136 Journal of the Asiatic Society of Bengal. [March, 1908. 
represents the process of finding the greatest common measure of 
the two numbers A and B. If the last pret is unity, Euclid 
states that the two numbers A an are prime ,inter se. His 
proof may be set down as follows :— 
r,=A-a,B 
Tg= A (—a,)+B (1—a,a,) 
rg=A (1+ 4,03) +B (—4,—a3—a,agd5) 
r,= A (—a,—4,—a,0,0,)+ B (lL +aja,+ aya, + a0, + a) 90304) 

ra=(—1)"*! (AL-—BM). 
Ir, =1 then A and B are prime to each other: for if not, let 
their common factor A such that d=af and B=bf and afL — 
bfM=1, ich, as all the terms are integral, is impossible. 
Traretene etc. 
Az — By= —— co ieee we get ALC — carrie and 
eee 
and mers as (Buelid VIL , 33), where ¢ is any integer. 
In solving 
Aa— By=C whenr;=1 
we have 
A =d,+ A, + M0304, 
= 1 + a)a2 + at azdyt A) AeAgh, 
2=Bi+ LC =t(1+ aja,+a,a, +430, + 4)4,0,0,) — O(a, + a3 44,4908) 
y= At + Mc=t(a,+ a,+a,a,0,)+ O(1 + a,a3). 
Now, following Aryabhata’s instructions, set down a, 
Gs 
i 
ta,=(. 
Add the lowest term to the product of the two preceding 
t+a,(ta,- M ); multiply this result by the next highest term (a,) 
and add to the product the penultimate term (ta,—(C) and so on. 
The final pion in this case oes 
t(1+ a,a,+a,4,+ aga, + a,0,a,0,) ~ O(a, + az + 4943) 
which equals LO + Bt=a as above. 


marvellous. Whether or not Euclid employed some sort of algebraic smy- 
bolism, we know that the later Alexandrian scholars did, and we also know 
that they translated Enclid’s proposition » into their new eyiiaionn: (See 
Gow, 83 and 104.) 
