Some authors, notably Newton (1977), claim that Ptolemy must have fudged his data so as to reproduce Hipparchus' parameter values, but in fact Ptolemy's errors are not large when one takes into account that an error of 6 h in the length of the spring season can lead to an error of about 7° in the solar apogee (Peterson and Schmidt (1967), Maeyama (1998)). In fact, this demonstrates that the great accuracy achieved by Hipparchus was fortuitous. North (1994) goes so far as to say 'a modern tradition that Ptolemy was little more than a plagiarist of Hipparchus is hardly worth refuting' and Hamilton and Swerdlow's review of Newton's work in the Journal for the History of Astronomy, 12 (1981), 59-63 is deeply critical of the latter's approach. On the other hand, Britton (1992) has demonstrated convincingly that Ptolemy must have had access to rather more observations than are mentioned in the Almagest and that he sometimes used these selectively so as to reproduce agreement with predetermined values. Sheynin (1973b) suggests that Ptolemy might have selected those observations he believed to be the least susceptible to random or systematic error. Whatever the truth of the matter - and the debate continues (see Thurston (2002), Gingerich (2002)) - astronomers from the eighteenth century onwards certainly realized that Ptolemy's observations were not to be relied upon (see, for example, Wilson (1984), also note 20 on p. 63 of this text).

Fig. 3.10. The prosthaphaeresis angle &.

(which never amount to more than about half an hour) were of little significance. Local time is determined by the position of the Sun with respect to meridians (great circles perpendicular to the celestial equator) and so the simplest way to appreciate the cause of the variation in the length of the day is to introduce the concept of the equatorial mean sun, a point that travels around the celestial equator at a uniform rate, once per tropical year. This then leads to the idea of the mean solar day, a concept Ptolemy introduced.

Thus, in Figure 3.11, when the actual Sun is at S, the equatorial mean sun will be at S which, because S moves non-uniformly around the ecliptic and a is nonlinearly related to X (see Figure 3.9), will at different times of the year be sometimes ahead of and sometimes behind A. The arc AS (the difference between the right ascension of the equatorial mean sun and that of the Sun itself) is known as the 'equation of time'. The discovery of the equation of time by Greek astronomers is just one example of the high level of sophistication they achieved, as it was deduced as a theoretical consequence of Hipparchus' solar theory, rather than from observation.

26 The equatorial mean sun should not be confused with the mean sun, which is another important astronomical concept. The latter rotates uniformly around the ecliptic once each tropical year. The two clearly are related since they move around their respective circles at precisely the same rate. In fact, the right ascension of the equatorial mean sun is equal to the ecliptic longitude of the mean sun.

mean sun.
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