compofed of pyramids the bafes of which are the aforefaid quadri- Book XII. lateral figures, and the triangle YRX, and thofe formed in the like manner in the rest of the sphere, the common vertex of them all being the point A. and the fuperficies of this folid polyhedron does not meet the leffer sphere in which is the circle FGH. for from the point A draw k AZ perpendicular to the plane of the quadrilateral k. 11. 11, KBOS meeting it in Z, and join BZ, ZK. and because AZ is perpendicular to the plane KBOS, it makes right angles with every ftraight line meeting it in that plane; therefore AZ is perpendicular to BZ and ZK. and because AB is equal to AK, and that the fquares of AZ, ZB, are equal to the fquare of AB; and the fquares of AZ, ZK to the fquare of AK; therefore the fquares of AZ, 47. 1. ZB are equal to the fquares of AZ, ZK. take from these equals the fquare of AZ, the remaining fquare of BZ is equal to the remaining fquare of ZK; and therefore the straight line BZ is equal to ZK. in the like manner it may be demonstrated that the straight lines drawn from the point Z to the points O, S are equal to BZ, or ZK. therefore the circle defcribed from the center Z, and distance ZB fhall pafs thro' the points K,O,S, and KBOS shall be a quadrilateral figure in the circle. and becaufe KB is greater than QV, and QV equal to SO, therefore KB is greater than SO. but KB is equal to each of the ftraight lines BO, KS; wherefore each of the circumferences cut off by KB, BO, KS is greater than that cut off by OS ; and these three circumferences together with a fourth equal to one of them, are greater than the fame three together with that cut off by OS; that is, than the whole circumference of the circle; therefore the circumference fubtended by KB is greater than the fourth part of the whole circumference of the circle KBOS, and confequently the angle BZK at the center is greater than a right angle. and because the angle BZK is obtufe, the fquare of BK is greater than 1. 12. 20 the fquares of BZ, ZK; that is, greater than twice the fquare of BZ. Join KV, and because in the triangles KBV, OBV, KB, BV are equal to OB, BV, and that they contain equal angles; the angle KVB is equal to the angle OVB. and OVB is a right angle; m. 4 1. therefore alfo KVB is a right angle. and becaufe BDis less than twice DV, the rectangle contained by DB, BV is lefs than twice the rectangle DVB; that is, the fquare of KB is lefs than twice the n. 8. é. fquare of KV. but the fquare of KB is greater than twice the fquare of BZ; therefore the fquare of KV is greater than the fquare of BZ. and becaufe BA is equal to AK, and that the fquares of

Book XII. BZ, ZA are equal together to the fquare of BA, and the fquares of KV, VA to the fquare of AK; therefore the fquares of BZ, ZA are equal to the fquares of KV, VA; and of these the square of KV is greater than the fquare of BZ, therefore the square of VA is lefs than the fquare of ZA, and the ftraight line AZ greater than VA. much more then AZ is greater than AG, because in the preceding Propofition it was fhewn that KV falls without the circle FGH. and AZ is perpendicular to the plane KBOS, and is therefore the shortest of all the straight lines that can be drawn from A the center of the sphere to that plane. Therefore the plane KBOS does not meet the leffer fphere.

And that the other planes between the quadrants BX, KX fall without the leffer fphere, is thus demonftrated. from the point A draw AI perpendicular to the plane of the quadrilateral SOPT, and join IO; and as was demonftrated of the plane KBOS and the point Z, in the fame way it may be fhewn that the point I is the center of a circle defcribed about SOPT, and that OS is greater than PT; and PT was fhewn to be parallel to OS. therefore because the two trapeziums KBOS, SOPT inscribed in circles have their fides BK, OS parallel, as alfo OS, PT; and their other fides 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 right angles, the fquares of AI, IO are equal to the fquare of AO or of AB; that is, to the fquares of AZ, ZB; and the fquare of ZB is greater than the square of IO, therefore the square of AZ is less than the fquare of AI; and the straight line AZ less than the straight line AI. and it was proved that AZ is greater than AG; much more then is AI greater than AG. therefore the plane SOPT falls wholly without the leffer fphere. in the fame manner it may be demonftrated that the plane TPRY falls without the fame fphere, as alfo the triangle YRX, viz. by the Cor. of 2d Lemma. and after the fame way it may be demonstrated that all the planes which contain the folid polyhedron fall without the leffer fphere. therefore in the greater of two spheres which have the fame center, a folid polyhedron is defcribed the fuperficies of which does not meet the leffer fphere. Which was to be done.

But the straight line AZ may be demonftrated to be greater than AG otherwife and in a fhorter manner, without the help of Prop. 6. as follows. From the point G draw GU at right angles to AG

and join AU. if then the circumference BE be bifected, and its Book X!!. half again bifected, and fo on, there will at length be left a circumference less than the circumference which is fubtended by a ftraight line equal to GU infcribed 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 obtufe, 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 fquare of GU than the square of BZ. and AU is equal to AB; therefore the fquare of AU, that is the fquares of AG, GU are equal to the fquare of AB, that is to the squares of AZ, ZB; but the fquare of BZ is less than the fquare of GU; therefore the square of AZ is greater than the fquare of AG, and the ftraight line AZ confequently greater than the ftraight line AG.

. COR. And if in the leffer sphere there be described a folid polyhedron by drawing ftraight lines betwixt the points in which the ftraight lines from the center of the sphere drawn to all the angles of the folid polyhedron in the greater sphere meet the fuperficies of the leffer; in the fame order in which are joined the points in which the fame lines from the center meet the fuperficies of the greater fphere; the folid polyhedron in the sphere BCDE has to this other folid poIyhedron the triplicate ratio of that which the diameter of the fphere BCDE has to the diameter of the other sphere. for if these two folids be divided into the fame number of pyramids, and in the fame order; the pyramids shall be similar to one another, each to each. because they have the folid angles at their common vertex, the center of the fphere, the fame in each pyramid, and their other folid angles at the bases equal to one another, each to each, because a. B. 11. they are contained by three plane angles equal each to each; and the pyramids are contained by the fame number of fimilar planes; and are therefore fimilar b to one another, each to each. but fimilar py- b. 11. Def. ramids have to one another the triplicate ratio of their homologous fides. 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 fphere has to the ftraight line from the fame center to the fuperficies of the leffer fphere. and in like manner each pyramid in the greater sphere has to each of the fame order in the leffer, the triplicate ratio of that which AB has to the femidiameter of the lesser sphere. and as one

c

II.

c. Cor.8.12.

B

Book XII. antecedent is to its confequent, fo are all the antecedents to all the confequents. Wherefore the whole folid polyhedron in the greater sphere has to the whole folid polyhedron in the other, the tripli cate ratio of that which AB the femidiameter of the firft has to the femidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other fphere.

2. 17. 12.

PROP. XVIII. THE OR,

SPHERES

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

Let ABC, DEF be two fpheres of which the diameters are BC, EF. the fphere ABC has to the fphere DEF the triplicate ratio of that which BC has to EF.

For if it has not, the fphere ABC fhall have to a sphere either lefs or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a lefs, viz. to the fphere GHK; and let the fphere DEF have the fame center with GHK; and in the greater sphere DEF defcribe a folid polyhedron the fuperficies of which does not meet the leffer sphere GHK; and in

the sphere ABC defcribe another fimilar to that in the sphere DEF. therefore the folid polyhedron in the fphere ABC has to Cor. 17. the folid polyhedron in the fphere DEF, the triplicate ratio 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 fphere ABC to the fphere GHK, fo is the folid polyhedron in the fphere ABC to the folid polyhedron in the fphere DEF. but the fphere 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 alfo lefs, because it is contained within it, which is impoffible. therefore the sphere ABC has not to c. 14 any sphere lefs than DEF, the triplicate ratio of that which BC has to EF. In the fame manner it may be demonstrated that the sphere DEF has not to any fphere lefs than ABC, the triplicate ratio of that which EF has to BC. Nor can the fphere ABC have to any fphere 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 inverfion, 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, fo is the sphere DEF to fome fphere, which must be less than the sphere ABC, because c. 14. 5. the fphere LMN is greater than the fphere DEF. therefore the fphere DEF has to a fphere lefs than ABC the triplicate ratio of that which EF has to BC; which was fhewn to be impoffible. 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 fphere less than DEF. Therefore the fphere ABC has to the fphere DEF, the triplicate ratio of that which BC has to EF. QE. D.

FINI S.