Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/404

the firſt plane Aa in the direction of the line $\scriptstyle GH^{2}$ and in its whole paſſage through the intermediate ſpace let it be attracted or impelled towards the medium of incidence, and by that action let it be made to deſcribe a curve line HI, and let it emerge in the direction of the line IK. Let there be erected IM perpendicular to Bb the plane of emergence, and meeting the line of incidence GH prolonged in M and the plane of incidence Aa in R; and let the line of emergence KI be produced and meet HM in L. About the centre L, with the interval LI, let a circle be deſcribed cutting both HM in P and Q, and MI produced in N; and firſt, if the attraction or impulſe be ſuppoſed uniform, the curve HI (by what Galileo has demonſtrated) be a parabola, whoſe property is, that a rectangle under its given latus rectum and the line IM is equal to the ſquare of 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, than is, to $\scriptstyle HM^{2}$ in a given ratio. But the rectangle NMI is equal to the rectangle PMQ that is, to the difference of the ſquares $\scriptstyle ML^{2}$, and $\scriptstyle PL^{2}$ or $\scriptstyle LI^{2}$; and $\scriptstyle HM^{2}$ hath a given ratio to its fourth part $\scriptstyle ML^{2}$ therefore the ratio of $\scriptstyle ML^{2}-LI^{2}$ to $\scriptstyle ML^{2}$ is given, and by converſion the ratio of LI to ML, and its ſubduplicate, the ratio of LI to ML. But in every triangle as LMI, the lines of the angles are proportional to the oppoſite ſides. Therefore the ratio of the line of the angle of incidence LMR