Precession and Apogees in Almagest Vii.2

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In Almagest VII.2 (Toomer 1984, pp. 327–329), Ptolemy noted that the return of the Sun to the vernal equinox did not have the same duration as the return of the Sun to the fixed stars. He considered the length of the year to be the tropical year, that is, the period of the Sun’s return to the vernal equinox, and introduced a motion of the fixed stars with respect to the vernal equinox in the direction of increasing longitude—the difference between these two definitions of the year is due to what is now called the precession of the equinoxes. For Ptolemy, however, the equinoxes were fixed whereas the positions in longitude of the fixed stars increase uniformly at a rate of 1° in 100 Egyptian years (where an Egyptian year is exactly 365 days), and he saw no need to tabulate the accumulated motion of the fixed stars due to precession. There were also theories in antiquity in which the rate of precession varied over time (Ragep 1996), but no tables for such a motion survive (if there were any). In 9th-century Baghdad astronomers noticed that the length of the tropical year no longer had the value Ptolemy assigned to it in Almagest III.1 (365;14,48d; see Toomer 1984, p. 140), whereas the sidereal year (the period of return of the Sun with respect to the fixed stars) remained essentially unchanged. Since they had no reason to deny that Ptolemy’s results were valid for his own time, they introduced models for variable precession, or trepidation (called in the Middle Ages “the motion in access and recess”), that could account for the data they collected as well as the data in the Almagest. The earliest treatise on trepidation accompanied by tables was ascribed in the Middle Ages to Thābit Ibn Qurra (erroneously in all probablity: see, e.g., Ragep 1996, pp. 267–268, n. 4), and it only survives in a Latin translation (see, e.g., Neugebauer 1962b). The tables in Pseudo-Thābit’s On the motion of the 8th sphere also appear in the Toledan Tables that survive in many Latin copies. In the Middle Ages all fixed stars were assumed to lie at the same distance from the center of the Earth on a sphere which was called the 8th sphere, that is, it was farther from the Earth than the spheres of the seven planets (Moon, Mercury, Venus, Sun, Mars, Jupiter, and 44 chapter four

Saturn). Some medieval astronomers (e.g., al-Battānī and Levi ben Gerson: see Nallino 1903–1907, 1:124; Goldstein 1975) did not appeal to a theory of trepidation; rather, they continued to use precession, that is, a uniform motion of the fixed stars, although the parameter for this motion often differed from the value that Ptolemy had proposed. For instance, al-Battānī’s value for precession was 1° in 66 years or about 0;0,0,9°/d.

1. Trepidation (Access and Recess)

Pseudo-Thābit’s description of trepidation is difficult to reconcile with his tables and the figure in the manuscripts is ambiguous but, accord- ing to one interpretation, the sidereally fixed point, Aries 0°, moves on a small circle, the center of which lies at the intersection of the equator and the mean ecliptic, both of them great circles fixed on the celestial sphere. The point Y at Aries 0° moves, together with the sphere of the fixed stars and the spheres of the planets, at a constant velocity on the small circle, through which a movable ecliptic passes. The second point on the movable ecliptic, M, which is required to fix its position on the celestial sphere, is defined as the point on the mean ecliptic 90° from its intersection with the equator (see Figure 7). For discussion of this model and different interpretations of it, see Neugebauer 1962b, p. 298; Goldstein 1965; North 1967 and 1976, 3:155–158; Mercier 1976, 1977, and Mercier 1996, pp. 303–306. Pseudo-Thābit’s tables (Neugebauer 1962b, pp. 296–298) also appear in the Toledan Tables (F. S. Pedersen 2002, pp. 1542–1545): the mean motion on the small circle is tabulated under the title “table for the mean motion of access and recess of the 8th sphere,” or variants of it; the table for equation of the motion in access and recess; and the table for the equation of radius of the small circle. The table for the mean motion of access and recess (i.e., trepidation) displayed in Table 4.1A is based on the Hijra as the epoch and Arabic years, and it gives the mean motion about O of Y, Aries 0°. The excerpt reproduced here is taken from Florence, Biblioteca Nazionale Centrale, MS Conv. Soppr. J.V.6 (San Marco 189), f. 48v. The mean motion, i, derived from the preceding table serves as argument for the table of the equation of access and recess of the 8th sphere, an excerpt of which is displayed in Table 4.1B, as it