240 G. Thibaiit— On the S'ulvasutras. [No. 3, 



I. 



2. 



1. 



2. 



8. 



9. 



3. 



18. 



16. 



4. 



32. 



36. 



5. 



50. 



49. 



6. 



72. 



64. 



7. 



98. 



100. 



8. 



128. 



121. 



9. 



162. 



169. 



10. 



200. 



196. 



11. 



242. 



256. 



12. 



288. 



289. 



13. 



338. 



324. 



14. 



392. 



400. 



15. 



450. 



441. 



16. 



512. 



529. 



17. 



578. 



576. 



18. 



648. 



625. 



19. 



722. 



729. 



20. 



800. 



784. 



How far the Sutrakaras went in their experiments we are of course 

 unable to say; the list up to twenty suffices for our purposes. Three 

 cases occur in which the number expressing the square of the diagonal 

 of a square differs only by one from a square-number ; 8 — 9 ; 50 — 49 ; 

 288 — 289 ; the last case being the most favourable, as it involves the 

 largest numbers. The diagonal of a square, the side of which was equal to 

 twelve, was very little shorter than seventeen ( \/ 289 — 17). Would 

 it then not be possible to reduce 17 in such a way as to render the square 

 of the reduced number equal or almost equal to 288 ? 



Suppose tbey drew a square the side of which was 17 padas long, 

 and divided it into 17 X 17 = 289 small squares. If the side of the 

 square could now be sbortened by so much, that its area would contain 

 not 289, but only 288 such small squares, then the measure of the side 

 would be the exact measure of the diagonal of the square, the side of which 

 is equal to 12 (12 2 -f 12 2 = 288). When the side of the square is shortened 

 a little, the consequence is that fromtwo sides of the square a stripe is cut off ; 

 therefore a piece of that length had to be cut off from the side that the 

 area of the two stripes would be equal to one of the 289 small squares. 

 Now, as the square is composed of 17 X 17 squares, one of the two stripes 

 cuts off a part of 17 small squares and the other likewise of 17, both together 

 of 34 and since these 34 cut-off pieces are to be equal to one of the squares, 

 the length of the piece to be cut off from the side is fixed thereby : it must 

 be the thirty-fourth part of the side of one of the 289 small squares. 



The thirty-fourth part of thirty-four small squares being cut off, one 



whole small square would be cut off and the area of the large square 



reduced exactly to 288 small squares ; if it were not for one unavoidable 



circumstance. The two stripes which are cut off from two sides of the square, 



let us say the east side and the south side, intersect or overlap each other 



in the south-east corner and the consequence is, that from the small square 



2 2 1 



in that corner not — are cut off, but only — — — — -. Thence the 



34 ' J 34 34 X 34 



