REVIEWS 503 



that Napier was also influenced by the greater appeal to intuition afforded by 

 the kinematical definition, and we must remember that fluxional ideas were 

 not uncommon among the Scholastic philosophers and were not wholly due to 

 Galileo, Roberval, Gregory of St. Vincent, Barrow, and Newton. 



Philip E. B. Jourdain. 



Indian Mathematics. By G. R. Kaye. [Pp. iv + 73.] (Calcutta and Simla: 

 Thacker, Spink & Co., 1915. Price 2s. 6d.) 



This is an excellent summary of what is known of ancient Indian mathematics. 

 The orientalists of about a century ago tended to antedate Indian discoveries 

 (pp. 1, 34). The history of Indian mathematics maybe divided into three periods 

 (p. 3): (0 The Sulvasiitra period with upper limit c. a.d. 200; (2) the 

 astronomical period, c. A.D. 400-600 ; (3) the Hindu mathematical period proper, 

 A.D. 600-1200. The first period (pp. 3-8, 46-7) is known by a name which 

 means " the rules of the cord," and the rules treat of the construction of sacrificial 

 altars. Their date has been variously fixed from 800 B.C. to A.D. 200; but as 

 a matter of fact it is quite unknown. They have not a mathematical but a 

 religious aim, no proofs of them are given, and there is in the presentation nothing 

 mathematical beyond the bare facts. They relate to the construction of squares 

 and rectangles, the relation of the diagonal to the sides, equivalent rectangles 

 and squares, and equivalent circles and squares ; and in them the Pythagorean 

 theorem is stated quite generally and illustrated by a number of examples. It 

 must of course be remembered that the Egyptians and Chinese were acquainted 

 with the theorem much earlier even than 800 B C. (p. 6). Later Indian mathe- 

 maticians completely ignored the mathematical contents of the Sulvasiltras. 

 The second period (pp. 9-13, 47) begins with the introduction of Western 

 astronomical ideas, and the chief names associated with this period are Varaha 

 Mihira and Aryabhata. Aryabhata's work contains one of the earliest records 

 known to us of an attempt at a general solution of indeterminate equations of 

 the first degree by the continued fraction process, and it may be considered as 

 forming an introduction to the later somewhat marvellous development of this 

 branch of mathematics in India (p. 12). The third period (pp. 14-26, 47-50) is 

 characterised by the names of Brahmagupta, Mahavira, Srldhara, and Bhaskara, 

 by the advanced treatment of indeterminate equations, the problem of finding 

 rational right-angled triangles, and the perfunctory treatment of pure geometry. 

 The question of our Hindu-Arabic place-value arithmetical notation (pp. 27-32) 

 here appears. It would seem that the modern place-value system was probably 

 not introduced earlier than about the ninth century. It seems to the reviewer 

 that two facts are not sufficiently considered by most historians of mathematics : 

 the Babylonians had place-value in their sexagesimal system ; and surely the 

 almost universal use of the abacus must have suggested the place-value in script. 

 There is mo evidence of the use of the abacus in India until quite modern times, 

 and there is evidence that the notation was introduced into India by a right- 

 to-left script (pp. 31-2). There seems to be evidence of an intimate connection 

 between Indian and Chinese mathematics (pp. 38-41). Only later did the Hindus 

 outstrip the Chinese in the development of indeterminate analysis. The mathe- 

 matics of the Arabs seems, contrary to the usual opinion, to be practically 

 independent of Indian influence, and shows a great advance on Indian work 

 in all branches, except perhaps indeterminate analysis (pp. 41-3). The Arabs 

 based their work almost wholly on Greek knowledge, and the Greeks may have 

 influenced the Indians by way of China or Persia (pp. 44-5). There is an 



