ISOPERIMETR1CAL PROBLEMS, are problems in the higher geometry, in which it is required to determine the nature of a curve, from some property which it is supposed to pos sess 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 restric tions.
The first proposal of such problems forms a remarkable epoch in mathematical history, on account of their present ing difficulties of a peculiar kind, to the surmounting the ordinary application of the differential calculus in questions of maxima and minima was at first supposed inadequate, and demanding more extensive views than had before been taken of the variations which magnitudes un dergo by a change, in the manner of their composition ; thus giving rise to a succession 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 differential 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 obvious to escape the notice of the most ordinary observer. That of the circle, by which it includes the greatest area under a given circumference, is demonstrated by Pappus, in the 5th book of his Mathematical Collections, with the greatest pre cision, (Prop. 10.) and though his mode of proceeding, founded on the inscription of regular polygons, will not ap ply to the sphere, on account of the impossibility of in scribing regular solids within it to an indefinite extent, yet that figure seems to have been generally regarded as the most capacious under a given surface. The first instance, however, of a problem of this kind resolved by direct in vestigation, was furnished by Newton, in the construction of a solid of revolution, which, \then moving in a fluid in the direction of its axis, shall be less resisted than any other of the same base and altitude. The demonstration, how ever, of the property by which he has characterized the figure in the 2d book of the Principia, (and which is mere ly its differential equation geometrically enunciated,) is suppressed, and no trace of the method by which it was ob tained appears. Nor does it appear (at least immediately,)
to have excited the curiosity of others, since, after a lapse of nine years, and upon another occasiol., 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 conduct ing them began to be distinctly seen.
No sooner had the newly acquired power of the differ ential calculus enabled John Bernoulli to resolve the pro blem of the catenary, which Galileo had in vain attempted, than another, proposed by the same philosopher, and whose true solution had in like manner eluded his penetration, of fered a fat they occasion of proving the force of the new me thods. It was the problem of the Brachystochronc, or curve, down which a body will descend in the least possible time from one given point to another, in a lerticai 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 is to become a maximum or minimum is given, or at least may be determined by pro per considerations, independent of the maximum or mini mum property, while, in the former, it is this very property which determines the nature of the cure in question, and by consequence, of the function to be made a minimum. It. was not, however, by any direct analysis, 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 Nv idelv 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 Ins tory of these problems, and, at the same time, serve to il lustrate their nature.