274 G. Thibaut — On the S'ulvasutras. [No. 3, 



sought for, and we know from the paribhasha rules that this could be easi- 

 ly managed. Accordingly, Baudhayana's rule has to be translated as fol- 

 lows : The tritiyakarani of that area which is made a square with the third 

 part of the mahavedi (i. e., of a square of 324 padas) is it (viz. the side of 

 a square of 108 padas) ; the result is the ninth part of the area (of the 

 mahavedi). 



Thus we see that the pre-conceived opinion of the commentator about 

 the method to be employed for the solution of the problem leads him to a 

 perfectly mistaken interpretation of the sutra. 



On the other hand, it is interesting to find some terms indicating a con- 

 nexion between the first rudiments of science as contained in the S'ulvasu- 

 tras and its later development. So for instance the term ' varga'. It is 

 true that we should be able to account for the meaning in which it is used 

 by later mathematicians — viz. that of the square of a number — without finding 

 earlier indications of the manner how it came to be used in that sense. The 

 origin of the term is clearly to be sought for in the graphical representation 

 of a square, which was divided in as many ' vargas', or troops of small 

 squares, as the side contained units of some measure. So the square drawn 

 with a side of five padas' length could be divided into five vargas, each con- 

 sisting of five small squares, the side of which was one pada long.. 



Nevertheless it is interesting to find this explanation of varga confirmed 

 by a passage in A'pastamba. 



^■RfSWrWT T^^TT^ff^fT^WT SfJrr^rTffT l 



As many measures (units of some measure) a cord contains, so many 

 troops or rows (of small squares) it produces (when a square is drawn on 

 it). 



But another case is more interesting still. The word ' karani' is one 

 of the most frequent mathematical terms in treatises as the Lilavati, Vija- 

 ganita, &c, and there it is invariably used to denote a surd or irrational 

 number ; as the commentators explain it, that of which when the square- 

 root is to be taken, the root does not come out exact. The square-roots of 

 two, three, five, &c, are karani's. How the word came by that meaning, we 

 are not told, but we are now able to explain it from the S'ulvasutras. As 

 we have seen above, in these it always means the side of a square. 



The connexion between the original and the derived meaning is clear 

 enough. Karani meant at first the side of any square, after that possibly 

 the square-root of any number. Possibly I say, for in reality the mathema- 

 tical meaning of karani was restricted. It was not used to denote the 

 square-roots of those numbers, the root of which can be exactly obtained, but 

 only of those the root of which does not come out exact, of those in fact 

 the root of which can be represented exactly only in a graphical way. It 

 was not possible to find the exact square-root of eight for instance, but it 



