The Latitudes of Venus and Mercury in the Almagest

R.C. RIDDELL

Communicated by B. L. VAN DER WAERDEN

1. Introduction In setting up his models for the latitudinal motions of the planets [Almagest XIII, 2], PTOLEMY describes some little circles, whose rotations are supposed to produce the transverse oscillations of the epicycle relative to the deferent needed to model the observed phenomena. NEUGEBAUER comments on this passage as follows: "In Chap. 2 of Book XIII Ptolemy makes a feeble attempt to describe a mechanical device which would cause all planes to move according to his model, assuming that all motions vary sinusoidally between the proper extrema. He does not give any detailed construction for such a device but it seems clear that all he meant to say was that in principle sinusoidal variations of the inclination of a plane could be obtained by the rotation of a vertical disk which leads on its circumference a point of the plane up and down. "This is the only instance in the whole Almagest where something like a mechanical model for the planetary (or lunar) motion is mentioned. And even here it concerns only one component in the latitudinal motion and it is obvious that Ptolemy does not think that any such mechanism actually exists in nature. Neither here nor anywhere else in the Almagest can we find a physical hypothesis like spherical shells driven by contacting spheres, etc., construction which later on became a favored topic of cosmological de- scriptions." a There is indeed no trace in the Almagest of any concern over whether the geometrical elements of the models have material counterparts in the sky. But there is every evidence of a concern over the feasibility of the models in a certain purely kinematic sense, having nothing to do with material bodies in the sky: namely, can the desired numerical effects be compounded out of circular motions

10. NEUGEBAUER, A History of Ancient Mathematical Astronomy (New York, Heidelberg, Berlin: Springer, 1975), vol. 1, p. 217. 96 R.C. RIDDELL

which satisfy, among other requirements, the condition that these motions preserve the constituent circles as plane curves of given magnitude. Our purpose is to point out a problem of feasibility in this sense, involving the latitudinal models for Venus and Mercury; and to show that PTOLEMY introduced the little circles in order to deal with this problem, in a manner entirely consistent with his procedure throughout the Almagest.

2. Ptolemy's problem PTOLEMY's basic technique to model the latitudinal motions is straightfor- ward. He tilts the planes of deferent and epicycle so as to match the maximum latitudes, and he interpolates between these cardinal configurations, if necessary, by making the plane inclinations oscillate between the extremes. This technique raises no particular problem with Mars, Jupiter, and Saturn, nor with the deferents of Venus and Mercury, because in these cases PTOLEMY's observations require only one tilt per circle. For the epicycle in the case of Venus or Mercury, however, he needs two distinct maximum tilts (el Figure 1, in which the deferent is shown without eccentricity): there is a maximum "inclination" or "obliquity" _+i 1 about the east-west diameter EW of the epicycle, to occur when the centre C of the epicycle is at elongation ~c0 = +90 ° from the apogee of the deferent, as seen by the earthbound observer O; and a maximum "slant" +_i 2 about the apogee-perigee diameter AP of the epicycle, to occur when C is on the apsidal line of the deferent. Then he wants each of the inclination and the slant to oscillate between its respective extremes with period equal to the longitudinal period of C, one mean solar year. And here is the difficulty: while an elastic circular loop can be made to perform this or any other combination of small transverse motions, it is not immediately clear whether, or how, a circle can do so and remain at all times a rigid plane figure. Before examining PTOLEMY's way of handling this problem, we shall consider a simple and plausible conjecture, suggested by Figure 1 itself. In the figurel the nodal axis of the epicycle has the same direction in space at the four cardinal points of extreme latitude; and one might suppose that the two oscillations about the two moving axes can be synthesized into one rocking motion about an axis which maintains that same direction at all intermediate values of the elongation tc0 of C from the apogee. We quote the source from which Figure 1 is taken: "Note that beginning at 90 ° from apogee [of the deferent] with an inclination il, the slanting of the epicycle from i~ to i 2 to i~ to i 2 and back to i~ takes one year. Aside from this slant, the plane of the epicycle remains parallel to itself as it moves about the earth .... [The] model, despite its reputation, is really quite simple and logical." 2 This supposed rocking, being a time-dependent rigid motion, certainly preserves

2 N.M. SWERDLOW, "The Derivation and First Draft of Copernicus's Planetary Theory. A Translation of the Commentariolus with Commentary," Proc. Amer. Phil. Soc. 117, 1973, pp. 495-6. (I have changed SWERDLOW'S "iota" to Roman "i'.) Venus and Mercury in the Almagest 97

A ,A

A Figure 1

the epicycle as a plane curve of given size, and so meets the feasibility criterion mentioned in § 1 above. But unfortunately this motion is not what PTOLEMY had in mind, for it is contradicted by his tables of latitude. We shall now demonstrate the contradiction. Independently of the values of i 1 and i 2, the supposed rocking about a self- parallel axis would place the descending node, for any value of ~c0, at the point of the epicycle with anomaly ~= 180°-~Co, reckoned counterclockwise from A. For instance, at the octant ~Co=45°, we would expect the points with e=135 ° and e=315 ° to be in the nodal axis, hence in the deferent plane. Therefore, whatever the optical correction might be for general c~, we would expect for c~ =135 ° and ~=315 ° no net contribution to the latitude from the tilt of the epicycle relative to the deferent. Expressing it in PTOLEMY's fashion, we would expect the latitudinal contributions from the inclination and the slant to be equal and opposite. However, entering the table for Venus [Alm. XIII, 5], beside 135 ° (=e) we find in column 3 the number 1;48 and in column 4 the number 2;30; and beside 135 ° (=~Co+90 °) and 45 ° (=~c0) we find in column 5 one and the same modulating factor 0;42,12. Combining these numbers with proper signs according to the latitudinal calculus [Alm. XIII, 6], we get contributions inclination = - 1 ; 16 °, slant = 1 ;45 °, sum = 0; 29 ° .

The errors in the tables, due to identification of small arcs with their chords, various linear interpolations, etc., amount to a few minutes of arc at most. Therefore the latitudinal contribution of 29 minutes, when our supposition calls for zero, rules out the supposition. In fact, as we see by a further look down the same table, for ~c0 =45 ° the contributions of inclination and slant are equal and opposite when e is, not 135 ° and 315 °, but about 144 ° and 321 °. Thus at •o=45 °, the epicycle is tilted about an axis which is displaced counterclockwise about 8 ° from its direction at the cardinal points. Further to the same idea, we recall that the supposed rocking 98 R.C. RIDDELL motion of the epicycle about a fixed direction in space would produce the same location of the nodes of the epicycle, for given ~co, whatever the particular values of the maximum inclination i t and maximum slant i 2. So let us try what happens with a hypothetical planet Venus 1 whose inclination is one-half that of the real Venus but whose slant is the same. Since the identification of arcs with their chords would then produce an even less appreciable error, and since the optical effects for any c~ are proportional to the vertical deflection there, the tables for Venus1 would be identical with PTOLEMY'S except that all entries in column 3 would be half those for Venus. Then, for ~0 = 45°, to find equal and opposite contributions from inclination and slant, we would have to go down the table to e= 159 ° or 338 °. Thus for Venus 1 at ~o=45 °, the epicycle is tilted about an axis which is displaced in space counterclockwise about 24 ° from its direction at the cardinal points. A further halving of the inclination produces in the same way Venus 2 with an axis through e = 169 ° and 349 °, a displacement of 34 °. The same experiment with Mercury is not decisive, on account of the near equality of il and i2, even if we disregard the small optical correction which PTOLEMY applies to the slant but, arbitrarily, withholds from the inclination. 3 But PTOLEMY'S tables for Mercury~ with half the inclination of real Mercury, at ~c0=45 °, would show the nodal axis of the epicycle at e=154 ° and e=334 °, a displacement of 19 ° from the supposed location; and the axis for Mercury 2 with one-quarter of Mercury's inclination would go through e = 166 ° and c~= 346 °, a displacement of 31 °. Thus the conjectured rocking motion is inconsistent with the numerical structure of PTOLEMY'S tables of latitude, and the rigidity criterion remains unmet. 3. Inclination and slant treated separately We shall now try to follow PTOLEMY's own attempt to deal with the matter. Since he treats the latitudinal contributions of deferent (="eccentric") and epicycle separately, and since the inclination of the deferent poses no problem in the present context, we go straight to his discussion of the epicyclic motions. He treats first the inclination about the EW-axis, referring to this inclination as a displacement of the diameter which we have denoted byAP: "[-The] diameters of the epicycles through their apparent apogees, starting somewhere in the eccentric's plane, are carried aside by little circles lying beside the perigeal limits, say, proportionate in size to the latitudinal deviation, perpendicular to the eccentrics' planes, and having their centres on them. These little circles are revolved regularly and in accord with the longitudinal passages starting from an intersection of the planes with the epicycles (say towards the north) and at the same time carrying the epicycles' planes in the first quadrant's turn to the northernmost limit, of course, then back again to the eccentric's plane; and in the third quadrant's turn to the southernmost limit; and, finally, at the return to the starting point. The beginning and return of this circuit are established-for Saturn, Jupiter, and

3 Cf the discussion at the end of § 3, below. Venus and Mercury in the Almagest 99

•"¢0 --" 90°~

.... ~\"'\,.~ 0

Figure 2

Mars-from the intersection at the ascending node; in the case of Venus, from the eccentric's perigee; and in the case of Mercury, from the eccentric's apogee." [Alm.XIII, 2] 4 Figure 2 depicts the situation for Venus at the end of "the third quadrant's turn.., from the eccentric's perigee", i.e. at ~0 =90°. The epicycle A iEP 1W is at maximum inclination il about EW from its mean position AEPVK. The centre G~ of the little circle is on OC '°beside the perigeal limit" P, and its radius q =lGiPl[ is determined by r i =r sin i l, (3.1) where r is the radius of the epicycle. 5 The signed vertical displacements of P~ and A~ from the deferent plane are -r~ and +r 1 respectively. The general point X at anomaly ~ is rotated about EW to X l, situated at a signed vertical displacement hl(c0 from its projection X~ in the deferent plane, where

h i (c0 = r i cos ~. (3.2)

Figure 3 depicts the situation for Venus with C at a general elongation ~o, taken for definiteness in the interval 0<1%<90 °. The position Gl=Gl(K0) of the centre of the little circle and the position Pl=Pi(Ko) of the perigee of the epicycle are jointly determined, by an easy straightedge and compass construction in the vertical plane through CP, from the conditions

ICPil=r , . [GiPl[=ri, angle CGiPi=180°+~Co, (3.3)

4 Alrnagest, trans. R.C. TALIAFERRO, Great Books of the Western World (Chicago: Encyclopaedia Brittanica, 1952), vol. 16, p. 428. Subsequent quotations are from the same source. 5 rl~ril (il in radians), provided i 1 is small. 100 R.C. RIDDELL

x, --~

Figure 3 where rl has been fixed in (3.1), and the angle is fixed by PTOLEMY's stipulation that the little circle begins its northward rotation "from the eccentric's perigee." Now P1 and A 1 are situated respectively below and above their vertical projections P~ and A~ in the deferent plane, according to

P~P1 = -rl sin ~:o, A'IA 1 =r 1 sin ~:o. (3.4)

The general point X at anomaly ~ has been rotated about EW to X 1, vertically displaced relative to its projection X] in the deferent plane by an amount X~ X 1 =hi(e, ~Co) given by

h i (c(, Ko) = A i A 1 cos ~ -~- (t"I sin ~Co)cos (3.5) =(r 1 cos c~) sin ~co.

The last equality in (3.5) is obvious to the modern reader because of the modern notation; but its computational content would have been equally obvious to PTOLEMY, who was completely fluent in such things without benefit of modern notation. The significance of the last expression for hi(e, ~:0) in (3.5) is that one may work out, as a function of ~, the optical effect at 0 of the vertical displacement X~ X 1 when C is at the cardinal point ~co = 90 ° (cf. (3.2) above); and then interpolate to any general value of ~:o simply by multiplying the cardinal latitude by sin ~co. And that is exactly PTOLEMY'S procedure: he determines i 1 by observations at •o=90 °[Alm. XIII, 3]; works out sample latitudes in accordance with (3.2) and (3.5) [Alm. XIII,4]; tabulates cardinal latitudes as a function of e in accordance with (3.2) [Alto. XIII, 5, column 3]; and prescribes for arbitrary tco an adjustment which amounts to multiplication by sin ~c0 [Alm. XIII, 6]. The reader might wish to consult NEUGEBAUER'S exposition of this Venus and Mercury in the Almagest 101 procedure,6 especially to see how, in such computations from a fixed geometrical configuration, PTOLEMY does without the modern conveniences of sine and cosine by controlling the signs of corrections modulated by ratios of chords. It must in any case be emphasized that (3.1) to (3.5) are exact in the following sense. Quite apart from the approximations made by PTOLEMY in computing from these relations the optical effects at O, his quoted description dictates the geometry of Figures 2 and 3, and this geometry in turn dictates, without any approximations, the relationships expressed in (3.1) to (3.5). Furthermore, in the same sense of exactness, the geometry introduces a shift in the longitude of each point X other than E and W, since the projection X'1 of X 1 onto the deferent plane does not lie on OX. PTOLEMY has anticipated this shift when setting up his longitudinal models, asserting however that it introduces only negligible differences in the longitudes [Alto. IX, 6]. He computes these differences explicitly for sample cases with tc0=45 ° [Aim. XIII, 4], arriving at quantities of the order of a few minutes of arc. Thus the model he has in mind really is a rotation about EW, and no mere vertical stretching of the epicycle above and below its mean position in the deferent plane. We now turn to PTOLEMY'S description of the second component of the latitudinal motion of the epicycle. This is the slant about the AP-axis, which he refers to as a displacement of the diameter perpendicular to the first diameter: "In the case of the three stars, the diameter perpendicular to these first diameters remain, as we said, always parallel to the ecliptic's plane or have an inappreciable slant to it. But in the case of Mercury and Venus, starting again from some point in the ecliptic's plane, they are carried aside by little circles lying beside, say, their eastern limits, again proportionate in size to the latitudinal deviation, perpendicular to the ecliptic's plane, and having their centres on diameters parallel to the ecliptic's plane. These little circles are revolved at the same speed as the others starting from an intersection of the planes and the epicycles again northwards (let us say), carrying the eastward extremities of these diameters in the same order as before. And again for these stars, the beginning and return of the similar circuit are established, for Venus, from the node in the additive semicircle, and, for Mercury, from the node in the subtractive semicircle" [Alm. XIII, 2]. Here one must recall that, in XIII, 1, PTOLEMY has prescribed a small inclination of the deferent about the line through O in the ecliptic plane, perpendicular to the apsidal line; and so "the node in the additive semicircle" means the point on the deferent with ~c0 = 270 °. Figure 4 depicts the situation for Venus at the end of a quarter-turn from that starting point, i.e., at the apogee of the deferent, where ~c0=0. The epicycle AE2PW2 is at maximum slant i 2 about AP from its mean position AEPW. The centre G 2 of the little circle is on EC "on a diameter [of the mean position of the little circle] parallel to the ecliptic's plane," and its radius r 2-- IG2Ezf is determined by

r 2 = r sin i2, (3.6)

6 NEUGEBAUER, 1975, vol. 1, pp. 216~26. 102 R.C. RIDDELL

Figure 4

r being the radius of the epicycle, as before. The signed vertical displacements of E 2 and W2 from the deferent plane are +r 2 and -r 2 respectively. The general point X at anomaly e is rotated about AP to X 2, situated at a signed vertical displacement h2(~ ) from its projection X; in the deferent plane, where

h 2 (~) = r 2 sin e. (3.7)

Figure 5 depicts the situation for Venus with C at a general elongation ~co, again taken for definiteness in the interval 0 < ~c0 < 90 °. The centre G 2 = G 2 (~o) and the eastern limit E 2 =E2(~Co) are jointly determined by the conditions

ICE2l=r , ]G2E2I=r2, angle CG2E2=90O +go, (3.8)

where r 2 has been fixed in (3.6), and the angle is fixed by PTOLEMY's stipulation that the little circle begins its northward rotation from "the node [of the deferent] in the additive semicircle." Now E 2 and W 2 are situated respectively above and below their vertical projections E 2 and W~ in the deferent plane according to

E~E 2 =r 2 cos 1¢o, W 2 W 2 = -- r 2 COS t¢ o. (3.9)

The general point X at anomaly e has participated in the general rotation about AP, arriving at X 2 which is vertically displaced from its projection X; in the deferent plane by an amount X'2X 2 =h2(cq Ko), given by

h2 (~, Ko) = E~ E 2 sin e = (r2 cos Ko) sin ct (3.1o) = (r2 sin ~) cos Ko. Venus and Mercury in the Almagest 103

N 2

(12 o c

Figure 5

Again the last expression underlies PTOLEMY's subsequent procedure, in this case leading to the tabulation of latitudes predicted at O due to (3.7) when ~co = 0 [Aim. XIII, 5, column 4], and the prescription of an adjustment for general ~c0 amounting to multiplication by cos tco [Aim. XIII, 6]. Again it must be emphasized that (3.6) to (3.10) express exact relationships dictated by PTOLEMY'S description, and that the rotation about AP, as in Figures 4 and 5, is to be taken literally as such. The foregoing analysis for Venus applies verbatim to Mercury, with two exceptions. First, all vertical displacements for Mercury are opposite to those for Venus. This requires only the introduction of minus signs in front of the relevant expressions, particularly those for h 1 in (3.5) and h 2 in (3.10). Second, in Almagest XIII, 6, PTOLEMY prescribes a further optical correction to the contribution of h 2 to the predicted latitudes: namely, decreasing the contribution from column 4 by one-tenth for - 90 ° =< tco < 90 ° and increasing it by one- tenth for 90 ° <~c 0_-<270°. Here PTOLEMY is considered the cardinal locations of C, at apogee of the deferent (~:o=0 °) and perigee of the deferent (~co=180°), whose distances from O differ from each other in the case of Mercury by an amount sufficient to warrant such an optical correction at O. Of course at these locations, h 1 due to the inclination about EW vanishes, so he is justified in omitting a similar correction to the contribution of column 3 when ~co = 0 ° or 180°. But it has apparently escaped his notice that, for intermediate values of ~co, where h 1 is no longer 0, column 3 ought to be treated just as he treats column 4. This is the reason, and the justification, for our ignoring the ___10 ~o correction to column 4 whenever we are concerned, as in the considerations of § 2 above, only with questions of the relative contributions of columns 3 and 4 to the latitudes predicted by the models. 104 R.C. RIDDELL

4. Synthesis of inclination and slant At this point, for each of Venus and Mercury, PTOLEMY has committed himself to two distinct rotations of the plane of the epicycle, one to produce the inclination about EW and the other to produce the slant about AP. The net deflection of the epicycle at any instant must therefore be compounded out of these two rotations; and since they take place about distinct axes, the problem of visualizing their composite effect is by no means trivial. In particular, the result will depend upon the order in which the two constituent rotations are applied. Since PTOLEMY has described the inclination first, then the slant, we shall follow that order. To carry out the composition for Venus, imagine the time frozen with C at elongation ~:0. Imagine the setup of Figure 3 as a rigid configuration, in which the solid circle A1EP1W represents the epicycle after the first rotation has been carried out. Now perform, on this configuration, the rotation from dotted to solid circle in Figure 5. This carries AEPW of Figure 3 onto AEePW2 of Figure 5, and carries the actual epicycle AIEP1W of Figure3 into a new situation A2EiPzW2, shown as the solid circle of Figure 6. The triangle G~PP2 and the little circle enclosing it in Figure 6 are no longer in a plane perpendicular to the deferent plane, but have been carried by the second rotation through an angle E 2 CE with the vertical. The role of the little circle beside P is to remember the position of P~ relative to P from Figure 3 during the second rotation, and so to define the final position P2. The role of the little circle beside E is to remember the final position E 2. Jointly the little circles determine two points P2, E2 in space equidistant from C; and the triad E2, P2, C then determines the final configuration of the epicycle as a unique circle in space, namely A2E2P2W2 . This is the geometrical content of the two paragraphs from Aim. XIII, 2 quoted in §3 above. In Figure 6, the general point X on the undeflected epicycle AEPW has been carried to Y on A2E2P2W2, where the vector XY is the vector sum of two terms: the vector obtained ~rotating XX~ in Figure 3 through the angle E2CE about AP, and the vector XX 2 of Figure 5. Thus to compute the precise location of Y without modern techniques would be rather a complicated process. But since the angles il and i 2 of rotation are small, of the order of a few degrees, one may compute approximately by ignoring the rotation of ~, identifying X with X~ in Figure 3, and identifying X with X~ in Figure 5. Then the vertical displacement of Y in Figure 6 is approximately the algebraic sum h(a,t%) of the displacements X',X 1 and X'2X 2 given in (3.5) and (3.10) respectively, i.e. approximately h(c~, too) = r, cos ~ sin Ko + r2 sin c~ cos Ko- (4.1)

And indeed, PTOLEMY's calculus of latitudes [Alm. XIII, 6] tells us simply to add the latitudinal contributions already computed by optical correction of (3.5) and (3.10), just as (4.1) indicates. One can nowadays estimate the error in the approximation (4.1) by a routine exercise in setting up and multiplying rotation matrices. Defining the quantities

a=sin ii sin tco, b=sini2 cosK o , Venus and Mercury in the Almagest 105

0 Figure 6 one finds that the precise vertical displacement of Y in Figure 6 differs from h(e, ~o) in (4.1) only by a factor (1 -b2) ~ multiplying the second term. This factor is not less than 0.998 for Venus and 0.992 for Mercury, and so the use of (4.1) cannot introduce into the tables an error exceeding 1 ~o. One also finds that the coordinates of the projection of Y in the deferent plane differ from those of X by factors of the order of (l-a2) ~ and (1-b2)~. 7 In other words, the fractional error in latitudes introduced by the approximation (4.1) is of the same order as the fractional error in longitudes, already appraised by PTOLEMY as being negligible. How might PTOLEMY, who lacked rotation matrices, have arrived at our appraisal of the error in the summing procedure (4.1)? He was certainly capable of computing the exact location of Y for any particular values of c~ and ~c0. By doing so for a representative sample of e's with ~o = 45°, where the compound- ing of the two rotations will tend to have its largest effect on the result, he could have satisfied himself that the latitudinal fractional error was of the same order as the longitudinal, hence negligible. Alternatively, or in addition, he might have experimented with a mechanical apparatus consisting of concentric rings piv- oted about various axles, by means of which he could set up and visually appraise the effects of various compound space-rotations. We know that he had just such an apparatus, of his own design and construction, for he tells us so [Alm. V, 1]. He says he used it to take angular observations of the sun and the moon, and no doubt he did; but it is hardly conceivable that anyone capable of making such a device could have refrained from playing with it. We therefore take (4.1) as the computational content of the two paragraphs quoted from XIII, 2, whose geometrical content is represented in Figure 6. The concordance between (4.1) and the latitudinal calculus is clear, and its approxi- mate agreement with the exact model, properly appraised, is factually clear and is at least plausibly attributable to PTOLEMY. It is interesting, as a check on these considerations, to subject (4.1) to the same test by which we discredited the conjecture in § 2 above. To this end, we

7 Plus an even smaller correction, of order a b. The reader who checks these calculations will also find that if the rotations are applied in the reverse order, i.e. inclination after slant, the image of X is different from Y but again has essentially X as its horizontal component and (4.1) as its vertical component. Of course this just illustrates that multiplication of small rotations commutes up to errors of smaller order. 106 R.C. RIDDELL rewrite (4.1) as

h(cq ~Co)= a(t%) sin (c~ + 6(Xo)), where a(~:o) z = r~ sin 2 ~:o + r~ cos 2 ~co, (4.2) tan 6(~Co) =--rl tan ~co. r 2

Then (4.1) predicts a zero contribution to latitude from the tilt of the epicycle just when c~ is given by

e=-6(~Co)=360°-6(~:o), or e=lS0°-6(~Co), where ~(~:o) is determined by (4.2). Using (3.1) and (3.6) for r 1 and r2, and recalling that sini~/sini2~il/i 2 when i 1 and i 2 are small, we arrive at the prediction that zero epicyclic latitude will be shown at epicyclic anomalies e = ~- (~o) and ~ + = ~ + (~Co), given approximately by

~ ~180°-tan -* tank o , c%~180°+~_. (4.3)

Here are the results for ~:o = 45°:

Planet i 1 i 2 tan-1(ix~J2) ~T(4.3) ~(Alm. XIII, 5)

Venus 2;30 3;30 35;32 144 324 144 321 Venus 1 1;15 3;30 19;39 160 340 159 338 Venus 2 0;37,30 3;30 10;7 170 350 169 349 Mercury 6;15 7;0 41;46 138 318 135 315 Mercury~ 3;7,30 7;0 24;3 156 336 154 334 Mercury z 1;33,45 7;0 12;35 167 347 166 346

In the case of Mercury and his hypothetical offspring, we have treated columns 3 and 4 of PTOLEMY'S tables on an equal footing, as discussed at the end of § 3 above. It is evident that the epicyclic nodes predicted by our approximations (4.1) and (4.3) agree with those in the tables, at least within the margin of the 3 ° step size used by PTOLEMY.

5. Regular motion of the little circles To this point we have held ~0 fixed. As time flows, the oscillations of the epicycle are supposed to be in phase with the true elongation ~:o of C from the apogee of the deferent. But on account of the eccentricities of the deferent, which we have been ignoring until now, Ko does not advance uniformly with the time. Thus as seen from their geometrical centres G~ and G2, the revolutions of the little circles fail to satisfy a second methodological demand, namely that all Venus and Mercury in the Almagest 107 circular motions be controlled by the uniform advance of some angle. This second problem, incidentally, presents itself equally in the case of the single oscillation of the epicycle of each of Saturn, Jupiter, and Mars, and of the deferent of each of Venus and Mercury. Here is what PTOLEMY says in the paragraph following those quoted above: "Now, concerning these little circles by which the oscillations of the epicycles are effected, it is necessary to assume that they are bisected by the planes about which we say the swayings of the obliquities take place; for only in this way can equal latitudinal passages be established on either side of them. Yet they do not have their revolutions with respect to regular movement effected about the proper centre, but about another which has the same eccentricity for the little circle as the star's longitudinal eccentricity for the ecliptic. For the returns on the ecliptic and the little circle are supposed isochronous and the quarterly passages on each apparently agree. But if the little circle's revolutions take place about its proper centre, that which is required can in no way come about, since the passages for the. little circle travel each quadrant in equal time, but the epicycle's passages with respect to the ecliptic do not because of the eccentricity assumed in each case. But if they take place about a centre similar in position to that of the eccentric, and of the quadrants, the returns of the obliquities will traverse corresponding parts of the ecliptic and of the little circles in equal times" [Aim. XIII, 2].

We take this to mean the following. Since, in Figure 6, P~ and E 2 are required to pass the quarter-points of their respective little circles when C passes the points of the deferent with ~c0=0°, 90 °, 180 °, and 270 °, the best arrangement will be to ascertain, from the tables of longitudinal correction, the mean elongation ffQ =90°+t/Q at which ~Co=90°; to define points H~ and H 2 in the little circles so that (see Figure 7):

angleQaH1Sl=fCf~, angle N2HzQ2=~Q; and to stipulate that P1 and E 2 shall revolve about G 1 and G 2 uniformly as seen from H 1 and H i respectively, i.e., in such a way that at all times (see Figure 8 and compare Figures 3 and 5):

angleQ1H1P1 =~, angleNzH2Ee=g.. -@°, N2 $1 Q2@~

Figure 7 108 R.C. RIDDELL

Co @p01

Figure 8

With these stipulations, the angles (Figure 8)

angle Q ~ G~ P~ = tc 1 angle N 2 G 2 E z = ~:2 will be determined by ~, and will be equal to each other with common value ~c~ ~- ~2 ~ K'G, where

~=~-c, where c --- sin- l (6 sin ~), 6=sint/Q. (5.1)

Then ~cG advances consistently with the methodological requirement, since it is produced by the uniform increase of ff at the "equant" points H 1 and H2; and ~c~ agrees with the angle ~co required by the latitudinal phenomena at the cardinal points, i.e. multiples of 90 °, since that is how ~:G was constructed. But how close is xa to the desired value ~o, assumed through (4.1) in the latitude calculations at all intermediate elongations? To check this, we let ~ assume a sequence of values, and we compare the corrections c, giving xG by (5.1), with the longitudinal corrections t/from Aim. XI, 11, giving ~co by ~¢0--if-q. Results:

?~ Venus r/,~=2;23 °, c5=0.042): Mercury (r/a= 3;2 °, 6=0.053): c (5.1) ~ q (Alm. XI, i1) c (5.1) q (Alto. XI, 11)

30 1;11 1;11 1;31 1;17 60 2;4 2;3 2;38 2;25 90 2;23 2;24 3;2 3;1 120 2;4 2;4 2;38 2;41 150 1;11 1;12 1;31 1;32

Further computation with other values of ff serves only to confirm that ~c~ is never more than about 14 minutes of arc away from Ko. Noting that in the tables of latitude, a longitudinal step of 3 ° never produces a change of more than about 5 ~, or 12 minutes, in latitude, we see that the discrepancy between ~cc and ~co would produce a difference of less than (14'/3°) • 12', i.e. 56 seconds of arc, in any predicted latitude of Venus or Mercury. Hence the use of t¢o in (4.1) and in the relations leading up to it is justified for computational purposes, even Venus and Mercury in the Almagest 109 though the strict kinematic model described by PTOLEMY demands ~:G in all those relations. To summarize our analysis of PTOLEMY'S "feeble attempt," we may say that his little circles serve (a) primarily to fix the epicycle as a circle in space, determined at each instant as the image of the undeflected epicycle under the composition of two space-rotations about a pair of orthogonal axes; and (b) subordinately, to carry the time-variation of the constituent space-rotations in a manner consistent with the uniform advance of an angle about suitable equant points. PTOLEMY seems willing to take into the bargain the small horizontal shuttlings of the centres G 1 and G 2 of the little circles, as dictated by (3.3) and (3.8), so long as the whole motion is determined, as it is, by the constancy of the radii r, rl, and r 2 and the uniformity of increase of ~ about H I and H 2. Incidentally, all that has been said in connection with the inclination (Figure 3) of course applies, with the obvious modification of the phases prescribed in the first passage quoted in § 3, to the epicycles of Mars, Jupiter, arid Saturn. What is not clear is why PTOLEMY omits to mention any little circle to control the oscillations prescribed for the deferents of Venus and Mercury. It is true that each of these deferents is called upon to execute only one oscillation, about a fixed axis in the ecliptic plane, and that such a motion presents no threat to its integrity as a proper circle. But the methodological requirement of uniform angular advance about suitable equants remains, for these deferent motions, not explicitly answered by his description. Perhaps, having shown how do it for the oscillations of the epicycles, he considered the required procedure to be obvious.

6. PTOLEMY's general treatment of circles There is of course no other place in the Almagest, besides XIII, 2, where little circles act to procedure transverse motions of other circles. Indeed, the methodological superiority of the eccentric-epicyclic models over the EUDOXAN nested spheres is precisely that, with epicycles and eccentrics, one need never consider compound rotations in space except in modelling the latitudes. Howev- er, the use of little circles to control the motion of previously chosen big circles is by no means confined to the latitudinal models. For comparison, we recall PTOLEMY's model for the second anomaly of the lunar motion [Alm.V].He finds that the first anomaly, modelled by the moon's epicycle, is somewhat larger at solar quadratures than it is at the syzygies. Referring to this discrepancy and to PTOLEMY's corresponding perturbation of the simple lunar model, NEU- GEBAUER writes: "Ptolemy realized that exactly this type of discrepancies from the simple theory would result from an increase of the size of the epicycle when in quadrature as compared with its size at the syzygies, as previously determined from eclipses. A change in actual size would contradict the whole spirit of cinematical models of Greek astronomy but the same effect could be obtained by bringing the epicycle nearer to the observer when in quadrature, thus increasing its apparent size. Consequently Ptolemy gave the deferent of the lunar orbit an eccentric motion of its own, depending on the elongation from 110 R.C. RIDDELL

the sun. At conjunction and opposition the center C of the epicycle had to remain at the original distance R=60 from the observer O, whereas at elongations 90 ° and 270 ° the distance O C had to be reduced to a value required by the greatest observed increment of the epicyclic equation. "The cinematic device invented by Ptolemy to achieve such a periodic variation of distance consists in a crank mechanism (cf. Fig. 78) in which the leg OM of length e rotates backwards .... Obviously this type of motion accounts for all the above-mentioned empirical data .... There remains only the determination of the eccentricity e on the basis of properly selected observations." 8 A change in actual size of the epicycle would indeed go against PTOLEMY's practice with circles, as would any other change in its shape. The figure referred to by NEUGEBAUER has the elbow-point M carried on a little circle of radius e about O. It is clear that this little circle, as well as its analogue in the longitudinal model for Mercury, has precisely the same methodological role as the little circles controlling the latitudinal tilt of the epicycles of the planets. If the ones are kinematic devices to achieve definite periodic variations of some previously determined quantities, while maintaining the size (and the circularity) of some previously determined big circles, then so are the others; and none of them has the slightest connection with any question of whether the models are or are not realized by material objects. One might object that the little latitudinal circles nonetheless differ in various ways from the little longitudinal ones-the former are vertical, their centres slide back and forth, etc.-and are on that account not really to be taken seriously. But PTOLEMY'S stated aim is to demonstrate all the appearances as the product of regular and circular motions, and he presumably meant all the circles introduced to that end to be taken equally seriously. Once one begins to pick and choose among the circles, it is difficult to see which ones would survive. For instance, the centres of the epicycles in the longitudinal models slide back and forth on the radii of the equant circles. Shall one then declare the epicycles to be a mere afterthought to the main business of interpolating sinusoidally between observed extrema? One could, conceivably, take the position that all the circles in the Almagest are supererogatory. Indeed, from a certain rather remote standpoint, the Al- magest appears as a sequence of numerical tables, headed by a master-table of chords versus arcs in a generic circle, and separated from one another by passages of description and sample calculation. One may say, viewing the work in this way, that apart from the head-table of chords and the catalogue of star positions, all the tables consist simply of sinusoidal interpolations between various observed extremes; and that the whole appratus of circular models is an elaborate heuristic device to guide the choice of proportionality constants in the various linear subregimes of these tables-for every table of discrete values, of whatever provenance, is in itself a piecewise-linear table. There is nothing absurd in this position. PTOLEMY's Handy Tables, though largely based on the Almagest, contains no constructions with circles; and it is perhaps no accident that precisely in this work one can find tables of predicted positions corrected by sinusoidal interpolation pure and simple, with no demonstrated connection s NEUGEBAUER, 1975, VO1. 1, p. 85; italics added. Venus and Mercury in the Almagest 111 to a coherent underlying geometrical model. 9 Self-consistent though it might be, this position would be the wrong position to take in trying to understand the Almagest. The notion, or the practice, of "sinusoidal interpolation" as such, without the dictatorship of definite circles moving regularly about definite points, cannot be found in it. The PTOLEMY of the Almagest can get along without the modern notation "sin ~o", but he cannot manage the notion of "sine" as a function, without visualizing the movement of a point on a circle where we would think of a bound variable ranging over its domain. It is worth recalling that the method of HIPPARCHUS-PTOLEMY, as brought to full power in the Almagest, combines two earlier methods of mathematical astronomy: the speculative geometric model-building of the classical Greeks, and the strictly numerical table-building of the contemporary Babylonians. The former appears to have involved at most a requirement of qualitative agreement with the phenomena; 1° while the latter consisted in ad hoc piecewise-linear interpolations between various cardinal phenomena, resulting in tables which gave rather accurate predictions, but which were mathematically unrelated to one another even when they concerned different appearances of the same celestial body. 11 By contrast, the geometric-kinematic models in the Almagest are constructed by a sequence of complications designed to remove discrepancies with the numerical phenomena; and all the particular tabulated predictions are computed from, and articulated by, the finished models. While there is no telling what reasons PTOLEMY had for adopting this method, it is a fact that the resulting theory can be grasped and criticized in terms which make no sense for qualitative models as such or for numerical tables as such. For instance, one can inquire whether, for a given degree of accuracy in the predictions, the models are the best possible ones according to some methodological criterion. Or, one can ask whether certain geometrical elements which initially lack phenomenal counterparts-such as the radii of the planetary deferents-can in fact be invested with precise observational significance. It is not my present business to pursue such questions, nor would I claim in any case that all of them would produce equally fruitful lines of investigation. I simply point out that they are the questions upon which subsequent developments were to hinge; that one cannot ask these questions about ARISTOTLE'S cosmology, however neat, nor about anybody's handy tables, however accurate; and that one can ask them about the Almagest just because of its peculiar method. It is the first complete work of applied rational kinematics; and the consistency of its use of circles, big and little, is not an embellishment but is the crux of its method.

Department of Mathematics University of British Columbia Vancouver, B. C. V6T lW5 Canada

(Received December 29, 1977)

9 Cf. NEUGEBAUER, 1975, vol. 2, p. 1014. lo Cf. NEUGEBAUER, 1975, vol. 2, pp. 675, 679-680. 11 Cf. NEUGEBAUER, 1975, vol. 1, pp. 373, 387, 397.