234 THE MATHEMATICAL PRINCIPLES [BOOK 1 \vill be still less increased by it, if the attraction, in the recess of the body attracted, decreases in a still less proportion. The proposition, therefore, is evident concerning attractive spheres. And the case is the same of con cave sphaerical orbs attracting external bodies. And much more does it appear in orbs that attract bodies placed within them, because there the attractions diffused through the cavities of those orbs are (by Prop. LXX) destroyed by contrary attractions, and therefore have no effect even in the place of contact. Now if from these spheres and sphoerical orbs we take away any parts remote from the place of contact, and add new parts any where at pleas ore, we may change the figures of the attractive bodies at pleasure ; but the parts added or taken away, being remote from the place of contact, will cause no remarkable excess of the attraction arising from the contact of the two bodies. '1 herefore the proposition holds good in bodies of all figures. Q.E.I). PROPOSITION LXXXV1. THEOREM XLIII. If the forces of the particles of which an attractive body is composed de crease, in. the recess of the attractive body, in a triplicate or more than a triplicate ratio of the distance from the particles, the attraction will be vastly stronger in the point of contact than when the attracting and attracted bodies are separated from each other, though by never so small an interval. For that the attraction is infinitely increased when the attracted corpus cle comes to touch an attracting sphere of this kind, appears, by the solu tion of Problem XLI, exhibited in ££e second and third Examples. The same will also appear (by comparing those Examples and Theorem XLI together) of attractions of bodies made towards concavo-convex orbs, whether the attracted bodies be placed without the orbs, or in the cavities within them. And by aiding to or taking from those spheres and orbs any at tractive matter any where without the place of contact, so that the attrac tive bodies may receive any assigned figure, the Proposition will hold good of all bodies universally. Q.E.D. PROPOSITION LXXXVII. THEOREM XI. IV. If two bodies similar to each other, and consisting of matter equally at tractive attract separately two corpuscles proportional to those bodies, and in a like situation to them, the accelerative attractions of the cor puscles towards the entire bodies will be as the acccleratire at tractions of the corpuscles towards particles of the bodies proportional to the wholes, and alike situated in them. For if the bodies are divided into particles proportional to the wholes, and alike situated in them, it will be, as the attraction towards any parti cle of one of the bodies to the attraction towards the correspondent particle