Page:Indian mathematics, Kaye (1915).djvu/27

gupta, while both are given by Bhāskara (five centuries later). The latter designates (a) the 'method by composition' and (b) the 'cyclic method.' These solutions are alone sufficient to give to the Indian works an important place in the history of mathematics. Of the 'cyclic method' (i.e., the combination of (a) and (b)) Hankel says, "It is beyond all praise: it is certainly the finest thing achieved in the theory of numbers before Lagrange." He attributed its invention to the Indian mathematicians, but the opinions of the best modern authorities (e.g., Tannery, Cantor, Heath) are rather in favour of the hypothesis of ultimate Greek origin.

The following conspectus of the indeterminate problems dealt with by the Indians will give some idea of their work in this direction; and although few of the cases actually occur in Greek works now known to us the conspectus significantly illustrates a general similarity of treatment.

^[1](1) $\scriptstyle {ax\pm by=c}$
(2) $\scriptstyle {ax+by+cz+\cdots =l}$
(3) $\scriptstyle {x\equiv a_{1}Mod.b_{1}\equiv \dots \equiv a_{a}Mod.b_{4}}$
(4) $\scriptstyle {Ax+By+Cxy=D}$
(5) $\scriptstyle {Du^{2}+1=t^{2}}$
(6) $\scriptstyle {Du^{2}-1=t^{2}}$
(7) $\scriptstyle {Du^{2}\pm s=t^{2}}$
(8) $\scriptstyle {D^{2}u^{2}\pm s=t^{2}}$
(9) $\scriptstyle {u^{2}+s=at^{2}}$
(10) $\scriptstyle {Du^{2}\pm au=t^{2}}$
(11) $\scriptstyle {s-Du^{2}=t^{2}}$
(12) $\scriptstyle {Du+s=t^{2}}$

(13) $\scriptstyle {x\pm a=s^{2},\quad x\pm b=t^{2}}$

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1. Of these only numbers 1—5, 7, 8, 12—14 occur before the twelfth century.