The geometricians of the last century paid great attention to the Indeterminate Analysis, or what is commonly called the Diophantine Algebra; but Bachet and Fermat alone can properly be said to have added any thing to what Diophantus himself has left us on that subject.

To the former, we particularly owe a complete method of resolving, in integer numbers, all indeterminate problems of the first degree: the latter is the author of some methods for the resolution of indeterminate equations, which exceed the second degree; of the singular method, by which we demonstrate that it is impossible for the sum, or the difference of two biquadrates to be a square; of the solution of a great number of very difficult problems; and of several admirable theorems respecting integer numbers, which he left without demonstration, but of which the greater part has since been demonstrated by M. Euler in the Petersburg Commentaries.

In the present century, this branch of analysis has been almost entirely neglected; and, except M. Euler, I know no person who has applied to it: but the beautiful and numerous discoveries, which that great mathematician has made in it, sufficiently compensate for the indifference which mathematical authors appear to have hitherto entertained for such researches. The Commentaries of Petersburg are full of the labors of M. Euler on this subject, and the preceding Work is a new service, which he has rendered to the admirers of the Diophantine Algebra.