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

THE RULE OF THREE 179
antiquity of the term. Bhāskara in his commentary of the
Āryabhaṭīya gives a justification of the use of this term for the
Rule of Three thus :
Here three quantities are needed (in the statement and
calculation) so the method is called trairāśika (meaning
thereby the "rule of three terms").
The problem of the Rule of Three has the form :
If p (pramāṇa) yields f (phala), what will i (icchā)
yield ?
Āryabhaṭa II (the author of the Mahāsiddhānta, 950 A.D.)
uses the terms māna, vinimaya and icchā, instead of pramāṇa,
phala and icchā respectively. It has also been pointed out by
several authors that the first and third terms are similar, i.e., of
the same denomination.
We shall give here the Rule of Three as given by Āryabhaṭa
I and Brahmagupta :
In the Rule of Three, the phala ("fruit"), being multi-
plied by the icchā ("requisition") is divided by the
pramāṇa ("argument"). The quotient is the fruit
corresponding to the icchā The denominators of one
being multiplied with the other give the multiplier
(i.e. numerator) and the divisor (i.e. denominator).¹
In the Rule of Three pramāṇa ("argument"), phala
("fruit") and icchā ("requisition") are the (given)
terms; the first and the last terms must be similar. The
icchā multiplied by the phala and divided by the
pramāṇa gives the fruit (of the demand).²
Śrīdhara also gives the Rule of Three almost in the same
words. Bhāskara II, Nārāyaṇa and others follow Brahmagupta
and Śrīdhara in the Trairāśika operation. Śrīdhara in his Pāṭīga-
nita says :

1. त्रैराशिकफलराशिं तमथेच्छाराशिनाहतं कृत्वा ।
   लब्धं प्रमाणभजितं तस्मादिच्छाफलमिदं स्यात् ।।
   छेदाः परस्परं हता भवन्ति गुणकार भागहाराणां ।
   छेदगुणं सच्छेदं परस्परं तत्सवर्णत्वम् ।। —Arya. II 26-27.
2. त्रैराशिके प्रमाणं फलमिच्छाद्यन्तयोः सदृशराशी ।
   इच्छाफलेन गुणिता प्रमाणभक्ता फलं भवति ।। —BrSpSi XII. 10