geometry, mechanics, hydraulics, &c. I equally omit the reflections of Leibnitz on the mode of resolving problems of quadratures by the construction of certain curves, which he describes by motions subjected to given laws. The description of the tractrix is an example of these motions: and it was in fact on occasion of this curve, the nature of which Claude Perrault had inquired of him, that Leibnitz made those remarks, in which his usual subtilty is observable. The same thing may be said of a new application, which Leibnitz made of his differential calculus, for the construction of curves from a condition of the tangents. About the same time also other geome tricians published various works, or solutions of pro blems, which it would be tedious to enumerate.

Geometricians seemed to have forgotten the problem of the paracentric isochronal curve, which Leibnitz had proposed in 1689, and the solution of which he still kept secret. The cause of this apparent forgetfulness was no doubt the difficulty of separating the indeterminates from the equation found when the curve is referred to perpendicular coordi nates. In 1694 James Bernoulli surmounted this difficulty, by taking for ordinates parallel right lines, and for abscisses the chords of an infinite number of circles, all of which have for their centre the given point. Thus he obtained a separate equation, which he constructed at first by the rectification of the clastic curve, and afterward by that of an algebraic curve. Soon after John Bernoulli likewise resolved the same problem. He gave a complete and minute analysis of it, on which I should bestow much praise,

if he had not saved me the trouble, and at the same time had refrained from unjustly criticising the constructions of his brother, to which even this analysis may be referred in substance. At the same time Leibnitz published his own solution, which does not essentially differ from that of the Bernoullis, but which is accompanied with reflections conducive to the progress of geometry.

In the Commercium epistolicum of Leibnitz and John Bernoulli, which was not published till 1745, we learn, that in the year 1694 each of them had separately discovered that branch of the new analysis, which is called the exponential calculus. Leibnitz has the priority in point of date: but John Bernoulli made the discovery himself; and in 1697 he published the rules and use of this calculus, whence he is commonly taken for the original and even sole inventor of it.

In this same Epistolary Correspondence we find an important remark by Leibnitz in the year 1695, on the analogy that subsists between the powers of a polynomial composed of variable terms, and the fluxions (of the same order) of the product of these terms. From this John Bernoulli deduced a method for summing up the differential formula of every order, in certain cases.

Among the most curious problems of this time must be reckoned that of the curve of equilibration in draw-bridges, solved by the marquis de l'Hopital in 1695. It chiefly merited the attention of geometricians on account of the practical utility expected from it. John Bernoulli observed, that the required

eurve, of which the marquis de l'Hopital had given the general equation, was an epicycloid, or that it could be generated by a point fixed in a circle revolving round another circle. Leibnitz and James Bernoulli likewise gave solutions of this problem.

About the same period we find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circum. stances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that,

as quantities in geometry increase by degrees, so' every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived,

This memoir concluded with an invitation to mathematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions.

In 1696 a great number of works appeared, which gave a new turn to the analysis of infinites. Such

were

were the Memoir on the quadratures of spheroidal surfaces by James Bernoulli, in which we find problems analogous to those of Viviani, but more general, and more complicated; several elegant theorems on the same quadratures by John Bernoulli; the third part of James Bernoulli's Treatise on Series; and above all the celebrated work of the marquis de l'Hopital, entitled: The Analysis of Infinites, for the understanding of curve Lines,' to which I shall spare a few moments.

Such a work had long been a desideratum.

'Hi

therto,' says Fontenelle, in his eulogy on the marquis, the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed: and even when it was exhibited, it was but a faint gleam of the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more properly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened; and means were found to obtain their applause, while the information, with which it should have been repaid, was withheld.'

The work of the marquis de l'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the same manner the inverse method of fluxions, which

is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infiniti, was to comprise both the direct and inverse methods of fluxions: but this, which would have been of great utility at that time, never appeared.

CHAP.