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The Latitudes of Venus and Mercury in the Almagest

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The Latitudes of Venus and Mercury in the Almagest 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 consid- er 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 E 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.
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