DISSERTATION.

His notation, as before observed, is scanty and inconvenient. In the whole science, he is very far behind the Hindu writers: notwithstanding the infinite ingenuity by which he makes up for the want of rule: and although presented to us under the disadvantage of mutilation; if it be, indeed, certain that the text of only six, or at most seven, of thirteen books which his introduction announces, has been preserved. 1 It is sufficiently clear from what does remain, that the lost part could not have exhibited a much higher degree of attainment in the art. It is presumable that so much as we possess of his work is a fair specimen of the progress which he and the Greeks before him (for he is hardly to be considered as the inventor, since he seems to treat the art as already known) had made in his time.

The points in which the Hindu Algebra appears particularly distinguished from the Greek are, besides a better and more comprehensive algorithm: 1st, The management of equations involving more than one unknown term. (This adds to the two classes noticed by the Arabs, namely simple and compound, two, or rather three, other classes of equation.) 2d, The resolution of equations of a higher order, in which, if they achieved little, they had, at least, the merit of the attempt, and anticipated a modern discovery in the solution of biquadratics. 3d, General methods for the solution of indeterminate problems of 1st and 2d degrees, in which they went far, indeed, beyond Diophantus, and anticipated discoveries of modern Algebraists. 4th, Application of Algebra to astronomical investigation and geometrical demonstration: in which also they hit upon some matters which have been reinvented in later times.

This brings us to the examination of some of their anticipations of modern discoveries. The reader’s notice will be here drawn to three instances in particular.

The first is the demonstration of the noted proposition of Pythagoras, concerning the square of the base of a rectangular triangle, equal to the squares of the two legs containing a right angle. The demonstration is given two ways in Bhāscara’s Algebra (Víj.-gań. § 146). The first of them is the same which is delivered by Wallis in his treatise on angular sections (Ch. 6) and, as far as appears, then given for the first time. 2

A mathematical diagram illustrates the proportionality of segments. B:C

1 Note M.
2 He designates the sides C. D. Base B. Segments κ, δ.