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

164 BRAHMAGUPTA AND ARITHMETIC bhaṭīya : The continued product of three equals and also the solid having twelve (equal) edges are called ghana.¹ A method of cubing applicable to numbers written in the decimal place-value notation, has been in use in this country from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had the familiarity with this method; he, however, does not give the method of cubing in his treatise, though he describes the inverse process of extracting the cube-root. Brahmagupta gives the method of cubing in the following verse : Set down the cube of the last (antya); then place at the next place from it, thrice the square of the last multi- plied by the succeeding; then place at the next place thrice the square of the succeeding multiplied by the last, and (at the next place) the cube of the succeeding. This gives the cube.² The rule may be illustrated by an example. Example : To cube 1357. The given number has four places, i.e., four portions. First we take the last digit 1 and the succeeding digit 3, i.e. 13 and apply the method of cubing thus : (i) Cube of the last (1³) = 1 (ii) Thrice the square of the last (3.1²) multiplied by the succeeding (3) gives (3.3.1²) = 9 (placing at the next place) (iii) Thrice the square of the succeeding, multiplied by the last gives (3.3².1) = 27 (placing at the next place) (iv) Cube of the succeeding (3³) = 27 (placing at the next place) ———————————————————————————————————————————————— Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रयस्यात् ।। — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसङ्गुणा च तत्प्रथमात् । उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.