Page:A History of Mathematics (1893).djvu/128

 $$\scriptstyle{q=5}$$, $$\scriptstyle{r=71}$$, and $$\scriptstyle{a=220}$$, $$\scriptstyle{b=284}$$. Tabit also trisected an angle.

Foremost among the astronomers of the ninth century ranked Al Battani, called Albategnius by the Latins. Battan in Syria was his birthplace. His observations were celebrated for great precision. His work, De scientia stellarum, was translated into Latin by Plato Tiburtinus, in the twelfth century. Out of this translation sprang the word 'sinus,' as the name of a trigonometric function. The Arabic word for "sine," dsckiba, was derived from the Sanscrit jiva, and resembled the Arabic word dschaib, meaning an indentation or gulf. Hence the Latin "sinus."[3] Al Battani was a close student of Ptolemy, but did not follow him altogether. He took an important step for the better, when he introduced the Indian "sine" or half the chord, in place of the whole chord of Ptolemy. Another improvement on Greek trigonometry made by the Arabs points likewise to Indian influences. Propositions and operations which were treated by the Greeks geometrically are expressed by the Arabs algebraically. Thus, Al Battani at once gets from an equation $$\scriptstyle{{{\sin \theta} \over {\cos \theta}}=D}$$, the value of $$\scriptstyle{\theta}$$ by means of $$\scriptstyle{\sin \theta={D \over \sqrt{1+D^2}}}$$,—a process unknown to the ancients. He knows, of course, all the formulas for spherical triangles given in the Almagest, but goes further, and adds an important one of his own for oblique-angled triangles; namely, $$\scriptstyle{\cos a=\cos b \cos c+\sin b \sin c \cos A}$$.

At the beginning of the tenth century political troubles arose in the East, and as a result the house of the Abbasides lost power. One province after another was taken, till, in 945, all possessions were wrested from them. Fortunately, the new rulers at BagdadBaghdad [sic], the Persian Buyides, were as much interested in astronomy as their predecessors. The progress of the sciences was not only unchecked, but the conditions