Elements of Algebra

464 ADDITIONS. 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 labours 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. Before the publication of it, there was no work in which this science was treated methodically, and which enumerated and explained the principal rules hitherto known for the solution of indeterminate problems. The preceding Treatise unites both these advantages: but in order to make it still more complete, I have thought it necessary to make several Additions to it, of which I shall now give a short account. The theory of Continued Fractions is one of the most useful in arithmetic, as it serves to resolve problems with facility, which, without its aid, would be almost unmanageable; but it is of still greater utility in the solution of indeterminate problems, when integer numbers only are sought. This consideration has induced me to explain the theory of them, at sufficient length to make it understood. As it is not to be found in the chief works on arithmetic and algebra, it must be little known to mathematicians; and I shall be happy, if I can contribute to render it more familiar to them. At the end of this theory, which occupies the first Chapter, follow several curious and entirely new problems, depending on the truth of the same theory; but which I have thought proper to treat in a distinct manner, in order that the solution of them may become more interesting. Among these will particularly be remarked a very simple and easy method of reducing the roots of equations of the second degree to Continued Fractions, and a rigid demonstration, that those fractions must necessarily be always periodical. The other Additions chiefly relate to the resolution of in- scattered through the Remarks of M. Fermat on the Questions of Diophantus; and through his Letters printed in the Opera Mathematica, &c. and in the second volume of the works of Wallis. There are also to be found, in the Memoirs of the Academy of Berlin, for the year 1770, & seq. the demonstrations of some of this author's theorems, which had not been demonstrated before.