1875.] G-. Thibaut— On the S'ulvasutras. 239 



Increase the measure by its third part and this third by its own fourth 

 less the thirty-fourth part of that fourth ; (the name of this increased mea- 

 sure) is savis'esha. 



A'pastamba gives the rule in the same words. 



Katyayana : 



^<jff a^tiN ^wi ^g3??ffiir^gf^»u^r ^feii«r Tf?r fkizm i 



The sutras themselves are of an enigmatical shortness, and do not state 

 at all what they mean by this increasing of the measure ; but the com- 

 mentaries leave no doubt about the real meaning ; the measure is the 

 karani, the side of a square and the increased measure the diagonal, the 

 dvikarani. If we take 1 for the measure, and increase it as directed, we get 



the following expression : 1 + - 4- - — and this turn- 



O O X 4 O X 4 X 34 



ed into a decimal fraction gives : 1"4142156 Now the side of a 



square being put equal to 1, the diagonal is equal to \/ 2 = 1 '414213 .. 

 Comparing this with the value of the savis'esha we cannot fail to be 

 struck by the accuracy of the latter. 



The question arises : how did Baudhayana or A'pastamba or whoever 

 may have the merit of the first investigation, find this value ? Certainly 

 they were not able to extract the square root of 2 to six places of decimals ; 

 if they had been able to do so, they would have arrived at a still greater 

 degree of accuracy. I suppose that they arrived at their result by the 

 following method which accounts for the exact degree of accuracy they 

 reached. 



Endeavouring to discover a square the side and diagonal of which 

 might be expressed in integral numbers they began by assuming two as 

 the measure of a square's side. Squaring two and doubling the result 

 they got the square of the diagonal, in this case = eight. Then they tried to 

 arrange eight, let us say again, eight pebbles, in a square ; as we should say, 

 they tried to extract the square root of eight. Being unsuccessful in this 

 attempt, they tried the next number, taking three for the side of a square ; 

 ~hut eighteen yielded a square root no more than eight had done. They 

 proceeded in consequence to four, five, &c. Undoubtedly they arrived soon 

 at the conclusion that they would never find exactly what they wanted, 

 and had to be contented with an approximation. The object was now to 

 single out a case in which the number expressing the square of the diago- 

 nal approached as closely as possible to a real square number. I subjoin 

 a list, in which the numbers in the first column express the side of the 

 squares which they subsequently tried, those in the second column the 

 square of the diagonal, those in the third the nearest square number. 



