
Vol. IV, No. 3.} Notes on Indian Mathematics. 137 
LW.8.] 
As ¢ is any integer we res? substitute any other integer for it. 
Set ¢’=ta,—C then t=(t’ + O)/a, and we have the series a, a, As, 
26 Ay, , (t'+C)/a, which may = treated as before 
If in Ax-By=C we have set e=Bta', We then 
have A+ (Az’0)/B=y. “Therefore if (ae C)/B is integral so is 
(Aw’~O)/B integral, and 2’ is a solution of Avx—By=C.. These 
tench: which are vaguely ahs Ban in Aryabhata’s rule, may, of 
course, be put ina perfectly general form 
Although there is ample evidence in Greek mathematics as to 
the existence of the preliminary notions necessary for ig se hon 
tion of the particular rule under consideration, yet w where 
find in extant Greek works the rule itself applied in n just this 
manner. On the other hand we do find that the Greeks carried 
the treatment of indeterminate equations much Pr ae than 
did Aryabhata, and there is no doubt that they were able to 
manipulate indeterminates of the first degree in the manner 
indicated in the rule of Aryabhata. 
t is interesting to note that discord in the early Christian 
church possibly had a significant connection with the development 
of Hindu mathematics. The Alexandrian Christians appear to 
have been much given to wrangling, and one of the points they 
chose to quarrel about was the ecclesiastical calendar. As early 
as the second century of our era great disputes had arisen about 
the proper time of celebrating Easter. At the Council of Nice 
(A.D. 326) a decision on this point was arrived at, but it was left 
to the scholars of Alexandria to find the exact date each year. 
Diophantus lived about A.D. 300-350, and Hypatia, who wrote a 
commentary on the works of Diophantus, was murdered by these 
quarrelsome Alexandrian Christians in A.D. 415. Aryabhata was 
born in A.D. 476. 
It is in connection with questions on the calendar that the 
most ane applications of pocine on gan of the first de- 
gree arise. The following examp a very marked manner 
illustrates reac points of Femur jem rule that at first seemed 
inexplicable 
“The year 1 of the Christian era was in the Solar cycle the thes 10 and in 
the Metonic cycle it was 2. What was it in the Dionysian cyc 
The Solar cycle a of 28 years, the Metonic cycle of 19 
years, and the Dio onysian of 28x19 years, Let n ste the date in 
years in the Dionysian cycle, then nj28 and n/19 must give r ieee 
tively 10 and 2 as remainders, or »/28=2x+ 10/28 aud nf19=y+ 
2/19, whence 282~—19y= -8. 
In accordance with Aryabhata’s rule we go through the pro- 
cess of finding the G.C.M. of 19 and 28, and obtain the series 1, 
tw 
this result che the preceding term and adding to the product the 
penultimate term, we get 35 x1+17=52. These results, 35 and 
52, are values for z and y which satisfy the equation ; but they 
are not the sicolent results, so we divide the 35 by the smaller 
