40 ASTRONOMY IN ANCIENT NATIONS Though the somewhat confused collection of essays entitled the Libros del Saber would not, if published in the thirteenth century, have advanced astronomical science, it cannot be denied that the Alfonsine Tables were very useful in their day. The actual elements are not given, nor is any thing said about any observations by which somewhat more correct values of the mean motions must have been found.¹ Arabs on motions of fixed stars. Thus we finish our review of the planetary theories of the Arabs. Now we shall say a few words about their ideas as to the nature and motion of the fixed stars. The exaggerated notion which prevailed before the invention of the telescope with regard to the apparent angular diameters of the stars natu- rally, led to erroneous estimates of their actual size, founded on the assumption that the sphere of the fixed stars (the eighth sphere) was immediately outside that of Saturn. The stars of the first magnitude were supposed to have an apparent diameter equal to of that of the Sun, from which it followed that their actual diameters were about 42 times that of the Earth, or about (Continued from previous page) propter motus supradictos non (ut in alijs planetis fit) circumferentiam deferentis circularem, sed potius figuræ, habentis similitudinem cumplana ovali peripheriam describere." Next by Albert of Brudzew in 1452 in his Commentariolum super theoricas novas, printed at Milan in 1495 (ed. Cracow, 1900, p. 124), where it is remarked that the centre of the lunar epicycle describes a similar figure. This is also stated by E. Reinhold in his commentary to Purbach, 1542, fol. p. 7 verso (ed, of Paris, 1558, fol. 78.) by Vurstisius in his Questiones novae in theoricas, & c.. Basle, 1573, p. 233; and in Riccioli's Almagestum novum T. I. p. 564. The last three writers (who give a figure) also take the equable angular motion round the centre of the equant into account, which centre lies on the point of the circumference of the small circle nearest the Earth. The curve described by the centre of the epicycle thus becomes egg-shaped, and not like an ellipse. 1. The tables in vol, v. of the Libros del Saber are quite different from the Alfonsine Tables, and are apparently only intended for astrological purposes. 2. Al Battani (cap. 50) gives the greatest distance of Saturn-18,094, and the distance of the fixed stars-19,000 semidiameters of the Earth, Al Fargeni (p. 82) puts them exactly equal. Al Kasgi gives the diameters in parasangs, of the concavity of the stellar sphere-33,509,180 of the ninth sphere 33,524,309, of its convexity "no one but God knows" (Shah Cholgi, p. 97).
