THE MATHEMATICAL PRINCIPLES [BoOK 1 tangle under its given latiis rectum and the line IM is equal to the squarrf cf HM ; and moreover the line HM will be bisected in L. Whence if to MI there be let fall the perpendicular LO, MO, OR will be equal; and adding the equal lines ON, OI, the wholes MN, IR will be equal also. Therefore since IR is given, MN is also given, and the rectangle NMI is to the rectangle under the latus rectum and IM, that is, to HMa in a given ratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, to the difference of the squares ML2, and PL2 or LI2 ; and HM2 hath a given ratio to its fourth part ML2; therefore the ratio of ML2 — LI2 to ML2 is given, and by conversion the ratio of LI2 to ML ', and its subduplicate, theratrio of LI to ML. But in every triangle, as LMI, the sines jf the angles are proportional to the opposite sides. Therefore the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given. QJE.lr). CASE 2. Let now the body pass successively through several spaces ter minated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a \ . force which is uniform in each of them separ\ a ately, but different in the different spaces ; and B \ fr by what was just demonstrated, the sine of the c ^^ c angle of incidence on the first plane Aa is to the sine of emergence from the second plane Bb in a given ratio ; and this sine of incidence upon the second plane Bb will be to the sine of emergence from the third plane Cc in a given ratio ; and this sine to the sine of emergence from the fourth plane Dd in a given ra tio ; and so on in infinitum ; and, by equality, the sine of incidence on the first plane to the sine of emergence from the last plane in a given ratio. I ,et now the intervals of the planes be diminished, and their number be in finitely increased, so that the action of attraction or impulse, exerted accord ing to any assigned law, may become continual, and the ratio of the sine of incidence on the first plane to the sine of emergence from the last plane being all along given, will be given then also. QJE.D. PROPOSITION XCV. THEOREM XLIX. The same thing's being supposed, I say, that the velocity of the body be fore its incidence is to its velocity after emergence as the sine of emer gence to the sine of incid nee. 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. And (by Cor. 2 of the