Page:A History of Mathematics (1893).djvu/126

 Then form $$\scriptstyle{f(a)=A}$$ and $$\scriptstyle{f(b)=B}$$, and determine the errors $$\scriptstyle{V-A=E_a}$$ and $$\scriptstyle{V-B=E_b}$$; then the required $$\scriptstyle{x={{bE_a-aE_b} \over {E_a-E_b}}}$$ is generally a close approximation, but is absolutely accurate whenever $$\scriptstyle{f(x)}$$ is a linear function of x.

We now return to Hovarezmi, and consider the other part of his work,—the algebra. This is the first book known to contain this word itself as title. Really the title consists of two words, aldshebr walmukabala, the nearest English translation of which is "restoration" and "reduction." By "restoration" was meant the transposing of negative terms to the other side of the equation; by "reduction," the uniting of similar terms. Thus, $$\scriptstyle{x^2-2x=5x+6}$$ passes by aldshebr into $$\scriptstyle{x^2=5x+2x+6}$$; and this, by walmukabala, into $$\scriptstyle{x^2=7x+6}$$. The work on algebra, like the arithmetic, by the same author, contains nothing original. It explains the elementary operations and the solutions of linear and quadratic equations. From whom did the author borrow his knowledge of algebra? That it came entirely from Indian sources is impossible, for the Hindoos had no such rules like the "restoration" and "reduction." They were, for instance, never in the habit of making all terms in an equation positive, as is done by the process of "restoration." Diophantus gives two rules which resemble somewhat those of our Arabic author, but the probability that the Arab got all his algebra from Diophantus is lessened by the considerations that he recognised both roots of a quadratic, while Diophantus noticed only one; and that the Greek algebraist, unlike the Arab, habitually rejected irrational solutions. It would seem, therefore, that the algebra of Hovarezmi was neither purely Indian nor purely Greek, bat was a hybrid of the two, with the Greek element predominating.

The algebra of Hovarezmi contains also a few meagre