162 



A SHORT HISTORY OF SCIENCE 



inevitable that individuals in this active-minded race should fall 

 under the spell of Greek mathematical science. Their religion 

 was in fact more tolerant towards science than was contemporary 

 Christianity. 



It would appear that by 900 A.D. the Arabs were familiar on the 

 one hand with Brahmagupta's arithmetic and algebra, including 

 the decimal system, and on the other hand with the chief works of 

 the great Greek mathematicians, some of which have come down 

 to us only through Arabic translations. 



The Algebra of Alkarismi written about 830 was based on the 

 work of Brahmagupta, and served in turn as the foundation for 

 many later treatises. From its title is derived our word " algebra," 

 from the author's name our " algorism." The book begins : 



The love of the sciences with which God has distinguished Al- 

 Mamun, ruler of the faithful, and his benevolence to scholars, have 

 encouraged me to write a short work on computation by completion 

 and reduction. Herein I limited myself to the simplest matters, and 

 those which are most needed in problems of distribution, inheritance, 

 partnership, land measurement, etc. 



The first book contains a discussion of five types of quadratic 

 equations : 

 ax 2 = bx, ax 2 = c, ax 2 + bx = c, ax 2 + c = bx, ax 2 = bx + c ; 



only real positive roots are accepted, but, unlike the Greeks, he 

 recognizes the existence of two roots. He gives a geometri- 



J A cal solution of the quadratic equation 



analogous to those of Euclid. Suppose 

 C x 2 + Wx = 39 and let AB = BC = x, 

 AH = CF = 5; then the areas are 

 AC = x 2 , AK = BF = 5z. 



The sum of these is x 2 + Wx. Com- 

 plete the square HF by adding KE = 25. 

 ~G F HF = (x + 5) 2 = 64, whence x = 3. 

 The series l n + 2 n + 3 n + ... + m n was summed for n = 1,2,3,4, 

 and about 1000 A.D. Alkayami is said to have asserted the im- 

 possibility of finding two cubes whose sum should be a cube. Even 



