History Online - Almagest

Here, we discuss the accuracy of the Almagest's star coordinates and the way Ptolemy measured them. Recall that the division value was 10 arc min., or one-sixth degree. Much work on identifying the stars in the Almagest catalogue with the stars of modern time was done by Bayer, Flamsteed, Bode, Baily, Peters, Knobel, and others in the 16-20th century. This was a very nontrivial job (for details, see [108], which gives a large number of versions of identification, with different authors identifying the same stars differently in the different Almagest manuscripts). Since not all the Almagest stars have been identified with certainty, we only consider the zodiacal stars. The zodiacal constellations were especially important for ancient and medieval astronomy and astrology. It is, therefore, reasonable to assume that their coordinates were measured more often than the coordinates of other stars. Hence, the coordinates of the zodiacal stars apparently make a homogeneous sample, whereas the sample of all the Almagest star coordinates is inhomogeneous, in particular, due to the different accuracy of determining longitudes in different latitudes (however, there may be other reasons, too; e.g., astrological, more important stars probably having been measured in a more thorough manner).

The Almagest's zodiac consists of 350 stars. The average error in longitude is zero if A.D. 60 is taken as the observation epoch (more precisely, this is the epoch to which the catalogue is related, with it possibly being much different from the time when the actual observations were made; see above). The average error in latitude is -2.4°, the sample variance in the longitudes (r^ = 472 min. (recall that 1°= 60 min.). Ten values exceeding 130 minutes were not taken into account. The sample variance in latitude was a^ o=. 262 min., with one value of 136 minutes being rejected. We can draw two conclusions from comparing these and the division value of 10 minutes in the Almagest.

Apparently, the longitudes were recalculated after actual observations. In fact, since the accuracy of the longitudes is much worse than that of the latitudes (remember that this difference cannot be due to the proximity of the stars to the poles, .considering only the zodiacal stars with latitudes within 20°), this circumstance is another argument for the recalculation of the longitude. In antiquity and the Middle Ages, calculations involved complex literal designations of integers and especially fractional parts, constructed from* fractional units denoted in the same way as their denominator, but primed (see [102], [105]). It is natural that, in each recalculation, errors arose and the accuracy fell.

The variances of the catalogue errors are not consistent with the division value of 10 minutes. It turns out that, as a rule, Ptolemy's mistakes consisted of several divisions at once, which confirms Morozov's standpoint that the Almagest coordinates were obtained by a method quite usual for the Middle Ages and, possibly, for antiquity, namely, by measuring equatorial coordinates and subsequently recalculating them into ecliptical ones, done graphically on large atlases or special terrestrial globes with grids of both coordinates (see [I]). However, it is assumed traditionally that Ptolemy observed ecliptical coordinates directly by means of a complex device, the astrolabe, adjusting it by the sun in the afternoon, and by calculating the correction for the shift of its plane due to the earth's rotation by a clepsydra. If we accept Morozov's viewpoint, then we derive at a natural and simple account for these "strange" average error graphs in determining the zodiacal star coordinates as the longitudinal function (stars are grouped with respect to their longitude, and the average error in longitude and latitude is calculated in each group; see the graphs in [108]). The graphs resemble two sine curves shifted with respect to each other approximately by one-fourth of the period. The fact can be easily explained by a small error (within 0.5°) in specifying the ecliptic's slope to the equator on the terrestrial globe (see also the discussion in [I]).

Thus, the coordinates of the Almagest stars are, most probably, a total of the following: (1) Measuring equatorial star coordinates, (2) recalculating equatorial into ecliptical coordinates graphically, and (3) adding a certain constant difference to the longitudes without altering the latitudes (possibly, it was done several times). In connection to the latter item, we once again stress the important investigation by Newton of the degree fractional parts distribution for the catalogue coordinates, showing that an integral multiple (possibly negative) of degrees and 40 minutes was added to the original longitudinal values (see [102]). This operation could have made the catalogue more ancient and shift it from the 10-15th century toward the turn of the first millennium.