Let the centre of the scale (in the same third figure) be A, the beam A B; and let any ponderant C, having B G for its moment, be applied to the point B; also let any other ponderant D, whose moment is E I, be applied to the point E. Through the point I let I K be drawn parallel to the beam A B, cutting A G in K; also let K L be drawn parallel to B G, K L will then be the moment of the ponderant D; and by the last article, it will be as B G, the moment of the ponderant C in the point B, to L K the moment of the ponderant D in the point E, so A B to A L. On the other side of the centre of the scale, let A N be taken equal to A L; and to the point N let there be applied the ponderant O, having to the ponderant C the proportion of A B to A N. I say, the ponderants in B and N will be equally poised. For the proportion of the moment of the ponderant O, in the point N, to the moment of the ponderant C in the point B, is by the 5th article, compounded of the proportions of the weight O to the weight C, and of the distance from the centre of the scale A N or A L to the distance from the centre of the scale A B. But seeing we have supposed, that the distance A B to the distance A N is in reciprocal proportion of the weight O to the weight C, the proportion of the moment of the ponderant O, in the point N, to the moment of the ponderant C, in the point B, will be compounded of the proportions of A B to A N, and of A N to A B. Wherefore, setting in order A B, A N, A B, the moment of O to the moment of C will be as the first to the last, that is, as A B to A B. Their moments therefore are equal; and consequently the plane which passes through A will (by the fifth definition) be a plane of equiponderation. Wherefore they will be equally poised; as was to be proved. Now the converse of this is manifest. For if there be equiponderation and the proportion of the weights and distances be not reciprocal, then both the weights will always have the same moments, although one of them have more weight added to it or its distance changed. Coroll. When ponderants are of the same species, and their moments be equal; their magnitudes and distances from the centre of the scale will be reciprocally proportional. For in homogeneous bodies, it is as weight to weight, so magnitude to magnitude. [Sidenote: If the parts of any ponderant press the beams of the scale every where equally, all the parts cut off, reckoned from centre of the scale, will have their moments in the same proportion with that of the parts of a triangle cut off from the vertex by strait lines parallel to the base.]