1875.] G-. Thibaut — On the S'ulvasutras. 271 



long before they were embodied in the Kalpasutras which have come down 

 to us. Besides, the quaint and clumsy terminology often employed for the 

 expression of very simple operations — for instance in the rules for the 

 addition and subtraction of squares — is another proof for the high antiquity 

 of these rules of the cord, and separates them by a wide gulf from the pro- 

 ducts of later Indian science with their abstract and refined terms. 



This leads to another consideration. Clumsy and ungainly as these 

 old sutras undoubtedly are, they have at least the advantage of dealing 

 with geometrical operations in really geometrical terms, and are in this point 

 superior to the treatment of geometrical questions which we find in the 

 Lilavati and similar works. They tell us that the diagonal of a square or 

 of an oblong produces an area equal to double the area of the square or to 

 the squares of the sides of the oblong — not that the square of the number of 

 units into which the diagonal is divided is equal to double the square of the 

 number expressing the side of the square or to the sum of the squares of 

 the two numbers which represent the sides of the oblong. 



Let us see how Bhaskara words the proposition about the rectangular 

 triangle (instead of which the sutras speak of the square and the oblong). 

 "We read in the chapter on kshetravyavahara in the Lilavati the following : 



— rTf§rWT*n3rc< ^nr: i 



The square root of the sum of the squares of these (of the two shorter 

 sides of a rectangular triangle) is the diagonal. 



£-T!3»'Jl'33T%fWcJT:T 5 3T'^l ^jfz: | 



The square root of the difference of the squares of the diagonal and one 

 of the short sides (called " doh") is the other short side (kotih), etc. 



It is apparent that these rules are expressed with a view to calculation, 

 and we find indeed that Bhaskara immediately proceeds to examples which 

 are exercises in arithmetic, not in geometry. 



^t!^ ^T: W?r: ^T^fffKlf ^ *gf ^ II 



A geometrical truth interests the later Indian mathematicians but in 

 so far as it furnishes them with convenient examples for their arithmetical 

 and algebraic rules ; purely geometrical constructions, as the samasa and 

 nirhara of squares, described in the S'ulvasutras, find no place in their 

 writings. 



It is true that the exclusively practical purpose of the S'ulvasutras 

 necessitated in some way the employment of practical, that means in this 

 case, geometrical terms, and it might be said that the later mathematicians 

 would have employed the same methods when they had had to deal with 

 the same questions. 



