The History of Mathematical Proof in Ancient Traditions

Algorithms in Bhāskara’s commentary on Āryabhat. īya 499

for the diagonals. Th e procedure is made of a multiplication followed by a
division:^24

Ab.2.8. Th e two sides, multiplied by the height 〈and〉 divided by their sum, are the
‘two lines on their own fallings’.|

When the height is multiplied by half the sum of both widths, one will know the
area.||

In other words, with the labels used in Figure 14.6 , we have:

EF AB EG

AB CD

FG

CD EG

AB CD

=

+

=

+

×

×

;

.

Likewise, the area 풜 is:

A=

+

EG

(AB CD)

2

×

On the fi rst part of the verse, Bhāskara comments: 25

Th e size of the ‘lines on their own fallings’ should be explained ( pratipādayitavya ) with
a computation of the Rule of Th ree on a fi gure drawn by 〈a person〉 properly instructed.
Th en, by means of just the Rule of Th ree with both sides, a computation of 〈the lines
whose top is〉 the intersection of the diagonals and a perpendicular 〈is performed〉.

Th is explanation consists of ‘reinterpreting’ the procedure – which is a
multiplication followed by a division – according to the Rule of Th ree. Th e

24 āyāmagun. e pārśve tadyogahr. te svapātalekhe te|
vistarayogārdhagun. e jñeyam. ks. etraphalam āyāme|| (Shukla 1976 : 63).

Figure 14.6 Inner segments and fi elds in a trapezoid.

HGI D

A E B

F

C

25 samyagādis. t.ena (rather than samyaganādis. t.ena of the printed edition) ālikhite ks. etre
svapātalekhāpraman. a m. trairāśikagan. itena pratipādayitavyam/ tathā trairāśikenaivobhaya
pārśve karn. āvalambakasampātānayanam/ (Shukla 1976 : 63).