With this confirmation of the supposition, Newton proceeded to the purely geometrical calculation of the law of centripetal[198] force necessary to make a moving body describe an ellipse round its focus, which Kepler's observations had established to be the form of the orbits of the planets round the sun. The result of the inquiry shewed that this curve required the same law of the force, varying inversely as the square of the distance, which therefore of course received additional confirmation. His method of doing this may, perhaps, be understood by referring to the last figure but one, in which C_d_, for instance, representing the space fallen from any point C towards S, in a given time, and the area CSD being proportional to the corresponding time, the space through which the body would have fallen at C in any other time (which would be greater, by Galileo's law, in proportion to the squares of the times), might be represented by a quantity varying directly as C_d_, and inversely in the duplicate proportion of the triangular area CSD, that is to say, proportional to C_d_/(SC × D_k_)², if D_k_ be drawn from D perpendicular on SC. If this polygon represent an ellipse, so that CD represents a small arc of the curve, of which S is the focus, it is found by the nature of that curve, that C_d_/(D_k_)² is the same at all points of the curve, so that the law of variation of the force in the same ellipse is represented solely by 1/(SC)². If C_d_, &c. are drawn so that C_d_/(D_k_)² is not the same at every point, the curve ceases to be an ellipse whose focus is at S, as Newton has shewn in the same work. The line to which (Dk)²/Cd is found to be equal, is one drawn through the focus at right angles to the longest axis of the ellipse till it meets the curve;—this line is called the _latus rectum_, and is a third proportional to the two principal axes.