पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

lxxii INTRODUCTION
(xiii) भुजस्य कृतिरिष्टस्य भक्तोनेष्टेन तद्दलं ।
कोटिरिष्टाधिका कर्णश्चतुरश्रायतस्य ते ॥
(p. 159, lines 4-5.)
(xiv) गुणके वर्गयोर्मध्ये तत्पदाधो भुजश्रुती ।
केचित्प्राक्कथिते तत्र वज्रकेणाहती तयोः ॥
अन्तरस्य कृतिः क्षेपस्तत्कोटिः प्रथमं पदम् ।
ऋजुहत्यन्तरं ज्येष्ठं रूपक्षेपेन्तरोद्धृते ॥ (p. 159, lines 9-12)
Quotation (xiv) above deserves special attention. It
gives the rational solution of the equation Nx² + 1 = y² in the
form¹
k Ah − Bb
x = ─────── , y = ───────── ,
Bh ~ Ab Bh ~ Ab
where b, k, and h are respectively the base, upright, and
hypotenuse of a right-angled triangle (which are determined
by the rule given in quotation xiii), A, B, being two numbers
such that A² − B² = N.
This form of solution of Nx² + 1 = y² is unique in so far as
it does not find its parallel in any other work on mathematics,
ancient or modern. It is at the same time the most general
rational solution of that equation. The solutions given by
other mathematicians, such as Brahmagupta (628 A.D.),
Śrîpati (1039 A.D.), Bhâskara II (1150 A.D.), Nârâyaṇa (1356
A.D.), Jñânarâja (1503 A.D.), and Kamalâkara (1658 A.D.)
and also those given by John Wallis and W. Brouncker (1657
A.D.) are easily deducible from it.
The hemistich of quotation (vii) is also of great interest.
For, as we have already mentioned, it is found to occur as a
quotation in the Bakhshâlî Manuscript also. There are two
places in the Bakhshâlî Manuscript where it is seen to be quoted.
At one place we have
तयोयोगवियो......... (BM, 54 recto)
─────────────────────────────────────────────────────────────
¹ For details, see my paper, entitled 'On Śrîdhara's Rational Solution
of Nx² + 1 = y²', Gaṇita Vol. I, No. 2, pp. 1-12. [It must be noted that the
above rational solution is due to some unknown mathematician, who
was the author of the verses of quotation (xiv), and not to Śrîdharâcârya
as was supposed in that paper.]