Book XII PRO P. XV. THE OR. THE bases and altitudes of equal cones and cylinders are reciprocally proportional, and cones and cylinders whofe base and altitudes are reciprocally proportional, are equal to one another. Let the circles ABCD, EFGH, be the bafes of the equal cones and cylinders, AC, EG, their diameters, and KL, MN, their axes; compleat the cylinders AX, EO; then, as ABCD is to EFGH, fo is the altitude MN to KL. For the altitudes KL, MN, are either equal or not. If equal, the cylinders AX, EO, are likewife equal; then the bases ABCÓ, EFGH, are equal a; therefore the bafes ABCD, EFGH, a 11: are to one another as their altitudes: But, if the altitudes KL, MN, are not equal, let one of them, as MN, be the greater, and cut off PM equal to LK, and let the plain TYS, parallel to the opposite plains; cut the cylinder EO in the point P, and compleat the cylinder ES; then the cylinder AX is to the cylinder ES as the cylinder EO is to the cylinder ES ; but the cylinder AX is to the cylinder ES, as the bafe ABCD to the bafe EFGHa; and, as the cylinder EO is to the cylinder ES, fo is the altitude MN to the altitude MP; therefore the base ABCD c 13. is to the base EFGH as the altitude MN is to the altitude KL; therefore the bafes and altitudes of the cylinders AX, EO, are reciprocally proportional. b 7. 5. And, if the bafes and altitudes of the cylinders AX, EO, are reciprocally proportional, then the cylinders are equal; for, the fame conftruction remaining, the bafe ABCD is to the bafe EFGH, as the altitude MN is to the altitude KL; and the altitudes KL, MP, are equal; therefore the base ABCD is to the base EFGH as the cylinder AX is to the cylinder ES 2, and the altitude MN is to the altitude MP as the cylinder EO is to ES; therefore the cylinder AX is to ES as EO is to ES; therefore the cylinder AX is equal to EO, and, because cones are e 9. 5. one third of the cylinder of the fame bafe f and altitude, and fro. parts have the fame proportions as their like multiples ; there-g 15. 5. fore the bafe and altitudes of equal cones and cylinders are reciprocally proportional. And, &c. PROP. Book XII a 16. 3. b lem. € 29. 3. TWO PRO P. XVI. PROB. IVO circles about the fame center, to infcribe in the greater a polygon of equal fides, even in number, that shall not touch the leffer circle. Let ABCD, EFGH, be two given circles, about the fame center K; it is required to infcribe a polygon in the circle ABCD, of equal fides, even in number, that fhall not touch the leffer circle EFGH. Through the center K draw the right line BD, and through the point G draw AG at right angles to BD: produce AG to C; then AC is a tangent to the circle EFGH, in the point G"; bifect the circumference BAD, and, again, the half thereof, and doing this, till a circumference is found lefs than AD, which let be LD; draw LM perpendicular to BD, and produce it to N; join LD, ND. Now, because LN is parallel to AC, the tangent of the circle EFGH, LN will not touch the circle EFGH, and, much lefs, the lines LD, ND; and, if right lines be applied in the circle, each equal to LD, there will be a polygon infcribed in the circle ABCD, of equal fides, and even in number, that will not touch the leffer circle EFGH; which was to be done. II. b 15. 3, PRO P. XVII. PRO B. To defcribe a folid polyhedron in the greater of two spheres, having the fame center, which shall not touch the fuperficies of the leffer Sphere. Let two spheres be supposed about the center A, it is required to describe a folid polyhedron in the greater sphere, not touching the fuperficies of the leffer fphere. Let the sphere be cut by fome plain paffing through the cena def. 14. ter, then the fections will be circles, and the circle defcribed by the half fection will be a great circle ; which let be BEDC; and FGH that of the leffer; and BD, CE, two of their diameters, drawn at right angles to each other; let BD meet the leffer circle in the point G; and draw GL a tangent to the leffer circle in the point G; and join AL: In the greater circle BEDC infcrike a polygon that will not touch the leffer circle FGH ; let the fides of the polygon, in the quadrant BE, be the right C 16 lines II. lines BK, KL, LM, ME, fuch that each will fubtend a lefs arch Book XII than a line equal to the tangent GL, then the right lines BK, KL, LM, ME, will each be lefs than the tangent GL; and produce the lines joining the points K, A, o N; and from the point A raife. AX perpendicular to the plain of the circle BADC, meeting the fuperficies of the fphere in X d. Let phains be drawn through a 12. 11. AX, BD, and AX, KN, which will make circles in the fuperficies of the fphere, and let 8XD, KXN, be femicircies on the diameters BD, KN; then, becaufe XA is perpendicular to the plain of the circle BEDC, the femicircles BXD, KXN, are perpendicular to the fame plain ; but, the femicircles BED, e 18. 11. BXD, KXN, are equal, for they ftand upon equal diameters BD, KN, their quadrants BE, BX, KX, fhall likewife be equal; therefore, as many fides of the polygon as are in the quadrant BE, so many equal fides may be in the quadrants BX, KX; let thefe fides be BO, OP, PR, RX, KS, ST, TY, YX; and join SO, TP, YR; from the points O, S, draw the perpendiculars OV, SQ, which will fali on BD, KN, the common fection f of the plain; join ; then, face the equal cir- f 33. 11. cumferences BO, SK, are taken in the equa femicircles BXD, KXN; and, because OV, SQ, are drawn perpendiculars from them, they are equal; as alfo, BV, KQ; but BA, KA, are equal; therefore AV is to VB as AQ is to OK; therefore VQ is parallel to BK, and OV is parallel to SQ; but it is proved e- & 2. 6. qual; therefore ZV, SO, are equal and parallel i; therefore OS, BK, are parallel ; but, because BK is parallel to VQ, and AB k 9. 11. equal to AK, AB is to BK as AV is to VQ ; and altern. AB is to AV as BK is to VK; but AB is greater than AV; therefore BK is greater than VK; but VK is equal to OS; therefore BK is greater than OS; join BO, KS, then OBKS is a quadrilateral figure in one plain. For the fame reafon, each of the quadrilateral figures SOPT, TPRY, and triangle YRS, are each in one plain; therefore, if from the points O, S, P, T, K, Y, to the point A, right lines are fuppofed drawn, will conftitute a polyhedrous figure within the circumferences BX, KX, confifting of pyramids, whofe bafes are KBOS, SOPT, TPRY, YRX, and vertex the point A; and if, in the fame manner, pyramids be conftructed on the fides KL, LM, ME, and on the other three quadrants and oppofite hemifphere, there will be conftructed a polyhedrous figure described in the fphere, compofed of pyramids whose bases are equal and fimilar to the forefaid quadrilateral figures, and triangle YRX, and vertex the point A. h 6 ¡I. i 33. 1. I. 7. II. But the polyhedron does not touch the fuperficies of the fphere in which the circle FGH is. For, because the quadrilateral figure KBSO is in one plain, and from the point A be drawn a right line AZ perpendicular to the plain", it will be at right m 11. 11, angles T II. 0 47. I. Book XII angles to all the right lines drawn in that plain "; join BZ, ZK, then AZ will be perpendicular to BŹ, ZK. But the n def. 3. fquares of AK, AB, are equal, and the fquares of AZ, ZB, are equal to the fquare of AB °, and the fquares of AZ, ZK, are equal to the fquare of AK"; therefore the fquares of AZ, ZB, are equal to the fquares of AZ, ZK. Take the common square of AŻ from both, then the fquares of EZ, ZK, are equal; that is, BZ equal to ZK. In like manner, lines drawn from Z to the points O, S, may be proved equal to BZ, ZK; therefore a circle defcribed about the center Z, with either of these distances, will país through the points O, S, K, B; and, becaufe KBSO is a quadrilateral figure infcribed in a circle, and OB, FK, KS, are equal, and OS lefs than BK, the angle BZK will be obtufe; therefore BK is greater than BZ; but GL is greater than KB, and therefore much greater than BZ; and the squares of AG, GL, are equal to the fquare of AL, AB, or AK; therefore the fquares of BZ, ZA, are equal to the fquare of AL; and the fquares of AZ, ZB, are equal to the fquares of AG, GL; but the fquare of GL was proved greater than the square of BZ; therefore the fquare of AZ is greater than the fquare of AG; that is, the right line AZ greater than AG; but AZ is perpendicular to one of the bafes of the polyhedron; and AG reaches the fuperficies of the leffer fphere; therefore the polyhedron does not touch the fuperficies of the leffer fphere. Wherefore, &c. b 12. P 15. S. r 12. 5. COR. If a folid polyhedron is infcribed in another sphere, fimilar to that in BCDE, they fhall be to one another in the triplicate ratio of the fquares of their diameters; for, the folids being divided into pyramids, equal in number, and of the fame order, they will be fimilar; and therefore to one another in the triplicate ratio of their homologous fides P; that is, as AB drawn from the center of the sphere BCDE, to the femidiameter of the other sphere; but the femidiameters of spheres are as their diameters; and one of the antecedents is to one of the confequents as all the antecedents to all the confequents ; therefore, the polyhedron in the one fphere is to the fimilar polyhedron in the other fphere, in the triplicate ratio of their diameters. S PROP. XVIII. THEOR. PHERES are to one another in the triplicate ratio of their diameters. Let |