COR. And if there be an isosceles triangle, the sides of which are equal to AD, BC, but its base less than AB, the greater of the two sides AB, DC; the straight line KA may, in the same manner, be demonstrated to be greater than the straight line drawn from the centre to the circumference of the circle described about the triangle. PROPOSITION XVII. PROBLEM. In the greater of two spheres which have the same centre, to inscribe a solid polyhedron, the superficies of which shall not meet the lesser sphere. Let there be two spheres about the same centre A. It is required to inscribe in the greater a solid polyhedron, the superficies of which shall not meet the lesser sphere. Let the spheres be cut by a plane passing through the centre; the common sections of it with the spheres shall be circles; because the sphere is described by the revolution of a semicircle about the diameter remaining unmoveable; so that in whatever position the semicircle be conceived, the common section of the plane in which it is with the superficies of the sphere is the circumference of a circle; and this is a great circle of the sphere, because the diameter of the sphere, which is likewise the diameter of the circle, is greater than any straight line in the circle or sphere. (III. 15.) Let then the circle made by the section of the plane with the greater sphere be BCDE, and with the lesser sphere be FGH; and draw the two diameters BD, CE at right angles to one another; and in BCDE, the greater of the two circles, inscribe a polygon of an even number of equal sides not meeting the lesser circle FGH; (XII. 16.) and let its sides in BE, the fourth part of the circle, be BK, KL, LM, ME; join KA, and produce it to N; and from A draw AX at right angles to the plane of the circle BCDE, meeting the superficies of the sphere in the point X: (xI. 12.) and let planes pass through AX, and each of the straight lines BD, KN, which from what has been said, shall produce great circles on the superficies of the sphere, and let BXD, KXN be the semicircles thus made upon the diameters, BD, KN: therefore, because XA is at right angles to the plane of the circle BCDE, every plane which passes through XA is at right angles to the plane of the circle BCDE; (XI. 18.) wherefore the semicircles BXD, KXN are at right angles to that plane: and because the semicircles BED, BXD, KXŇ upon the equal diameters BD, KN, are equal to one another; their halves BE, BX, KX, are equal to one another, therefore as many sides of the polygon as are in BE, so many are there in BX, KX, equal to the sides BK, KL, LM, ME: let these polygons be described, and their sides be BO, OP, PR, RX; KS, ST, TY, YX; and join OS, PT, RY; and from the points O, S, draw OV, SQ perpendiculars to AB, AK: and because the plane BOXD is at right angles to the plane BCDE, and in one of them BOXD, OV is drawn perpendicular to AB the common section of the planes, therefore OV is perpendicular to the plane BCDE: (x1. def. 4.) for the same reason SQ is perpendicular to the same plane, because the plane KSX is at right angles to the plane BCDE. Join VQ: and because in the equal semicircles BXD, KXN, and OV, SQ are perpendicular to their diameters, wherefore VQ is parallel to BK: (vi. 2.) and because OV, SQ are each of them at right angles to the plane of the circle BCDE, OV is parallel to SQ: (xI. 6.) and it has been proved, that it is also equal to it; (1. 33.) and therefore BO, KS, which join them are in the same which these parallels are, plane in 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 shewn to be parallel to the same KB; wherefore TP is parallel to SO, (xI. 9.) 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: (xI. 2.) therefore, if from the points O, S, P, T, R, Y, there be drawn straight lines to the point A, there will 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 will be formed a solid polyhedron inscribed 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. Also the superficies of this solid polyhedron shall not meet the lesser sphere in which is the circle FGH. For, from the point A draw AZ perpendicular to the plane of the quadrilateral KBOS, meeting it in Z, and join BZ, ZK: (xr. 11.) 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; (1.47.) therefore the squares of AZ, ZB are equal to the squares of AZ, ZK: take from these equals the square of AZ, and 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, will pass through the points K, O, S, and KBOS will 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; the 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, square of BK is greater than the squares of BZ, ZK; (11.12.) that is, greater than twice the square of BZ. Join KV: and because in the triangle KBV, OBV, KB, BV are equal to OB, BV, and that they contain equal angles ; the angle KBV is equal to the angle OVB: (1. 4.) 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 BD, BV is less than twice the rectangle ᎠᏤ, ᏤᏴ ; that is, the square of KB is less than twice the square of KV: (v1.8.) but the square of KB is greater than twice the square of BZ; VA; and that the squares of BZ, ZA are equal together to the square of BA, and the straight line AZ greater than VA: it was shewn that KV falls without the circle FGH: and is therefore the shortest of all the straight lines that can be 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 Z, in the same way it may be shewn that the point I is the centre 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 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 than IO. (XII. Lem. 2.) Join AO, which will be equal to AB; and because AIO, AZB are right angles, squares of AI, IO are equal to the square of AO or of AB; that is, to the squares of AZ, ZB; the and the square of ZB is greater than the square of 10, 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 XII. Lem. 2. Cor. 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 two spheres, which have the same centre, a solid polyhedron is inscribed, the superficies of which does not meet the lesser sphere. Q.E.F. But the straight line AZ may be demonstrated to be greater than AG otherwise, and in a shorter manner, without the help of Prop. XVI., 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; of AG, 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 and the straight line AZ consequently greater than the straight line AG. COR.-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 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 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 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 to one another, each to each: (xI. def. 11.) but similar pyramids have to one another the triplicate ratio of their homologous sides: (XII. 8. Cor.) 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 less, the triplicate ratio of that which AB has to the semi-diameter of the less 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. |