Let A (in [fig. 2]) be the centre of the scale; and let the equal ponderants D and E press the beam A B in the points B and C; also let the strait lines B D and C E be diameters of equiponderation; and the points D and E in the ponderants D and E be their centres of equiponderation. Let A G F be drawn howsoever, cutting D B produced in F, and E C in G; and lastly, upon the common centre A, let the two arches B H and C I be described, cutting A G F in H and I. I say, the moment of the ponderant D to the moment of the ponderant E is as A B to A C, or as B H to C I, or as B F to C G. For the effect of the ponderant D, in the point B, is circular motion in the arch B H; and the effect of the ponderant E, in the point C, circular motion in the arch C I; and by reason of the equality of the ponderants D and E, these motions are to one another as the quicknesses or velocities with which the points B and C describe the arches B H and C I, that is, as the arches themselves B H and C I, or as the strait parallels B F and C G, or as the parts of the beam A B and A C; for A B. A C:: B F. C G:: B H. C I. are proportionals; and therefore the effects, that is, by the 4th definition, the moments of the equal ponderants applied to several points of the beam, are to one another as A B and A C; or as the distances of those points from the centre of the scale; or as the parallel bases of the triangles which have a common angle at A; or as the concentric arches B H and C I; which was to be demonstrated.
The moments of unequal ponderants have their proportion to one another compounded of the proportions of their weights and distances from the centre of the scale.
5. Unequal ponderants, when they are applied to several points of the beam, and hang at liberty, that is, so as the line by which they hang be the diameter of equiponderation, whatsoever be the figure of the ponderant, have their moments to one another in proportion compounded of the proportions of their distances from the centre of the scale, and of their weights.
Let A (in [fig. 3]) be the centre of the scale, and A B the beam; to which let the two ponderants C and D be applied at the points B and E. I say, the proportion of the moment of the ponderant C to the moment of the ponderant D, is compounded of the proportions of A B to A E, and of the weight C to the weight D; or, if C and D be of the same species, of the magnitude C to the magnitude D.
Let either of them, as C, be supposed to be bigger than the other, D. If, therefore, by the addition of F, F and D together be as one body equal to C, the moment of C to the moment of F + D will be (by the [last article]) as B G is to E H. Now as F + D is to D, so let E H be to another E I; and the moment of F + D, that is of C, to the moment of D, will be as B G to E I. But the proportion of B G to E I is compounded of the proportions of B G to E H, that is, of A B to A E, and of E H to E I, that is, of the weight C to the weight D. Wherefore unequal ponderants, when they are applied, &c. Which was to be proved.
6. The same figure remaining, if I K be drawn parallel to the beam A B, and cutting A G in K; and K L be drawn parallel to B G, cutting A B in L, the distances A B and A L from the centre will be proportional to the moments of C and D. For the moment of C is B G, and the moment of D is E I, to which K L is equal. But as the distance A B from the centre is to the distance A L from the centre, so is B G, the moment of the ponderant C, to L K, or E I the moment of the ponderant D.
If two ponderants have their weights and distances from the centre of the scale in reciprocal proportion, they are equally poised; and contrarily.
7. If two ponderants have their weights and distances from the centre in reciprocal proportion, and the centre of the scale be between the points to which the ponderants are applied, they will be equally poised. And contrarily, if they be equally poised, their weights and distances from the centre of the scale will be in reciprocal proportion.
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.