154 Journal of the Asiatic Society of Bengal. [March, 1908. 
where a+b+c-a=™m., 
atb+c—b=n. 
at+b+c—c=p. 
Cantor maintains that this is a Greek theorem, disguised in 
form by Aryabhata, in order to ela his plagiarism, The rule 
is given by Jamblichus as fo “When any defined or 
undefined quantities amount to a given: sum, and the sum of one of 
them plus every other (in pairs) is given, the sum of these pairs 
minus the first-given sum is (if there be 3 quantities) equal to the 
quantity which was added to all the rest (in the pairs) : or (if 
there be 4 quantities) to } of it: (if 5) to4; (if 6) toZ; ete. 
That is, if a,+a,+42,=S be given, and #,+#,=8), an oF a= Qs 
then #,=8,+S8,—S8. And, gene if ik tt Bs oe ea 
an gt Bare ® + %3 =o, a+a,=8,_;, then a= 
(S)+S,+. . Sp 1~—-8)/(n —2). The proposition is known as 
ep enanthomn | or after blossom’ (Gow 
odet maintains that this mine Ce between the Greek 
and Indian rules is not very close ; but the following two examples 
given ot ® Diophantus (i., 16 and 17), ad oie noticed neither by 
Rodet nor Cantor in this connection, dispel all doubt as to the 
origin of  Kegabhate’s rule. In ordinary notation the examples 
given by Diophantus are 
Gi) a@,+%,+2,=9, 2,+%,=20, a+2,=30, 2,+42,=40 
whence 
$= (20+ 30+ 40)/2=45. 
(ii) a+e,+agta=$, ej +e,+%,=20, a+as+a,=22, 
Ut, +2,=24, w,+%,+a,=27 whence >= 
30. The difference between the objects divides the difference 
between the money possessed by two persons. The quotient is the 
original value of an object the wealth being equa 
The rule states that z=(b—a)/(m—p) when mz+a=petb. 
Ahmes designated the unknown quantity by haw or ‘ heap,’ 
and, according to Rodet, the word af@art here used is a direct 
translation of hau. See Diophantus, i., 4f. 
31. Division of the space when going tn opposite sagt by 
the sum of their speeds; when going in the same direction by the 
difference of their speeds, The two quotients are the times of 
meeting as they pass by sought for. 
T=8/(v+v’) T’=8/(V-V’). 
32-33. The Ainge original divisor is divided by the lesser origi- 
nal divisor and the rest divide one another. An assumed number to- 
gether with the poner lowe is thrown in. The lower is multi- 
d 
