The Reflexions and Refractions of Light

Proposition 96 Theorem 50

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.

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.

SCHOLIUM

These attractions bear a great resemblance to the reflexions and refractions of light made in a given ratio of the secants.

• This was discovered by Snellius.
• It was consequently in a given ratio of the sines, as was exhibited by Descartes.

From Jupiter’s Satellites, we know that light:

• is propagated in succession
• requires 7-8 minutes to travel from the sun to the earth

Moreover, the rays of light that are in our air (as lately was discovered by Grimaldus, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inflected round those bodies as if they were attracted to them.

Those rays which in their passage come nearest to the bodies are the most inflected, as if they were most attracted: which thing I myself have also carefully observed.

Those which pass at greater distances are less inflected; and those at still greater distances are a little inflected the contrary way, and form three fringes of colours. In the figure s represents the edge of a knife, or any kind of wedge AsB; and gowog, fnunf, emtme, dlsld, are rays inflected towards the knife in the arcs owo, nvn, mtm, lsl; which inflection is greater or less according to their distance from the knife.

Since this inflection of the rays is performed in the air without the knife, it follows that the rays which fall upon the knife are first inflected in the air before they touch the knife.

The case is the same of the rays falling upon glass.

The refraction, therefore, is made not in the point of incidence, but gradually, by a continual inflection of the rays: which is done partly in the air before they touch the glass, partly (if I mistake not) within the glass, after they have entered it; as is represented in the rays ckzc, biyb, ahxa, falling upon r, q, p, and inflected between k and z, i and y, h and x.

Therefore, because of the analogy there is between the propagation of the rays of light and the motion of bodies, I thought it not amiss to add the following Propositions for optical uses: not at all considering the nature of the rays of light, or inquiring whether they are bodies or not.

But only determining the trajectories of bodies which are extremely like the trajectories of the rays.