Page:Indian mathematics, Kaye (1915).djvu/26

532). They were naturally disappointed but the effect of their visit may have been far greater than historical records show.

13. The state of knowledge regarding indeterminate equations in the west at this period is not definitely known. Some of the works of Diophantus and all those of Hypatia are lost to us; but the extant records show that the Greeks had explored the field of this analysis so far as to achieve rational solutions (not necessarily integral) of equations of the first and second degree and certain cases of the third degree. The Indian works record distinct advances on what is left of the Greek analysis. For example they give rational integral solutions of ${\begin{aligned}\\&\scriptstyle {(A)\quad ax\pm by=c}\\&\scriptstyle {(B)\quad Du^{2}+1=t^{2}}\\\end{aligned}}$

{{p|aj}The solution of (A) is only roughly indicated by Āryabhata but Brahmagupta's solution (for the positive sign) is practically the same as $\scriptstyle {x=\pm cq-bt,\quad y=\mp cp+at}$ where t is zero or any integer and p/q is the penultimate convergent of a/b.

The Indian methods for the solution of $\scriptstyle {Du^{2}+1=t^{2}}$ may be summarised as follows:

If $\scriptstyle {Da^{2}+b=c^{2}}$ and $\scriptstyle {D\alpha ^{2}+\beta =\gamma ^{2}}$ then will ${\begin{aligned}\\&\scriptstyle {(a)\quad D(c\alpha \mp \gamma a)^{2}+b\beta =(c\gamma \pm a\alpha D)^{2}}\\&\scriptstyle {(b)\quad D({\frac {c+\alpha r}{b}})^{2}\pm {\frac {r^{2}-D}{b}}=({\frac {Da+cr}{b}})^{2}}\\\end{aligned}}$ where r is any suitable integer.

Also $\scriptstyle {(c)\quad D({\frac {2n}{D-n^{2}}})^{2}+1=({\frac {D+n^{2}}{D-n^{2}}})^{2}}$ where n is any assumed number.

The complete integral solution is given by a combination (a) and (b) of which the former only is given by Brahma-