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.