greater than A G. But A Z is perpendicular to the plane K BOS, and is, therefore, the shortest of all the straight lines that can be drawn from A, the centre of the sphere, to that plane. Therefore the plane K BOS does not meet the less sphere.

Also the other planes between the quadrants BX and KX do not meet the less sphere. From the point A draw AI perpendicular to the plane of the quadrilateral S O PT, and join IO. In the same manner, as it was proved of the plane KB OS and the point Z, it may be shown that the point I is the centre of a circle described about SOPT; and that OS is greater than PT. But P T was shown to be parallel to OS; and because the two trapeziums K BOS and SOPT inscribed in circles have their sides B K and OS parallel, as also OS and PT; and their other sides BO, KS, OP, and ST all equal to one another, but BK is greater than OS, and OS greater than PT. Therefore the straight line ZB is greater (XII. Lemma 2) than IO. Join A O. In the triangles A B Z and AIO, it may be shown, as above, that A Z is less than A I. But it was proved that A Z is greater than AG; much more then is AI greater than A G. Therefore the plane SOPT does not meet the less sphere. In the same manner it may be proved, that the plane TPRY does not meet the same sphere, as also the triangle YR X (XII. Lem. 2, Cor.) In the same manner it may be demonstrated, that all the planes, or faces the polyhedron, do not meet the less sphere. Therefore in the greater of two spheres, which have the same centre, a polyhedron is inscribed, the superficies of which does not meet the lesser sphere. Q. E. F.

COROLLARY.-If in the less sphere a similar polyhedron be inscribed by joining the points in which the radii of the sphere drawn to the angles of the polyhedron in the greater sphere meet the superficies of the less, the polyhedron in the sphere BCDE shall have to this polyhedron, the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the less 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 angles at the bases, equal to one another, each to each (XI.B), 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 (XI. Def. 11) to one another, each to each. But similar pyramids have to one another the triplicate (XII. 8, Cor.) ratio of their homologous sides. Therefore the pyramid of which the base is the quadrilateral K BO S, and vertex A, has to the pyramid in the inner sphere of the same order, the triplicate ratio of their homologous sides, that is, of that ratio which the radius AB, the greater sphere, has to the radius of the less sphere. In like manner, each pyramid in the greater sphere has to each of the same order in the less, the triplicate ratio of that which AB has to the radius of the less sphere. But as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole polyhedron in the greater sphere has to the whole polyhedron in the less, the triplicate ratio of that which A B has to the radius of the less; that is, which the diameter BD of the greater has to the diameter of the less sphere.

Corollary 2.-Every face of a polyhedron inscribed in a sphere is inscribable in a circle.

PROP. XVIII. THEOREM.

Spheres have to one another the triplicate ratio of that which their diameters have.

Let ABC and DEF be two spheres, of which the diameters are B C and EF. The sphere A B C has to the sphere DEF the triplicate ratio of that which B C has to EF.

For, if 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 G H K; and let the sphere DEF have the same centre with GHK. In the greater sphere DEF inscribe (XII. 17) a polyhedron, the superficies of which does not meet the lesser sphere GHK; and in the sphere A B C inscribe another similar to that in the sphere DEF. Therefore

B

A

D

L

G

CE

FM

the polyhedron in the sphere A B C has to the polyhedron in the sphere DEF the triplicate ratio (XII. Cor. 17) of that which BC has to E F. But the sphere A B C has to the sphere G HK, the triplicate ratio of that which BC has to EF. Therefore, as the sphere A B C is to the sphere GHK, so is the polyhedron in the sphere A B C to the polyhedron in the sphere DEF. But the sphere ABC is greater than the inscribed polyhedron. Therefore (V. 14), also the sphere GHK is greater than the 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 E F. 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 D E F, the triplicate ratio of that which B C has to EF; for, if it can, let it have that ratio to a greater sphere I. MN. Therefore, by inversion, the sphere L M N has to the sphere A B C, the triplicate ratio of that which the diameter EF has to the diameter B C. But the sphere L M N is to A B C as the sphere DEF is to some sphere, which must be less (V. 14) than the sphere A B 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 E F has to BC; which was shown to be impossible. Therefore, the sphere A B C has not to any sphere greater than DE F, the triplicate ratio of that which B C has to EF; and it was demonstrated, that neither has it that ratio to any sphere less than DEF. Therefore, the sphere A B C has to the sphere DEF, the triplicate ratio of that which BC has to EF. Wherefore, spheres have to one another, &c.

Every sphere is two-thirds of its circumscribed cylinder.-Archimedes.

FINIS.