Page:A History of Mathematics (1893).djvu/112

 Passing now to algebra, we shall first take up the symbols of operation. Addition was indicated simply by juxtaposition as in Diophantine algebra; subtraction, by placing a dot over the subtrahend; multiplication, by putting after the factors, bha, the abbreviation of the word bhavita, "the product"; division, by placing the divisor beneath the dividend; square-root, by writing ka, from the word karana (irrational), before the quantity. The unknown quantity was called by Brahmagupta yâvattâvat (quantum tantum). When several unknown quantities occurred, he gave, unlike Diophantus, to each a distinct name and symbol. The first unknown was designated by the general term "unknown quantity." The rest were distinguished by names of colours, as the black, blue, yellow, red, or green unknown. The initial syllable of each word constituted the symbol for the respective unknown quantity. Thus yâ meant x; kâ (from kâlaka = black) meant y; yâ kâ bha, "x times y"; $$\scriptstyle{ka~15~ka~\overset{.}{10}}$$, "$$\scriptstyle{\sqrt{15}-\sqrt{10}}$$."

The Indians were the first to recognise the existence of absolutely negative quantities. They brought out the difference between positive and negative quantities by attaching to the one the idea of 'possession,' to the other that of 'debts.' The conception also of opposite directions on a line, as an interpretation of + and — quantities, was not foreign to them. They advanced beyond Diophantus in observing that a quadratic has always two roots. Thus Bhaskara gives $$\scriptstyle{x=50}$$ and $$\scriptstyle{x=-5}$$ for the roots of $$\scriptstyle{x^2-45x=250}$$. "But," says he, "the second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots." Commentators speak of this as if negative roots were seen, but not admitted.

Another important generalisation, says Hankel, was this, that the Hindoos never confined their arithmetical operations to rational numbers. For instance, Bhaskara showed how,