Epicycle centered on 10° of Libra center of Eccentre orbit, -Z-

Epicycle centered on 10° of Aries

Epicycle centered on 10° of Libra center of Eccentre orbit, -Z-

D maximum elongation at this part of Eccentre

D maximum elongation at this part of Eccentre

------observei(center of Ecliptic)

E maximum elongation at this part of Eccentre

Figure 7.16. The geometry of Mercury's epicycle and eccentric deferent, according to Ptolemy. Drawing by E.F. Milone, adapted from Toomer (1984, Fig. 9.5).

ment came only many centuries later when Tycho Brahe discovered the variation, with 40-arc-min amplitude). More refined treatment by Ptolemy resulted in a further reduction in residuals to ~±172°. Here, however, one of the several mysteries surrounding Ptolemy's work arises: The improvement is said to be based on only two observations that modern authorities agree are very unlikely to have produced the refinement indicated. This is characteristic: Brilliant results and exposition and unclear observational evidence. We illustrate further.

The star catalog, although useful for several purposes, does not seem to demonstrate great ability on Ptolemy's part as an observer. Some authorities maintain that the catalog positions are merely those of Hipparchos with a constant (2°40', according to Newton 1977) added to all celestial longitudes to approximate a precession correction. Ptolemy's determination of the value of the precession was 36 arc-sec/year, exactly 1° per century, certainly no improvement over Hipparchos's work. This sort of result has caused some modern scholars to question the methods and motives of Ptolemy's data treatment, which often disagree with modern calculations using his mathematical methods. Toomer (1984, p. viii) suggested that a critical summary dealing with all aspects of this question is yet to be done. Gingerich (1993) has provided part of the response. Some of the other difficulties and modern scholarly responses are indicated below.

Several recent studies bear on this question, but it is likely to persist for some time. Concerning the star catalog, reevaluation by Shevchenko (1990) suggests that the source of this additive constant is an uncertainty in solar longitude, to which the stellar positions are ultimately linked through reference stars, and appears to be based on observation, according to the methods described in the Almagest. Moreover, Shevchenko (1990) discusses Ulugh Beg's catalog, the observations for which were carried out with the instrumentation and methods of the Almagest; the accidental and systematic errors are shown to be comparable to those of Ptolemy's catalog. Concerning errors that are found in some of Ptolemy's tables for use in calculating lunar and planetary (celestial) longitudes, Van Brummelen (1994) demonstrates that those in auxiliary lunar tables are due to roundoff errors in the linear interpolations of functions of two variables, errors in angular measurements resulting in increases in the errors in the resulting longitudes. Work by Nevalainen (1996) suggests that errors due to linear interpolation are not important in the case of Mercury data, but he also indicates that Ptolemy's model for Mercury has a smaller mean motion than is actually the case, resulting in systematic discrepancies in predicted position. Nevalainen attributes this circumstance to the use by Ptolemy of values of the stellar longitudes that are too small by an average of -1°7; moreover, because Ptolemy compared these too small values with those of Hipparchos, he obtained too small a value of the precession. Newton (1982) also criticizes three of the four observational lunar eclipse records of Ptolemy (see Table 5.2). The maximum deviation is about 50 minutes according to sources used by Schove (1984, pp. 25-27), but the match seems better than this, for at least the last three. For the first, there is some question about whether Ptolemy quotes the time of maximum eclipse in equinoctial or equatorial hours. See Toomer (1984, pp. 198, 205-208) for a fuller discussion.

In the cases of solar, lunar, and planetary observations, Gingerich (1993, pp. 55-73) notes several cases in which Ptolemy's reported observations do not agree with modern theory, but they do agree with Ptolemy's theory. Fraud on Ptolemy's part (as argued, for example, by Newton 1977) is unlikely on several grounds. As Gingerich notes, "How can Ptolemy's parameters, which seem generally more accurate than his data base, be derived from observations that are simply fabricated?" We are left with an incomplete knowledge of how he was able to formulate those theories. It appears likely, at least in some cases, that Ptolemy selected the observations as illustrative of the correctness of his theories rather than as evidence of independent, positive proof. The situation may be analogous to a lecturer working out an illustrative example in a modern science classroom. This is in fact the viewpoint already taken by Delambre (1817; cited in Pannekoek 1989, pp. 149-150). Clearly, Ptolemy had sufficient confidence in his theories of motions of the planets to provide ephemerides, and perhaps, this was sufficient enough reason to create "virtual facts"—i.e., illustrative examples.)

A subsequent work by Ptolemy, entitled TpoOeoeZ twv niavro|i£va>v, Planetary Hypotheses, was meant to provide improvements to results obtained in the Almagest because they were based on "more continual observations" (see Murschel 1995 for explicit details). In this work, much of which was lost until this century, Ptolemy argues for a solar system distinctly different from the rolling spheres of Eudoxos and Aristotle. There are a set of outer, concentric shells to provide the diurnal motion, the precessional motion, and additional "mover" shells to move the planets. Thus, Ptolemy's planets participate in the diurnal motion of the stellar sphere, but in order to explain their peculiar motions, and to avoid the many countermotion spheres of Eudoxos, they each require a different character than do the passive objects of earlier Greek astronomy. The planets have souls that enable them to move with "voluntary motion" and, thus, perform their individual epicyclic and deferent motions.

Ptolemy was an encyclopedist with a wide range of interests. Astronomers tend to regard the Almagest as his most important work, whereas historians and geographers think of him for his Geographia. He also wrote major works on optics and on musical theory. Another of Ptolemy's works, the Tetrabiblos, had much the same impact on the acceptance of astrological views that the Almagest had on epicycli-cal astronomy (see §7.4.3). The first distinction between the terms "astronomy" and "astrology" appears to have been made by Ptolemy in these works.

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