194 THE MATHEMATICAL PRINCIPLES [BOOK I with the interval TV let the projection of that little circle in the plane AOP be the ellipsis pQ. And because the magnitude of that little circle T/, and TN or PO its distance from the axis CO is also given, the ellipsis pQ, will be given both in kind and magnitude, as also its position to the right line PO. And since the area PO/? is proportional to the time, and therefore given because the time is given, the angle POp will be given. And thence will be given jo the common intersection of the ellipsis and. the right line Op, together with the angle OPp, in which the projection APp of the tra jectory cuts the line OP. But from thence (by conferring Prop. XLI, with Us 2d Cor.) the mariner of determining the curve APp easily appears. Then from the several points P of that projection erecting to the plane AOP, the perpendiculars PT meeting the curve superficies in T, there will be £iven the several points T of the trajectory. Q.E.I. SECTION XL f 'f the motions of bodies tending' to each other with centripetal forces. I have hitherto been treating of the attractions of bodies towards an im movable centre; though very probably there is no such thing existent in nature. For attractions are made towards bodies, and the actions of the bodies attracted and attracting are always reciprocal and equal, by Law III ; BO that if there are two bodies, neither the attracted nor the attracting body is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it were mutually attracted, revolve about a common centre of gravity. And if there be more bodies, which are either attracted by one single one which is attracted by them again, or which all of them, attract each other mutu ally , these bodies will be so moved among themselves, as that their common centre of gravity will either be at rest, or move uniformly forward in a right line. I shall therefore at present go on to treat of the motion of bodies mutually attracting each other ; considering the centripetal forces as attractions ; though perhaps in a physical strictness they may more truly be called impulses. But these propositions are to be considered as purely mathematical; and therefore, laying aside all physical considerations, I make use of a familiar way of speaking, to make myself the more easily understood by a mathematical reader. PROPOSITION LVII. THEOREM XX. Two bodies attracting each other mutually describe similar figures about their common centre of gravity, and about each other mutually. For the distances of the bodies from their common centre of gravity are leciprocally as the bodies; and therefore in a given ratio to each other: *nd thence, bv composition of ratios, in a given ratio to the whole distance