of the globe, and M the center of gravity of the weight K. In the first place let the weight hang so, that a line drawn from L to M shall lie horizontally; and I say, if the globe be moved either up or down the plane D F, the weight will so move along with it, that the center of gravity common to both the weights shall continue in this line L M, and therefore shall in no case descend. To prove this more fully, I shall depart a little from the method of this treatise, and make use of a mathematical proportion or two: but they are such, as any person, who has read ~EUCLID’S ELEMENTS~, will fully comprehend; and are in themselves so evident, that, I believe, my readers, who are wholly strangers to geometrical writings, will make no difficulty of admitting them. This being premised, let the globe be moved up, till its center be at G, then will M the center of gravity of the weight K be sunk to N; so that M N shall be equal to G L. Draw N G crossing the line M L in O; then I say, that O is the common center of gravity of the two weights in this their new situation. Let G P be drawn perpendicular to M L; then G L will bear the same proportion to G P, as D F bears to D E; and M N being equal to G L, M N will bear the same proportion to G P, as D F bears to D E. But N O bears the same proportion to O G, as M N bears to G P; consequently N O will bear the same proportion to O G, as D F bears to D E. In the last place, the weight of the globe A bears the same proportion to the other weight K, as D F bears to D E; therefore N O bears the same proportion to O G, as the weight of the globe A bears to the weight K. Whence it follows, that, when the center of the globe A is in G, and the center of gravity of the weight K is in N, O will be the center of gravity common to both the weights. After the same manner, if the globe had been caused to descend, the common center of gravity would have been found in this line M L. Since therefore no motion of the globe either way will make the common center of gravity descend, it is manifest, from what has been said above, that the weights A and K counterpoize each other. 56. I SHALL now consider the case of pendulums. A pendulum is made by hanging a weight to a line, so that it may swing backwards and forwards. This motion the geometers have very carefully considered, because it is the most commodious instrument of any for the exact measurement of time.