By Roshdi Rashed (ed.)
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The final expression is as follows: He concludes that for the crescent will be visible. This theory of the visibility of the crescent thus involves six elements: an observation, A=25; a constant relation k between the ‘increase’ of α1 and the ‘decrease’ of α2; a term-by-term correspondence of two progressions, one REGIS MORELON 47 arithmetical and the other geometrical; the situation of the three main variables with respect to their limit values, α0 ≤ α1 ≤ A, 0 ≤ α2 ≤ α0 , 0 ≤ α3 ≤ A; a simple interpolation formula taken from Ptolemy; the formula of the Phaseis to modify the result according to the position of the moon over the horizon.
This purely mathematical proof permits him to analyse precisely the apparent motion and the mean uniform motion relative to one another and to situate two axes: AC, an axis of symmetry for the mean uniform motion as observed from point E; and BF, an axis of symmetry for the apparent motion on the ecliptic. Thus for Thabit it became possible to analyse theoretically a geometric model as such that had been postulated to account for the movement of a heavenly body, using all the resources offered by the development of mathematics, leading him here to carry out the first mathematical analysis of a movement.
Following the observations made in Baghdad between 830 and 832— some 700 years after the Almagest and 950 years after Hipparchus—the author of the ‘Book on the solar year’ notes that the sun’s apogee in his era is at 20; 45° of Gemini and that this shift of 15; 15° since Hipparchus’ observations is analogous to that due to the precession of the fixed stars, which had been measured on Regulus as 13; 10°, excepting observational errors of which the author was well aware. 1 and to conclude that the apogee of the eccentric orb of the sun is subject to precessional motion.