Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c13.35

c13.35

The notation or algorithm of Algebra is so essential to this art, as to deserve the first notice in a review of the Indian method of analysis, and a comparison of it with the Grecian and Arabian algebras. The Hindu algebraists use abbreviations and initials for symbols: they distinguish negative quantities by a dot;⁴⁰ but have not any mark, besides the absence of the negative sign, to discriminate a positive quantity. No marks or symbols indicating operations of addition, or multiplication, & c. are employed by them: nor any announcing equality⁴¹ or relative magnitude (greater or less).⁴² But a factum is denoted by the initial syllable of a word of that import,⁴³ subjoined to the terms which compose it, between which a dot is sometimes interposed. A fraction is indicated by placing the divisor under the dividend,⁴⁴ but without a line of separation. The two sides, of an equation are ordered in the same manner, one under the other,⁴⁵ and, this method of placing terms under each other being likewise practised upon other occasions,⁴⁶ the intent is in the instance to be collected from the recital of the steps, of the process in words at length, which always accompanies the algebraic process. The recital is also requisite to ascertain the precise intent of vertical lines interposed between the terms of a geometric progression, but used also upon other occasions to separate and discriminate quantities. The symbols of unknown quantity are not confined to a single one: but extend to ever so great a variety of denominations: and the characters used are initial syllables of the names of colours,⁴⁷ excepting the first, which is the initial of *yavat-tavat*, as much as; words of the same import with Bombelli’s *tanto*; used by him for the same purpose. Colour therefore means unknown quantity, or the symbol of it: and the same Sanscrit word, *varna*, also signifying a literal character, letters are accordingly employed likewise as symbols; either taken from the alphabet;⁴⁸ or else initial syllables of words signifying the subjects of the problem; whether of a general nature,⁴⁹ or specially the names of geometric lines in algebraic demonstrations of geometric propositions or solution of geometric problems.⁵⁰ Symbols too are employed, not only for unknown quantities, of which the value is sought; but for variable quantities of which the value may be arbitrarily put, (*Vij* Ch.6, note on commencement of §153—156) and, especially in demonstrations, for both given and sought quantities. Initials of the terms for square and solid respectively denote those powers; and combined they indicate the higher. These are reckoned not by the sums of the powers; but by their products.⁵¹ An initial syllable is in like manner used to mark a surd root.⁵² The terms of a compound quantity are ordered according to the powers; and the absolute number invariably comes last. It also is distinguished by an initial syllable, as a discriminative token of known quantity.⁵³ Numeral coefficients are employed, inclusive of unity which is always noted, and comprehending fractions;⁵⁴ for the numeral divisor is generally so placed, rather than under the symbol of the unknown: and in like manner the negative dot is set over the numeral coefficient: and not over the literal character. The coefficients are placed after the symbol of the unknown quantity.⁵⁵ Equations are not ordered so as to put all the quantities positive; nor to give precedence to a positive term in a compound quantity: for the negative terms are retained, and even preferably put in the first place. In stating the two sides of an equation, the general, though not invariable, practice is, at least in the first instance, to repeat every term, which occurs in the one side, on the other: annexing nought for the coefficient, if a term of that particular denomination be there wanting.