It is well known that Fermat had early signalized him self by the discovery of a method of maxima and minima, which has procured him, and with reason, the reputation of having invented this application of the future differential calculus. Of the various results afforded by his method, the following was not the least remarkable; that, on the Huygenian hypothesis of the refraction of light, where its velocity before refraction is to that after in the inverse pro portion of the refractive densities of the media, its course is necessarily such, that in passing from a given point on one side of a refracting surface to one on the other, the time occupied is a minimum. This singular conclusion had, however, been anticipated upon a metaphysical principle, (if it deserve the name) that, as nature always operates in the most direct and simple way, therefore, by some neces sity, the ray must shape its course so as to arrive at its des tined object in the least time possible. The principle, at the instance of Clerselerius, and the preponderance of natu ral good sense, was given up by Fermat, as soon as he had learned to regard the fact as a consequence rather than a cause of the laws of refraction ; but Leibnitz and Huygens strongly adhered to it, the former defending it from his pe culiar views of final causes, while John Bernoulli profes sed himself convinced by their arguments, so that, without farther consideration, it 'became a received principle, that, under all circumstances, light performs its course, how ever interrupted, from point to point, however distant, in the least time that circumstances will permit ; and in this form it was laid down by Leibnitz, in the Act. Erudito rum, 1682, as the foundation of optical science, and attribut ed by him to the immediate fiat of the Deity.
Bernoulli had proposed to himself to determine the path of a ray through a medium, whose refractive density (and consequently the velocity of the ray) varied according to a given law, in which investigation .no difficulty occurred. Having (from the pre-established dependence of the sine of refraction upon the refractive density, and without any any element, and, if possible, let B c D be an arc of some other curve, the time of describing which is less than that of describing BCD. Since the velocity at D is the same, whether the body fall down AB c D or ABCD, the time through DE will be the same on either supposition, and therefore the whole time through AB c DE less than through ABCDE, contra hypothesin.
Suppose,_now, the curve ABCDE,- according to the spirit of the differential calculus, resolved into an infinite consideration of the velocity,) ascertained the curve de scribed, and satisfied with the metaphysical prineiple above stated. he then abstracted altogether from optical consider ations, and regarding the variation of velocity as produced by any cause, as, for instance, the force of gravity, he thus concluded the brachystochrone on any hypothesis of gravi tation.
It is surprising to observe what ascendency these consi derations of metaphysical propriety had, at that period, ob tained over the mind of this singular man. No sooner had he identified the brachystochrone, on the supposition of uniform gravity, with the cycloid, which had previously been identified with the tautochrone, on the same hypothe sis, than he celebrates it as a wise dispensation of Provi dence, or, at least, as a wonderful instance of the frugality of nature in her operations, thus to make one curve serve two purposes, observing, that this could not have happened, were not Galileo's hypothesis of uniform gravity agreeable to nature, although New ton's discoveries had long ago de monstrated its falsehood, and Bernoulli well knew the fact to be so.
It would not be difficult to clear Bernoulli's solution from any objection on the score of rigour ; and, in fact, he speedily obtained a more direct one, upon which, in June 1696, he proposed his problem in the Lcipsic acts, under the title, Problema novum, ad cujus solutionem Ma thcmatici invitantur, allowing six months bar the solution, which, however, at the request of Leibnitz, who, as well as his brother James Bernoulli, had resolved the problem, was prolonged to a year. At the expiration of this period, a multitude of the chief mathematicians of the age were found to have been successful, but the only direct analysis which appeared was that of James Bernoulli. The prin ciple on which this analysis turns is of very extensive ap plication, and it once reduces this, and other problems of the same nature, to questions of ordinary maxima. It is
this, that the maximum or minimum property, which belongs to the whole curve, belongs also to every elementary, or in finitely small portion of it. It is true, this principle is not absolutely general, and therefore must be verified before it is applied in any particular case. In the present it is easily shewn to hold good.
Let ABCDE be the curve (Fig. I.) required, BCD number of rectilinear elements, of which BC. CD, arc two, corresponding to two elements PQ, QR. (equal to each other,) of the vertical abscissa AP (31, the arc AB being called s, and the ordinate PB, y. On this supposi tion, the velocity muse he reg.rded as uniform through each of the elements BC, CD, and if we call v the velo city of describing BC, then v dv (or v') will be the con secutive veto: ty with which the arc CD = d s + 8 d •') is described. Now, to discover what function y is of x, we must endeavour to estahlish some relation be tween the differentials d s and d y or d s, by means of the proposed milititurn property ; and to this effect, regarding the points B, D, as fixtd, and C moveable along the line QC. we must inquire what must be the position of C upon that line, (or what relation FC and BP must bear to each other.) that the time through BC, with the uniform velo city v. plus the time through CD, with the uniform velo city shall be a minimum. Now, this is evidently an ordinary question of maxima and minima ; it is, in fact, identical with Fermat's problem concerning refraction above mentioned, and the solution is precisely similar ; BCD will be described in a minimum of time, when the two lines BC, CD, make angles with the vertical, whose sines are to each other as the velocities v and with which they are described. This gives at once This is the relation required between the differentials, or the differentia! equation of the curve, for v is given in functions of .r On the supposition of uniform gravity, we have v = iJ x, and writing v a for a to make both sides homogenous, differential of FC (d y), on the supposition that C changes its place on QC, cannot be expressed by d d y, because it would thus be confounded with the variation it undergoes, when (by the shifting of the ordinate QC to its consecutive position RD) FC attains its consecutive value ID — d at d d y. There are two ways in which this difficulty may be avoided. The first is the one we have above taken, viz. resolving the question of the maximum or minimum, as a separate and independent problem, and then adapting the conclusion so obtained to the particular case in ques tion ; accordingly this is the course pursued at first by the Bernoullis ; but, not to mention the want of analytical neat ness in such a mode of proceeding, the questions of maxima and minima, which thus arise, would, in most cases, be found of extreme difficulty, and their subsequent adaptation to infinitely small quantities a matter of uncommon deli cacy'. The other method consists in looking on the diffi culty (in its true light) as merely one of notation, and ob viating it by a refinement in that point, viz. by employing a new characteristic ;' for that hypothesis of differentiation, by which the point C shifts, as it were, from one curve, BCD, to another infinitely near it, along the fixed ordinate, and a new name variation for this peculiar change in the value of d y. We have here the origin of the calculus of variations ; and the manner in which this simple artifice enables us to combine the question of the minimum, with the peculiar circumstances of the ease where it arises, cannot be better shewn than by the very instance be fore us It is, first of all, evident, that as a means nothing more than differentiation, on another hypothesis, all the rules of the calculus must be attended to in the management of the new symbol, only making the varia tions of such quantities as do not change on this hypothesis, zero. Thus, — 0, cl"v — 0, a" — 0, because the length cf BF is given, and the velocities through BF and CD are uniform. Again, d y') because GD = d y +dy' is invariable. Schy, d s d d 8 ( =0, Or -+ = 0 ; . . . . (a) by the condition of the minimum. Now, the equation of a cycloid, whose base is the hot izontal line AK. Such is the solution of James Bernoulli, cleared of the geometrical form which embarrasses and obscures it, and expressed, as he probably would have expressed it, in the present state of symbolic reasoning.