tions *. John Bernoulli then perceived how far he differed from his brother as to their results: but not discovering the principle of the true solution, and still persuaded, that his own method was accurate, he gave it at large in a paper, which was sent under a seal to the Academy of Sciences at Paris, in the month of february 1701, on condition, that it should not be opened without his consent, and after his brother had published his analysis.

He

As soon as James Bernoulli was informed of this, he had no longer any reason to keep his method secret. Accordingly he made it public, and maintained it by way of a thesis at Basil, in march 1701, with a dedication to the four illustrious geometricians, l'Hopital, Leibnitz, Newton, and Fatio de Duillier. likewise printed it separately at Basil, and in the Leipsic transactions for may 1701, under the following title: Analysis magni Problematis isoperimetrici. It was considered as a prodigy of sagacity and invention and indeed, if the time be considered, it will not be too much to assert, that a more difficult problem never was resolved. The marquis de l'Hopital wrote to Leibnitz, that he had read it with avidity, and that he had found it very direct, and very accurate. This testimony Leibnitz transmitted to John Bernoulli himself, though he was much prejudiced in his favour. Com. Epist. f. II, p. 48.

The journalists suppressed the preceding part of the letter; and through the influence of John Bernoulli it was likewise excluded. from the edition of his brother's works published in 1744. I have had it reprinted in the Journal de Physique for September 1792.

After so much bustle it was to be expected, that John Bernoulli would cither criticise the solutions of his brother, or that he would publicly avow their accuracy. But from this time he maintained a profound silence: no observations, no criticisms, appeared on his part; and instead of setting his own method in opposition to that of his rival, he allowed it to rest quietly for five years in the repository of the academy. At length, in 1705, James Bernoulli died; and, soon after, this method appeared in the Memoirs of the Academy for 1706.

What must we think of this strange conduct? Can we suppose, contrary to every appearance, that a man of so ardent and impetuous a temper as John Bernoulli was desirous of dropping a dispute, of which he had grown weary? Is it not much more probable, that, suspecting some defect in his method, he was afraid to submit it to the judgment of his brother; and that, this brother being dead, the shame of appearing vanquished in the eyes of all Europe induced him to publish this memoir, sent in 1701; in the hope that no person would enter so deeply into the question, as to decide between the two methods, and that, with some of the learned world he should at least obtain the credit of having likewise resolved the problem? This conjecture is strengthened by the reflection, that Fontenelle, in his eulogy of James Bernoulli, and forty-three years afterward Fouchi, in that of John, have both spoken of their solutions as equally accurate, and equally general.

The most profound geometricians passed a very different judgment, and the palm of victory was de

creed

creed to the methods of James Bernoulli. In spite of all the shifts of John, and all the specious means he employed for giving his method an appearance of truth, it was really defective, as his brother had constantly maintained. The radical errour of it was, that John Bernoulli considered only two elements of the curve, instead of which it is requisite to consider three, or employ an equivalent condition. In problems of the same kind as that of the line of swiftest descent, where it is simply required to fulfil the condition of the maximum or minimum, the applying of this condition to two elements is sufficient, to find the differential equation of the curve. But when, beside the maximum or minimum, the curve must possess a farther property, that of being isoperimetrical to another, this new condition requires that a third element of the curve shall have a certain inclination with respect to the other two; and every determination founded simply on the first consideration will give false results; except in cases, where a curve cannot satisfy one of the two conditions, without at the same time fulfilling the other. In vain John Bernoulli imagined, that he had fulfilled the condition of it's being isoperimetrical, without derogating from the maximum and minimum, by considering two elements of the curve as two small right lines, drawn to an intermediate point between the two foci of an infinitely small ellipsis: this supposition did not introduce a new condition into the calculus; it had no other effect than that of rendering the differential of the absciss constant or variable. James Bernoulli had explicitly employed three ele

ments of the curve, and by this he had obtained precise, general, and complete solutions.

This consideration of the three elements was then so essential, that at length John Bernoulli made it the basis of a new solution, more than thirteen years after his brother's death, confessing himself deceived in his first. Mem. de l'Acad. 1718. This was a tardy avowal, but at least it would have done him honour, had he at the same time acknowledged, that his new solution was in substance the same as his brother's, given in a form which considerably abridged the calculation; and had he not sought with some degree of affectation, to point out certain superfluities in that of James; which, however, though useless, were no way detrimental to it's accuracy or generality.

I have thought it incumbent on me to give a connected account of the dispute between these two brothers on the subject of isoperimetrical figures: and before I take leave of it, I cannot avoid expressing my astonishment, that no other geometrician of that time, at least publicly, undertook to solve these problems; for, though James Bernoulli challenged his brother in particular, every person was at liberty to enter the lists, and the question proposed united every circumstance capable of exciting emulation ; novelty of the subject, great difficulties to be surmounted, and an addition to the treasures of geometry,

CHAP.

CHAP. IV.

Solutions of various problems. Leibnitz invents the method of differencing de curva in curvam. Justi fication of the marquis de l'Hopital.

Newton's

works, Account of some other geometricians.

THE dispute, of which I have just given an account, has led me to anticipate a little on the order of time, and to pass by several interesting and remarkable problems, to which I shall now return.

When James Bernoulli proposed the problem of isoperimetrical figures in 1697, he added to it that of the cycloid of swiftest descent to a line of a given position, in order to complete in some degree the theory of the brachistochronon. He demonstrated, that the cycloid sought is that which cuts the given line at right angles: and he taught generally how to find among similar curves, terminating at a line the position of which is given, that which possesses some property of a maximum or a minimum.

John Bernoulli had arrived at similar results by a method a little indirect, but very ingenious, which gave birth to a signal enlargement of the infinitesimal analysis. In this research he employed the consideration of the synchronous curve, or that which cuts a series of similar curves placed in similar positions, so that the arcs of the latter, included between a given point and the synchronous curve,

shall