The curved surface of a cone is called also the convex surface: the point through which the straight line always passes is called the vertex or summit; the circle is called the base; and the line which is drawn from the vertex to the centre of the base, the axis of the cone. 5. A cone is said to be right or oblique, according as the axis is perpendicular to the base, or inclined to it. C is a right, and C' an oblique cone. The slant side of a right cone is a straight line which is drawn from the vertex to any point in the circumference of the base. 6. Similar Cones respective bases, in a cone, when its vertex is the same with that of the cone, and its base is inscribed in the base of the cone and in like manner a pyramid is said to be circumscribed about a cone, when its vertex is the same with that of the cone, and its base circumscribed about the base of the cone. The right cylinder, right cone, and sphere, are sometimes styled, by way of pre-eminence, the three round bodies. They are also termed solids of revolution, because each of them may be conceived to be generated by the revolution of a plane figure about a fixed straight line taken in its plane. Thus we have seen (IV. def. 21.) that a sphere is generated by the revolution of a semicircle about its diameter. If a right angle triangle VCA revolve about one of the sides containing the right angle, as V C, that side will be the axis of a right cone, of which the other side AC will describe the base and in the same ratio to the radii of (IV. 3. Cor. 2.) and those bases. 7. If a cone be divided into two parts by a plane parallel to its base, the part next the base is called a frustum of a cone, or sometimes a truncated cone. The axis of a frustum is that part of the axis of the whole cone, which is intercepted between the cutting plane and the base of the cone. 8. A prism is said to be inscribed in a cylinder, when its bases are inscribed in the bases of the cylinder; and in like manner a prism is said to be circumscribed about a cylinder, when its bases are circumscribed about the bases of the cylinder. 9. A pyramid is said to be inscribed α V the hypotenuse VA the convex surface. And if a rectangle Ac revolve about one of its sides C c, that side will be the axis of a right cylinder, of which the two adjoining sides will describe the two bases (IV. 3. Cor. 2.), and the side opposite, Aa, the convex surface. Before we can proceed to consider the convex surfaces of the cone and cylinder, it will be necessary to establish the following lemmas concerning convex surfaces in general. dimension B D of the plane is less than the dimension B PD of the other surface (I. 10. Scholium). But if the plane surface were equal to or greater than the other, the dimension of the plane surface could not be, in every direction, less than the dimension of the other. Therefore, the plane surface O ABCD is neither equal to nor greater than the surface PABCD; that is, it is less than the surface PAB CD. Therefore, &c. Lemma 2. If a convex surface A B is enveloped on all sides by another surface MN; whether they have any points, lines, or planes in common, or have no point at all in common, the enveloped surface will always be less than the surface which envelops it. For it is the nature of a convex surface that there is no point of it through which a plane cannot be drawn touching, or at least not cutting, the surface. There C B M D fore, if such a surface A B be enveloped by any other MN, and if A be any point in the former which is not also in the latter surface, a plane CD may be drawn through A, cutting the surface M N, and not cutting the surface AB. And because the plane CD is less than the surface C MD by the preceding lemma, the whole surface CND is less than the whole MN (I. ax. 6.) Therefore, of all the surfaces which envelop the space A B, and are in any part exterior to the surface A B, there is none than which a less cannot be found enveloping the same space AB. But of the surfaces enveloping this space, there must be some (one or more) less than any of the others; for none of them can be less than of some certain magnitude. Therefore, since one or more of the surfaces must be less than any of the others, and since a less may be found than any which is in any part exterior to the surface AB, such least surface is none other than the surface A B. Therefore, the surface AB is less than the surface MN. Therefore, &c. PROP. 1. A right cylinder is greater than any inscribed prism, and less than any circumscribed prism: also the convex surface of the cylinder is greater than the convex surface of any inscribed prism, and less than the convex surface of any circumscribed prism. The first part of the proposition is manifest: we have only, therefore, to demonstrate that which relates to the surfaces. Let A a be the axis of a right cylinder, and BCD, bed its two bases; and, first, let E F, ef be the bases of any prism inscribed in the cylinder: the convex surface of the cylinder shall be greater than the convex surface of the inscribed prism. A B F be less than the latter surface sively with the parts of the larger cylinder, it may be shown that the convex surface of the latter is the same multiple of the convex surface of the former that A a' is of A a, or Q of P; and it is evident that the convex surface of the prism inscribed in it is the same multiple of the convex surface of the prism inscribed in the former cylinder: therelesser cylinder together with P is equal to fore, because the convex surface of the the convex surface of its inscribed prism, the convex surface of the larger cylinder together with Q is likewise equal to the convex surface of its inscribed prism. cles B CD, bcd taken together and the But the difference between the two cirtwo polygons E F, ef taken together (or, which is the same thing, between the and the two polygons EF, ef' taken two circles B CD, b'c'd taken together together) is less than Q. Therefore, the convex surface of the larger cylinder together with this difference is less than which is impossible (Lem. 2.), because the convex surface of its inscribed prism; the convex surface of the cylinder, together with this difference and with the two polygons EF, e'f', envelops the convex surface of the prism, together with the same two polygons EF, ef'. Therefore, the convex surface of the cylinder which has the axis A a is not less than the convex surface of the inscribed prism. Neither can the convex surface of the cylinder be equal to the convex surface of the inscribed prism. For, if this be supposed, then, because a polygon may be inscribed in the circle BC D, which shall have a greater perimeter than the polygon E F, a prism may be inscribed in the cylinder which shall have (IV. 29. Scholium) a greater convex surface than the prism upon the base EF, that is, than the cylinder has. But this is impossible, as has been already demonstrated. Therefore, the convex surface of the cylinder must be greater than the convex surface of the inscribed prism. And by a similar course of reasoning applied to the adjoined figure, it may be demonstrated that the convex surface of a cylinder is less than the convex surface of any circumscribed prism. Therefore, &c. PROP. 2. เ B E 'a A D Any right cylinder being given, two prisms may be the one inscribed in it, and the other circumscribed about it, such that the difference of their convex surfaces, or of their solid contents, shall be less than any given difference. Let ABG, abg be the bases of a given right cylinder; C, c their centres. And, first, let P be the given difference of surfaces, and let Q be the convex surface of some circumscribed prism. Then, because (as in III. 31.) a regular polygon may be inscribed in the circle ABG, the apothem CE of which approaches to the radius CD within any given difference, it is evident that a polygon may be inscribed such that k CD-CE shall be to CE in a ratio less than any assigned, and, therefore, in a ratio less than that of P to Q. Let ABFGH be such a polygon, and CE its apothem. Join CA, CB, &c., and let the planes ACc, BCc, &c. cut the upper base in the radii ca, cb, &c., and join A a, Bb, &c., ab, bf, &c.; then (def. 8.) it is evident that the polygons ABF GH, abfgh are the bases, and the straight lines A a, B b, &c., the principal edges of a prism inscribed in the given cylinder. Again, let the polygon KLMNO, similar to ABFGH, be circumscribed about the circle A B D, so that one of its sides, K L, may touch the circle in D (III. 27. Cor. 2.): let the plane D Cc cut the circle abd in the radius cd, and let a similar polygon klmno be circumscribed about this circle, so that the side kl may touch it in d, and join K k, Ll, &c.; then (def. 8.) it is evident that KLMNO, klmno are the bases, and the straight lines Kk, Ll, &c. the principal edges of a prism circumscribed about the given cylinder. Now, the convex surfaces of these prisms, inscribed and circumscribed, are the sums of their rectangular faces. And, since these rectangles have all of them the same altitude, the sums of the rectangular faces are as the sums of the bases of the rectangles (II. 35.), which sums are the perimeters of the inscribed and circumscribed polygons, and are, therefore, as the apothems CE, CD (III. 30.). Therefore, the convex surfaces of the prisms are to one another as CE, CD (II. 12.); and their difference is to the convex surface of the inscribed prism as CD CE to CE (II. 20.), that is in a less ratio than that of P to Q. But even had the ratio of their difference to the convex surface of the inscribed prism been the same with that of P to Q, the difference would have been less than P (II. 18. Cor.), because the surface of the inscribed prism is less than Q, which is the surface of some circumscribed prism (IV. 29. Scholium): much more, therefore, being less, is the difference less than P. In the next place, let S be the given difference of solid contents, and let T be the solid content of some circumscribed prism. Then, as before, a polygon may be inscribed in the lower base such that its apothem CE shall approach to the radius CD within any given difference; and, therefore, such also that CD2-CE2 may be to CE2 in a ratio less than any assigned; less, therefore, than that of S to T. Let ABFGH be such a polygon, and let the prisms be inscribed and circumscribed as before. Then, because these prisms have the same altitude, their solid contents are to one another as their bases (IV. 29. Cor. 2.), which bases are the inscribed and circumscribed polygons, and, therefore, are to one another as CE2, CD2. Therefore, the contents are to one another as CE, CD2 (II.12.); and their difference is to the content of the inscribed prism as CD - CE to CE2 (II. 20.), that is, in a less ratio than that of S to T. Therefore, as before, because the content of the inscribed prism is less than T, much more is the difference of contents less than S. Therefore, &c. Cor. 1. Any right cylinder being given, a regular prism may be inscribed (or circumscribed) which shall differ from the cylinder in convex surface, or in solid content, by less than any given difference. For the difference between the cylinder and either of the prisms, whether in surface or in content, is less than the difference of the two prisms (1.). Cor. 2. Any two similar right cylinders being given, similar regular prisms may be inscribed (or circumscribed) which shall differ from the cylinders in convex surface or in solid content, by less than any the same given difference. PROP. 3. The convex surface of a right cylinder is equal to the product of its altitude by the circumference of its base. lium); therefore, the circumference of the base of the cylinder is less than the perimeter of this inscribed polygon, which is impossible. Therefore the product in question is neither greater nor less than the convex surface of the cylinder; that is, it is equal to it. a Therefore, &c. Cor. If R is the radius of the base of right cylinder, and A its altitude, the convex surface of the cylinder is 2 RA (III. 34. Scholium.). PROP. 4. The solid content of a right cylinder is equal to the product of its base and altitude. This proposition is demonstrated in exactly the same manner as the preceding. If the product in question exceed the content of the cylinder, it must likewise exceed the content of some circumscribed prism (2. Cor. 1.): but this is impossible, because the prism (IV. 29. Cor. 1.) is equal to the product of its altitude, which is the same with that of the cylinder, by its base, which is greater than the base of the cylinder. If, on the other hand, it be less than the content of the cylinder, it must likewise be less than the content of some inscribed prism (2. Cor. 1.); but this is impossible, because the prism (IV. 29. Cor. 1.) which is the same with that of the cyis equal to the product of its altitude, linder, by its base, which is less than the base of the cylinder. Therefore, the product in question cannot but be equal to the content of the cylinder. Therefore, &c. Cor. 1. If R is the radius of the base For, if this product be not equal to the convex surface of the cylinder, it must either be greater or less than that surface. If greater, as by a difference D, it must be greater also than the convex surface of some circumscribed prism; for a prism may be circumscribed, the convex surface of which approaches to Cor. 2. If a right cylinder and a that of the cylinder within the difference D (2.Cor.1.): but the convex surface of such prism have equal bases and altitudes, a prism is equal to the product of its alti-the cylinder shall be equal to the prism tude (which is the same with that of the cylinder) by the perimeter of some circumscribed polygon (IV. 29. Scholium); therefore, the circumference of the base of the cylinder is greater than the perimeter of this circumscribed polygon, which is impossible. And, in the same manner, if less, as by a difference D, it must be less also than the convex surface of some inscribed prism (2. Cor. 1.); but the convex surface of such a prism is equal to the product of its altitude (the same with that of the cylinder) by the perimeter of some inscribed polygon (IV. 29. Scho of a right cylinder, and A its altitude, the solid content of the cylinder is R2 A (III. 34. Scholium.). (IV. 29. Cor. 1.). Cor. 3. (Euc. xii. 11. and 14.) Right cylinders which have equal altitudes are to one another as their bases; and right cylinders which have equal bases are to one another as their altitudes: also any two right cylinders are to one another in the ratio which is compounded of the ratios of their bases and altitudes (IV. 29. Cor. 2.). PROP. 5. (Euc. xii. 12.) The surfaces of similar right cylinders are as the squares of the axes; and their solid contents are as the cubes of the axes. For, in the first place, there may be inscribed in the cylinders similar prisms, the convex surfaces of which approach more nearly to the convex surfaces of the cylinders than by any the same given difference (2. Cor. 2.): and it may easily be shown that the convex surfaces of these prisms are to one another always in the same ratio, viz., as the squares of their principal edges (II. 43. Cor. 1.), which are equal respectively to the axes of the cylinders: therefore, the convex surfaces of the cylinders are to one another in the same ratio (II.28.), viz., as the squares of their axes. And, in the same manner, because there may be inscribed in the cylinders, similar prisms, the solid contents of which approach more nearly to the solid contents of the cylinders than by any the same given difference (1.Cor. 2.); and because the solid contents of these prisms are to one another always in the same ratio, viz., as the cubes of their principal edges (IV. 35.), which are equal respectively to the axes of the cylinders, the solid contents of the cylinders are to one another in the same ratio (II. 28.), viz., as the cubes of their axes. Otherwise: Let A, a represent the axes of two similar right cylinders, and R, r the radii of their bases. Then (3. Cor.) 2 RA, 2ra will represent their convex surfaces respectively, and (4. Cor. 1.) RA, ra their solid contents. But, because the cylinders are similar, (def. 3.) R: Ara; therefore (p. 47, Rule ii.) RA: A2 :: ra: a2, alternando (II. 19.) RA:ra:: A2: a2, and (p. 47, Rule ii.) 2 RA: 2 ra:: A2: a, that is, the convex surfaces of the cylinders are to one another as the squares of their T axes. Again, because R: A::r: a, R2 : A2 : ; r2; a2 (II. 37. Cor. 4.), and (p.47, Rule ii.)] R2 A: A3 :: r2 a: a3; therefore, alternando (II. 19.) R2 A : r2 a :: A3 a3, and (p. 47, Rule ii.) R2 A: Tra:: A3 a3; that is, the solid contents of the cylinders are to one another as the cubes of their axes. Therefore, &c. PROP. 6. A right cone is greater than any inscribed pyramid, and less than any circumscribed pyramid; also the convex surface of the cone is greater than the convex surface of any inscribed pyramid, and less than the convex surface of any circumscribed pyramid. The former part of the proposition is manifest: the latter respecting the surfaces may be demonstrated as follows: Let VA be the axis and B C D the base of a right cone; and, first, let EF be the base of any pyramid inscribed in the cone : the convex surface of the cone shall be greater than the convex surface of the inscribed pyramid. B E D For, if the axis VA be produced to V', so that A V' may be equal to A'V,'and if a right cone be described which shall have the axis V'A and the base B CD, it may be shown by coincidence that the convex surface of the latter cone V' BCD is equal to the convex surface of the first cone V B CD; and, because the triangles which form the convex surface of the inscribed pyramid V' E F have their sides equal respectively to the sides of the corresponding triangles which form the convex surface of the inscribed pyramid V E F (I. 4.), the former triangles are equal to the latter, each to each, and the whole convex surface of the pyramid VEF is equal to the whole convex surface of the pyramid V E F. Therefore the convex surface of the cone VB CD is equal to half the convex surfaces of the two cones taken together; and the convex surface of the pyramid VEF is equal to half the convex surfaces of the two pyramids taken together. But the convex surfaces of the two cones taken together are greater than the convex surfaces of the two pyramids taken together (Lemma 2), because the former envelop the latter. Therefore (I. ax. 8.), the convex surface of the cone VBCD is greater than the convex surface of the inscribed pyramid VEF. And by a similar demonstration applied to the adjoined figure, it may be shown that E the convex surface of the cone is less than the convex surface of any circumscribed pyramid. Therefore, &c. PROP. 7. B V D F Any right cone being given, two pyramids may be the one inscribed in the cone, and the other circumscribed about it, such that the difference of their con |