A Route to the Ancient Discovery of Non-Uniform Planetary Motion

In the description of the Moon's model in his later Planetary hypotheses , Ptolemy apparently repudiates this, now defining the apogee of the epicycle as lying on the radius from the centre of the cosmos through the epicycle's centre ( Heiberg J. L. , Claudii Ptolemaei Opera quae exstant omnia , ii: Opera astronomica minora (Leipzig, 1907), 82–84).

Swerdlow N. M. , “Ptolemy's theory of the inferior planets”, Journal for the history of astronomy , xx (1981), 29–60.

“The origin of Ptolemaic planetary theory”, cited by Toomer G. J. , Ptolemy's Almagest (London, 1984), 480 n. 24. A revised version of Swerdlow's paper recently appeared as “The empirical foundations of Ptolemy's planetary theory”, Journal for the history of astronomy , xxxv (2004), 249–71.

Evans J. , “Fonction et origine probable du point équant de Ptolémée”, Revue d'histoire des sciences , xxxvii (1984), 193–213, and “On the function and the probable origin of Ptolemy's equant”, American journal of physics , lii (1984), 1080–9.

Presumably influenced by this hypothesis, Toomer actually translates Ptolemy's word proegeseis as “retrograde arcs”, although the Greek word is not that specific. (The literal meaning is “advancings”, reflecting the notion that a planet is advancing when it is travelling in the same direction relative to the stars as the stars travel in their revolutions relative to the horizon.).

If Ptolemy meant his description to apply to an equantless model, it is in fact accurate only for Mars, and even then requires a reversal of the apogee and perigee. These facts are integral to Evans's reconstruction.

I use Ptolemy's final eccentricities for illustrative purposes only. As Swerdlow remarks, the eccentricity derived from a set of three oppositions assuming an equantless model is not constant, varying in the case of Mars by roughly ± 10+.

This offset is due in part to the large angle between the apsidal lines of the Earth's and Jupiter's orbits. If it could have been detected in Antiquity (which I doubt), it would have revealed a defect in Ptolemy's equant model, which could be corrected by giving an eccentricity and equant to the epicycle as well as to the deferent. The phenomenon is noted by Aaboe A. , “A Late-Babylonian procedure text for Mars, and some remarks on retrograde arcs”, in From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E.S. Kennedy (Annals of the New York Academy of Sciences , d (1987)), 1–14, pp. 11–13.

To take a single example, in Almagest 3.4 Ptolemy justifies the use of an eccentre model for the Sun by the ostensible phenomenon that the time the Sun takes in passing from least to mean speed is always greater than the time from mean speed to greatest. In fact the alternative model that Ptolemy rejects, an epicycle with the Sun revolving in the same direction around the epicycle as the epicycle revolves around the Earth, differs from Ptolemy's model in true daily motion by a maximum of about 13′, with a maximum accumulated difference in longitude (at the octants) less than 6′. Hence it is highly questionable whether anyone in Antiquity could have verified Ptolemy's assertion from observations, and no one would suggest that this is how the eccentre model was arrived at historically. The introduction of the second lunar model in Almagest 5.1 is an exceptional instance where Ptolemy writes, using the first person, that he discovered a phenomenon in a particular stated manner.

Neugebauer O. , A history of ancient mathematical astronomy (3 vols, Berlin, 1975), 792.

See Jones A. , “A likely source of an observation report in Ptolemy's Almagest”, Archive for history of exact sciences , liv (1999), 255–8, discussing P. Oxy. 61.4133. Others among the handful of early planetary observations preserved in the Almagest catch a planet near a significant stage of its synodic cycle; thus the ancient observation of Mercury used in Almagest 9.10 has been identified by Toomer (note ad loc.) as a station, and the Babylonian observation of Saturn used in 11.7 very nearly coincided with opposition. Again Ptolemy does not make use of these facts, so that they probably tell us something about the nature of the channels by which Ptolemy got access to third-century b.c. planetary observations.

Ptolemy tells us in Almagest 9.2 ( Toomer , op. cit. (ref. 3), 421–2) that such models had been employed by his predecessors or contemporaries. Elder Pliny , Naturalis historia 2.63–64, has an obscure and muddled discussion of planetary apsidal lines that evidently refers to eccentre-and-epicycle models.

Although the procedure outlined here presumes only an approximate knowledge of the apsidal line, it deserves to be remarked how accurate most of the apsidal lines in Ptolemy's models are. About a.d. 140 the tropical longitudes of the apparent apogees of Saturn, Jupiter, Mars, and Venus were respectively (to the nearest half-degree) 236.5°, 160.5°, 116.5°, and 57.5°. (The apparent geocentric apsidal line is the line through the centres of the Earth's and the planet's orbit, which I have computed from the elements in Meeus J. , Astronomical algorithms (Richmond, 1991), 197–201; for discussion see Aaboe A , Episodes from the early history of astronomy (New York, 2001), 160–8.) Ptolemy's, in his tropical frame of reference by which longitudes are systematically about 1° too low for his time, are 233°, 161°, 115.5°, and 55°. Ptolemy's apogee for Mercury, at 190°, is notoriously distant from the actual apparent apogee (222.5°), undoubtedly as a consequence of the patchy observational record for this planet.

Babylonian observations of planets passing ‘Normal Stars’ (bright stars in the zodiacal belt) give a lower bound of what naked eye observation could achieve; see Jones A. , “A study of Babylonian observations of planets near Normal Stars”, Archive for history of exact sciences (forthcoming, 2004) for general discussion. I know of 71 Babylonian reports of Jupiter, 106 reports of Mars, and 83 reports of Venus passing stars within two degrees of the ecliptic (I exclude reports that had, or may once have had, a statement that the planet was some distance ahead of or behind the star). The median absolute difference in longitude for the reported dates in the Jupiter observations is approximately 0.20°, i.e. a little more than two days of mean motion; the median for the Mars observations is approximately 0.46°, i.e. less than one day of mean motion; and the median for the Venus observations is approximately 0.56°, i.e. again less than one day of mean motion. Observations made carefully for theoretical applications could certainly have improved on these standards.