Elements of Algebra

578 ADDITIONS. CHAP. VIII. CHAP. VIII. Remarks on Equations of the form $p^2 = Aq^2 + 1$, and on the common method of resolving them in Whole Numbers. 85. The method of Chap. VII. of the preceding Treatise, for resolving equations of this kind, is the same that Wallis gives in his Algebra (Chap. 98), and ascribes to Lord Brouncker. We find it, also, in the Algebra of Ozanam, who gives the honor of it to M. de Fermat. Whoever was the inventor of this method, it is at least certain, that M. de Fermat was the author of the problem which is the subject of it. He had proposed it as a challenge to all the English mathematicians, as we learn from the Commercium Epistolicum of Wallis; which led Lord Brouncker to the invention of the method in question. But it does not appear that this author was fully apprised of the importance of the problem which he resolved. We find nothing on the subject, even in the writings of Fermat, which we possess, nor in any of the works of the last century, which treat of the Indeterminate Analysis. It is natural to suppose that Fermat, who was particularly engaged in the theory of integer numbers, concerning which he has left us some very excellent theorems, had been led to the problem in question by his researches on the general resolution of equations of the form, $x^2 = Ay^2 + B$, to which all quadratic equations of two unknown quantities are reduced. However, we are indebted to Euler alone for the remark, that this problem is necessary for finding all the possible solutions of such equations *. The method which I have pursued for demonstrating this proposition, is somewhat different from that of M. Euler; but it is, if I am not mistaken, more direct and more general. For, on the one hand, the method of M. Euler naturally leads to fractional expressions, where it is required to avoid them; and, on the other, it does not appear very evidently, that the suppositions, which are made in order to remove the fractions, are the only ones that could have taken place. Indeed, we have elsewhere shewn, that the finding of one solution of the equation $x^2 = Ay^2 + B$, is not always sufficient to enable us to * See Chap. VI. of the preceding Treatise, Vol. VI. of the Ancient Commentaries of Petersburg, and Vol. IX. of the New.