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

PRINCIPLE OF COMPOSITION 249
Principle of Composition
The above results have been technically known amongst
Indian algebraists as Bhāvanā (demonstrated or proved, hence
theorem or lemma). The word bhāvanā also means "composition
or combination" in algebra. Bhāvanā may be of two types :
Samāsa Bhāvanā (or addition Lemma, or additive composition)
and Antara Bhāvanā (or subtraction Lemma or subtractive
composition). Whenever, again, the bhāvanā is made with two
equal sets of roots and interpolators, it is technically named as
Tulya Bhāvanā (or composition of equals), and when with two
unequal sets of values then it is known as Atulya Bhāvanā (or
composition of unequals).
Proof of Brahmagupta's Lemmas
It is significant to be indicated that Brahmagupta's Lemmas
were rediscovered by Euler in 1764 and by Lagrange in 1768,
and a considerable importance was attached to them. Kṛṣṇa,
(1580 A.D.) the commentator on the Bījagaṇita of Bhāskara II
gives the following proof of Brahmagupta's Lemmas :
Let (α,β) and (α',β') be the two solutions of the equation
nx² + k = y².
we have
Nα² + k = β²
Nα'² + k' = β'²
Multiplying the first equation by β'², we get
Nα²β'² + kβ'² = β²β'²
Now, substituting the value of factor β²' of the interpolator
from the second equation, we get
Nα² β'² + k (Nα'² + k') = β²β'²
or N(α²β'² + Nkα'² + kk' = β²β'²
Again, substituting the value of k from the first equation
in the second term of the left-hand side expression, we have
Nα²β'² + Nα'²(β² - Nα²) + kk' = β²β'²
or N(α²β'² + α'²β²) + kk' = β²β'² + N²α²α'²
Adding ±2Nαβα'β' to both sides, we get
N(αβ' ± α'β)² + kk' = (ββ' ± Nαα')²