Technical modifications to Ptolemaic astronomy

In Ptolemy's theory, the motion of the Sun plays a fundamental role in determining the motion of all the other heavenly bodies, and so all of the calculations in the Almagest are based on the parameter values Ptolemy used for his solar theory. Ptolemy failed to improve on the values of Hipparchus and so used 23° 51' 20" for the obliquity of the ecliptic, 365; 14, 48 days for the length of the tropical year, 65° 30' for the longitude of the solar apogee, and 1° per century for the value of precession. Islamic astronomers had one great advantage over their Greek predecessors: they could compare their observations with those of others who had lived over 1000 years previously. Thus, they had a much better chance of discovering small irregularities only discernible from observations collected over a long period of time. Two of these irregularities discovered in the ninth century were the variation in the obliquity of the ecliptic and the change

The size of the cosmos, as described by al-Farghani, can be found in a number of medieval works of popular literature (see van Helden (1985), pp. 37-9).

in the ecliptic longitude of the solar apogee, and the astronomers responsible for the discoveries were al-BattanT(known in Latin as Albategnius), and Thabit ibn Qurra, two of the most influential of the early Islamic astronomers. Both were quoted by a number of later Latin writers.

The interest of Islamic astronomers in the variability of the obliquity goes back to the mid eighth century - which is quite surprising, since the change (approximately 1 /2'' per year) was of no practical use. Al-BattanTmeasured the obliquity in his day as 23° 35', which is in line with modern theory, and Thabit used 23° 33'. If the value used by Ptolemy was accurate, and most astronomers believed that it was, the obliquity clearly had decreased since the second century, but the implied rate at which it had changed was exaggerated by the fact that Ptolemy's value was about 10' too large.

In 831, Thabit ibn Qurra found the longitude of the solar apogee to be 82° 45' and recognized that the change in apogee could be attributed to precession. Since the increase in longitude of the stars since Ptolemy's time as a result of precession was similar to the change in longitude of the solar apogee, he believed that the solar apogee, like that of all the planets, remains fixed with respect to the stars, and Thabit thus used the sidereal year rather than the tropical year as the basic period of his solar theory. Similar conclusions were reached by al-Battani, who found 82° 17' for the solar apogee. Both astronomers also found new values for the eccentricity of the deferent of the Sun and constructed new solar tables.

Due to the fact that different astronomers had computed different values for precession, and in particular due to the inaccurate value used by Ptolemy, Thabit believed that the precession of the equinoxes was not a linear function of time as Ptolemy had supposed, but that it varied periodically. This led him to propose his influential 'trepidation' theory, a consequence of which is that, not only is the rate of precession variable, but the obliquity of the ecliptic is also a periodic function of time.9

Thabit's theory is illustrated schematically in Figure 4.1. The celestial equator is ARB, A and B being diametrically opposite points, and the fixed mean ecliptic is AQB, Q being midway between A and B . The mean ecliptic makes an angle e = 23° 33' with the equator (it is exaggerated greatly in the figure).

8 The solar apogee actually is not fixed with respect to the stars but possesses a very slow steady motion. This was discovered in the eleventh century (see p. 97).

His theory is described in his treatise entitled On the Motion of the Eighth Sphere, that has been translated into English with a commentary by Neugebauer (1962) and is discussed in Goldstein (1964). Thabit was not the first to suggest that the rate of precession was variable, theories incorporating non-constant precession were known to, for example, Theon of Alexandria (fourth century) and Proclus (fifth century) and were incorporated into early Indian astronomy (Pingree 1972).

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