to FB, FD reſpectively, and therefore QS to be always equal to CE; and (by cor. 2. prop. 97.) PD will be to QD as M to N, and therefore as DL to DK. or FB to FK; and by diviſion as DL−FB or PH−PD−FB to FD or FQ−QD; and by compoſition as PH−FB to FQ, that is, (becauſe PH and CG, QS and CE are equal) as CE+BG−FR to CE−FS. But (becauſe BG is to CE, as M−N to N) it comes to paſs alſo that CE+BG is to CE as M to N; and therefore, by diviſion, FR is to FS as M to N; and therefore (by cor. 2. prop. 97.) the ſuperficies EF compels a body, falling upon it in the direction DF to go on in the line FR to the place B. Q. E. D.
Scholium.
In the ſame manner one may go on to three or more ſuperficies. But of all figures the ſphærical is the moſt proper for optical uſes. If the object glaſſes of teleſcopes were made of two glaſſes of a ſphærical figure, containing water between them; it is not unlikely that the errors of the refractions made in the extreme parts of the ſuperficies of the glaſſes, may be accurately enough corrected by the refractions of the water. Such object-glaſſes are to be preferred before elliptic and hyperbolic glaſſes, not only becauſe they may be formed with more eaſe and accuracy, but becauſe the pencils of rays ſituate without the axis of the glaſs would be more accurately refracted by them. But the different refrangibility of different rays is the real obſtacle that hinders optics from being made perfect by ſphærical or any other figures. Unleſs the errors thence ariſing can be corrected all the labour ſpent in correcting the others is quite thrown away.
The End of the Firſt Volume.
