+ =e 
Vol. IV, No. 3.] Notes on Indian Mathematics. 131 
~ [N.S.] 
The second rule (19) appears to be out of place, and can 
hardly have been intended to apply only to the particular case of 
19a when p=0 as, of course, it does, 
20. The number of terms: Multiply (the sum) by eight times 
the common difference, add the square of the difference between twice 
the first term and the common difference : the square-root diminished 
by twice the first term and divided by the common difference with one 
added take one half. 
The rule means 
n=} fee are aE 


which may be obtained directly from 
n—1 
2 

*S=n(at d). 
Diophantus in his Polygonal Numbers gives this rule in the 
form 8dS+(d—2)*={d(2n—1)+2}% which is identical with 
Aryabhata’s formula except that the first term is unity, Alkarkhi, 
whose work is based on that of Diophantus, gives a good number 
of solutions of which the following are particular examples : 
(4)3454+7+..... (na terms)=255, n=15; (i) 104154204 . 
- . « (v terms) =325, n=10. Brahmagupta (XII, iii., 18) and 
Bhaskara (Lil. v., 125) give the same rule, but give no examples. 
21. (a) The common difference and the first term being unity, take 
the number of terms for the first factor and one for the increase an 
multiply together thrice and divide by six: ct vs the volwme of the 
pile. (b) Or the cube of the number of terms plus one minus its root. 
e are rules for finding the contents of a pile with a tri- 
angular base, which may be expressed thus: 
+1)8§~(n+1 
(a) P=n (n+1)(n+2)/6. @) Pa GF). 
As Rodet remarks, it appears strange that Aryabhata should 
give the correct formula here, while he gives an incorrect rule for 
finding the volume of a pyramid (§ 6). The only conclusion is 
that Aryabhata did not recognise the connection between the two 
rules. 
These and similar problems were favourites with the Greeks 
(cf. Nicomachus, p. 89f, ed. Hoche ; Boetius, p. 107, ed, Friedlein ; 
Archimedes On Spirals, prop. x.; Alkarkhi, p. 60, etc.). Alkarkhi 
points out the identity between the formula (a) in (6). Brahma- 
gupta (XIL., iii., 19) gives the rule for a pile with a square base and 
connects this with the sum of the squares of the natural numbers. 
He also gives the rule for the sum of the cubes. Alkarkhi gives 
elegant demonstrations of these rules. 
