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

FRACTIONS 175 A.D.) gives twenty-six variations of this class. We shall illus- trate it by the following example from Śrīdhara : What is the result when half, one-fourth of one-fourth, one divided by one-third, half-plus half of itself, and one-third diminished by half of itself, are added together ? (Triśatikā, p. 12). A modern writer would have written it as : ½+(¼ of ¼)+(1÷⅓)+(½+½ of ½)(⅓—½ of ⅓) In the old Indian notation, it is written as · | 1 | 1 | 1 | 1 | 1 | 1 | | 2 | 4 | 4 | 1 | 2 | 3 | | | | | 3 | 1 | .1 | | | | | | 2 | 2 | The defect of the notation is obvious: | 1 | 1 | can be read | 4 | 4 | also as ¼+¼ and | 1 | can also be read as 1 ⅓. | 1 | | 3 | And therefore the original meaning is inferred from the context or from the enunciation of the problem. The rules for reduction of the first two classes (bhāga and prabhāga) are those of addition or subtraction and multiplica- tion. The rule for the reduction of the third (bhāgānubandha) and fourth (bhāgāpavāha) classes are given by Brahmagupta in the Brāhmasphuṭa-siddhānta thus : The (upper) denominator is multiplied by the deno- minator and the upper numerator by the same (deno- minator) increased or diminished by its own numera- tor.¹ “Numerator” is known as “aṁśa” and the “denominator” as “cheda.” We give here from Śrīdhara's Pāṭīgaṇita (about 900 A.D., according to K.S. Shukla, 750 A.D. according to Datta and Singh) a rule for reducing a fraction of the bhāgānubandha class (i.e., a whole number increased by a fraction or a fraction incre- ased by a fraction itself) :

1. ऊर्ध्वां शारच्छेदगुखास्तृतीयाजातौ द्वयोः पृथक्परयोः । छेदैश्छेदा गुणिताः स्वांशयुतोनैरुपरीमांशाः ॥ —BrSpSi. XII. 9.