Page:Popular Science Monthly Volume 64.djvu/331

Rh 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 clumsy 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 formulæ involving sines only. Abul-Wafa of Bagdad (940–948) gave the formulæ 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 doctrine. 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 calculated 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/m⋅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 physiology of seeing, and it remained the standard work until the time of Roger 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 observatories and supplied them generously. The grandson of Jhenghiz-Khan maintained a splendid establishment of the sort at Meraga on