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

178 BRAHMAGUPTA AND ARITHMETIC
on the commentaries of Sūryadēva and Bhāskara I (the commen-
tary of Parameśvara on this line is vague and misleading).
Sūryadeva in this connection says :
Here by the word guṇakāra is meant the multiplier
and multiplicand, i.e., the phala and icchā quantities
that are multiplied together. By Bhāgahāra is meant
the pramāṇa quantity. The denominators of the phala
and icchā are taken to the pramāṇa. The denominator
of the pramāṇa is taken with the phala and icchā. Then
multiplying these, i.e., (the numerators of) the phala
and icchā and this denominator, and dividing by (the
product of) the numbers standing with the pramāṇa
the result is the quotient of the fractions.
Brahmagupta gives the method of division as follows :
The denominator and numerator of the divisor having
been interchanged, the denominator of the dividend is
multiplied by the (new) numerator. Thus division of
*proper fractions is performed.*¹
Square and Square-Root of Fractions
Brahmagupta says as follows in this connection:—
The square of the numerator of a proper fraction divi-
ded by the square of the denominator gives the
square².
This rule of Brahmagupta has been followed by other
authors also. The rule regarding the square-root as given by
Brahmagupta is as follows :
The square-root of the numerator of a proper fraction divi-
ded by the square-root of the denominator gives the square-
root.³
The Rule of Three :
The Indian term in Sanskrit for the Rule of Three is Trai-
rāśika (literally, "three terms"). The term occurs in the Bakh-
śālī Manuscript also, and also in the Āryabhaṭīya, indicating the

1. परिवर्त्य भागहारच्छेदांशौ छेद संगुणच्छेदः ।
   अंशोंशगुणो भाज्यस्य भागहारः सवर्णितयोः ॥ —BrSpSi: XII. 4
2. संवर्गितांशवर्गश्छेदकृतिविभाजितो भवति वर्गः । —BrSpSi. XII. 5 (1)
3. संवर्गितांशमूलं छेदपदेनोद्धृतं मूलम् । —BrSpSi. XII. 5 (2)