320 



ISOPERIMETRICAL PROBLEMS. 



Problem*. 



Imptnme- ISOPERIMETRICAL PROBLEMS, are problems in the 

 " icj higher geometry, in which it is required to deter- 

 mine the nature of a curve, from some property which 

 it is supposed to possess in a greater degree than any 

 other curve, either drawn between given points, having 

 an equal length or perimeter, comprising the same area, 

 or under other similar restrictions. 



The first proposal of such problems forms a remark- 

 able epoch in mathematical history, on account of their 

 presenting difficulties of a peculiar kind, to the sur- 

 mounting which the ordinary application of the diffe- 

 rential calculus in que'stions of. maxima and minima 

 was at first supposed inadequate, and demanding more 

 extensive views than had before been taken of the va- 

 riations which magnitudes undergo by a change in the 

 manner of their composition ; thus giving rise to a suc- 

 cession of profound researches, which terminated at 

 length in the invention of the calculus of variations, 

 one of the greatest discoveries in the modern analytics, 

 and tending remotely to the establishment of the diffe- 

 rential calculus itself on principles purely analytical. 



The property'of a straight line, by which it measures 

 the least distance between two given points, is too ob- 

 vious to escape the notice of the most ordinary obser- 

 ver. That of the circle, by which it includes the 

 greatest area under a given circumference, is demon- 

 strated by Pappus, in the 5th book of his Mathematical 

 Collections, with the greatest precision, (Prop. 10.) and 

 though his mode of proceeding, founded on the inscrip- 

 tion of regular polygons, will not apply to the sphere, 

 on account of the impossibility of inscribing regular 

 solids within it to an indefinite extent, yet that figure 

 seems to have been generally regarded as the mest ca- 

 pacious under a given surface. The first instance, 

 however, of a problem of this kind resolved by direct 

 investigation, was furnished by Newton, in the con- 

 struction of a solid of revolution, which, when moving 

 in a fluid in the direction of its axis, shall be less re- 

 sisted than any other of the same base and altitude. 

 The demonstration, however, of the property by which 

 he has characterized the figure in the 2d book of the 

 Princlpia, (and which is merely its differential equa- 

 tion geometrically enunciated,) is suppressed, and no 

 trace of the method by which it was obtained appears. 

 Nor does it appear (at least immediately,) to have ex- 

 cited the curiosity of others, since, after a lapse of nine 

 years, and upon another occasion, the attention of the 

 mathematical world was first fixed upon the subject ; 

 and from that period researches of this nature assumed 

 a regular and definite character, and the method of con- 

 ducting them began to be distinctly seen. 



No sooner had the newly acquired power of the dif- 

 ferential calculus enabled John Bernoulli to resolve 

 the problem of the catenary, which Galileo had in vain 

 attempted, than another, proposed by the same philo- 

 sopher, and whose true solution had in like manner 

 eluded his penetration, offered a farther occasion of 

 proving the force of the new methods. It was the 

 problem of the Bracfiystochrone, or curve, down which 

 a body will descend in the least possible time from one 

 given point to another, in a vertical plane. This was 

 evidently a question of far greater difficulty than any 

 ordinary problem of maxima or minima. In the latter, 

 the form of the function which is to become a maximum 



or minimum is given, or at least may be determined by Isopcrims. 

 proper considerations, independent of the maximum or trical 

 minimum property, while, in the former, it is this very i_ ble| 

 property which determines the nature of the curve in ">~^ 

 question, and by consequence, of the function to be made 

 a minimum. It was not, however, by any direct ana- 

 lysis, setting out from this property as his datum, and 

 following it as his directing principle, that Bernoulli 

 first resolved his problem. The minimum property of 

 his curve appears to have struck him as a collateral 

 view, in the course of investigation of a widely different 

 nature ; and a succinct account of the course pursued by 

 him, and the progress of his thoughts, may materially 

 assist us in our inquiry into the early history of these 

 problems, and, at the same time, serve to illustrate their 

 nature. 



It is well known that Fermat had early signalized 

 himself by the discovery of a method of maxima and mini- 

 ma, which has procured him, and with reason, the repu- 

 tation 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 af- 

 ter, in the inverse proportion 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 mini- 

 mum. This singular conclusion had, however, been 

 anticipated upon a metaphysical principle, (if it de- 

 serve the name) that, as nature always operates in the 

 most direct and simple way, therefore, by some necessi- 

 ty, the ray must shape its course so as to arrive at its 

 destined object in the least time possible. The princi- 

 ple, at the instance of Clerselerius, and the preponder- 

 ance of natural good sense, was given up by Fermat, 

 as soon as he had learned to regard the fact as a conse- 

 quence rather than a cause of the laws of refraction; but 

 Leibnitz and Huygens strongly adhered to it, the for- 

 mer defending it from his peculiar views of final causes, 

 while John Bernoulli professed himself convinced by 

 their arguments, so that, without farther consideration, 

 it became a received principle, that, under all circum- 

 stances, light performs its course, however 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. Eruditorum, 1G82, as 

 the foundation of optical science, and attributed 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 den- 

 sity (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 consideration of the velocity,) 

 ascertained the curve described, and satisfied with the 

 metaphysical principle above stated, he then abstract- 

 ed altogether from optical considerations, and regarding 

 the variation of velocity as produced by any cause, a?, 

 for instance, the force of gravity, he thus concluded 

 the brachystochrone on any hypothesis of gravitation. 



It is surprising to observe what ascendency these 

 considerations of metaphysical propriety had, at that 



