Page:Newton's Principia (1846).djvu/251

 Aa, Bb, Cc, &c, and another parallel to them. The force of attraction or impulse, acting in directions perpendicular to those planes, does not at all alter the motion in parallel directions; and therefore the body proceeding with this motion will in equal times go through those equal parallel intervals that lie between the line AG and the point H, and between the point I and the line dK; that is, they will describe the lines GH, IK in equal times. Therefore the velocity before incidence is to the velocity after emergence as GH to IK or TH, that is, as AH or Id to vH; that is (supposing TH or IK radius), as the sine of emergence to the sine of incidence. Q.E.D.


 * The same things being supposed, and that the motion before incidence is swifter than afterwards; I say, that if the line of incidence be inclined continually, the body will be at last reflected, and the angle of reflexion will be equal to the angle of incidence.

For conceive the body passing between the parallel planes Aa, Bb, Cc, &c., to describe parabolic arcs as above; and let those arcs be HP, PQ, QR, &c. And let the obliquity of the line of incidence GH to the first plane Aa be such that the sine of incidence may be to the radius of the circle whose sine it is, in the same ratio which the same sine of incidence hath to the sine of emergence from the plane Dd into the space DdeE; and because the sine of emergence is now become equal to radius, the angle of emergence will be a right one, and therefore the line of emergence will coincide with the plane Dd. Let the body come to this plane in the point R; and because the line of emergence coincides with that plane, it is manifest that the body can proceed no farther towards the plane Ee. But neither can it proceed in the line of emergence Rd; because it is perpetually attracted or impelled towards the medium of incidence. It will return, therefore, between the planes Cc, Dd, describing an arc of a parabola QRq, whose principal vertex (by what Galileo has demonstrated) is in R, cutting the plane Cc in the same angle at q, that it did before at Q; then going on in the parabolic arcs qp, ph, &c., similar and equal to the former arcs QP, PH, &c., it will cut the rest of the planes in the same angles at p, h, &c., as it did before in P, H, &c., and will emerge at last with the same obliquity at h with which it first impinged on that plane at H. Conceive now the intervals of the planes Aa, Bb, Cc, Dd, Ee, &c., to be infinitely diminished, and the number in finitely increased, so that the action of attraction or impulse, exerted according to any assigned law, may become continual; and, the angle of emergence remaining all along equal to the angle of incidence, will be equal to the same also at last. Q.E.D.