ARABIAN AND MEDIEVAL SCIENCE. 6r 



a dread of contaminating their geometry with any consideration of 

 numerical quantities. They therefore pre- 

 served to their geometry its ideal perfection, 

 and they would have shrunk from the line of 

 investigation which gives trigonometry a style 

 so different from that of the Euclidean geo- 

 metry. The foundation of trigonometry is the 



simple idea of giving names to the ratios be- c 



tween the length of the sides of a right-angled FlG - 22 - 



triangle. In Fig. 22 let A B c be a right-angled 



triangle, having the right angle at B ; and let a b c represent the re- 

 spective lengths of the sides opposite to the angles marked by corre- 

 sponding letters. The ratio a : b, that is, a-^-b or |, is called the sine 

 of the angle CAB; | is named the tangent; j the co-sine; - the secant; 

 c - the co-tangent; and ^ the co-secant of the same angle. The Arabian 



mathematician appears first to have introduced the sine in the calcu- 

 lation of angles; and about A.D. 1000 ABOUL WEFA, an astronomer of 

 Baghdad, constructed tables of the values of the tangents and co- 

 tangents. By a still later Arabian astronomer of Spain, Geber by 

 name, we first find the cosine employed. The introduction cf tri- 

 gonometrical formulae has placed in the hands of mathematicians a 

 most powerful and fertile invention. 



Though trigonometry is founded on geometrical considerations, it is 

 practically a branch of algebra, a kind of universal arithmetic which 

 was diligently cultivated by the Arabs. We have already seen that 

 the Alexandrian Diophantus has some claims to be considered the 

 first to use a system of algebraical signs. But it is usually considered 

 that algebra had its origin in the East, and that from thence the Arabs 

 obtained their knowledge of it. It is on record that Mahomet Ben 

 Musa of Khqrasan, who is distinguished as the author of the earliest 

 treatise on algebra, travelled into India for the express purpose of im- 

 proving his acquaintance with this subject. The same writer is also 

 noted as being the first to introduce among his countrymen the well- 

 known characters i, 2, 3, etc., in which we usually write numbers. These 

 ArabiaiT'characters, as we still call them, probably also came from the 

 East. Use has made these characters so familiar that we seldom think 

 what arithmetic would be without them ; but we recommend the reader 

 to find for himself how much they facilitate calculation, by trying to 

 go through a sum using only the Roman numerals I., II., III., IV., etc. 

 The quadrature of the circle engaged the attention of the Arabian 

 geometers, and their calculation made the ratio of the circumference 

 to the diameter to be 3,927 to 1,250, which is a very close approxima- 

 tion. 



AmciQg_Lhe_Arabians most distinguished in science we must not for- 

 get to name ALHAZEN, astronomer and mathematician, who flourished 



