2. 12. than the pyramid EFG-H. Neither is it greater; for, if possible, Book XII. let it be equal to some pyramid PFG-H, of which the base PFG M is greater than EFG: and because PFG is greater than the circle ABC, rectangles can be made about the circle which together are lets 4 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 less 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. 1. * Hence, a cone is the third part of a cylinder of the fame altitude, and upon the same base with it; for the pyramid EFG-H is the third part of a prism on the base EFG, and of the fame altitude with it d: and this prism is equal to a cylinder d 1. Cor. upon the base ABC and of the same altitude with the cone ABC-D ; therefore the cone is the third part of the cylinder. Cor. 2. Hence, cones of the same altitude, are to one another as their bases. 5. IX. e 6. 12. * This is the 10th Proposition of Book XII. of Euclid. PROP. VIII. THEOR. L ET ABCD be afquare, and join BD, and from the centre B, at the distance BA, describe 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 constructed revolve about AB as an axis, the cylinders described by the rectangles in the sector ABC, together with those described by the rectangles about the triangle ABD, are equal to the cylinder described by the square 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 EO and because BEL is a right angle, the squares of BE, EL b 5. 1. C 6. 4. are equal d to the square of BL: and BL is equal to BC or EF; and BE to EO; therefore the squares of OE, EL are equal to d 47. 1. the square of EF; and because the cylinders described by the revolution of BF and BL about the axis BE have the same altitude BE, they are to one another as e the circles described by e 1. Cor. the revolution of EF and EL about the point E: and these circles a 29. 1. 6. 121 D Book XII. circles are as the squares' of their diameters or radii ; therefore the cylinder described by BL is to the cylinder described by BF, f 3. 12. as the square of EL to that of EF: For the same reason, the cylinder described by BO is to the cylinder described by BF, as the square of EO to that of EF; therefore the cylinders de scribed by BL, BO together are to the cylinder described by BF, 8 24. 1. as 8 the squares of EL, EO to the square of EF: but the squares of EL, EO together are equal to the square of EF; therefore the cylinders described by h A. 5. BL, BO are equal a to the cylinder described by BF. In the same manner, it may I K e be demonstrated, that the cylinders descri. G PAIN bed by EM, EP are equal to that described 1Η by EH, and the cylinders described by GN, E GQ equal to that described by GK: Wherefore all the cylinders described 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 square ABCD. E. D. Cor. Hence, because the hemisphere 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 mani. fest, that the hemisphere, together with any series of cylinders about the cone, is greater than the cylinder described by the square AC: and that the cone, together with any series of linders in the hemisphere, is less than the said cylinder described cy by AC. PROP. IX. THEOR. cylinder. Let ACR be the semicircle by the revolution of which about AR the sphere 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 sector ABC shall describe a hemisphere, the square AC a cylinder about the hemisphere, and the triangle ABD a cone: The hemisphere is two thirds of the cylinder described by AC; that is, because a I. Cor. the cone described by ABD is the third part 2 of the cylinder 7. 12. described by AC, the hemisphere and the cone together are equal to the cylinder. b 10. I. If not, let them, first, be less than it by some folid X: and C31. 1. bisect b AB in G, and draw o GH parallel to AD: Then, if AH, HB 12. G-TP IT AK; HB revolve about AB, the cylinders described by them shall be Book XII. equal d; therefore the cylinder described by BH is the half of m that described by AC: In the same manner, may the half of the d Cor. 6. cylinder described by AH be cut off from it, and so on : Therefore, if this be done continually, there shall at length remain a cylinder less than the solid X*: let this be the cylinder described + 1. 12. by AK; and let the other parts of AB equal to AI, be IG, GE, EB ; and let the rectangles BO, EP, GO be completed about the triangle ABD; and produce SO to T: and because ES is equal to ET, the cylinders described by them are egual d. For the same reason, if the rectangles IP, AQ be completed, the cylinders described by them in the cone are equal to those descri. bed about it by EP, GQ: confequently, the cylinder described by AK is the ex D greater NDIO therefore the cone and X are greater E than the cylinders about the cone: To ko B! these unequals, add the hemisphere, and S the hemisphere and cone, with X, are greater than the hemisphere and the cylinders about the cone : But the hemi. sphere and cone, with X, are equal to the cylinder described by AC; therefore the cylinder described by AC is greater R than the hemisphere and the cylinders about the cone: and it is also less than them f; which is im- f Cor-8.12. possible: Therefore the cylinder described by AC is not greater than the hemisphere and the cone. Neither is it lefs ; for, if possible, let it be less than them by some solid Z*: Then, as before, a rectangle may be cut off from AC, which shall describe a cylinder less than the folid Z: let this be the rectangle EC: and, as before, construct 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 hemisphere, and therefore much more are they greater than the hemisphere: To these unequals, add the cone; and the cylinders in the hemisphere, together with the cone and Z, are greater titan the hemisphere and cone ; that is, greater than the cylinder described by AC, together with Z: Take away the solid Z, and the cylinders in the hemisphere, together with the cone, are greater than the cylinder described by AC: and they are also less *; which is impossible: Therefore the cylinFf der * The solids X and Z are not represented in the figure of this proposition. BO-XII. der described by AC is not less than the hemisphere and the cone: and it has been shown, that it is not greater than them; therefore the cylinder is equal to the hemisphere and the cone : a 1. Cor. 7. But the cone described by ABD is the third part a of the cylin. der ; therefore the hemisphere described by the sector ABC is two thirds of the cylinder described by AC: Wherefore the whole sphere is two thirds of the cylinder described about it be the rectangle AV. Therefore, &c. Q. E. D. PROP. X. THEOR*. PHERES have to one another the triplicate ratio of that which their diameters have. Let A be the diameter of the greater sphere, and B the dia. a 11.6. meter of the leffer; and as A is to B, so make a B to C, and Ç bui.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 sphere to the lesser. Let cylinders be described about the spheres; then, the altitudes of the cylinders, and likewise 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. square of A is to that of B, as A to C: A and circles are as the squares d of their Bd 3. 12. diameters; therefore, as A is to C, so is Cthe base of the greater cylinder to that of D. the lesser; and the altitude of the greater is to that of the lesser as C to D: Wherefore the greater cy€ 2. Cor. linder has to the leffer the ratio e which is compounded of the ratios of A to C, and of C to D; that is, the ratio of A to D: Def. A.5. and cach of the spheres is two thirds of its circumscribing cyh 9. 12. linder h; therefore, as A is to D, to 8 is the greater sphere to the & C. S. lesser sphere. Wherefore, &c. Q. E. D. 20. 6. 6. 12. * This is Prop. XVIII. Book XII. of Euclid. THE THE ELEMENTS OF PLANE TRIGONOMETRY. A LEMMA I. angles, as the arch on which it stands to the whole circumference. Let ABC be an angle at the centre B of the circle ACD, standing 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 С are all equal a ; therefore the circumference is four times the arch AE: a 27. 39 and because the angles ABC, ABE D А B В are at the centre, the angle ABC is to the right angle ABE, as the arch b 33. 6. AC to the arch AE: and quadru. pling the consequents, the angle ABC is to four right angles, as c Cor.4.S, the arch AC to four times the arch AE; that is, to the whole circumference. LEMMA |