composed of pyramids the bases of which are the aforesaid quadri- Book XII. Book XII. BZ, ZA are equal together to the square of BA, and the squares of KV, VA to the square of AK; therefore the squares of BZ, ZA are equal to the squares of KV, VA; and of these the square of KV is greater than the square of BZ, therefore the square of VA is less than the square of ZA, and the straight line AZ greater than VA. much more then AZ is greater than AG, because in the preceding Proposition it was shewn that KV falls without the circle FGH. and AZ is perpendicular to the plane KBOS, and is therefore the Mortest of all the straight lines that can be drawn from A the center of the sphere to that planc. Therefore the plane KBOS does not meet the lesser sphere. And that the other planes between the quadrants BX, KX fall without the lesser sphere, is thus demonstrated. from the point A draw AI perpendicular to the plane of the quadrilateral SOPT, and join 10; and as was demonstrated of the plane KBOS and the point 2, in the same way it may be shewn that the point I is the center of a circle described about SOPT, and that OS is greater than PT; and PT was shewn to be parallel to OS. therefore because the two trapeziums KBOS, SOPT inscribed in circles have their fides BK, OS parallel, as also OS, PT; and their other sides BO, KS, OP, STall equal to one another, and that BK is greater than OS, and OS 0. 2. Lem. greater than PT, therefore the straight line ZB is greater than IO. Join AO which will be equal to AB; and because AIO, AZB are But the straight line AZ may be demonstrated to be greater than and join AU. if then the circumference BE be bisected, and its Book X!T. Cor. And if in the lesser sphere there be described a solid poly- py b. 11. Def. ramids have to one another the triplicate 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 fame order, the triplicate ratio of their homologous fides; that is, of that ratio which AB from the center of the greater sphere has to the straight line from the same center to the superficies of the leffer 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 femidiameter of the lesser sphere, and as one po c.Cor.8.12. Book XII. antecedent is to its consequent, fo are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole tolid polyhedron in the other, the triplicate ratio of that which AB the femidiameter of the first has to the femidiameter of the other ; that is, which the diameter BD of the greater has to the diameter of the other sphere. PRO P. XVIII. THEOR, SPHERES 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 center with GHK; 2. 17. 12. and in the greater sphere DEF describe : a folid polyhedron the superficies of which does not meet the lefser sphere GHK; and in D B CEH K FM N the sphere ABC describe another similar to that in the sphere DEF. therefore the solid polyhedron in the sphere ABC has to . Cor. 14. the solid polyhedron in the sphere DEF, the triplicate ratio b 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 GHK, fo is the folid polyhedron in the sphere ABC to the solid polyhedron in the sphere DEF. but the sphere ABC is greater than the folid polyhedron in it therefore • also the sphere GHK is greater than the folid polyhe-Book XII. dron in the sphere DEF. but it is also less, because it is contained within it, which is imposible. therefore the sphere ABC has not to c. 14 S. 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 liave that ratio to a greater sphere LMN. therefore, by inversion, the fphere 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 e than the sphere ABC, because c. 14. 5. 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 shewn 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 sriplicate ratio of that which BC has to EF. 0. E. D. FINI S. |