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

DIVISION OF FRACTIONS 177
duct of the denominators is the (result of) multiplica-
tion of two or more fractions.¹
While all other writers give the rule in the same way as
Brahmagupta, Mahāvīra in the Gaṇitasārasaṃgraha refers to
cross reduction in order to shorten the work :
In the multiplication of fractions, the numerators are
to be multiplied by the numerators and the denomi-
nators by denominators, after carrying out the pro-
cess of cross reduction, if that be possible.²
Division of Fractions
The Āryabhaṭīya does not explicitly give the rule of divi-
sion, but under the Rule of Three, we have an indication of
this operation. The Rule of Three states the result as (f × i) / p,
where f stands for phala i.e. "fruit", i for icchā, i.e., demand or
requisition, and p for pramāṇa i.e. argument. When these
quantities are fractional, we get an expression of the form
(a/b × c/d) / (m/n)
for the evaluation of which Āryabhaṭa I states :
The multipliers and the divisor are multiplied by the
denominators of each other.
These quantities are written in the following way
┌───────┐
│ a m │
│ b n │
├───────┤
│ c │
│ d │
└───────┘
Transferring the denominators we have
┌───────┐
│ a m │
│ n b │
│ c d │
└───────┘
Performing multiplication, the result is anc / mbd. The above
interpretation of the obscure line in the Āryabhaṭīya is based

1. रूपाणिच्छेद गुण्यान्यंशयुतानि द्वयोर्बहूनां वा ।
   प्रत्युत्पन्नो भवति च्छेदवधेनोद्धृतोऽशवधः ॥ —BrSpSi. XII. 3
2. GSS. p. 25. (2)