234 G. Thibaut — On the S'ulvasutrafs. [No. 3, 



instead of "cord"); this was therefore the name for the diagonal of a 

 square. Other compounds with karani will occur further on ; the change 

 of meaning which the word has undergone in later times will be consider- 

 ed at the end of this paper. 



The authors of the sutras do not give us any hint as to the way in 

 which they found their proposition regarding the diagonal of a square ; 

 but we may suppose that they, too, were observant of the fact that the 

 square on the diagonal is divided by its own diagonals into four triangles, 

 one of which is equal to half the first square. This is at the same time 

 an immediately convincing proof of the Pythagorean proposition as far as 

 squares or equilateral rectangular triangles are concerned. 



The second proposition is the following : 



Baudhayana : 



3TTrf?r i 



The cord stretched in the diagonal of an oblong produces both (areas) 

 which the cords forming the longer and the shorter side of an oblong pro- 

 duce separately. 



That is : the square of the diagonal of an oblong is equal to the 

 square of both its sides. 



^pastamba : 



Katyayana gives the rule in the same words as Baudhayana. 



The remark made about the term samachaturasra applies also to 

 " dirghachaturasra" " the long quadrangle" meaning the long quadrangle 

 with four right angles. " Pars'vamani (rajju)" is the cord measuring the 

 pars'va or the long side of the oblong or simply this side itself ; tiryanrnani, 

 the cord measuring the horizontal extent or the breadth of the oblong, in 

 other words its bhorter side, which stands at right angles to the longer 

 side. Noteworthy is the expression " prithagbhute ;" for as one of the 

 commentators observes it is meant as a caution against taking the square 

 of the sum of the two sides instead of the sum of their squares (prithag- 

 grahanam samsargo ma bhud ity evamartham). 



It is apparent that these two propositions about the diagonal of a 

 square and an oblong, when taken together, express the same thing that 

 is enunciated in the proposition of Pythagoras. 



But how did the sutrakaras satisfy themselves of the general truth of 

 their second proposition regarding the diagonal of rectangular oblongs '? 



Here there was no such simple diagram as that which demonstrates 

 the truth of the proposition regarding the diagonal of a square, and other 

 means of proof had to be devised. 



