QA; as therefore BV is to VA, so is KQ to QA, wherefore VQ is parallel (2. 6.) to BK; and because OV, SQ are each of them at right angles to the plane of the circle BCDE, OV is parallel (6. 11.) to SQ; and it has been proved that it is also equal to it; therefore QV, SO are equal and parallel (33. 1.): and because QV is parallel to SO, and also to KB, OS is parallel (9. 11.) to BK; and therefore BO, KS which join them are in the same plane in which these parallels are, and the quadrilateral figure KBOS is in one plane; and if PB, TK be joined, and perpendiculars be drawn from the points P, T to the straight lines AB, AK, it may be demonstrated, that TP is parallel to KB in the very same way that so was shown to be parallel to the same KB; wherefore (9. 11.) TP is parallel to so, and the quadrilateral figure SOPT is in one plane: for the same reason, the quadrilateral TPRY is in one plane; and the figure YRX is also in one plane (2. 11.). Therefore, if from the points O, S, P, T, R, Y X R there be drawn straight lines to the point A, there shall be formed a solid polyhedron between the circumferences BX, KX composed of pyramids, the bases of which are the quadrilaterals KBOS, SOPT, TPRY, and the triangle YRX, and of which the common vertex is the point A: and if the same construction be made upon each of the sides KL, LM, ME, as has been done upon BK, and the like be done also in the other three quadrants, and in the other hemisphere; there shall be formed a solid polyhedron described in the sphere, composed of pyramids, the bases of which are the aforesaid quadrilateral figures and the triangle YRX, and those formed in the like manner in the rest of the sphere, the common vertex of them all being the point A; and the superficies of this solid polyhedron does not meet the lesser sphere in which is the circle FGH: for, from the point A draw (11. 11.) AZ perpendicular to the plane of the quadrilateral KBOS, meeting it in Z, and join BZ, ZK: and because AZ is perpendicular to the plane KBOS, it makes right angles with every straight line meeting it in that plane; therefore AZ is perpendicular to BZ and ZK; and because AB is equal to AK, and that the squares of AZ, ZB are equal to the square of AB; and the squares of AZ, ZK to the square of AK (47. 1.): therefore the squares of AZ, ZB are equal to the squares of AZ, ZK : take from these equals the square of AZ, the remaining square of BZ is equal to the remaining square 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 described from the centre Z, and distance ZB, shall pass through the points K, O, S, and KBOS shall be a quadrilateral figure in the circle: and because KB is greater than QV, and, QV equal to so, therefore KB is greater than SO; but KB is equal to each of the straight 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 same three together with that cut off by OS; that is, than the whole circumference of the circle; therefore the circumference subtended by KB is greater than the fourth part of the whole circumference of the circle KBOS, and consequently the angle BZK at the centre is greater than a right angle: and because the angle BZK is obtuse, the square of BK is greater (12. 2.) than the squares of BZ, ZK; that is, greater than twice the square 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 (4. 1.) to the angle OVB: and OVB is a right angle; therefore also KVB is a right angle: and because BD is less than twice DV, the rectangle contained by DB, BV is less than twice the rectangle DVB; that is (8. 6.), the square of KB is less than twice the square of KV: but the square of KB is greater than twice the square of BZ; therefore the square of KV is greater than the square of BZ: and because BA is equal to AK, and that the squares of 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 is AC greater than AG; because, in the preceding proposition, it was shown 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 centre of the sphere, to that plane. 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 IO; and, as was demonstrated of the plane KBOS, and the point Z, in the same way it may be shown that the point I is the centre of a circle described about SOPT; and that OS is greater than PT; and PT was shown to be parallel to OS; therefore, because the two trapeziums KBOS, SOPT inscribed in circles have their sides BK, OS, parallel, as also OS, PT; and their other sides BO, KS, OP, ST all equal to one another, and that BK is greater than Os, and OS greater than PT, therefore the straight line ZB is greater (2. lem. 12.) than 10. Join AO which will be equal to AB: and because AIO, AZB are right angles, the squares of AI, IO are equal to the square of AO or of AB; that is, to the squares of AZ, ZB: and the square of ZB is greater than the square of 10, therefore, the square of AZ is less than the square 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 lesser sphere: in the same manner it may be demonstrated that the plane TPRY falls without the same sphere, as also the triangle YRX, viz. by the cor. of 2d lemma. And after the same way it may be demonstrated that all the planes which contain the solid polyhedron, fall without the lesser sphere. Therefore in the greater of the two spheres which have the same centre, a solid polyhedron is described, the superficies of which does not meet the lesser sphere. Which was to be done. But 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 of GU than the square of BZ, and AU is equal to AB; therefore the square of AU, that is, the squares of AG, GU 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. 8. 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 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. PROP. XVII. THEOR. SPHERES have to one another the triplicate ratio of that which their diameters have. EF. Let ABC, DEF be two spheres, of which the diameters are BC, 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 triplibate 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 solid polyheA D L B H N dron, 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 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 (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 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 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, 30 |