ABU 'L WEFA AND HIS ALMAGEST 21 of astronomy.¹ Nor can any objection be raised to Abu'l Faraj (Bar Hebraeus), and it would be impossible to explain more clearly than he does the effect of the prosneusis. He says: The third inequality is the angle formed at the centre of the epicycle by two lines which are drawn, one from the centre of the univ- erse and the other from the point called the prosneusis, at the end of which is the apogee of the epicycle, at which commences the proper motion, and which is called the mean apogee. The apogee which is at the end of the line drawn from the centre of the uni- verse is called the apparent one. The point proseneusis is on the side of the perigee of the eccentric, 10 parts 17 minutes from the centre of the world which is itself at the same distance from the centre of eccen The axim value of this angle is 13 parts 9 minutes when the Moon is a crescent or gibbous, that is, near the hexagon or trigon with the Sun. In fact, when the epicycle is four or eight signs distant from the apogee of the eccentric, the Sun is itself two or four signs distant from [the centre of the epicycle] because it is half way between this centre and the apogee. In the tables, this inequality of the two apogees is called the first angle and is included in the motion of the centre." While this describes the construction of Ptolemy as clrealy as possible, at the same time the agreement of the account with that of Abu' I Wefa is perfect. Abu'l Faraj even (like Nasir ed- din) describes as a fourth inequality in longitude that caused by the motion along an orbit inclined to the ecliptic, so that he would not have neglected to describe the variation, if it had been found by an astronomer of Baghdad. We may add that the Jewish writer Abraham ben Chija (A. D. 1100), in his Sphaera Mundi, also describes the "aberration" of the apside of the epi- cycle, chiefly "in sexta et tertia parte mensis."¹4 1. Translated by Rudloff and Hochbeim, Zeitschrift der Deutschen Morgen- land Ges. XLVII pp. 213--275, He describes (p. 249) how the line of apsides is directed to a point called the corresponding point," and gives its position correctly. The inequality he calls the deviation. 2, Nasir ed-din gives 100 9. 12 14 1 3. Le livre de l'ascension, & c. T. II. PP. 29-30. Twe codices add after the word prosneusis: "This is the point mohazat," 4. Sphaera Mundi (1546, ed. Schreckentuchs), p. 75. Muuster's commentary to the Hebrew text (p. 116) has "cum centrum est in sextili aut trino aspectu [id est, quando abest a sole duobus signis aut quatuorl"; the words in brackets are not in the Hebrew original. The words "sixth" and "third" are unmistakable (shithith and shelishith). Apparently no one has hitherto thought of consulting Abraham ben Chija.
