192 THE MATHEMATICAL PRINCIPLES [BOOK 1 Q it be divided into innumerable equal parts, and let Dd be one of those parts. From the centre C, with the intervals CD, Cd, let the circles DT, dt be de scribed, meeting the curve line ST*R in T and t. And because the law of centripetal force is given. and also the altitude CS from which the body at first fell, there will be given the velocity of the body in any other altitude CT (by Prop. XXXIX). But the time in which the body describes the lineola Tt is as the length of that lineola, that is, as the secant of the angle /TC directly, and the velocity inversely. Lei, the ordinate DN, proportional to this time, be made perpendicular to the right line CS at the point D, and because Dd is given, the rectangle Dd X DN, that is, the area DNwc?, will be proportional to the same time. Therefore if PN/?, be a curve line in which the point N is perpetually found, and its asymptote be the right line SQ, standing upon the line CS at right angles, the area SQPJN D will be proportional to the time in which the body in its descent hath described the line ST ; and therefore that area being found, the time is also given. Q.E.I. PROPOSITION LV. THEOREM XIX. If a body move in any curve superficies, whose axis passes through the centre of force, and from the body a perpendicular be let fall iipon the axis \ and a line parallel and equal thereto be drawn from any given point of the axis ; I say, that this parallel line will describe an area proportional to the time. Let BKL be a curve superficies, T a body revolving in it, STR a trajectory which the body describes in the same, S the beginning of the trajectory, OMK the axis of the curve superficies, TN a right line let fall perpendic ularly from the body to the axis ; OP a line parallel and equal thereto drawn from the given point O in the axis ; AP the orthogra phic projection of the trajectory described by the point P in the plane AOP in which the revolving line OP is found : A the beginning of that projection, answering to the point S ; TO a right line drawn from the body to the centre ; TG a part thereof proportional to the centripetal force with which the body tends towards the centre C ; TM a right line perpendicular to the curve superficies ; TI a part thereof proportional to the force of pressure with which the body urges