ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

BRAHMAGUPTA FIRST TO USE SECOND DIFFERENCES 99
at the interval of 15° of arc of the mean anomaly, in the Uttara
Khaṇḍakhādyaka teaches, for the first time in the history of
mathematics, the improved rules for interpolation by using the
second difference. This very important feature I am reproducing
here from the translation by Senagupta of the verse ¹
(UKK. 8) :
Multiply the residual arc left after division by 900'
(i.e by 15°), by half the difference of the tabular
difference passed over and that to be passed over and
divide by 900' (i. e. 15°) : by the result increase or
decrease, as the case may be, half the sum of the same
two tabular differences ; the result which, whether less
or greater than the tabular difference to be passed, is
the true tabular difference to be passed over. (UKK. 8)
The rule given here applies to the case of all functions
hitherto considered in the Khaṇḍakhādyaka, which are tabulated
at the difference of 15° of arc of the argument. They are :
(i) the tabular differences of the Sun's equation,
(ii) the tabular differences of the Moon's equation.
(iii) the tabular differences of the 'sines'.
Sengupta has illustrated the rule by an example belonging
to the table of sines.
Illustration—To find the 'sine' of 57°.
Brahmagupta's table of sines in the Khaṇḍakhādyaka is as
follows :
Thirty increased severally by nine, six and one; twenty-
four, fifteen and five, are the tabular differences of
sines at intervals of half-a-sign. For any arc, the 'sine'
is the sum of the parts passed over, increased by the
proportional part of the tabular difference to be passed
over.² (*KK.*I.30 ; also III.6)

1. गतभोग्य खण्डकान्तरदलविकलवधात् शतैर्नवभिराप्त्या ।
   तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥ —UKK. IX. 8
2. त्रिंशत् सन्नवरसेन्दुजिन (तिथि) विषयागृहार्द्धं चापानां ।
   भर्द्ध्ज्या खण्डानि ज्या भुक्तैक्यं सभोग्यफलम् ॥ —KK. I. 30; also III. 6