238 G. Tliibaut — On the S'ulcasutras. [No. 3, 



So many " cognizable" measurements of the vedi exist. 

 That means : these are the measurements of the vedi effected by oblongs, 

 of which the sides and the diagonal can be known, i. e., can be expve-^ed 

 in integral numbers. 



In this manner A'pastamba turns the Pythagorean triangles known 

 to him to practical use (the fourth of those which Baudhayana enumerates 

 is not mentioned, very likely because it was not quite convenient for the 

 measurement of the vedi), but after all Baudhayana's way of mentioning 

 these triangles as proving his proposition about the diagonal of an oblong 

 is more judicious. It was no practical want which could have given the 

 impulse to such a research — for right angles could be drawn as soon as one 

 of the " vijneya" oblongs (for instance that of 3, 4, 5) was known— but the 

 want of some proof which might establish a firm conviction of the truth of 

 the proposition. 



The way in which the Sutrakaras found the cases enumerated above, 

 must of course be imagined as a very primitive one. Nothing in the 

 sutras would justify the assumption that they were expert in long cal- 

 culations. Most likely they discovered that the square on the diagonal 

 of an oblong, the sides of which were equal to three and four, could be 

 divided into twenty-five small squares, sixteen of which composed the 

 square on the longer side of the oblong, and nine of which formed the 

 area of the square on the shorter side. Or, if we suppose a more con- 

 venient mode of trying, they might have found that twenty-five pebbles or 

 seeds, which could be arranged in one square, could likewise be arranged 

 in two squares of sixteen and of nine. Going on in that way they would 

 form larger squares, always trying if the pebbles forming one of these 

 squares could not as well be arranged in two smaller squares. So they 

 would form a square of 36, of 49, of 64, &c. Arriving at the square form- 

 ed hy 13 X 13 = 169 pebbles, they would find that 169 pebbles could be 

 formed in two squares, one of 144 the other of 25. Further on 625 peb- 

 bles could again be arranged in two squares of 576 and 49, and so on. 

 The whole thing required only time and patience, and after all the number 

 of cases which they found is only a small one. 



Having found that, in certain cases at least, it was possible to express 

 the sides and the diagonal of an oblong in numbers, the Sutrakaras natu- 

 rally asked themselves if it would not be possible to do the same thing for 

 a square. As the side and the diagonal of a square are in reality incom- 

 mensurable quantities we can of course only expect an approximative 

 value ; but their approximation is a remarkably close one. 



Baudhayana : 



