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 reciprocally as the bodies; and therefore in a given ratio to each other: and thence, by composition of ratios, in a given ratio to the whole distance[Pg 195] between the bodies. Now these distances revolve about their common term with an equable angular motion, because lying in the same right line they never change their inclination to each other mutually. But right lines that are in a given ratio to each other, and revolve about their terms with an equal angular motion, describe upon planes, which either rest with those terms, or move with any motion not angular, figures entirely similar round those terms. Therefore the figures described by the revolution of these distances are similar. Q.E.D. PROPOSITION LVIII. THEOREM XXI. If two bodies attract each other mutually with forces of any kind, and in the mean time revolve about the common centre of gravity; I say, that, by the same forces, there may be described round either body unmoved a figure similar and equal to the figures which the bodies so moving describe round each other mutually. Let the bodies S and P revolve about their common centre of gravity C, proceeding from S to T, and from P to Q. From the given point s let there be continually drawn sp, sq, equal and parallel to SP, TQ; and the curve pqv, which the point p describes in its revolution round the immovable point s, will be similar and equal to the curves which the bodies S and P describe about each other mutually; and therefore, by Theor. XX, similar to the curves ST and PQV which the same bodies describe about their common centre of gravity C; and that because the proportions of the lines SC, CP, and SP or sp, to each other, are given.