DISSERTATION.

Which is as near an approach to a general solution of such problems as was made until the days of Lagrange, who first demonstrated that the problem on which the solutions of all questions of this nature depend is always resolvable in whole numbers. 1 The Hindus had likewise attempted problems of this higher order by the application of the method which suffices for those of the first degree; 3 indeed with very scanty success, as might be expected.

They not only applied algebra both to astronomy 4 and to geometry; 5 but conversely applied geometry likewise to the demonstration of Algebraic rules. 6 In short, they cultivated Algebra much more, and with greater success, than geometry; as is evident from the comparatively low state of their knowledge in the one, 7 and the high pitch of their attainments in the other: and they cultivated it for the sake of astronomy, as they did this chiefly for astrological purposes. The examples in the earliest algebraic treatise extant (Brahmegupta's) are mostly astronomical: and here the solution of indeterminate problems is sometimes of real and practical use. The instances in the later treatise of Algebra by Bháscara are more various: many of them geometric; but one astronomical; the rest numeral: among which a great number of indeterminate; and of these some, though not the greatest part, resembling the questions which chiefly engage the attention of Diophantus. But the general character of the Diophantine problems and of the Hindu unlimited ones is by no means alike: and several in the style of Diophantine are noticed by Bháscara in his arithmetical, instead of his algebraic, treatise. 8

To pursue this summary comparison further, Diophantus appears to have been acquainted with the direct resolution of affected quadratic equations; but less familiar with the management of them, he seldom touches on it. Chiefly busied with indeterminate problems of the first degree, he yet seems to have possessed no general rule for their solution. His elementary instructions for the preparation of equations are succinct. 9

1 Brahm. 18. § 29—49. Víj.-gań. § 75—99.
2 Mem. of Acad. of Turin: and of Berlin.
3 Víj.-gań. § 206—207.
4 Brahm. 18. passim. Víj.-gań.
5 Víj.-gań. § 117—127. § 146—152.
6 Víj.-gań. § 212—214.
7 Brahm. 12. § 21; corrected however in Líl. § 169—170.
8 Líl. § 59—61, where it appears, however, that preceding writers had treated the question algebraically. See likewise § 139—146.
9 Def. 11.