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

100 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
This can be shown in the tabular form thus :

Arc	'Sine'	Tabular diference	Second difference
0°	0
15°	39	39
30°	75	36	—3
45°	106	31	—5
60°	130	24	—7
75°	145	15	—9
90°	150	5	—10

Now 57° = 3420 minutes = 900' × 3 + 720'. Thus three of the
tabular differences are considered as passed over ; the last one
being 31 and the one to be passed over is 24.
The true tabular difference by the rule, for arc 57°,
= (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2
Hence the 'sine' of 57°
= 39 + 36 + 31 + (720 / 900) [ (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 ]
= 125.76
As worked out from the logarithm tables the same comes
out to be 125.80.
Again 'sine' of 57° from Brahmagupta's formula
= 106 + (720 / 900) × 24 + ((31 - 24) / 2) × (720 / 900) - [(720 / 900)] × (31 - 24) / 2
= 106 + (720 / 900) × 24 + (720 / 900) { (720 / 900) - 1 } × (24 - 31) / 2
This in fact is the modern form the interpolation equation
up to the term containing the second difference. Brahmagupta thus
takes a decidedly improved step here and is undoubtedly the first
man in the history of mathematics who has done this. One should
also remember that in the case where the function is not tabula-
ted at a constant interval, Brahmagupta's rule is remarkable.