Descartes would have conceived his philosophy to be disgraced if it had borrowed any general principle from experience, and he therefore derived, or affected to derive, the law of refraction from reasoning or from theory. In this reasoning, there were so many arbitrary suppositions concerning the nature of light, and the action of transpa rent bodies, that no confidence can be placed in the conclusions deduced from it. It is indeed quite evident, that, independently of experiment, 'Descartes 'himeelf could have put no trust in it, and it is impossible not to feel how much more it would have been for the credit of that philosopher to have fairly confessed that the knowledge of the law was from experiment, and that the business of theory was to deduce from thence some in ferences with respect to the constitution of light and of transparent bodies. This I con ceive to be the true method of philosophizing, but it is the reverse of that which Descartes pursued on all occasions.
The weakness of his reasoning was perceived and attacked by Fermat, who, at the same time, was not very fortunate in the theory which he proposed to substitute for that of his rival. The latter had laid it down as certain, that light, of which he supposed the velocity infinite, or the propagation instantaneous, meets' with less obstruction in dense than in rare bodies, for which reason, it is refracted toward the popendicular, in passing from the latter into the former. This seemed to Fermat a very improbable supposition, and he conceived the contrary to be true, viz. that light in rare bodies has less obstruction, and moves with greater velocity than in dense bodies. On this supposition, and appealing, not to physical, but to final causes, Fermat imagined to himself that he could deduce the true law of refraction. He conceived it to be a fact that light moves always between two points, so as to go from the one to the other in the least time possible. Hence, in order to pass from a given point in a rarer medium where it moves faster, to a given point in a denser medium where it moves slower, • so that the time may be a minimum, it must 'continue longer in the former medium than if it held a rectilineal course, and the bending of its path, on entering the latter, will therefore be toward the perpendicular. On instituting the calculus, according to his own doctrine of marima and Wain*, Fermat found, to his surprise, that the path of the ray must be such, that the sines of the angles of incidence and refraction have a constant ratio to .one another. Thus did these philosophers, setting out from suppositions en
tirely contrary, and following routes which only agreed in being quite unp.hilOsophical and arbitrary, wive, by a very unexpected coincidence, at the same conclusion. Fermat could no longer deny the law of refraction, as laid down by Descartes, but he was less than ever disposed to admit the justness.of his reasoning.
Descartes proceeded from this to a problem which, though suggested by optical con siderations, was purely. geometriqal, and in which his researches were completely success ful. It was well known, that, in. the ordinary cases of refraction by spherical and other surfaces, the rays are not collected into one point, but have their foci spread over a cer tain surface, the sections of which are the curves called caustic curves, and that the focus of opticians is only a point in this surface, where the rays are more condensed, and, of course, the illumination more intense than in other parts of it. It is plain, however, that if refraction is to be employed, either for the purpose of producing light or heat, it would be a great advantage to have all the rays which come from the same point of an object united accurately, after refraction, in the same point of the image. This gave rise to an inquiry into the figure which the superficies, separating two transparent media of different refracting powers, must have, in order that all the rays diverging from a given point ntight, by refraction at the said superficies, be made to converge to another given point. ' The problem was resolved by Descartes in its full eftent ; and he proved, that the curves, proper: for generating such superficies by their revolution, are all comprehended under one general character, viz. that there are always two given points, from which, if straight lines be drawn to any point in the curve, the one of these, plus or minus, that which has a given ratio to .the other, is equal to a: given line.
It is evident, when given ratio here mentioned is a ratio of equality, that the curve is a conic section, and the•two given points its.two foci. The curves, in general, are of the fourth.or the second order, and have been distinguished by the name of the ovals of Descartes.
From this very ingenious investigation no practical result of advantage in the con struction of lenses has been.derived. The mechanical difficulties of working a super Mcies into any figure but a spherical one are so great, that, notwithstanding all the ef.