


Vol. ae No. 3.] Notes on Indian Mathematics. 125 
N.S.] 
be obtained by its EK and that with the seventh sine begins a 
discordance between the table and the result of calculation by the 
rule which finally amounts to as much as 70 minutes. It follows, 
merit: that either the rule was used, but corrections were 
y the aid of other tablnns or the table was copied whole- 
sale. 
In the ee ee is given a table that was ee 
deduced from Ptolemy’s table of chords (J. Burgess in Ind. A 
1891, 228). Ptolemy” s table increased by half degrees mae nes 
divided the radins into sixty equal parts and subdivided it 
sexagesimally, The Panchascddhantika table is obtained _ by 
simply dividing Ptolemy’s chords by two, and hence the term 
half-chord. This table i ‘er only for twenty-four angles at intervals 
of 3¢ degrees. Aryabhata and the compilers of the Surya Sid- 
dhanta express their results in a sort of cir pues measure, and to 
obtain them from Ptolemy’s chords it is simply necessary to 
rote ae by ae igh a ones eae the radius equal 
o 60°, the Hindus qual to 90°x2x7. In Ptolemy’s 
measure J90° = ee 80°72 = "120: */2, while in Aryabhata’s measure 
J90=90° x 27; therefore to change chord 180° to J90° we have 
120° x C=90° x 2) x m or the change ratio C =3/27 
Aryabhata has J90°= 3438’ * therefore 10400’ /r= 3438’ which 
gives r=3'141361 .. . Rodet puts the matter thus : 10800/3: ons 
=3437°7 =3438 nearly and concludes that the —— 31416 wa 
used ; but this is not quite ingenuous. We might replace this 
value by Ptolemy’s value and then we should have 10800 x 120/377 
3437°66 = 3438 nearly and just as forcibly conclude that Ptolemy’s 
value was used. Indeed, Ptolemy’s value was most probably used 
in the reducing mea but when the reduction took an is not 
easy to determine. There were two stages in the process: first, 
as in the Paiichasidahantika ! the chords of Ptolemy were spy 
halved, and the old measure for the radius (=60°) retained ; 
, 
secondly, the new measure for the radius ( =3438’) was intro- 
duced. This new measure first occurs in the Pulisa-siddhanta, 
for Albiruni writes (i., 275) : ur calculation is based on this, 
that = sinus totus is 3438’ The source of this calculation 
rae s the Pulisa Siddhanta, wioekk divides the are of the quarter 
of wisieabs into 24 kardajat. He says: ‘If anybody asks for the 

Aryabhata may be put as ‘sin A’=chord 24/2. The Hindu ‘sine’ isa 
projection of the pea, Putting J for the so-called Hindu sine a correct 
chd 2A 
relationis J A= 
| This portion (Ch. iv) of the Seissieagrcmtvee is ganerale allotted bade the 
Pauliga Siddhanta, but there is an element of doubt abou Dr. rhe, ay 
(p. x.) : “I am more doubtful abont the siooikaots of chapter ~ — in the 
-Romaka, Panulisa and Sirva tigen ag ai 
& ter follows and precedes iis ev 
oy mpossible. that ashe contents are meant to sum up the teaching of 
Siddhinte only.” 
