describe a series of cylinders circumscribed about the cone VAB, and another series of cylinders inscribed in it. As in the case of the pyramid, and the inscribed and circumscribed prisms (Prop. 65), it is evident that the sum of the series of circumscribed cylinders will exceed the sum of the series of inscribed cylinders by the cylinder described by the rectangle Chy. Since the cone VAB is intermediate to the two series of cylinders, it will differ from the sum of either series by a solid less than the cylinder described by the rectangle Ch. Let the solid S be equal to a cylinder of which the base is the base of the cone, and the altitude is a. Then, if the axis VC be divided into such a number of equal parts that Ce, is less than a, it is evident that the sum of the series of circumscribed cylinders will exceed the sum of the series of inscribed cylinders by a solid (the cylinder described by the rectangle Ch.), less than the solid S: and consequently that the sum of either series will differ from the cone VAB, which is intermediate to the two, by a solid less than the given solid S, however small this may be. Wherefore a series of cylinders, &c. : which was to be proved. Cor. 1. Since the proposition applies to any cone cut from the cone VAB, by a plane parallel to the base, it applies also to the remaining portion of the cone, towards the base, called a frustum of the cone; that is, a series of cylinders all having the same altitude may be described about any frustum of a cone, and another series of cylinders all of the like altitude may be inscribed in the frustum, such that the sum of either series shall differ from the frustum by a solid less than any given solid, however small. Cor. 2. In like manner, as was shown in Prop. 67, that pyramids which have equal bases and altitudes are equal to one another, it may be demonstrated from this proposition that cones having equal bases and altitudes are equal to one another. Cor. 3. Any cone is a third part of a cylinder of the same base and altitude: for it is equal to a right cone of the same base and altitude. Cor. 4. Cones are to one another in the compound ratio of their bases and altitudes; for the cylinders of which they are equal parts are in that ratio (Prop. 70. Cor. 3). Cor. 5. For the like reason (Prop. 70. Cor. 4) similar cones are to one another as the cubes of the diameters of their bases, or as the cubes of their altitudes. PROP. LXXIV. THEOR. If a sphere be cut by a plane, the section will be a circle. Let ABC (fig. 81) be a sphere, and let it be cut by the plane BDC: the section BDC is a circle. From E, the centre of the sphere, draw EF perpendicular to the plane BDC (Prop. 5); through the point F draw the straight line BFC; in the line of section BDC of the plane with the sphere take any point D; and join DF, DE, BE. Because EF is perpendicular to the plane BDC, each of the angles EFD, EFB is a right angle; and because DE is equal to BE (Defs. 16. 18), the square of DE is equal to the square of BE; but the square of DE is equal to the squares of DF and FE (I. 47), and the square of BE is equal to the squares of BF and FE; therefore the squares of DF and FE are equal to the squares of BF and FE. From each of these equals take away the common square of FE, and there remains the square of DF equal to the square of BF; consequently DF is equal to BF. In like manner it may be shown, that any straight line drawn from a point in the line of section BDC, to the point F is equal to BF or DF': therefore the section BDC is a circle (I. Def. 15); and the point F is its centre. Wherefore if a sphere be cut by a plane, &c. : which was to be proved. PROP. LXXV. THEOR. A series of cylinders all of the same altitude may be described about a hemisphere, and another series of cylinders all of the like altitude may be inscribed in the hemisphere, such that the sum of either series shall differ from the hemisphere by a solid less than any given solid, however small. Let ADB (fig. 82) be a semicircle of which the diameter is AB, the centre C, and CD a radius at right angles to AB: if either of the sectors ACD, BCD revolve about DC, it will describe a hemisphere, having C for its centre. Also, if DC be divided into any number of equal parts in the points C1, C2, C3, C4; through these points straight lines be drawn parallel to AB, and to the semicircle, circumscribed and inscribed rectangles be drawn, as in the figure; if the figure revolve about CD, either half of these rectangles will describe series of cylinders described about and in the hemisphere described by either of the sectors ACD, BCD: then CD may be so divided that the sum of either series of cylinders shall differ from the hemisphere by a solid less than any given solid S, however small. It is evident that here, as in the case of the cone (Prop. 73), the sum of the series of circumscribed cylinders exceeds the sum of the series of inscribed cylinders by the cylinder described by the rectangle Ce, or Cf. If then the given solid S be equal to a cylinder of which the base is the circle described by CA or CB, and the altitude is a; and CD be divided into such a number of equal parts that Cc, is less than a; cylinders will exceed the sum of the series of inscribed cylinders by a solid (the cylinder described by the rectangle Ce) less than the solid S; and consequently that the sum of either series will differ from the hemisphere described by the sector ACD, which is intermediate to the two, by a solid less than the given solid S, however small this may be. Wherefore a series of cylinders, &c. : which was to be proved. Cor. The proposition is true for any segment of the hemisphere; for the demonstration will be the same, whether it be CD which is divided into equal parts, and rectangles are described about and in the semicircle, or that it is a portion of CD which is so divided, and rectangles are described about and in the segment cut off. PROP. LXXVI. THEOR. If a hemisphere and a cone have equal bases and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes, such that their sum shall differ from the sum of the hemisphere and the cone by a solid less than any given solid, however small. Let ADB (fig. 83) be a semicircle of which the diameter is AB, the centre C, and CD a radius at right angles to AB; let CE and CF be squares described on CD; and join CE, CF: then if the figure revolve about CD, either of the sectors ACD, BCD will describe a hemisphere having C for its centre, and either of the triangles CDE, CDF will describe a cone having its vertex at C, and having for its base the circle described by DE or DF, equal to that described by CA or CB, which is the base of the hemisphere. Let S be any given solid. A series of cylinders may be inscribed in the hemisphere, and another series described about the cone, such their sum shall differ from the sum of the hemisphere and the cone, by a solid less than s, however small S may be. Let the given solid Š be equal to a cylinder of which the base is equal to the base of the hemisphere or of the cone, and the altitude is a. Divide CD into such a number of equal parts in C1, C2, C, &c. that Cc, is less than a; through C1, C, Cg, &c. draw straight lines parallel to AB or EF; and complete the rectangles inscribed in the semicircle, as in the last proposition, and the rectangles described about the triangle ECF as in Proposition 73: then if the figure revolve about CD, the rectangle Caj, Cyan, Czaz, &c. will describe cylinders in the hemisphere ADB, and the rectangles cıkı, Cake, Cgka, &c. will describe cylinders about the cone ECF. Since, by the last proposition, the hemisphere ADB exceeds the sum of the series of cylinders inscribed in it, by a solid less than S, let this excess be a solid T which is less than S; so that the sum of the series of cylinders inscribed in the hemisphere together with the solid T is equal to the hemisphere. And since, by Proposition 73, the sum of the series of cylinders described about the cone exceeds the cone by a solid likewise less than S, let this excess be a solid V which is less than S; so that the sum of the series of cylinders described about the cone is equal to the cone together with the solid V. Hence, adding equals to equals, the sum of the two series of cylinders, the one inscribed in the hemisphere, and the other described about the cone, together with the solid T, is equal to the sum of the hemisphere and the cone together with the solid V. From each of these equals take away the solid T, then the sum of the two series of cylinders will differ from the hemisphere and cone by a solid which is the difference of the solids T and V; and since T and V are each less than S, their difference must be less than S: consequently the sum of the two series of cylinders, the one inscribed in the hemisphere, and the other described about the cone, differ from the sum of the hemisphere and the cone by a solid less than the solid S, however small S may be. Which was to be proved. Cor. The proposition is true for any segment of the sphere, which would be described by the revolution of the figure DIG about DC, and any portion of the cone, which would be described by the revolution of the figure DFHG, likewise about DC. For the demonstration in the proposition applies equally to the segment of the sphere and the portion of the cone, making use of the corollaries to Propositions 75 and 73, instead of the propositions themselves. PROP. LXXVII. THEOR. The same things being supposed as in the last proposition, the sum of the two series of cylinders, the one inscribed in the hemisphere, and the other described about the cone, is equal to a cylinder having the same base and altitude as the hemisphere, or the cylinder circumscribing the hemisphere. Let the figure be constructed as in the last proposition (fig. 83), and let it revolve about CD; the sum of the two series of cylinders, the one described by the rectangles Caj, Cy@a, caz, &c., and which are inscribed in the hemisphere ADB, and the other described by the rectangles ck, coką, czky, &c., and which are described about the cone ECF, is equal to the cylinder described by the square DA or the square DB, that is, to the cylinder circumscribing the hemisphere. Let a straight line passing through one of the points of division in CD, cut the semicircle in I, CF in H, and FB in K; join CI. Then because CGI is a right angle, the circles described with the distances CG and GI are equal to the circle described with the distance CI or GK (Supp. Prop. Io. Cor. 2); now CG is equal to GH, because GH is parallel to DF, and CD is equal to DF; therefore the circles described with the distances GH and GI are equal to the circle described with the distance GK, that is, the circles described by the revolution of GH and GI about the axis CD are equal to the circle described by the revolution of GK about the same axis. And since cylinders having the same altitude are to one another as their bases (Prop. 70. Cor. 2), the cylinder described by the rectangle GM, is to the cylinder described by GO, as the circle described by GH is to the circle described by GK; and the cylinder described by GN is to the cylinder described by GO, as the circle described by GI is to the circle described by GK; therefore the cylinders described by GM and GN together, are to the cylinder described by GO, as the circles described by GH and GI together, are to the circle described by GK (V. 24): but the circles described by GH and GI are together equal to the circle described by GK; therefore the cylinders described by GM and GN are together equal to the cylinder described by GO (V.A). In like manner it may be shown, that each of the cylinders described about the cone, together with the corresponding cylinder inscribed in the hemisphere, is equal to a cylinder of the same altitude, and having its base equal to the base of the cone or of the hemisphere; therefore the sum of the two series of cylinders inscribed in the hemisphere and described about the cone is equal to the sum of all the cylinders having the same altitude, and having their bases equal to the base of the cone or of the hemisphere, that is, to the cylinder described by the square DB, or the cylinder circumscribing the hemisphere. Which was to be proved. Cor. The sum of the two series of cylinders, the one inscribed in a segment of the sphere, and the other in the corresponding frustum of the cone, is equal to the cylinder having the same base as the cone, and the same altitude as the frustum. This is shown in the demonstration of the proposition. PROP. LXXVIII. THEOR. Every sphere is two-thirds of the circumscribing cylinder. Let the figure be constructed as in the last two propositions (fig. 83); then the hemisphere ADB, described by the sector DCB, and the cone ECF, described by the triangle FDC, are together equal to the cylinder ABFE, described by the square DB: for if they be not equal, let them be unequal, and let the hemisphere ADB, and cone ECF together, differ from the cylinder ABFE by a solid S. Then however small S may be, a series of cylinders may be inscribed in the hemisphere, and described about the cone, such that their sum shall differ from the sum of the hemisphere ADB and the cone ECF by a solid less than 8 (Prop. 76); but the sum of the series of cylinders inscribed in the hemisphere ADB, and described about the cone ECF, is equal to the cylinder ABFE circumscribing the hemisphere (Prop. 77): therefore the sum of the hemisphere ADB and the cone ECF differs from the cylinder ABFE circumscribing the |