THE PREDECESSORS OF COPERNICUS. 327 



30° (the true value is 30° 2'). Like all the Arabs he adopted the 

 theories of the Almagest without change;* but his observations were 

 materially better than Ptolemy's and his numerical results were, 

 consequently, much more accurate. What is said of Ibn Yunus is, in 

 general, true of the whole school of Arab and Moorish astronomers. 



Ibn Yunus was acquainted with the Indian numerals 1, 2, 3, 4, 

 5, 6, 7, 8, 9, and used them occasionally in place of the climisy Greek 

 system, and he also introduced tangents and secants into trigonometry, 

 as well as auxiliary angles (which latter were not used in Europe till 

 the eighteenth century), but he continued to calculate triangles by 

 formulae involving sines only. Abul-Wafa of Bagdad (940-948) gave 

 the formulae relating to tangents and cotangents, and also to secants 

 and cosecants, and even calculated tables of tangents; though he also 

 stopped short of useful applications that were well within his reach. 

 The science of trigonometry was, however, built up by Arabs, and the 

 way was prepared for Vieta, who is the founder of the accepted doc- 

 trine. Abul-Wafa is the discoverer of the third inequality of the 

 moon — the variation. Observing at a time when the first and second 

 inequalities (discovered by Hipparchus and Ptolemy) had no effect, 

 he noticed that the moon was a degree and a quarter from her calcu- 

 lated place. ''Hence," he says, "I perceived that this inequality 

 exists independently of the two first." This discovery remained 

 unknown in Europe for six centuries until Tycho Brahe independently 

 came to the same result. 



Alhazen was an Arabian mathematician and astronomer of the 

 eleventh century who is noteworthy for his treatment of physical 

 problems, especially that of refraction. Ptolemy had experimented 

 on the refraction of glass and of water and had made out the law that 

 the angle of refraction is a fixed submultiple of the angle of incidence 

 (r= 1/to -i). This was denied by Alhazen, but the true law was not 

 discovered till the time of Willebrod Snell in 1621, who found the 

 relation sine r = 1/m • sine i, where m has a different value for each 

 different substance. Alhazen 's 'Optics' treats of the anatomy of the 

 eye, and of vision, and has several propositions relating to the physiol- 

 ogy of seeing, and it remained the standard work until the time of 

 Eoger Bacon and Vitello (thirteenth century). 



The astronomical instruments of the Arabs were greatly superior to 

 those of the Greeks. The caliphs of Bagdad and of Cairo founded ob- 

 servatories and supplied them generously. The grandson of Jhenghiz- 

 Khan maintained a splendid establishment of the sort at Meraga on 



* It is to be noted, however, that the theories of Ptolemy, as understood 

 by the Arabs, made some of the crystal spheres of the planets clash; and that 

 Ptolemy's place for Mercury was consequently changed arbitrarily to allow 

 room for its motion! This is not a change of theory; but it illustrates how 

 slavishly the doctrine of spheres was followed by some of its votaries. 



