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 figure are which the bodies ſo moving deſcribe round each other mutually.
Let the bodies S and P (Pl. 20. Fig. 7.) 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. 20. 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.
Case 1.. The common centre of gravity C (by cor. 4 of the laws of motion) is either at reſt, or moves uniformly in a right line. Let us firſt ſuppoſe it at reſt, and in s and p let there be placed two bodies, one immovable in s, the other
