The History of Mathematical Proof in Ancient Traditions

Algorithms in Bhāskara’s commentary on Āryabhat. īya 497

employs the Pythagorean Th eorem to compute OS ′ u. In order to identify
the right-angled triangle, Bhāskara renames the Rsine of latitude ( aksajyā ,
SuS ′ u) as the base of a right-angled triangle ( bhujā ) and he identifi es the
radius of the celestial sphere as the hypotenuse. Th us, the Rsine is identi-
fi ed with the upright side of a right-angled triangle. Th is identifi cation
implicitly explains how the computation is carried out. However, Bhāskara
immediately adds: 22

With the Rule of Th ree also 13, 12, 3438; what has been obtained is the Rsine of the
colatitude, 3174. 23

In this way, Bhāskara again computes OS ′ u by using the similarity
of OSuSu ′ and OGC. Bhāskara thus computes the same value twice, using
two diff erent methods. Th e most likely explanation is that he verifi es
the results obtained with one algorithm by using another independent
process.

Figure 14.5 Latitude and co-latitude on an equinoctial day.

Z′

P′

90 – φ

φ

Z

P

S

O

G

C

Q

Q′ = Su

Su′

N

22 trairāśikenāpi 13/ 12/ 3438/ labdham avalambakah. 3174/ (Shukla 1976 : 90).
23 Th is value is an approximation again.