The Empirical Foundations of Ptolemy's Planetary Theory

The bisection of the eccentricity was originally treated by the present author in a paper called “The origin of Ptolemaic planetary theory”, written for Scientific American in 1979 but never published. Copies have circulated ever since, in particular because it was referred to by Toomer Gerald in his translation of the Almagest, which led to any number of inquiries. In 1984 Evans James published a somewhat different explanation of the bisection in “On the function and probable origin of Ptolemy's equant”, American journal of physics , lii (1984), 1080–9, which led me to do nothing further about publication until a paper by Jones Alexander on the same subject was submitted to this journal. Jones's paper, which will appear in the November issue, referred to my 1979 text. In adapting the 1979 text in what follows, I have stayed close to the original paper, but have excluded background material designed for a reader of Scientific American, treated the principal subject somewhat more technically and with a few details I have learned since, and added the observational confirmations of epicyclic and eccentric motion, also written many years ago but not published, which complement the principal subject. All quotations and paraphrases from the Almagest are from Toomer G. J. , Ptolemy's Almagest (New York, 1984). Toomer gave me copies of the pertinent parts of his translation several years before publication, and if my memory does not fail me, he also commented on the original version of my paper, both of which actions were very helpful. In addition to Toomer's annotated translation, the principal sources I have consulted are Neugebauer O. , A history of ancient mathematical astronomy (3 vols, New York, 1975; hereafter HAMA), and Pedersen O. , A survey of the Almagest (Odense, 1974; hereafter Pedersen , Survey).

This demonstration excludes a model for the planets with motion on the epicycle in the negative direction at apogee. Such models seem to have existed; cf. HAMA on Keskinto Inscription, 703, Pliny , Natural history 2.74, 803, Michigan P. 149, 807–8, following Aaboe A. , “On a Greek qualitative planetary model of the epicyclic variety”, Centaurus, ix (1963), 1–10.

Pliny , Natural history 2.63 gives the apogees of the planets, with Jupiter in Virgo, and all apogees in the middle of their signs, thus Virgo 15° for Jupiter. In Babylonian texts, the slowest motion of Jupiter in System B is at Virgo 15° and in System A the middle of the slow zone is Virgo 12;30°, which is probably only an adjustment from Virgo 15° due to beginning the fast zone at Sagittarius 0°.

This demonstration excludes a circle concentric to the Earth with uniform motion about an eccentric equant point, for in such a model the time from least speed to mean speed, atκ = 90°, is equal to the time from mean speed to greatest speed. This model has been found in Indian astronomy and was presumably based upon an earlier Greek source. See Pingree David , “Concentric with equant”, Archives internationales d'histoire des sciences, xxiv (1974), 26–29. If such a model were in use prior to Ptolemy and known to him, neither of which is certain, it means that he did not invent equant motion itself, but did discover the bisection of the eccentricity.

This proof is in Pedersen , Survey , 331–2, and HAMA , 191. The proof in 12.1, for both epicyclic and eccentric hypotheses, is in Pedersen , Survey, 332–8, and HAMA, 267–70. There is a very fine study of Ptolemy's theory of retrogradation by MacMinn Donn , “An analysis of Ptolemy's treatment of retrograde motion”, Journal for the history of astronomy, xxix (1998), 257–70.

The method is described with derivations for the superior planets in HAMA, 270–3.

In Natural history 2.72–73, Pliny gives Venus 46° and Mercury 23°. A variety of elongations from various sources may be found in HAMA , 804–5. The radius of the epicycle may also be derived from the times used to show the direction of motion on an epicycle in Figure 1. Let ν_p = 0;37°^/d and, to the nearest 10 days, α₂ = 70^d = 43;10°, from which η_max = 46;50° and r = 43;46. But since finding α₂ requires measuring η_max, it makes more sense to use η_max directly.

Conversely, when –λ and t are derived from the final model, as Ptolemy does in 12.2–6, –λ is far less sensitive to changes in parameters than t. Ptolemy understands this completely, for he gives –λ very precisely to seconds but t only roughly to integer days and 1/4, 1/3, and ½ day, and since the time of station can be found by observation only within a few days, anything more precise would be meaningless. It is also notable that when –λ is fixed and t is varied in deriving r, the nearly correct values of r derived here are all close to the minimum. The reason is that retrograde arc and time are related, indirectly through Kepler's third law, by a fixed ratio between relative distances and mean motions, which are inversely as periods; cf. my “On the retrogradation of planets”, Journal for the history of astronomy, xv (1984), 30–32.

This relation is contained in Babylonian System A theory of Mars, the longer retrograde arc in the slowest zone and the shortest in the fastest, although the range of the arcs, 15° to 18°, is too small.

The eccentricity could be found directly from the times between apogee, 90° of true motion from apogee, and perigee shown in Figure 2. This requires assuming a direction of the apsidal line, which may be satisfactory for the demonstration of motion on an eccentric, as shown in the figure, but seems too uncertain for finding the eccentricity. Still, it could be useful for a preliminary analysis, which is what seems to be described here. For example, when applied to Jupiter using ν_c = 0;5°^/d and the times given above in the demonstration of the eccentric, κ₁ = 1140^d = 95;0° and κ₂ = 1020^d = 85;0°, from which c_1max = 5;0° and e = R sin 5;0° = 5;14 where R = 60; Ptolemy finds e = 5;30. The result comes no closer to Ptolemy's when more accurate times are used, but our intention is to show only a preliminary result with round numbers. The method has something in common with Ptolemy's derivations of parameters for the inferior planets, which also require specific locations of the centre of the epicycle, at apogee, 90° of mean motion from apogee, and perigee. For many reasons, the derivation of the eccentricity from three oppositions is far superior.

This note is written for the curious, to spare them the trouble of doing the computations; reading it at all is optional. To examine and eliminate the effect of the approximation OC_b' = OC_b = 80;56, we note first that OC_b' < OC_b cos (κ_b — κ_a) = 80;48. If we repeat the computation using OC_a = 98;41 and OC_b' = 80;48, we find e' = 8;57, e = 9;23, R = 89;47, and where R = 60, e = 6;16 and r = 40;6. This already is a more accurate result, but we may also recompute OC_b and OC_b' separately and iterate the computation of e, R, and r. From R = 89;51 and e = 9;19, found using OC_b' = 80;56, we find OC_b = 81;13 and OC_b' = 81;1. Thus where OC_b = 80;56, OC_b' = 81;1 (80;56/81;13) = 80;44. Repeating the preceding steps using OC_a = 98;41 and OC_b' = 80;44, we find e' = 8;59, e = 9;25, R = 89;45, and where R = 60, e = 6;18 and r = 40;7, hardly significant differences. A second iteration using R = 89;45 and e = 9;25 again gives OC_b' = 80;44, so there will be no further changes in e, R, and r. If instead we begin with R = 89;47 and e = 9;23, found using OC_b' = 80;48, after the steps just shown, we again find OC_b' = 80;44, so also where R = 60, e = 6;18 and r = 40;7. Obviously, these iterations can be worried to death with no significant change. If the concentric with equant motion described in ref. 4 is used, which gives a different equation of centre, c₁ = sin⁻¹(e' sin κ/R), then OC_a = 100;53 and OC_b = 83;11, from which, without iteration, e = 6;2 and r = 39;6. But as explained in ref. 4, such a model is excluded by the demonstration of the eccentric. I mention this only to exclude the possibility that Ptolemy made use of it. Finally, if the entire computation is done using c₁ from Ptolemy's tables, thus from the bisected eccentricity of the final model, then OC_a = 99;38 and OC_b = 82;7, from which, without iteration, e = 6;4 and r = 39;36.

The range of the retrograde arcs in Babylonian System A is much too large, for Saturn 6;40° to 8° and about 7;2° to 8;26°, for Jupiter 8;20° to 10° and 10° to 12°. Far from confirming the bisection, these could even give e > e'. Likewise, for Venus in System A₁ and A₂, the range is 16;10° to 20;20°; correctly it is far smaller, about 15;20° to 16;40°. It is most likely that Ptolemy applied the bisection to Saturn and Jupiter only by analogy and because there was no reason not to do so.

Ptolemy's derivation of the models and parameters for Venus and Mercury, with explanations of the points mentioned here, are treated in my “Ptolemy's theory of the inferior planets”, Journal for the history of astronomy , xx (1989), 29–60. The parameters for Mercury differ in the earlier Canobic inscription, in which the model might also differ, and the later Planetary hypotheses.