Elements of Algebra

ADDITIONS BY M. DE LA GRANGE. ADVERTISEMENT. 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||. * See Chap. 3, in these Additions. I do not here mention his Commentary on Diophantus, because that work, properly speaking, though excellent in its way, contains no discovery. † These are explained in the 8th, 9th, and 10th chapters of the preceding Treatise. Père Billi has collected them from different writings of M. Fermat, and has added them to the new edition of Diophantus, published by M. Fermat, junior. ‡ This method is explained in the 13th chapter of the preceding Treatise; the principles of it are to be found in the Remarks of M. Fermat, on the XXVIth Question of the VIth Book of Diophantus. ‡ The problems and theorems, to which we allude, are