CHAP. III.

Extraordinary progress in the theory of maxima and minima. Dispute between the two Bernoullis on the problem of isoperimetrical figures.

THE sole object of all the problems of maxima and minima, that had been resolved previous to the time at which we are arrived, A. D. 1696, had been, to find in the number of explicit functions, which contain but one variable quantity, or are reducible to one, those which, among their similars, could become maxima or minima. Des Cartes, Fermat, Sluze, Hudde, and others, had contrived particular methods for these problems: but that of fluxions had supplanted them all by it's simplicity and generality.

There remained another class of problems of the same kind, but far more complicated and profound, where both the direct and inverse methods of fluxions were necessary. This consisted in finding, among the implicit or affected functions of summatory signs, those which give maxima or minima: as for instance the curve which includes the greatest area according to given conditions, or which produces by it's revolution the greatest solid within given limits, &c.. Newton, Prin. lib. 11, prop. 34, after having determined, among all the truncated right cones, of the same base and the same altitude, that which, being moved in a fluid by the smallest (unknown) base, in

the

the direction of it's axis, experiences the least resistance possible (which was a problem of the ancient kind), had given, without any demonstration, a ratio, from which might be deduced the differential equation of the curve, that produces, by revolving on it's axis, the solid of least resistance: a problem relating to the second kind.

The principle of this solution, of which Newton as usual made a mystery, is, that, when a property of a maximum or a minimum pertains to a curve, or to a finite portion of a curve, it likewise pertains to a portion infinitely small. It has some analogy with means frequently employed in geometry: as, for instance, when we demonstrate the equality of a spherical zone with the corresponding area of the circumscribed cylinder, by the reciprocal equality of their elements. But even had Newton formally announced this principle, the general problem would still have had it's particular difficulty in each individual case, either in finding the differential equation of the curve, or in resolving it. The sciences, therefore, have an obligation of the highest importance to John Bernoulli, for having drawn the attention of geometricians to this general theory, by proposing to them in 1697 the celebrated problem of the brachystochronon, or that curve, along the concave side of which if a heavy body descend, it will pass in the least time possible from one point to another, the two points not being in the same vertical line.' It is certain, that, at the period in question, this problem was more difficult than that of the solid of least resistance; the solution of which Newton had even left

incomplete, since he had not resolved the differential equation of the generating circle.

At first view it would be imagined, that a right line, as it is the shortest path from one point to the other, must likewise be the line of swiftest descent: but the attentive geometrician will not hastily assert this, when he considers, that in a concave curve, described from one point to the other, the moving body descends at first in a direction more approaching to the perpendicular, and consequently acquires a greater velocity, than down an inclined plane; which greater velocity is to be set against the length of the path, and may cause the body to arrive at the end sooner by the curve, than by the straight line. Metaphysics alone, therefore, cannot solve the question, which must be examined by the most accurate calculation. Now the result of this calculation shows, that the line sought is in reality a curve, which is, in fact, an arc of a cycloid reversed: a new and very remarkable property of this curve, which the researches of Huygens and Pascal had already rendered so celebrated.

Leibnitz resolved the problem on the day on which he received it from John Bernoulli, and immediately informed him of it. They both agreed, to keep their solutions secret, and to give other geometricians a year to exercise their ingenuity on this curious question. This delay was announced in the periodical publications, and in a circular paper, which John Bernoulli sent to all quarters.

The year had not expired, when three other solutions appeared, the authors of which were Newton,

the marquis de l'Hopital, and James Bernoulli. That of Newton appeared without a name in the Philosophical Transactions of the Royal Society at London, but John Bernoulli guessed the author; tanquam, said he, er ungue leonem.

The marquis de l' Hopital had great difficulty in finding his solution: yet he might have obtained it readily enough from a principle, which he employed himself when seeking the path a traveller ought to pursue, in order to go from one place to another in the shortest time possible, if he have to cross two fields, in which he experiences obstacles to his progress, that occasion his pace to vary in a given ratio. For, if the two fields be considered as the two elements of a curve, situated in a vertical plane; and if we suppose, agreeably to the laws of the descent of heavy bodies, that the velocities of a body along any curve are as the square roots of the heights from which the body has descended, we shall immediately obtain the differential equation of a cycloid. But no person at that time made the remark, or associated together ideas, which now appear to us so closely connected.

Lastly, before the expiration of the time prescribed by his brother, James Bernoulli gave a solution, in which he demonstrated, that the curve sought is an arc of the cycloid. In the course of his investigation he had ascended to problems on isoperimetrical figures, requiring still more profound speculations; and after he had resolved these, he proposed them to the public at the conclusion of his method for the curve of swiftest descent.

All these solutions appeared at the same time, and without any possibility of one of the authors having derived any information from the others.

The rivalry in glory, which had long divided the Bernoullis, was fully displayed on this occasion. At first it was a little moderated by their habits of seeing each other, at least occasionally, and by the intervention of some common friends; but the younger brother having been appointed professor of mathematics at Groningen in 1695, all private intercourse between them soon ceased, and they no longer corresponded, except through the medium of periodical publications for the purpose of proposing to each other the most difficult problems. John Bernoulli was the aggressor: but perhaps his brother displayed a little too much haughtiness in the first answer he made him, a sketch of which I have given. Their minds being exasperated, John Bernoulli frequently returned to the charge, and his quondam master was not a man to endure for a long time attacks unjust in themselves, independent of the gratitude, by which they ought to have been restrained.

In this frame of mind James Bernoulli, desirous at length of avenging himself in a signal manner, which should at the same time be beneficial to geometry, challenged his brother by name, to resolve the following problem. To find, among all the isoperimetrical curves between given limits, such a curve, that, constructing a second curve, the ordinates of which shall be functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimum.' To this leading problem he

added