greater of two spheres which have the same center, a solid polyhedron is inscribed, the superficies of which does not meet the lesser sphere. SCHOLIUM. The straight line AZ may be demonstrated to be greater than AG otherwise, and in a shorter manner, without the help of Prop. 16, as follows. From the point G draw GU at right angles to AG, and join AU. If then the circumference BE be bisected, and its half again bisected, and so on, there will at length be left a circumference less than the circumference which is subtended by a straight line equal to GU, inscribed in 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 on GU than the square on BZ; and AU is equal to AB; therefore the square on AU, that is, the squares on AG, GU are equal to the square on AB, that is, to the squares on AZ, ZB: but the square on BZ is less than the square on GU; therefore the square on AZ is greater than the square on AG, and the straight line AŻ consequently greater than the straight line AG. COROLLARY. And if in the lesser sphere there be inscribed a solid polyhedron, by drawing straight lines betwixt the points in which the straight lines from the center 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 center meet the superficies of the greater sphere, the solid polyhedron in the sphere BCDE shall have 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 center of the sphere, the same in each pyramid, and their other solid angles at the bases, equal to one another, each to each (a), because they are contained by three plane angles, each equal to each; and the pyramids are contained by the same number of similar planes; and are therefore similar to one another, each to each (b): but similar pyramids have to one another, the triplicate ratio of their homologous sides (c): 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 center of the greater sphere, has to the straight line from the same center 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 semi-diameter 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 semi-diameter of the first has to the semi-diameter of the other; that is, which the diameter BD of the greater has to the diameter of the other sphere. PROPOSITION XVIII. THEOREM.-Spheres have to one another, the triplicate ratio of that which their diameters have. DEMONSTRATION. Let ABC, DEF be two spheres, of which the diameters are BC, EF: the sphere ABC shall have to the sphere DEF, the triplicate ratio of that which BC has to EF. For if it has not, the sphere ABC must 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: and in the greater sphere DEF inscribe a solid polyhedron, the su K perficies of which does not meet the lesser sphere GHK (a); and in the sphere ABC inscribe 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 of that which BC has to EF (6). But the sphere ABC has to the sphere GHK, the triplicate ratio of that which BC has to EF; therefore, as the sphere ABC is to the sphere GHK, 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 also the sphere GHK is greater than the solid polyhedron in the sphere DEF (c): 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 inversion, 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 is to ABC, so is the sphere DEF to some sphere which must be less than the sphere ABC (c), 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. A CLASSIFIED INDEX TO THE FOURTH, FIFTH, SIXTH, ELEVENTH, AND TWELFTH BOOKS OF THE ELEMENTS OF EUCLID. VI. 4. THEOREMS. C. Comparison of Triangles as to Equality. HYPOTHESES. If triangles are equiangular . If two triangles have one angle in each equal, and the sides about the equal angles proportional. If two triangles have their sides proportional. If two triangles have two sides in the one proportional to two sides in the other, And be joined at one angle so as to have their homologous sides parallel to one another. Or have the angles opposite to one pair of the homologous sides equal; and those opposite to the other pair, either both less, or both not less than a right angle. CONSEQUENCES. The sides about the equal angles are proportionals. The sides which are opposite to the equal angles are homologous. The triangles are equiangular, And have those angles equal which the equal sides subtend. They are equiangular, And the equal angles are subtended by the homologous sides. The remaining sides shall be in a straight line. The triangles are equiangular, And the angles contained by the proportional sides are equal. D. On the Relations between the Sides and Angles of Triangles. HYPOTHESES. VI. 31. If a triangle be right-angled. CONSEQUENCES. The rectilineal figure de- VI. 2. E. On the Relations of Lines drawn in Triangles. HYPOTHESES. If a straight line be parallel CONSEQUENCES. It cuts the other sides, or |