44 ASTRONOMY IN ANCIENT NATIONS the ninth sphere; the annual precession varies between 26".45 28.96, or from + 55.41. to -2.51. It was now necessary to assume the existence of a tenth sphere, which as primum mobile communicated the daily rotation to all the others, while the ninth produced the progressive and the eighth periodical motion on the small circles, which are situated in the conca- vity of the ninth sphere." This was a nice and comfortable theory on account of the long periods involved and the slow changes it produced in the amount of annual precession; and oblivious of the fact that the theory had no foundation except the circumstance that the obiquity of the ecliptic was now about 20' less than it had been stated to be by Ptolemy, and that he had given the amount of precession as 36" a year instead of about 50, and often shutting their eyes to several of the necessary consequences of it, such as the changes in the latitudes of stars which it ought to produce, astronomers continued to accept the theory until at last a real observer of the stars arose and wiped it out by showing that the obliquity of the ecliptic had steadily diminished, and that the amount of annual precession had never varied. We have in this place only alluded to it because it in- volved some rearrangement of the spheres and because it is eminently characteristic of the period during which no persistent observations were taken, and hardly an attempt was made to improve the theories of Ptolemy. The theory of trepidatio or titubatio, as it was sometimes called, was one attempt and it would have been better left alone. But it forms a not uninter esting chapter in the history of astronomy. (Continued from previous page) this development to a Jew of Toledo, Isaac Hassan (see above. p. 38). adding that Alfonso four years after the completion of the tables became convinced of the futility of the theory by reading the book on the fixed stars by Al Sufi, Riccioli, Almag. novum, I. p. 166. 1. In the Alfonsine Tables the maximum took place at the birth of Christ. In Essler's Speculum astrologicum, p. 224 (appended to Purbach's Theoricae novae, Basle, 1573) the epoch is A,D, 15, diebus 137 completis. Reinhold in his commentary to Purbach (Paris, 1558, f. 163b) explains that 26".45 is the space passed over by the Sun in 10 mins, 44 secs., by which amount the Alfonsine Tables made the tropical year smaller than 365 days, 2. Abraham ben Chija (p. 103, Schrackenfuchs) says that trepidation does not change the latitudes. Perhaps he refers to the earliest form of the notion, that described by Theon of Alexandria,
