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

260 BRAHMAGUPTA AS AN ALGEBRAIST
by n², then we are justified in saying that if we put nx=u, the
equation (i) becomes Mu²±c=y² (ii),
and clearly the first root of (i) is equal to the first root of (ii)
divided by n. The corresponding second root will be the same
for both the equations.
FORM a²x²±c=y² :
We find Brahmagupta giving the following rule in this
connection : This is a solution of a particular form of a
varga-prakṛti or Square-nature.
If the multiplier be a square, the interpolator divided
by an optional number and then increased and
decreased by it, is halved. The former (of these
results) is the second root; and the other divided by
the square-root of the multiplier is the first root.¹
Thus the solutions of the equation
a²x²±c=y²
are :
x = 1/(2a) (±c/m - m)
y = 1/2 (±c/m + m)
where m is an arbitrary number.
Bhāskara II and Nārāyaṇa have also given the same
solutions as proposed by Brahmagupta.
Rational Geometrical Figures
In the days of the Taittirīya Saṁhitā and the Śatapatha
Brāhmaṇa, Indian mathematicians got familiarity with the
solution of such equations
x²+y²=z²
and the results were arrived geometrically on the basis of the
law of rectangle as propounded by Baudhāyana in the Śulba Sūtras
and which goes by his name. The reader is referred to the Chapter
on Baudhāyana, the first Geometer in the author's "Founders
of Sciences in Ancient India". Baudhāyana (c. 800 B.C.) gave a

1. वर्गे गुणके क्षेपः केनचिदुद्धृतयुतोनितो दलितः ।
   प्रथमोऽन्यन्मूलमन्त्यो गुणकारपदोद्धृतः प्रथमः ॥ —BrSpSi. XVIII. 69