DISSERTATION.

is called by them mukábalah comparison. Hence the name of Tarík aljebr wa almukabala, "the method of restoration and comparison," which obtained among the Arabs for this branch of the Analytic art; and hence our name of Algebra, from Leonardo of Pisa’s exact version of the Arabic title: Fi istakhráju’l majhulát ba tarik aljebr wa almukábalah, "On the solution of certain questions according to the method of Algebræ and Almuchabalæ." 2

The two steps or operations, which have thus given name to the method of analysis, are precisely what is enjoined, without distinctive appellations for them, in the introduction of the arithmetics of Diophantus, where he directs that, if the quantities be positive on both sides, like are to be taken from like until one species be equal to one species; but, if on either side or on both, any species be negative, the negative species must be added to both sides, so that they become positive on both sides of the equation: after which like are again to be taken from like, until one species remain on each side. 3

The Hindu Algebra, not requiring the terms of the equation to be all exhibited in the form of positive quantity, does not direct the preliminary step of restoring negative quantity to the affirmative state; but proceeds at once to the operation of equal subtraction (samasódhana) for the difference of like terms, which is the process denominated by the Arabian algebraists comparison (mukábalah). On that point, therefore, the Arabian Algebra has more affinity to the Grecian than to the Indian analysis.

As to the progress which the Hindus had made in the analytic art, it will be seen that they possessed well the arithmetic of surd roots; 4 that they were aware of the infinite quotient resulting from the division of finite quantity by cipher; 5 that they knew the general resolution of equations of the second degree; and had touched upon those of higher denomination; resolving them in the simplest cases, and in those in which the solution happens to be practicable by the method which serves for quadratics; 6 that they had attained a general solution of indeterminate problems of the first degree; 7 that they had arrived at a method for deriving a multitude of solutions of answers to problems of the second degree from a single answer found tentatively.

1 Khulásatúl hisáb, c. 8. Calcutta.
2 Liber abbaci, 9. 15. 3. M.S. in Magliab. Libr.
3 Def. 11.
4 Brahm. 18. § 27—29. Víj.-gań. § 29—52.
5 Líl. § 45. Víj.-gań. § 15—16 and § 135.
6 Víj.-gań. § 129. and § 137—138.
7 Brahm. 18. § 3—18. Víj.-gań. 53—73. Líl. § 248—265.