The Sine of Incidence

Proposition 95 Theorem 49

The same things being supposed, the velocity of the body before its incidence is to its velocity after emergence as the sine of emergence to the sine of incidence.

Make AH and Id equal, and erect the perpendiculars AG, dK meeting the lines of incidence and emergence GH, IK, in G and K. In GH take TH equal to IK, and to the plane Aa let fall a perpendicular Tv.

By Cor. 2 of the Laws of Motion, let the motion of the body be resolved into two, one perpendicular to the planes 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.

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.