2. 12. than the pyramid EFG-H. Neither is it greater; for, if poffible, Book XII. let it be equal to fome pyramid PFG-H, of which the bafe PFG is greater than EFG: and because PFG is greater than the circle ABC, rectangles can be made about the circle which together are lefs than PFG: and, if pyramids be erected upon these a 1. Cor. rectangles, having their common vertex at D, it may be proved, as before, that they are together lefs than the pyramid PFG-H; that is, than the cone ABC-D: and they are also greater, because they contain the cone; which is impoffible; therefore the cone ABC-D is not greater than the pyramid EFG-H: and it has been proved not to be less than it; therefore the cone ABC-D is equal to the pyramid EFG-H. Wherefore, &c. Q. E. D. COR. I. * Hence, a cone is the third part of a cylinder of the fame altitude, and upon the fame bafe with it; for the pyramid EFG-H is the third part of a prifm on the base EFG, and of the fame altitude with it : and this prifm is equal to a cylinder d 1. upon the base ABC and of the fame altitude with the cone ABC-D; therefore the cone is the third part of the cylinder. COR. 2. Hence, cones of the fame altitude, are to one another as their bases. * This is the 10th Propofition of Book XII. of Euclid. Cor. 5. 12. e 6. 12. L' PROP. VIII. THEOR. ET ABCD be a fquare, and join BD, and from the centre B, at the distance BA, defcribe the arch AC; and let any number of straight lines EF, GH, IK be drawn parallel to BC, meeting the arch AC in L, M, N, and BD, in O, P, Q; and complete the rectangles BL, EM, GN in the circle, and the rectangles BO, EP, GQ about the triangle ABD: If, then, the figure thus conftructed revolve about AB as an axis, the cylinders described by the rectangles in the fector ABC, together with those defcribed by the rectangles about the triangle ABD, are equal to the cylinder defcribed by the fquare ABCD. Join BL and because EO is parallel to AD, the angle EOB is equal ADB, that is, to ABD; therefore EB is equal to : a d C EF; C 6.4. d 47. I. EO and becaufe BEL is a right angle, the fquares of BE, EL b 5. 1. are equal to the fquare of BL: and BL is equal to BC or and BE to EO; therefore the squares of OE, EL are equal to the fquare of EF; and because the cylinders defcribed by the revolution of BF and BL about the axis BE have the fame altitude BE, they are to one another as the circles defcribed by e 1. Cor. the revolution of EF and EL about the point E: and thefe circles 6.12. f 3. 12. BOOK XII. circles are as the fquares of their diameters or radii; therefore the cylinder defcribed by BL is to the cylinder defcribed by BF, as the fquare of EL to that of EF: For the fame reason, the cylinder defcribed by BO is to the cylinder defcribed by BF, as the fquare of EO to that of EF; therefore the cylinders defcribed by BL, BO together are to the cylinder defcribed by BF, g 24. 1. as the fquares of EL, EO to the fquare of EF: but the squares of EL, EO together are equal to the square I. of EF; therefore the cylinders described by may I G P H F I h A. 5. BL, BO are equal to the cylinder defcribed by BF. In the fame manner, it be demonstrated, that the cylinders defcribed by EM, EP are equal to that described by EH, and the cylinders defcribed by GN, E GQ equal to that defcribed by GK R Wherefore all the cylinders defcribed by the rectangles BL, EM, GN in the circle, and by BO, EP, GQ, ID about the triangle ABD, are together equal to the whole cylinder described by the fquare ABCD. QE. D. B COR. Hence, because the hemifphere described by the sector ABC is greater than the cylinders in it, and the cone described by the triangle ABD less than the cylinders about it; it is manifeft, that the hemifphere, together with any series of cylinders about the cone, is greater than the cylinder defcribed by the fquare AC and that the cone, together with any feries of cylinders in the hemisphere, is less than the faid cylinder described by AC. PROP. IX. THEOR. VERY fphere is two thirds of its circumfcribing E cylinder. Let ACR be the femicircle by the revolution of which about AR the fphere is described; and from the centre B draw BC at right angles to AR, and complete the square ABCD, and join BD: Then, if the whole revolve about AB, the fector ABC fhall defcribe a hemifphere, the fquare AC a cylinder about the hemisphere, and the triangle ABD a cone: The hemifphere is two thirds of the cylinder described by AC; that is, because a 1. Cor. the cone defcribed by ABD is the third part of the cylinder defcribed by AC, the hemifphere and the cone together are equal to the cylinder. 7.12. b To. I. 2 If not, let them, firft, be less than it by fome folid X: and 31. 1. bifect AB in G, and draw GH parallel to AD: Then, if AH, b c HB HB revolve about AB, the cylinders described by them fhall be BookXII. equal; therefore the cylinder defcribed by BH is the half of n that defcribed by AC: In the fame manner, may the half of the d Cor. 6. cylinder described by AH be cut off from it; and fo on: There e I D 12. fore, if this be done continually, there fhall at length remain a cylinder lefs than the folid X*: let this be the cylinder defcribed e 1. 12. by AK; and let the other parts of AB equal to AI, be IG, GE, EB; and let the rectangles BO, EP, GQ be completed about the triangle ABD; and produce SO to T: and because ES is equal to ET, the cylinders defcribed by them are equal d: For the fame reafon, if the rectangles IP, AQ be completed, the cylinders described by them in the cone are equal to those defcribed about it by EP, GQ: confequently, the cylinder defcribed by AK is the excefs of the cylinders about the cone above those within it: But the cone is greater than the cylinders in it, and the folid X is greater than the cylinder defcribed by AK; therefore the cone and X are greater E than the cylinders about the cone: To these unequals, add the hemifphere, and the hemifphere and cone, with X, are greater than the hemifphere and the cylinders about the cone: But the hemisphere and cone, with X, are equal to the cylinder defcribed by AC; therefore the cylinder described by AC is greater than the hemifphere and the cylinders Ꮐ B R T P L C about the cone and it is alfo lefs than them ; which is im- fCor-8.12. poffible: Therefore the cylinder defcribed by AC is not greater than the hemifphere and the cone. Neither is it lefs; for, if poffible, let it be less than them by some solid Z*: Then, as before, a rectangle may be cut off from AC, which shall defcribe a cylinder lefs than the folid Z: let this be the rectangle EC: and, as before, conftruct in the sector ABC the rectangles BL, EM, GN: and it may be proved, as in the former cafe, that the cylinders described by these rectangles in the hemisphere, together with Z, are greater than the cylinders about the hemifphere, and therefore much more are they greater than the hemifphere: To thefe unequals, add the cone; and the cylinders in the hemisphere, together with the cone and Z, are greater than the hemifphere and cone; that is, greater than the cylinder defcribed by AC, together with Z: Take away the folid Z, and the cylinders in the hemifphere, together with the cone, are greater than the cylinder defcribed by AC: and they are also lefs; which is impoffible: Therefore the cylinFf der The folids X and Z are not reprefented in the figure of this propofition. 225 THE ELEMENTS, &ė. Book XII. der defcribed by AC is not lefs than the hemifphere and the cone and it has been shown, that it is not greater than them; therefore the cylinder is equal to the hemifphere and the cone : a 1. Cor. 7. But the cone defcribed by ABD is the third part 2 of the cylin der; therefore the hemifphere defcribed by the sector ABC is two thirds of the cylinder defcribed by AC: Wherefore the whole fphere is two thirds of the cylinder defcribed about it by the rectangle AV. Therefore, &c. Q. E. D. 12. PHERES have to one another the triplicate ratio of that which their diameters have. SP Let A be the diameter of the greater fphere, and B the dia a 11. 6. meter of the leffer; and as A is to B, fo make a B to C, and C b11.Def.5 to D; then the ratio of A to D is triplicate of that of A to B: as A is to D, fo is the greater fphere to the leffer. 20. 6. : d A B C Let cylinders be defcribed about the spheres; then, the altitudes of the cylinders, and likewife the diameters of their bases, are equal to A and B and because A is to B, as B to C, the c 1. Cor. fquare of A is to that of B, as A to C: and circles are as the fquares of their d 3, 12. diameters; therefore, as A is to C, fo is the base of the greater cylinder to that of the leffer; and the altitude of the greater is to that of the leffer as C to D: Wherefore the greater cy2. Cor. linder has to the leffer the ratio which is compounded of the ratios of A to C, and of C to D; that is, the ratio f of A to D: and each of the spheres is two thirds of its circumfcribing cylinder h; therefore, as A is to D, to 8 is the greater fphere to the leffer fphere. Wherefore, &c. Q. E. D. 6.-12. f Def. A.5. h 9. 12. g C. 5. e *This is Prop. XVIII. Book XII. of Euclid. THE N angle at the centre of a circle is to four right Pl. Trig. A angles, as the arch on which it ftands to the whole circumference. Let ABC be an angle at the centre B of the circle ACD, ftanding on the arch AC. Draw BE at right angles to AB: and produce AB, EB to D, F: and because the right angles at B are equal, the arches AE, ED, DF, FA are all equal; therefore the circumference is four times the arch AE: and because the angles ABC, ABE D F C a 27.3. A B b 33.6. the arch AC to four times the arch AE; that is, to the whole c Cor. 4.5, circumference. Ff2 LEMMA |