THE MATHEMATICAL PRINCIPLES. [BOOK J a second attractive superficies EF, which may make those bodies con verge to a given place B. Let a line joining AB cut the first superficies in C and the second in E, the point D being taken any how at plea sure. And supposing the f sine of incidence on the first superficies to the sine of emergence from the same, and the sine of emergence from the second super ficies to the sine of incidence on the same, to be as any given quantity M to another given quantity N; then produce AB to G, so that BG may be to CE as M — N to N ; and AD to H, so that AH may be equal to AG ; arid DF to K, so that DK may be to DH as N to M. Join KB, and about the centre D with the interval DH describe a circle meeting KB produced in L, and draw BF parallel to DL; and the point F will touch the line EF, which, being turned round the axis AB, will describe the superficies sought. Q.E.F. For conceive the lines CP, CQ to be every where perpendicular to AD, DF, and the lines ER, ES to FB, FD respectively, and therefore QS to be always equal to CK; and (by Cor. 2, Prop. XCVII) 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 ; arid 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 XCVI1) 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 sphserical is the most proper for optical uses. If the ob ject glasses of telescopes were made of two glasses of a sphaerical 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 ob ject 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 accu rately refracted by them. But the different refrangibility of different raya is the real obstacle that hinders optics from being made perfect by sphaeri cal or any other figures. Unless the errors thence arising can be corrected, all the labour spent in correcting the others is quite thrown away.