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.
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.
8. If to the whole length of the beam there be applied a parallelogram, or a parallelopipedum, or a prisma, or a cylinder, or the superficies of a cylinder, or of a sphere, or of any portion of a sphere or prisma; the parts of any of them cut off with planes parallel to the base will have their moments in the same proportion with the parts of a triangle, which has its vertex in the centre of the scale, and for one of its sides the beam itself, which parts are cut off by planes parallel to the base.
First, let the rectangled parallelogram A B C D (in [figure 4]) be applied to the whole length of the beam A B; and producing C B howsoever to E, let the triangle A B E be described. Let now any part of the parallelogram, as A F, be cut off by the plane F G, parallel to the base C B; and let F G be produced to A E in the point H. I say, the moment of the whole A B C D to the moment of its part A F, is as the triangle A B E to the triangle A G H, that is, in proportion duplicate to that of the distances from the centre of the scale.
For, the parallelogram A B C D being divided into equal parts, infinite in number, by strait lines drawn parallel to the base; and supposing the moment of the strait line C B to be B E, the moment of the strait line F G will (by the [7th article]) be G H; and the moments of all the strait lines of that parallelogram will be so many strait lines in the triangle A B E drawn parallel to the base B E; all which parallels together taken are the moment of the whole parallelogram A B C D; and the same parallels do also constitute the superficies of the triangle A B E. Wherefore the moment of the parallelogram A B C D is the triangle A B E; and for the same reason, the moment of the parallelogram A F is the triangle A G H; and therefore the moment of the whole parallelogram to the moment of a parallelogram which is part of the same, is as the triangle A B E to the triangle A G H, or in proportion duplicate to that of the beams to which they are applied. And what is here demonstrated in the case of a parallelogram may be understood to serve for that of a cylinder, and of a prisma, and their superficies; as also for the superficies of a sphere, of an hemisphere, or any portion of a sphere. For the parts of the superficies of a sphere have the same proportion with that of the parts of the axis cut off by the same parallels, by which the parts of the superficies are cut off, as Archimedes has demonstrated; and therefore when the parts of any of these figures are equal and at equal distances from the centre of the scale, their moments also are equal, in the same manner as they are in parallelograms.
Secondly, let the parallelogram A K I B not be rectangled; the strait line I B will nevertheless press the point B perpendicularly in the strait line B E; and the strait line L G will press the point G perpendicularly in the strait line G H; and all the rest of the strait lines which are parallel to I B will do the like. Whatsoever therefore the moment be which is assigned to the strait line I B, as here, for example, it is supposed to be B E, if A E be drawn, the moment of the whole parallelogram A I will be the triangle A B E; and the moment of the part A L will be the triangle A G H. Wherefore the moment of any ponderant, which has its sides equally applied to the beam, whether they be applied perpendicularly or obliquely, will be always to the moment of a part of the same in such proportion as the whole triangle has to a part of the same cut off by a plane which is parallel to the base.
The diameter of equiponderation of figures which are deficient according to commensurable proportions of their altitudes and bases, divides the axis, so that the part taken next the vertex is to the other part as the complete figure to the deficient figure.
9. The centre of equiponderation of any figure, which is deficient according to commensurable proportions of the altitude and base diminished, and whose complete figure is either a parallelogram or a cylinder, or a parallelopipedum, divides the axis, so, that the part next the vertex, to the other part, is as the complete figure to the deficient figure.
For let C I A P E (in [fig. 5]) be a deficient figure, whose axis is A B, and whose complete figure is C D F E; and let the axis A B be so divided in Z, that A Z be to Z B as C D F E is to C I A P E. I say, the centre of equiponderation of the figure C I A P E will be in the point Z.
First, that the centre of equiponderation of the figure C I A P E is somewhere in the axis A B is manifest of itself; and therefore A B is a diameter of equiponderation. Let A E be drawn, and let B E be put for the moment of the strait line C E; the triangle A B E will therefore (by the [third article]) be the moment of the complete figure C D F E. Let the axis A B be equally divided in L, and let G L H be drawn parallel and equal to the strait line C E, cutting the crooked line C I A P E in I and P, and the strait lines A C and A E in K and M. Moreover, let Z O be drawn parallel to the same C E; and let it be, as L G to L I, so L M to another, L N; and let the same be done in all the rest of the strait lines possible, parallel to the base; and through all the points N, let the line A N E be drawn; the three-sided figure A N E B will therefore be the moment of the figure C I A P E. Now the triangle A B E is (by the [9th article] of chapter XVII) to the three-sided figure A N E B, as A B C D + A I C B is to A I C B twice taken, that is, as C D F E + C I A P E is to C I A P E twice taken. But as C I A P E is to C D F E, that is, as the weight of the deficient figure is to the weight of the complete figure, so is C I A P E twice taken to C D F E twice taken. Wherefore, setting in order C D F E + C I A P E. 2 C I A P E. 2 C D F E; the proportion of C D F E + C I A P E to C D F E twice taken will be compounded of the proportion of C D F E + C I A P E to C I A P E twice taken, that is, of the proportion of the triangle A B E to the three-sided figure A N E B, that is, of the moment of the complete figure to the moment of the deficient figure, and of the proportion of C I A P E twice taken to C D F E twice taken, that is, to the proportion reciprocally taken of the weight of the deficient figure to the weight of the complete figure.