124 Journal of the Asiatic Society of Bengal. [March, 1908. 
and. sin (w+ 1)y+sin (n—1)y =(sin ny. sin 2y)/siny 
from which we get — 
sin (n+ 1)y- sin xy=sin wy—sin (n— Lyaipin auahne 
by substituting the values of sin 2y and siny, when y=34", thee 
in the table. 
The last formula may be expressed— 
Kos oF D,,4.1,=D,,—sin y/siny, 
which is the rule given in the text (12b). 
Aryabhata gives the corresponding table of differences ‘of 
sines in the Gitiéka (§ 10). In the Sarya Siddhanta the matter is: 
expressed thus (II.,15): “The eighth part of the minutes of a 
sine is called the first sine; that, increased by the remainder left. 
a 
four tabular sines in order as follows’ ’: then follows the table of 
sti which corresponds exactly with Aryabhata’ s table of differ- 
ence: 
Zn the. Pajtchasiddhantika (iv., 3) the following methods are 
given : n order to find the remaining desired (sines) 
the doable of the arc, deduct it from the quarter, diminish t 
radius by the sine of the remainder, and add to the square of half 
of that the square of half the sine of double the arc. The square 
root of that sum is the desired sine .... Another method is 
also taught here. Lessen the radius by the sine of three signs 
m which double the required arc has been previously deducted 
and multiply the remainder by sixty ; the result is the square.” 
hese rules may be expressed thus :— 
4 sin#y=sin? 2y+(1—sin (90—2y)) 
and 
7 sin*y=r(7 -y sin (90—2y)). 
As Dr. Thibaut has shown, these methods are not essentially 
different. Ptolemy proved that 
(chord +)*=120 (60—14/120?— (chord y)* 
from which the second formula given in the Pafichasiddhantika is 
a derived. 
the formula given by Aryabhata and in the Sarya Sid- 
Banta, we find that only five of the sines! following the first can 






“1 The term sine is here not used quite in the modern sense. The term 
used by the Hindus was gry or half-chord, and like Ptolemy’s chord it is not 
aratio but alength. Strictly * sin A=chord 24/2, but the relation used by 
