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

158 BRAHMAGUPTA AND ARITHMETIC also occur in the Bakhaśālī Manuscript. Āryabhaṭa I does not mention the everyday methods of multiplication in his Āryabhaṭīya probably because they were too elementary to be included in a Siddhānta work. Brahma- gupta, however, in a supplement to the section on mathematics in his Siddhānta, gives the names of some methods with very brief descriptions of the processes:— The multiplicand repeated, as in gomūtrikā as often as there are digits in the multiplier, is severally multiplied by them and (the results) added according to places; this gives the product. Or the multiplicand is repeated as many times as there are component parts in the multiplier.¹ (the word bheda occurring in the verse has . been translated as "integrant portions" by Colebrooke p. 319. Again by the term bheda are meant portions which added together make the whole, or aliquot parts which multiplied together make the entire quantity. The multiplicand is multiplied by the sum or the differ- ence of the multiplier and an assumed quantity and, from the result the product of the assumed quantity and the multiplicand is subtracted or added.² (Colebrooke thinks that this is a method to obtain the true product when the multiplier has been taken to be too great or too small by mistake.³ Datta and Singh think, however, that this is not correct.⁴ Thus Brahmagupta mentions four methods of multiplica- tion: (i) gomūtrikā, (ii) khaṇḍa, (iii) bheda, and (iv) iṣṭa. The com- mon and the well known method of kapāṭa-sandhi has been omitted by him.

1. गुणकारखण्डतुल्यो गुण्यो गोमूत्रिकाकृतो गुणितः । सहितः प्रत्युत्पन्नो गुणकारक्रमेदतुल्यो वा ॥ — BrSpSi XII. 55
2. गुण्यो राशिर्गुणकारराशिनेष्टाधिकोनकेन गुणः । गुण्योष्टवधो न युतो गुण्येऽभ्यधिकोनके कार्यः ॥ — BrSpSi. XII. 56
3. Colebrooke, T. H., Hindu Algebra, p. 320.
4. Datta, B. and Singh, A. N., History of Hindu Mathematics Pt. I (Arithmetic), p. 135 (1962).