If the ellipsis, by having its centre removed to an infinite distance, degenerates into a parabola, the body will move in this parabola; and the[Pg 116] force, now tending to a centre infinitely remote, will become equable. Which is Galileo's theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or diminishing the ordinates in any given ratio, or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre; provided the periodic times remain equal; so also in all figures whatsoever, if the ordinates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same, the forces directed to any centre placed in the abscissa are in the several ordinates augmented or diminished in the ratio of the distances from the centre. SECTION III. Of the motion of bodies in eccentric conic sections. PROPOSITION XI. PROBLEM VI. If a body revolves in an ellipsis; it is required to find the law of the centripetal force tending to the focus of the ellipsis. Let S be the focus of the ellipsis. Draw SP cutting the diameter DK of the ellipsis in E, and the ordinate Qv in x; and complete the parallelogram QxPR. It is evident that EP is equal to the greater semi-axis AC: for drawing HI from the other focus H of the ellipsis parallel to EC, because CS, CH are equal, ES, EI will be also equal; so that EP is the half sum of PS, PI, that is (because of the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which taken together are equal to the whole axis 2AC. Draw QT perpendicular to SP, and putting L for the principal latus rectum of the ellipsis [Pg 117], we shall have to as QR to Pv, that is, as PE or AC to PC; and to GvP as L to Gv; and GvP to as to ; and by (Corol. 2, Lem. VII) the points Q and P coinciding, is to in the ratio of equality; and or is to as to , that is, as to , or (by Lem. XII) as to . And compounding all those ratios together, we shall have to as , or to , or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gv are equal. And therefore the quantities and , proportional to these, will be also equal. Let those equals be drawn into , and will become equal to . And therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as , that is, reciprocally in the duplicate ratio of the distance SP. Q.E.I. The same otherwise.