccentricities in order not to spoil the beauty of the Eudoxan arrangement of homocentric spheres.1 We do not know at what time epicycles were introduced to explain planetary phenomena but the brilliant mathematical treatment of both cinematic models as one common structure is undoubtedly the work of Apollonius. We can reconstruct the basic ideas of Apollonius' theory because of references by Ptolemy in the Almagest (mainly in XII,1 and IV,6) to the alternative use of eccenter and epicycle models, culminating in a theory of planetary stations expressly ascribed to Apollonius.

7 Vettius Valens, Anthol., ed. Kroll, p. 354, 4-7; Cumont [1910], p.161. Cf. also below p. 602. 8 On Sudines and Kidenas cf. below p. 611; on the norm with 8° below IV A4, 2A. 9 Cf. Cumont [1910], p. 163, n. 2; also Kroll, RE Suppl. V, col. 45 (No. 114) and Honigmann in Mich. Pap. III, p. 310. The date of the Myndian is extremely insecure, based on a huge web of very tenuous arguments. 10 CCAG 5,1, p. 204,16; 5, 2, p. 128, 16 and note I; CCAG I, p. 80,8 and p. 113, note 1. 1 Cf. below p. 658, n. 15.

264 I D 2, 1. Eccenters and Epicycles

There are two levels discernible in the discussion of the equivalence of eccenters and epicycles. One consists in a simple proof based on the parallelogram with epicycle radius r and eccentricity e as sides, shown, e.g., in AIm. III, 32 and innumerable times thereafter.3 The other approach does not keep the position of the observer unchanged but operates with two positions with respect to one fixed circle which carries the planet: for an exterior observer 0 the circle serves as epicycle, for an interior observer 0 as an eccenter, while 0 and 0 are related to one another by an inversion with reciprocal radii on the given circle. This relationship is then used to determine the position of two points on the circle which separate direct from retrograde motion of the planet, i.e. the "stations" (cf. below I D 3, 1). Ptolemy names Apollonius as the author of this procedure 4 ; it also fits in perfectly with the fact that the above-mentioned transformation 0-0 is a special case of the pole-polar relation discussed in full generality by Apollonius in Book III of his Conic Sections.5 Evidently Apollonius was also in possession of the preliminary considerations which lead up to this treatment of the stationary points, even if his name is not explicitly associated with each individual step.

In order to make clear the main ideas of Apollonius' theory I shall use modern notation and terminology, sometimes rearranging a little the material which is embedded in different sections of the Almagest or Theon's commentary. At no point, however, is it necessary to supply essential considerations which are not in one form or another explicitly attested.

Needless to say the cinematic models under consideration are always the "simple" models, i.e. epicycles observed from the center of the deferent or eccenters where the center of uniform rotation coincides with the center ofthe eccenter.

1. Transformation by Inversion

We begin with a seemingly useless generalization of the standard equivalence proof for epicycles and eccenters. Instead of making CSMO a parallelogram (cf. Fig. 264), i.e. instead of assuming r=e and R=R', we observe that all pheno­mena remain unchanged for an observer in 0 as long as the star S is located anywhere on OS, be it at S' farther away, or at S", nearer to 0. 1 Hence the equiva­lence requires only equal ratios:

(1)

2 Cf. above Fig. 51, p. 1220. 3 E.g. in Copernicus, De revol. III, 15. Theon of Smyrna (2nd cent. A.D.) says that Hipparchus con­sidered it worth the attention of mathematicians to investigate the cause of so greatly different ex­planations of the phenomena. Theon gives the impression that Adrastus (around A.D. 1(0) first proved the mathematical equivalence (ed. Hiller, p. 166,6-12; Dupuis, p.268/269). This only goes to show that even an ancient author may have an incorrect view of the chronological sequence of events. 4 Aim. XII, 1 (Manitius II, pp. 268, 1 and 272, 18). 5 Apollonius, Opera I, p. 402-413; ed. Heiberg; trsl. Ver Eecke, p.249-255. 1 Aim. III, 3 (Manitius I, p. 162).

I D 2, 2. Lunar Theory 265

If we in particular assume R'=r=M"S"

then we have (2 a)

or its algebraic equivalent R+r r+e --=-- (2 b) R-r r-e

which we find in Aim. XII, 1.2 The observer can be said to be at a distance R outside an epicycle of radius r, or, with the same right, to be inside an eccenter of radius r with eccentricity e = OM". Hence we can replace Fig. 264 by a new figure which operates with only one circle of radius r but with two positions, 0 and 0, for the observer (cf. Fig. 265). It follows from (2a) that ° is located at the inter­section of OM with the chord that connects the points T and U at which the tangents from 0 contact the circle. This is, of course, a special case of the pole-polar relation with respect to a conic section.

2. Lunar Theory

In connection with Book IV of the Almagest we have discussed the method by means of which Ptolemy determined from three lunar eclipses the radius of the lunar epicycle, l assuming the "simple" model which, as he had shown, remains valid in the case of eclipses.2 We furthermore know from an appendix (Chap. II) to Book IV that Hipparchus also had used triples of eclipses for the determination of the eccentricity of the lunar orbit,3 using both eccenter and epicycle models. As we shall see in the next section Apollonius developed the theory of planetary stations again for both models, making use of his principle of inversion on a fixed circle by means of reciprocal radii. It seems therefore plausible to assume that Hipparchus' method for finding the lunar eccentricity from three eclipses is simply the eccenter-equivalent of the epicycle procedure which we know in all details from the Almagest. Indeed Ptolemy refers explicitly to such an equivalent method4 and Theon in his commentary to Book IV 5 completes the details which belong to a figure in the Almagest. Ptolemy'S remark about another use of these transformations is obviously directed at Apollonius' theory of the planetary stations and it is therefore likely that the whole mathematical background for the determination of the basic parameters of the simple lunar theory belongs to Apollonius, representing a close parallel to his theory of planetary stations. Hipparchus' contribution would then be the selection of convenient empirical data from recorded lunar eclipses and the numerical execution ofthe quite involved trigonometric computations as we know them from the Almagest.

2 Manitius II, p. 270. The relation (2 a) motivates the term "reciprocal radii" since e= I/R for r= 1. 1 Above I B 3, 4 A. 2 Cf. above I B 4, 1. 3 Cf. for his results below p. 315. 4 AIm. IV, 6, Manitius I, p. 223. S Rome CA III, p.l053-1056.

266 I D 2, 2. Lunar Theory

We now repeat, step by step, the procedure which furnished us in I B 3,4 A the radius of the lunar epicycle, but using the terminology for an eccenter model (cf. Fig. 266 6). We assume that two lunar eclipses give us the difference .dA. of the true longitudes for two lunar positions P1 and Pl , a time interval .dt apart, such that also the corresponding increments .d1 and .doc of mean longitude and anomaly are known.

The center of the eccenter is moving during .dt from M1 to Ml by an angle 8=.d1-.doc. If we tum M1 into Ml the point P1 obtains a position p;. on the circle of center M2 and radius R. From 0 the arc P~ P 2 appears under the angle .dA. - 8 =.dA. -.d1 + .doc; from Ml it is seen under the angle OC2 - OC1 = .doc, that is under the same angle as in the epicycle model from its center C. If we deal with a third eclipse at P 3 in the same fashion we can formulate the following problem: three points are seen on a circle of radius R from its center M under the angles ~1 and ~2; one should find within this circle the position of a point 0 from which these three points appear under given angles b1 and bl (cf. Fig. 268); then MO=e is the sought for eccentricity of the model.

As in the case of the epicycle model the place of the observer cannot be chosen arbitrarily. The locus of all points which see an arc I II under the same angle as it is seen from M is a circle through the three points I, II, and M (cf. Fig. 267). If b1 <,51 then 0 must lie in the exterior of this circle. Using for an example the same numeri­cal data as before 7 we have for the three eclipses

1-+11 11-+ III

.dA. 349;15 169;30

.dI 345;51 170;7

hence

11-+1 I -+ III

III -+ II

seen fromM ~1 = 360 - 306;25 = 53;35 ~l = 150;26- 53;35 =96;51 ~3=209;34

.doc 306;25 150;26

8 39;26 19;41

LlA.-8 309;49 149;49

seen from 0 b1 =360-309;49=50;11 <~1 b2 = 149;49-50;11 =99;38>,52 b3 = 210;11 > ~3'

The comparison of these two sets of angles shows that 0 must be located in the zone numbered "4" in Fig. 267, exactly as 0 was found to belong to the exterior zone" 4" of the epicycle.8

After the position of 0 relative to the three points on the circle has been in principle determined it is only a problem of plane trigonometry to find the eccentricity MO, a problem which can be solved in the same fashion as before for an exterior position of O. The details of the computation are presented in Theon's commentary to Aim. IV, 6.9 We reproduce in the following the method of solution without the numerical details which lead Theon (with some fudging), as expected, to almost exactly Ptolemy'S result (e=5;13 as against 5;14 chosen by Ptolemy).

6 Fig. 266 is the exact equivalent of Fig. 63, p.1225. 7 Cf. above p.74. 8 Cf. Fig. 67, p.1227. 9 Rome, CA III, p. 1053-1056.

I D 2, 2. Lunar Theory 267

Given three points PI' P2, P3 on a circle of radius R and midpoint M (cf. Fig. 268 10). The arcs PI P2 and P2 P3 appear from M under the given angles;51,;52' from an eccentric point 0 under "1' "z. Find e = MO.

Let Q be a point on the circle and on the straight line 0 PI . In the right triangle QZP2 one has

PzQ=QO sin "dsin 1'1

with 1'1 known from "1 =1'1 + P1 =1'1 + 1/2;51' In the right triangle QHP3 one has

~ Q = QO sin(180 - "1 - "z)/sin 1'z

with 1'z known from "1+"2=1'Z+Pz=1'z+1/2(;51+;5z). In the right triangle Qep3 one has

p3e=P3 QsinP3 Qe=P3QCOSP3 with P3=1/2;5z.

Thus pze = Pz Q -Qe and therefore

Pz P3 =VPz e Z + P3 e Z

is known in terms ofQO. But Pz P3 is also known in terms of R =60 because

Pz P3 =Crd ;52'

Consequently QO and P3 Q are known in terms of R = 60. Therefore we also know

cx4=arcCrd P3Q· Hence

and P10=P1 Q-QO

are known and e can be found from

(R +e)(R-e)=QO· P10.

§ 3. Planetary Motion; Stationary Points

1. Apollonius' Theorem for the Stations

We know from the discussion about planetary retrogradations and stationary points in Book VII of the Almagest 1 that Apollonius had established for the simple epicyclic model a relation which characterizes a stationary point by the ratio

(1 )

10 For the sake of greater clarity the points on the circle in Fig. 268 have been spaced more conveniently than in Fig. 267. 1 Cf. above I C 6.

268 ID3, 1. Stationary Points

where vp and Vc are the angular velocities of P (the planet) and C (the center of the epicycle), respectively and T the midpoint of the chord PQ on the line OPQ (cf. Fig. 269). We have also remarked that (1) is simply the expression for the direc­tion of the velocity vector at P toward the observer 0, being the resultant of the rotation of P about C and ofC about 0.2

Similarly for the eccenter model (cf. Fig. 269): a point P is a station, seen from 6, when

(2)

where

is the angular velocity of the center M of the eccenter with respect to some fIxed direction from 0.3

Apollonius had no vectorial defInition of a stationary point at his disposal. He conceived of a station as the boundary between the set of points on the epicycle which appear to have direct motion and the set of points which are seen to move retrograde. In other words Apollonius applied to cinematics essentially the same mathematical concepts upon which Eudoxus had built the theory of irrational quantities, known to us from the Books V and XII of Euclid's Elements.4

It is clear that the simple relations (1) or (2) could not have been found by the set-theoretical considerations which Apollonius introduced for a rigid proof. Perhaps it is the special case in which the retrograde arc collapses that suggested a relation of the form (1). Indeed it is explicitly remarked in the Almagest 5 that stations occur for an epicyclic model only when

PC r Vc -=-->-OP R-r vp

(3)

Obviously rVp=(R-r)vc means that the rotation of P on the epicycle and the motion of C on the deferent compensate each other at the perigee of the epicycle. It therefore seems plausible to look for a direction OPTQ such that (3) appears to be the limiting case.

We now tum to Apollonius' proof that the points P which satisfy the relations (1) or (2) respectively are stationary points, i.e. that they separate points of direct motion from points of retrograde motion.

Thanks to the equivalence theorem in the form given it by Apollonius the same circle can be used as carrier of the planet,6 representing the epicycle for an outside observer 0, and eccenter for an inside observer 6, assuming that 0 and 6 are related to one another by a transformation by reciprocal radii. Only for the sake of simplicity of the fIgures do we represent the different cases in different drawings.

2 Cf. Fig. 195, p.1268. 3 Cf., e.g., Fig. 134, p. 1248. 4 Cf. for these problems and the role of Eudoxus: Hasse-Scholz, Die Grundlagenkrisis der Griechischen Mathematik, Charlottenburg 1928 (Pan BUcherei, Philosophie No.3). 5 Aim. XII, 1 (Manitius II, p. 277). Cf. also above p.191. 6 Cf. above p. 264f.

I D 3, 1. Stationary Points 269

The first step consists in locating a point P on the circle such that the relations (1) and (2) are satisfied. Make PB=BP' (cf. Fig. 269, p.1304) and define 0 on the diameter AB through the intersection with DP. Then we have

AO AO BO = OB

(4)

i.e. 0 and 0 are mapped on each other by inversion on the given circle. Further­more we have

and hence

DO DO --=-=--DP' OP

TP SO PO = OP

(5)

(6)

We now assume that P (and P') had been chosen such that (1) is satisfied, hence with (6)

and therefore

But

and thus

TP Vc SO PO= vp =OP

Vm SP vp =OP

(7)

(8)

which is the criterium (2). Hence we have shown: if 0 and 0 are related by inversion and P is chosen such that it satisfies (1) with respect to 0 it automatically also satisfies (2) with respect to 0, and, obviously, vice versa.

The next step makes use of the following lemma: if in a triangle

c~d<a (9 a) then

d y a-d>/f' (9b)

The proof compares the circular sectors of vertex A (cf. Fig. 270) with the corre­sponding triangles. 7

We now look at a point K located somewhere between P and Q (case of epicycle model). Hence (Fig. 271)

and from (9)

7 Aim. XII, 1 (Manitius II, p. 272 f.).

QK<QP<QO

QP y PO>/f'

270 I D 3, 1. Stationary Points

Therefore from (1)

(10)

IfN is a point on the deferent such that its angular distance from the line OPQ is y' > y, satisfying exactly

(11)

then we seen that the planet moves from K to P in the same time 2pvp in which the epicycle progresses (during 1'/vc) from T to N. Seen from 0 the arc KP appears under the angle y, the arc TN under the angle l' > y. Hence the planet is seen to move forward by the amount l' - y > O.

In the case of the eccenter model (cf. Fig. 272) we have as before QP/PO> y/P, hence

(12)

Since QF = DE both QF and DE are seen from K under the same angle b. Similarly KP is seen both from Q and from D under the angle p. Since QO/PO= DO/OP we have from (12)

(13)

But according to (2) we have

thus with (13)

(14)

If 8' is an angle such that exactly

(15)

we see that the corresponding point K' is at a greater distance from P than K. From 8'>8 it follows that seen from 0 the eccenter moves forward through a greater angle than K.·Hence K appears to have direct motion.

Similarly one can show for both models that a point on the arc PP' appears to be retrograde. Hence P and P' are stationary points satisfying (1) or (2).

2. Empirical Data

Ptolemy never connects numerical parameters with the theory of planetary motion of Apollonius, in marked contrast to what he tells us about Hipparchus. Nevertheless we have no proof that this omission is significant and one can argue

I D 3, 2. Stationary Points 271

that any theory of epicyclic or eccentric motion will naturally have to face the question ofthe relative size of the radii involved, in particularly since the maximum elongations of Venus and Mercury immediately lead to a numerical estimate for the size of the epicycle. Hence one seems bound to ask for methods to answer the same question for the outer planets. On the other hand there is much in the astronomy of Eudoxus, Aristarchus, and Archimedes (i.e. in the period just preceding Apollonius) that shows a lack of interest in empirical numerical data in contrast to the emphasis on the purely mathematical structure. 1 It would therefore be a perfectly defensible position, in view of Ptolemy's silence, to assume that Apollonius also was primarily interested only in the mathematical aspects of the theory of planetary motion and not in the numerical agreement with obser­vational facts. One could even imagine a very valid argument in favor of such an attitude. Apollonius' theory in all its mathematical elegance is nevertheless based on a model which is obviously insufficient to explain the phenomena more than qualitatively. A model based on a simple epicyclic motion, seen from its center, can only result in a strictly periodic repeiition of all phenomena, in flagrant contradiction, e.g., to the greatly variable shape and amplitude of the retrograde arcs of the planets.

I see no way to decide between these alternative attitudes in the interpretation of our sources. Nevertheless it may be useful to show that a geometric analysis of the simple epicyclic motion makes it possible to determine the order of magnitude of the radius of the epicycle for an outer planet. Then we can at least say that it was fully within the grasp of Apollonius' planetary theory to establish numerical data for its models on the basis of some simple observations. It seems to me obvious that either Apollonius, or at the latest Hipparchus, must have investigated such numerical consequences. Any improvement of the theory had to start from empiri­cally established deviations from earlier estimates based on the simple cinematic models.

The problem of determining the radius of the epicycle can be considered as mathematically similar to the determination of the radius r of the lunar epicycle from three eclipses.2 There the value of r can be found (for R = 60) from the relation 3

(1)

where the segments PIO and QO are trigonometrically determined by the given angles at the center C of the epicycle and at the observer O.

In the theory of the planetary stations we have a very similar configuration (cf. above p. 269, Fig. 269) with

(R+r)(R-r)=QO· PO. (2)

In this case, however, we do not know the two distances QO and PO but only their ratio: using (1), p.267 we have

PT 1/2PQ VC -=--- -OP OP vp

(3)

1 cr. below p. 643. 2 Cf. above I B 3,4 A and p. 267. 3 Cf. Fig. 68, p. 1227 (and similarly Fig. 268, p. 1303).

272 I D 3, 2. Stationary Points

hence OQ Vp+2vc OP Vp

(4)

The values of vp and Vc can be consider known since they are directly obtainable from the basic parameters for the synodic and the sidereal periods. If one further­more assumes the length 2')' of the retrograde arc PP' as seen from 0 one can also determine OP and OQ individually. From (3) we have

TO vc+vp OP =-vp-

and, assuming')' known (cf. Fig. 273)

TO=Rcos')' hence

and thus from (4)

Finally from (2)

which gives us r for R = 60.

v Op=--p -R cos')'

vc+vp

OQ

(5)

Unfortunately the angle')' is not directly observable because the center of the epicycle moves by the amount vcA t during the time A t of retrogradation. Hence

(6)

In this equation the retrograde arc -AA. can be easily observed but the time between two stations is certainly not a sharply defined quantity. Still, it is possible to get at least a rough idea what results for r are obtainable by using, e.g., Ptolemy's data for retrogradations which surely agree with estimates obtainable from observations.4 In Ie 6,1 p.195, 193, 196, respectively we have values for At which lead to some plausible estimates as follows:

Lit max. mean min. estimate Vc 1/2 vcLlt

Saturn 140 2/3d 138d 136d 14<f 0;2o /d 2;200 Jupiter 123 121 118 120 0;5 5 Mars 80 73 641/2 70 0;30 17;30

Similarly for AA.: for Saturn and Jupiter the retrograde arcs vary little and 7° and 100 respectively may be accepted as fair estimates. For Mars we have variations between about 100 and 20°; hence about 15° seems acceptable. This leads to the

4 Also Babylonian data for Lit and Llliead to essentially the same values (cf. ACT II, p. 303-315).

I D 3, 2. Stationary Points 273

following results

-1/2 LlJ. 1/2 veLlt }' cos }' k kR2 coS2 }' ,2

Saturn 3;30° 2;20 5;50° 0;59,40 1 59,20 40 6;30 Jupiter 5 5 10 0;59 0;59,30 57,30 2,30 12 Mars 7;30 17;30 25 0;54,20 0;45 36,50 23,10 37

as compared with Ptolemy's results r=6;30, 11 ;30, and 39;30, respectively. In other words: relatively crude estimates for the time and the amount of retro­gradation give quite correct values for the sizes of the epicycles.

Another possibility for determining the radius r of the epicycle of an outer planet has been suggested by A. Aaboe. 5 The method is algebraically simple but rather sensitive to changes in the empirical data which are neither easy to obtain nor attested in Greek sources. In Babylonian procedure texts, however, one finds estimates for the daily velocities of the outer planets 6 such that we may assume, at least in principle, as known for the sections of invisibility (.0 to r) and for the retrograde arcs (at opposition 8) the extremal values Vmax and Vmin for the angular velocity of the planet as seen from O. These extrema are related to Vc and vp by

vmax(R+r)=vc(R+r)+rvp at the apogee and

vmin(R-r)=vc(R-r)-rvp at the perigee.

Hence one can find r from

r=R Vc-Vmin or from r= R ----- (7)

Unfortunately the Babylonian parameters are adapted to arithmetical, not cine­matic models and do not result in identical values for r when substituted in (7).

Both methods for finding r, described in the preceding pages, require a relatively high degree of mathematical sophistication in arranging the steps which we have presented here in algebraic notation. Furthermore the required empirical data are not easily obtainable with the necessary accuracy, neither duration and amount of the retrogradation nor the extremal direct or retrograde velocity of the planet. Hence one can hardly assume that considerations of the foregoing type provided the first estimates for the relative sizes of the epicycles of the outer planets. Fortunately one finds parameters in early Greek astronomy which concern the visibility of stars and planets and which can be used to determine the radius of the epicycle by the simplest trigonometry.7 It seems plausible that this possibility had been exploited by the time of Apollonius.

5 Aaboe [1963], p. 8f. 6 ACT, No. 801, Sections 4 and 5 for Saturn, No. 810, Sections 3 and 4 for Jupiter. 7 Cf. below p. 832.