«ct. v.] DISSERTATION SECOND. 191 



instituting the calculus, according to his own doctrine of 

 maxima and minima, Fermat found, to his surprise, that 

 the path of the ray must be 3uch, that the sines of the an- 

 gles of incidence and refraction have a constant ratio to 

 one another. Thus did these philosophers, setting out 

 from suppositions entirely contrary, and following routes 

 which only agreed in being quite unphilosophical and arbi- 

 trary, arrive, 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 considerations, was purely 

 geometrical, and in which his researches were completely 

 successful. 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 certain surface, the sections of which are the curves 

 called caustick curves, and that the focus of opticians is 

 only a point in this surface, where the rays are more con- 

 densed, 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 ac- 

 curately, 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 might, by refraction at the said, 

 superficies, be made to converge to another given point. 1 

 The problem was resolved by Descartes in its full extent ; 



T 



! Cartesii Dioptrices, cap. 3vum ; Geometria, lib. 2dns, 



