160 THE MATHEMATICAL PRINCIPLES [BOOK I Prop. XIII) describe some conic section whose focus is A placed in the centre of force. Suppose that conic sec tion to be A RPB and its focus S. And, first, if the figure be an ellipsis, upon the greater axis thereof AB describe the semi-circle ADB, and let the right line I) PC pass through the falling body, making right angles with the axis; and drawing DS, PS, the area ASD will c be proportional to the area ASP, and therefore also to the time. The axis AB still reaiaining the same, let the breadth of the ellipsis be perpetually diminished, and s the area ASD will always remain proportional to the time. Suppose that breadth to be diminished in, in fruit um ; and the orbit APB in that case coinciding with the axis AB, and the focus S with the extreme point of the axis B, the body will descend in the right line AC1. and the area ABD will become proportional to the time. Wherefore the space AC will be given which the body describes in a given time by itsperpendicular fall from the place A, if the area ABD is taken proportional to the time, and from the point D the right line DC is let fall perpendic ularly on the right line AB. Q,.E.I. CASE 2. If the figure RPB is an hyperbola, on the same principal diameter AB describe the rectangular hyperbola BED ; and because the areas CSP, CB/P, SPy 13, are severally to the sev eral areas CSD, CBED, SDEB, in the given ratio of the heights CP, CD, and the area SP/B is proportional to the time in which the body P will move through the arc P/B. the area SDEB will be also proportional to that time. Let the latus rectum of the hyperbola RPB be diminished in infitiitum, the latus transversum remaining the same; and the arc PB will come to coincide with the right line CB, and the focus S, with the vertex B, Aand the right line SD with the right line BD. And therefore the area BDEB will be proportional to the time in which the body C, by its per pendicular descent, describes the line CB. Q.E.I. CASE 3. And by the like argument, if the figure RPB is a parabola, and to the same principal ver tex B another parabola BED is described, that may always remain given while the former para bola in whose perimeter the body P moves, by having its latus rectum diminished and reduced to nothing, comes to coincide with the line CB, the parabolic segment BDEB will be proportional if to the time in which that body P or C will descend to the centre S or B Q.K.T