shall be passed through in equal times by a heavy body. He demonstrated, that among all the cycloids thus intersected, that which is cut perpendicularly is passed through in less time than any other terminating equally at the synchronous curve. The only question therefore was, how to draw a tangent to the synchronous curve of the cycloids in a given direction and to solve the problem generally, it was requisite, that the solution should depend, not on the properties of the cycloid alone, but on principles applicable to every other series of similar curves similarly placed. John Bernoulli determined by a geometrical construction the synchronous curve corresponding to the cycloid of the shortest time: but he was unable to discover the analytical expression of the subtangent of synchronal curves for all kinds of similar curves. Having long sought the solution of this problem in vain, he proposed it to Leibnitz, who resolved it very readily, and on this occasion invented the celebrated method of differencing de curca in

curcam.

On the receipt of the letter which contained this method, John Bernoulli was transported with joy and admiration, and complained in a friendly manner, that 'the god of geometry had admitted Leibnitz farther than him into his sanctuary.' This first sentiment was just but we find with regret, that, after the death of Leibnitz, he endeavoured to make himself considered as the coinventor of this method, though in fact he could claim no merit further than that of having made some very ingenious applications of it, as may be seen in the second volume of his works.

Leibnitz

Leibnitz never published it himself; and it did not appear under his name till 1745, when his Correspondence with John Bernoulli was published.

We find by the posthumous works of James Bernoulli, that he had likewise discovered a similar method, and employed it for the solution of the problems which his brother had proposed to him during the course of their dispute on isoperimetrical figures: but he had contented himself with indicating his solutions by means of anagrams, wishing to avoid every thing that could lead to a diversion from the business of the isoperimetrical problem, till this was brought to a conclusion. These incidental problems related to the method of maxima and minima. I shall mention but one, which will be sufficient to give a general idea of them all. John Bernoulli demanded,

which of all the semiellipses, that can be described in the same vertical plane, and on the same horizontal axis, would be passed through in the least possible time by a heavy body, the motion of which should commence from one of the extremities of the given axis.'

An innumerable multitude of curious and difficult researches beside these occupied the attention of geometricians at this time. A. D. 1699, 1700, 1701, &c. These were the quadrature of certain cycloidal spaces; the indefinite section of circular arcs; the curve of equal pressure; the transformation of curves into others of the same length; new methods of approximation for the quadratures and rectifications of curves; how to find certain curves from the given ratios of their branches; &c. Leibnitz, the two Bernoullis,

Bernoullis, the marquis de l'Hopital, &c., appeared continually in the lists. These researches were not all of equal use, but every one contributed more or less to the progress of geometry. I should never have finished, were I to attempt to enter into them at any length: but I shall notice a work of John Bernoulli, because it attacks the memory of an illustrious frenchman, whom it is my duty to defend, as far as I am able.

The marquis de l'Hopital had given in his work on the analysis of Infinites a very ingenious rule, for finding the value of a fraction, the numerator and denominator of which should vanish at the same time, when a certain determinate value is given to the variable quantity that enters into it. No person thought proper to dispute his title to this while he lived; but about a month after his death, John Bernoulli, remarking that this rule, was incomplete, made a necessary addition to it, and thence took occasion to declare himself it's author. Several of the marquis de l'Hopital's friends complained loudly and with warmth of a claim, which ought to have been made sooner, if it had not been without foundation. Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Analysis of Infinites. The reader will indulge me in a brief examination of his pretensions.

In 1692 John Bernoulli came to Paris. He was received with great distinction by the marquis de l'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months

months in studying together the new geometry. Every attention, and every substantial mark of acknowledgment, were lavished on the learned foreigner. Soon after, the marquis de l'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometricians of the time. From the year 1693 he made one in the lists of mathematical science, in which he distinguished himself till his death. At this period he was ranked among the first geometricians of Europe; and it is particularly to be observed, that John Bernoulli was one of his most zealous panegyrists. Perhaps he was exalted too high during his lifetime: but the accusation brought against him by John Bernoulli after his death forms too weighty a counterpoise, and justice ought to restore the true balance.

Now I will boldly ask, is it probable, that a geometrician, who had given so many proofs of profound knowledge before his publication of the Analysis of Infinites, who had resolved, for example, the curve of equilibration in drawbridges, was a mere editor of all the difficult parts of that work?

Can we suppose him possessed of so little delicacy, as to ask or accept such humiliating assistance? Do we not know on the contrary, that he had great loftiness of mind? The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. Act. Leips. 1721. It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis de l'Hopital, but by no means that these lessons were the Analysis of Infi

.. nites :

1

nites: the pupil, a man of genius, had become master of his art, and soared on his own wings. We see

too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli, with the confidence of friendship, explanations relative to certain questions, which are treated in it: but we have not the answers of John Bernoulli, and we know not whether these explanations were furnished by him, or found by the marquis himself after farther reflecting on them.

Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli [aux lumières de J. B.]; and to presume, that if he had any obligations to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment. If, notwithstanding all these reasons, any one should think proper to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality at least will never absolve him, for having disturbed the ashes of a generous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed sufcient scientific wealth besides. This example, however, may afford a striking lesson to those ambitious men, who are desirous of posting too hastily to the goal of reputation: it warns them, to reject officious services, more frequently the offers of vanity than of kindness; and to bear this strongly in mind, that true and solid glory is never to be obtained but by our own exertions.

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