the circle BCDE: let this be the circumference KB; therefore the straight line KB is less than GU; and because the angle BZK is obtuse, as was proved in the preceding, therefore BK is greater than BZ: but GU is greater than BK; much more then is GU greater than BZ, and the square of GU than the square of BZ, and AU is equal to AB; therefore the square of AU, that is, the squares of AG, CU are equal to the square of AB, that is, to the squares of AZ, ZB; but the square of BZ is less than the square of GU; therefore the square of AZ is greater than the square of AG, and the straight line AZ consequently greater than the straight line AG. COR. And if in the lesser sphere there be described a solid polyhedron, by drawing straight lines betwixt the points in which the straight lines from the centre of the sphere drawn to all the angles of the solid polyhedron in the greater sphere meet the superficies of the lesser; in the same order in which are joined the points in which the same lines from the centre meet the superficies of the greater sphere; the solid polyhedron in the sphere BCDE has to this other solid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other sphere; for if these two solids be divided into the same number of pyramids, and in the same order, the pyramids shall be similar to one another, each to each; because they have the solid angles at their common vertex, the centre of the sphere the same in each pyramid, and their other solid angle at the bases equal to one another, each to each (B. 11.), because they are contained by three plane angles equal each to each: and the pyramids are contained by the same number of similar planes; and are therefore similar (11. def. 11.) to one another, each to each but similar pyramids have to one another the triplicate (Cor. S. 12.) ratio of their homologous sides. Therefore the pyramid of which the base is the quadrilateral KBOS, and vertex A, has to the pyramid in the other sphere of the same order, the triplicate ratio of their homologous sides, that is, of that ratio which AB from the centre of the greater sphere has to the straight line from the same centre to the superficies of the lesser sphere. And in like manner, each pyramid in the greater sphere has to each of the same order in the lesser, the triplicate ratio of that which AB has to the semidiameter of the lesser sphere. And as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole solid polyhedron in the other, the triplicate ratio of that which AB, the semidiameter of the first, has to the semidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other sphere. PROP. XVIII. THEOR. SPHERES have to one another the triplicate ratio of that which their diameters have. Let ABC, DEF be two spheres, of which the diameters are BC, EF. The sphere ABC has to the sphere DEF the triplicate ratio of that which BC has to EF. For, if it has not, the sphere ABC shall have to a sphere either less or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a less, viz. to the sphere GHK; and let the sphere DEF have the same centre with GHK; and in the greater sphere DEF describe (17. 12.) A L G D B CE K F M a solid polyhedron, the superficies of which does not meet the lesser sphere GHK; and in the sphere ABC describe another similar to that in the sphere DEF; therefore the solid polyhedron in the sphere ABC has to the solid polyhedron in the sphere DEF, the triplicate ratio (Cor. 17. 12.) of that which BC has to EF. But the sphere ABC has to the sphere GHK, the triplicate ratio of that which BC has to EF: therefore, as the sphere ABC to the sphere GIIK, so is the solid polyhedron in the sphere ABC to the solid polyhedron in the sphere DEF: but the sphere ABC is greater than the solid polyhedron in it; therefore (14.5.) also the sphere GHK is greater than the solid polyhedron in the sphere DEF; but it is also less, because it is contained within it, which is impossible: therefore the sphere ABC has not to any sphere less than DEF, the triplicate ratio of that which BC has to EF. In the same manner, it may be demonstrated that the sphere DEF has not to any sphere less than ABC, the triplicate ratio of that which EF has to BC. Nor can the sphere ABC have to any sphere greater than DEF, the triplicate ratio of that which BC has to EF: for, if it can, let it have that ratio to a greater sphere LMN: therefore, by inver-, sion, the sphere LMN has to the sphere ABC the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, so is the sphere DEF to some sphere, which must be less (14. 5.) than the sphere ABC, because the sphere LMN is greater than the sphere DEF. Therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC; which was shown to be impossible: therefore the sphere ABC has not to any sphere greater than DEF the triplicate ratio of that which BC has to EF: and it was demonstrated, that neither has it that ratio to any sphere less than DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D. FINIS. NOTES, CRITICAL AND GEOMETRICAL; CONTAINING AN ACCOUNT OF THOSE THINGS IN WHICH THIS EDITION DIFFERS FROM THE GREEK TEXT; AND THE REASONS OF THE ALTERATIONS WHICH HAVE BEEN MADE. AS ALSO OBSERVATIONS ON SOME OF THE PROPOSITIONS. BY ROBERT SIMSON, M. D. IMERITUS PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW. |