132 Journal of the Asiatic Society of Bengal. {March, 1908. 
22, Take the sixth part of the product of the be ee factors made 
up of the last term, the last term plus one, and of this plus the num- 
ber of terms: the result is the volume of the pile oo as The 
square of the pile is the volume of the pile of cubes, 
it first part means P= meal cr atl bay om the second 
(14243444 ..... + )8= 18 + 23 + 384+ n’, 
See Alkarkhi (p. 61); Eitianagd pts’ GAIL A *30) ; etc., etc. 
If from the square of a sum is taken the sum of the squares 
the hat? of the result is the product of the factors. 
From a product multiplied by the square “ two and 
increased by the square of the difference extract the root: add and 
subtract the difference. The two factors are obtained by dividing by 
two. 
noe eka: of these in our notation is (a + b)* — (a* + b*) =2ab, 
the latt 
van ae Tite or / 4ab+(b—a)? + (b—a) =2b. 


This appears to a fragmentary section on identitie Tt 
corresponds ipaielat 5 a fuller section of Alkarkhi’s entitled 
* Theorems that help to solve difficulties,” which contains a num- 
ber of identities alin stirred pretty closel y to the second book of 
Euclid. The chief use of these identities was to help to solve 
indeterminate sokations of the second degre 
The first of the above ola ( (93) i an expression of 
Buclid IL, 4, The second (a—b)? + 4ab = oa by bia IL, 8) 
is used by Diophantus, in his ii on Polygonal Numbers 
25. The interest on the original sum plus the interest on that is 
multiplied by the time (and the original sum) and increased by the 
square of half the original sum : take the square root, deduct half the 
original sum and divide by the time. The result is the interest on the 
amount. 
We have the relation Discount + Interest on discount = Interest 
on amount or Pri+ Prt. rt=M where is what is termed the 
r= 4/M/PH+ (1/26) -1/2t 

Pr =v MP + Pi4—P/2 _ 
t 
/ Pt( Pr+ Pr. rt) + P2/4— P/2 
t 



which is the rule given by Aryabhata. 
