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A HISTORY OF MATHEMATICS.

quantities.' This is: If $\scriptstyle {PP_{1}}$ and $\scriptstyle {QQ_{1}}$ be two arcs of great circles intersecting in $\scriptstyle {A}$ and if $\scriptstyle {PQ}$ and $\scriptstyle {P_{1}Q_{1}}$ be arcs of great circles drawn perpendicular to $\scriptstyle {QQ_{1}}$ , then we have the proportion

$\scriptstyle {\sin AP:\sin PQ=\sin AP_{1}:\sin P_{1}Q_{1}}$ .

From this he derives the formulas for spherical right triangles. To the four fundamental formulas already given by Ptolemy, he added a fifth, discovered by himself. If a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, right-angled at A, then $\scriptstyle {\cos B=\cos b\sin C}$ . This is frequently called "Geber's Theorem." Radical and bold as were his innovations in spherical trigonometry, in plane trigonometry he followed slavishly the old beaten path of the Greeks. Not even did he adopt the Indian 'sine' and 'cosine,' but still used the Greek 'chord of double the angle.' So painful was the departure from old ideas, even to an independent Arab! After the time of Gabir ben Aflah there was no mathematician among the Spanish Saracens of any reputation. In the year in which Columbus discovered America, the Moors lost their last foot-hold on Spanish soil.

We have witnessed a laudable intellectual activity among the Arabs. They had the good fortune to possess rulers who, by their munificence, furthered scientific research. At the courts of the caliphs, scientists were supplied with libraries and observatories. A large number of astronomical and mathematical works were written by Arabic authors. Yet we fail to find a single important principle in mathematics brought forth by the Arabic mind. Whatever discoveries they made, were in fields previously traversed by the Greeks or the Indians, and consisted of objects which the latter had overlooked in their rapid march. The Arabic mind did not possess that penetrative insight and invention by which mathematicians in Europe afterwards revolutionised the science.