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

312 the firt plane Aa in the direction of the line $$\scriptstyle GH^2$$ and in its whole paage through the intermediate pace let it be attracted or impelled towards the medium of incidence, and by that action let it be made to decribe 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 decribed cutting both HM in P and Q, and MI produced in N; and firt, if the attraction or impule be uppoed uniform, the curve HI (by what Galileo has demontrated) be a parabola, whoe property is, that a rectangle under its given latus rectum and the line IM is equal to the quare of HM; and moreover the line HM will be biected 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 alo. Therefore ince IR is given MN is alo 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 quares $$\scriptstyle ML^2$$, 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 converion the ratio of LI to ML, and its ubduplicate, the ratio of LI to ML. But in every triangle as LMI, the lines of the angles are proportional to the oppoite ides. Therefore the ratio of the line of the angle of incidence LMR