Fig. 252. Fig. 253. Fig. 254. 510. Two cylinders or two cones are similar, when their axes are to each other as the diameters of their bases. 511. If, in the circle ACD (fig. 252), considered as the base of a cylinder, a polygon ABCDE be inscribed, and upon the base ABCDE a right prism be erected equal in altitude to the cylinder, the prism is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism. It is manifest that the edges AF, BG, CH, &c., of the prism, being perpendicular to the plane of the base, are comprehended in the convex surface of the cylinder; therefore the prism and cylinder touch each other along these lines. 512. In like manner, if ABCD (fig. 253) be a polygon circumscribed about the base of a cylinder, and upon the base ABCD a right prism, equal in altitude to the cylinder, be constructed, the prism is said to be circumscribed about the cylinder, or the cylinder inscribed in the prism. Let M, N, &c., be the points of contact of the sides AB, BC, &c., and through the points M, N, &c., let the lines MX, NY, &c., be drawn perpendicular to the plane of the base; it is evident that these perpendiculars will be in the surface of the cylinder and in that of the circumscribed prism at the same time; therefore they will be lines of contact. N. B. The cylinder, the cone and the sphere are the three round bodies, which are treated of in the elements. Preliminary Lemmas upon Surfaces. 513. 1. A plane surface QABCD (fig. 254) is less than any other surface PABCD terminated by the same perimeter ABCD. Demonstration. This proposition is sufficiently evident to be ranked among the number of axioms; for we may consider the plane among surfaces what the straight line is among lines. The straight line is the shortest distance between two given points; in like manner the plane is the least surface among all those which have the same perimeter. Still, as it is proper to make the number of axioms as small as possible, I shall present a process of reasoning which will leave no doubt with regard to this proposition. As a surface is extension in length and breadth, we cannot conceive one surface to be greater than another, except the di mensions of the first exceed in some direction those of the second; and if it happens that the dimensions of one surface are in all directions less than the dimensions of another surface, it is evident that the first surface will be less than the second. Now, in whatever direction the plane BPD be made to pass, as it cuts the plane in BD, and the other surface in BPD, the straight line BD will always be less than BPD; therefore the plane surface OABCD is less than the surrounding surface PABCD. 514. 11. A convex surface OABCD (fig. 255) is less than any Fig. 255 other surface which encloses it by resting on the same perimeter ABCD. Demonstration. We repeat here, that we understand by a convex surface a surface that cannot be met by a straight line in more than two points; still it is possible that a straight line may apply itself exactly to a convex surface in a certain direction; we have examples of this in the surfaces of the cone and cylinder. It should be observed, moreover, that the denomination of convex surface is not confined to curved surfaces; it comprehends polyedral faces, or surfaces composed of several pianes, also surfaces that are in part curved and in part polyedral. This being premised, if the surface OABCD is not smaller than any of those which enclose it, let there be among these last PABCD the smallest surface which shall be at most equal to OABCD. Through any point O suppose a plane to pass touching the surface OABCD without cutting it; this plane will meet the surface PABCD, and the part which it separates from it wi'l be greater than the plane terminated by the same surface; there fore, by preserving the rest of the surface PABCD, we can sub stitute the plane for the part taken away, and we shall have a new surface, which encloses the surface OABCD, and which would be less than PABCD. But this last is the least of all, by hypothesis; consequently this hypothesis cannot be maintained; therefore the convex surface OABCD is less than any which encloses OABCD and which is terminated by the same perimeter ABCD. 515. Scholium. By a course of reasoning entirely similar it may be shown, 1. That, if a convex surface terminated by two perimeters, ABC, DEF (fig. 256), is enclosed by any other surface termi- Fig. 256. nated by the same perimeters, the enclosed surface will be less than the other. Fig. 257. 2. That, if a convex surface AB (fig. 257) is enclosed on all sides by another surface MN, whether they have points, lines, or planes in common, or whether they have no point in common, the enclosed surface is always less than the enclosing surface. For among these last there cannot be one which shall be the least of all, since in all cases we can draw the plane CD a tangent to the convex surface, which plane would be less than the surface CMD; and thus the surface CND would be smaller than MN, which is contrary to the hypothesis, that MN is the smallest of all. Therefore the convex surface AB is less than any which encloses it. Fig. 258. THEOREM. 516. The solidity of a cylinder is equal to the product of its base by its altitude. Demonstration. Let CA (fig. 258) be the radius of the base of the given cylinder, H its altitude; and let surf. CA represent the surface of a circle whose radius is CA; we say that the solidity of the cylinder will be surf. CA× H. For, if surf. CA × H is not the measure of the given cylinder, this product will be the measure of a cylinder either greater or less. In the first place let us suppose that it is the measure of a less cylinder, of a cylinder, for example, of which CD is the radius of the base and H the altitude. Circumscribe about the circle, of which CD is the radius, a regular polygon GHIP, the sides of which shall not meet the circumference of which CA is the radius (285); then suppose a right prism having for its base the polygon GHIP, and for its altitude H; this prism will be circumscribed about the cylinder of which the radius of the base is CD. This being premised, the solidity of the prism is equal to the product of its base GHIP multiplied by the altitude H; and the base GHIP is less than the circle whose radius is CA; therefore the solidity of the prism is less than surf. CAX H. But surf. CAx H is, by hypothesis, the solidity of the cylinder inscribed in the prism; consequently the prism would be less than the cylinder; but the cylinder, on the contrary, is less than the prism, because it is contained in it; therefore it is impossible that surf. CA × H should be the measure of a cylinder of which the radius of the base is CD and the altitude H; or, in more general terms, the product of the base of a The same construction being supposed as in the first case, the prism circumscribed about the given cylinder will have for its measure GHIP × H; the area GHIP is greater than surf. CD; consequently, the solidity of the prism in question is greater than surf. CD × H; the prism then would be greater than the cylinder of the same altitude whose base is surf. CA. But the prism, on the contrary, is less than the cylinder, since it is contained in it; therefore it is impossible that the product of the base of a cylinder by its altitude should be the measure of a greater cylinder. We conclude, then, that the solidity of a cylinder is equal to the product of its base by its altitude. 517. Corollary 1. Cylinders of the same altitude are to each other as their bases, and cylinders of the same base are to each other as their altitudes. 518. Corollary II. Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of the bases. For the bases are as the squares of their diameters; and, since the cylinders are similar, the diameters of the bases are as the altitudes (510); consequently the bases are as the squares of the altitudes; therefore the bases multiplied by the altitudes, or the cylinders themselves, are as the cubes of the altitudes. 519. Scholium. Let R be the radius of the base of a cylinder, H its altitude, the surface of the base will be ☛ R2 (291), and the solidity of the cylinder will be R2 x H, or a RH. T LEMMA. 520. The convex surface of a right prism is equal to the perimeter of its base multiplied by its altitude. Demonstration. This surface is equal to the sum of the rectangles AFGB, BGHC, CHID, &c. (fig. 252), which compose it. Fig. 252, Now the altitudes AF, BG, CH, &c., of these rectangles are each equal to the altitude of the prism. Therefore the sum of these rectangles, or the convex surface of the prism, is equat the perimeter of its base multiplied by its altitude. 521. Corollary. If two right prisms have the same altitude, the convex surfaces of these prisms will be to each other as the perimeters of the bases. LEMMA. 522. The convex surface of a cylinder is greater than the convex surface of any inscribed prism, and less than the convex surface of any circumscribed prism. Demonstration. The convex surface of the cylinder and that Fig. 252. of the inscribed prism ABCDEF (fig. 252) may be considered as having the same length, since every section made, in the one and the other parallel to AF is equal to AF; and if, in order to obtain the magnitude of these surfaces, we suppose them to be cut by planes parallel to the base, or perpendicular to the edge AF, the sections will be equal, the one to the circumference of the base, and the other to the perimeter of the polygon ABCDE less than this circumference; since, therefore, the lengths being equal, the breadth of the cylindric surface is greater than that of the prismatic surface, it follows that the first surface is greater than the second. By a course of reasoning entirely similar it may be shown that the convex surface of the cylinder is less than that of any cirFig. 253. cumscribed prism BCDKLH (fig. 253). Fig. 258. THEOREM. 523. The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude. Demonstration. Let CA (fig. 258) be the radius of the base of the given cylinder, H its altitude; and let circ. CA be the circumference of a circle whose radius is CA; we say that circ. CA × H will be the convex surface of the cylinder. For, if this proposition be denied, circ. CA × H must be the surface of a cylinder either greater or less; and, in the first place, let us suppose that it is the surface of a less cylinder, of a cylinder, for example, of which the radius of the base is CD and the altitude H. |
