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

218 BRAHMAGUPTA AS AN ALGEBRAIST Thus in the above example the product of pramāṇa-dhana and pramāṇa kāla divided by parakāla is (pt₁/t₂)—is (500 × 4) / 10 = 200. This is first multiplied by the total interest accrued (A); it becomes 200 × 78 = 15600. To this is now added square of half of 200 (which is 10000) ; it becomes 15600 plus 10000 = 25600. Its square-root is taken which is 160. From this is subtracted half of the quantity (i.e. half of 200 which is 100). Thus 160–100 = 60, which is the answer. It was the interest which first accrued (x). Another Quadratic Problem : Brahmagupta refers to an astronomical problem which involves the quadratic equation (72 + a²)x² ∓ 24 apx = 144 ( R² / 2 - p² ), where a = agra (the sine of the amplitude of the Sun), b = palabha (the equinoctial shadow of a gnomon 12 aṅguli long), R = radius, and x = koṇaśaṅku (sine of the altitude of the Sun when his altitude is 45°). Dividing out by (72 + a²) we have x² ∓ 2mx = n, where m = 12 ap / (72 + a²), n = 144(R²/2 - p²) / (72 + a²) Therefore we have x = √(m² + n) ± m, as stated by Brahmagupta. We find the same result in the Sūrya-siddhānta and in the text of Śrīpati. Āryabhaṭa II (1150) also followed the method of Āryabhaṭa I and Brahmagupta in solving a quadratic equation in connection with finding out the number of terms in an arithmetical progression (A.P.) whose first term is (a), common difference is b and the sum is s. The number of terms n is given by¹ n = [√(2bs + (a - b/2)²) - a + b/2] / b Two Roots of a Quadratic Equation and Brahmagupta A quadratic equation has two roots. This must have been known to Indian algebraists even at a very early stage. Bhāskara II in his Bījagaṇita has quoted a rule ascribed to an ancient writer Padmanābha whose works are not available now :

1. Mahāsiddhānta. Bhāskara II, XV. 50