Ptolemy's Theory of the Inferior Planets

My general observations on this subject, for whatever they are worth, can be found in “Ptolemy on trial”, American scholar, xlviii (1979), 523–31. The present article is not entirely off the point. A truly valuable examination of the problems of Ptolemy's observations and derivations can be found in Britton J. P. , The quality of Ptolemy's solar and lunar observations and parameters (Yale University dissertation, 1966), soon to be published in a revised form.

Hamilton N. T. Swerdlow N. M. and Toomer G. J. , “The Canobic inscription: Ptolemy's earliest work”, From ancient omens to statistical mechanics: Essays on the exact sciences presented to Asger Aaboe , ed. by Berggren J. L. and Goldstein B. R. (Acta historica scientiarum naturalium et medicinalium, xxxix; Copenhagen, 1987), 55–73.

See, for example, the discussion in Van Helden A. , Measuring the universe (Chicago, 1985), 17–18.

All passages from the Almagest are quoted from Toomer G. J. , Ptolemy's Almagest (New York, Berlin, Heidelberg, Tokyo, 1984).

This interesting point is made in Goldstein B. R. and Sawyer F. W. , “Remarks on Ptolemy's equant model in Islamic astronomy. Appendix: On Ptolemy's determination of the apsidal line for Venus”, ΠPIΣMATA: Festschrift für Willy Hartner, ed. by Maeyama Y. and Saltzer G. (Wiesbaden, 1977), 165–81.

It is not the only interesting relation between the motion of Venus and the Earth. The sidereal period of Venus's retrograde rotation on its axis is about 243 days, and since its heliocentric period is about 225 days, its period of rotation with respect to the Sun, the length of the solar day, is 116.8 days, which is exactly one-fifth Venus's synodic period of 584 days. Hence when Venus is at inferior conjunction, it always turns the same hemisphere to the Earth.

Good accounts of Ptolemy's procedure in the Almagest may be found in Pedersen O. , A survey of the Almagest (Odense, 1974), 295ff., and Neugebauer O. , A history of ancient mathematical astronomy (New York, Heidelberg, Berlin, 1975), 152ff. Analyses of Ptolemy's theory of the inferior planets along with Copernicus's closely related theory, may be found in Wilson C. , “The inner planets and the Keplerian Revolution”, Centaurus, xvii (1972), 205–48; Moesgaard K. P. , “Success and failure in Copernicus' planetary theory”, Archives internationales d'histoire des sciences, xxiv (1974), 73–111, 243–318; Swerdlow N. M. and Neugebauer O. , Mathematical astronomy in Copernicus's De revolutionibus (New York, Berlin, Heidelberg, Tokyo, 1984), 369ff.

In the Planetary hypotheses (trans. by Goldstein B. R. , Transactions of the American Philosophical Society, n.s., lvii/4 (1967), 8) the diameter of Venus is the solar diameter, about 3′, still much too high but perhaps what one would guess with the unaided eye due to scattering of Venus's bright image.

The computations according to modern theory were done with a computer program written to duplicate the positions in Tuckerman B. , Planetary, lunar, and solar positions at five-day and ten-day Intervals, [I] 601 B.C. to a.d. 1, [II] a.d. 2 to a.d. 1649, Memoirs of the American Philosophical Society, lvi (1962), lix (1964). All computations were checked and, if necessary, corrected using the nearest positions in Tuckerman. The elongations, the differences in the longitudes of Venus and the mean Sun, are independent of the systematic error of about −1 ° in Ptolemy's tropical longitudes, and so may be directly compared. The true greatest elongations are the maximum values of the differences in longitude.

A point that has been made in Goldstein and Sawyer (op. cit. (ref. 5)).

Ptolemy , Opera astronomica minora, ed. by Heiberg J. L. (Leipzig, 1907), 150, where the text is emended, correctly I believe, from 22½ to 22;30 (Δ′ corrected to Δ). See also Hamilton Swerdlow Toomer , op. cit. (ref. 2), 65.

Much the same analysis has been made in Gingerich O. , “The Mercury theory from Antiquity to Kepler”, Actes du XII Congrès international d'histoire des sciences , iiiA (1971), 57–64; “Ptolemy and the maverick motion of Mercury”, Sky and telescope, lxvi (1983), 11–13.

This observation was first made in Hartner W. , “The Mercury horoscope of Marcantonio Michiel of Venice: A study in the history of Renaissance astrology and astronomy”, Vistas in astronomy, i (1955), 84–138, p. 117 — reprinted in Hartner W. , Oriens-occidens (Hildesheim, 1968), 440–94, p. 473 — and the correct location of the minimum distance in Hartner W. , “Ptolemy, Azarquiel, Ibn al-Shātir, and Copernicus on Mercury: A study of parameters”, Archives internationales d'histoire des sciences, xxiv (1974), 5–25, p. 7.

Text in Heiberg , op. cit. (ref. 11), 86.

Ibid., 72.

See Hamilton Swerdlow Toomer , op. cit. (ref. 2), 66–67, where for 206°, 210°, 206° read 186°, 190°, 186° (entirely my fault).