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Fig. 4.1. Thabit ibn Qurra's theory of trepidation.

Two small circles, each with a radius of 4° 18' 43'', are centred at the points A and B and a point C is chosen on the circumference of the circle centred at A , with the diametrically opposed point D on the other small circle. The great circle CQD is the movable ecliptic and, as C rotates around A, the ecliptic plane oscillates, Q and Q' being fixed points. The points at which the movable ecliptic crosses the equator are the equinoxes, and the position of the equinox, the mean position of which is A, varies between E and E'. Similarly, the obliquity has its mean value when the equinox is at A and has its maximum value e' when the equinox is at E'. The stars have fixed longitudes with respect to the movable ecliptic.

The theory of trepidation was used extensively by later Islamic and medieval astronomers, but it was not to everybody's liking. Al-Battani devoted a special chapter of his zij to refuting it, instead arguing for the traditional linear theory of precession at 1° in 66 years, or 54.5'' per year. Al-BattanTs zij is regarded by some as one of the most important works on astronomy between Ptolemy and Copernicus, and was translated into Latin by Robert of Chester in the twelfth century. The first part is modelled closely on the Almagest, the second part on

10 Implementing such a scheme in astronomical tables was no easy task and it appears that most astronomers, Thabit included, used procedures which only approximated the true geometrical picture (see North (1967)).

the Handy Tables. As was fairly typical in Islamic astronomy, the solar theory of Ptolemy was refined considerably, but the lunar and planetary theories were left untouched. Al-Battani improved on Ptolemy's value for the obliquity of the ecliptic, the solar mean motion (and, hence, the precession of the equinoxes), the eccentricity of the Sun, and the longitude of its apogee. He also introduced developments in trigonometrical techniques by replacing Greek chords with the sines (imported from India) and introducing cosines.

The method used by Ptolemy for finding the distance of the Sun in terms of the radius of the Earth was the basis of all future attempts for well over 1000 years. The earliest surviving redetermination of this distance is that of al-Battani, and his result for the mean Earth-Sun distance is 1108 Earth radii, which compares with Ptolemy's value of 1210.

The influence of al-Battani's work, which has probably been studied more carefully than that of any other Islamic astronomer, can be seen from the fact that Peurbach in the fifteenth century and Copernicus in the sixteenth quote him often, particularly on matters of solar motion and precession. References to al_ _ 12

Battani can be found also in the works of Tycho Brahe, Kepler and Galileo.

The close relation that still existed between astronomy and astrology is evident from the fact that al-Battani also wrote a commentary on Ptolemy's Tetrabiblos.

The work of Thabit ibn Qurra and al-Battam raised the level of awareness among Islamic astronomers. In particular, they began to appreciate the inadequacies of the parameters used in the Almagest. This led to numerous attempts to improve on Ptolemy's values so as to produce more accurate tables, and also to a much greater interest in the theoretical aspects of Ptolemy's geometrical schemes.

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