surface PABCD, less than all the rest, and at most, equal to PABCD. Through any point O, extend a plane, touching the surface OABCD, without cutting it; this plane will meet the surface PABCD, and (Lemma 1. VIII.) the part which it cuts off from that surface will be greater than the plane which terminates in the same boundary: hence, retaining the rest of the surface PABCD, we might substitute the plane instead of the part cut off from it, and so have a new surface, still enveloping OABCD, and less than PABCD.

But by hypothesis, PABCD is the least of all; hence the hypothesis was false; hence the convex surface OABCD is less than any other surface enveloping it, and terminating in the same contour ABCD.

Scholium. By a mode of reasoning entirely similar, we could shew,

1. That, if a convex surface terminated by two contours ABC, DEF, is enveloped by any other surface, terminated by the same contours, the enveloped surface will be the smaller of the two.

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2. That, if a convex surface AB, 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 envelopes it.

For, among the enveloping surfaces, there cannot be any one less than all the rest because in every case a plane CD may be drawn so as to touch the enveloped convex surface, and (Lemma 1. VIII.) this plane will always be less than the surface CMD; whence the surface CND would be less than MN; which is contrary to the supposition of MN being the least of all. Hence the convex surface AB is less than all those

which envelope it.

PROPOSITION I. THEOREM.

The solidity of a cylinder is equal to the product of its base by

its altitude.

Let CA be a radius of the given cylinder's base; H the altitude; let surf. CA, represent the area of the circle whose radius is CA: we are to shew that the solidity of the cylin- H der is surf. CA× H. For, if surf. CAXH is not the measure of the given cylinder, it must be the measure of a greater cylinder, or of a smaller one. Suppose it first to be the measure of

a smaller one; of a cylinder, for example, which has CD for the radius of its base, H being the altitude.

About the circle whose radius is CD, circumscribe a regular polygon GHIP (10. IV.), the sides of which shall not meet the circumference whose radius is CA. Imagine a right prism having the regular polygon GHIP for its base, and H for its altitude; this prism will be circumscribed about the cylinder, whose base has CD for its radius. Now, (14. VII.) the solidity of the prism is equal to its base GHIP, multiplied by the altitude H; the base GHIP is less than the circle, whose radius is CA; hence the solidity of the prism is less than surf. CA × H. But by hypothesis, surf. CA × H is the solidity of the cylinder inscribed in the prism; hence the prism must be less than the cylinder: whereas in reality it is greater, because it contains the cylinder; hence it is impossible that surf. CA X H can be the measure of the cylinder whose base has CD for its radius, H being the altitude; or in more general terms, the product of the base, by the altitude of a cylinder, cannot measure à less cylinder.

We must now prove that the same product cannot measure a greater cylinder. To avoid the necessity of changing our figure, let CD be a radius of the given cylinder's base; and, if possible, let surf. CDx H, be the measure of a greater cylinder, for example, of the cylinder, whose base has CA for its radius, H being the altitude.

The same construction being performed as in the first case, the prism, circumscribed about the given cylinder, will have GHIPXH for its measure: the area GHIP is greater than surf. CD; hence the solidity of this prism is greater than surf. CDxH: hence the prism must be greater than the cylinder

having the same altitude, and surf. CA for its base. But on the contrary, the prism is less than the cylinder, being contained in it; hence the base of a cylinder, multiplied by its altitude, cannot be the measure of a greater cylinder.

Hence finally, the solidity of a cylinder is equal to the product of its base by its altitude.

Cor. 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.

Cor. 2. Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases. For the bases are as the squares of their diameters; and the cylinders being similar, the diameters of their bases (Def. 4) are to each other as the altitudes: hence the bases are as the squares of the altitudes; hence the bases, multiplied by the altitudes, or the cylinders themselves, are as the cubes of the altitudes.

Scholium. Let R be the radius of a cylinder's base; H the altitude: the surface of the base (12. IV.) will be R2; and the solidity of the cylinder will be Rx H, or R2 H.

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PROPOSITION II. LEMMA.

The convex surface of a right prism is equal to the perimeter of its base multiplied by its altitude.

For this surface is equal to the sum of the rectangles AFGB, BGHC, CHID, &c. (see fig. of Def. 5.) which compose it. Now the altitudes AF, BG, CH, &c. of those rectangles, are equal to the altitude of the prism; their bases AB, BC, CD, &c. taken together, make up the perimeter of the prism's base. Hence the sum of these rectangles, or the convex surface of the prism, is equal to the perimeter of its base, multiplied by its altitude.

Cor. If two right prisms have the same altitude, their convex surfaces will be to each other as the perimeters of their bases.

PROPOSITION III. LEMMA.

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.

For (see the fig. of Def. 5.), the convex surface of the cylinder and that of the prism may be considered as having the same length, since every section made in either parallel to AF is equal to AF; and if these surfaces be cut, in order to obtain the breadths of them, by planes parallel to the base, or perpendicular to the edge AF, the one section will be equal to the circumference of the base, the other to the contour of the polygon ABCDE, which is less than that circumference: hence, with an equal length, the cylindrical surface is broader than the prismatic surface; hence the former is greater than the latter.

By a similar demonstration, the convex surface of the cylinder might be shewn to be less than that of any circumscribed prism BCDKLKH. (See the fig. of Def. 6.)

PROPOSITION IV. THEOREM.

The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

Let CA be the radius of the given cylinder's base, H its altitude; the circumference whose radius is CA, being represented by circ. CA, we are to shew that circ. CAXH will be the convex surface of the cylinder. For, if this proposition is not true, then circ. CAx H must be the surface of a greater cylinder, or of a less one. Suppose it first to be the surface of a less cylinder; of the

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cylinder, for example, the radius of whose base is CD, and whose altitude is H.

About the circle whose radius is CD, circumscribe a regular polygon GHIP, the sides of which shall not meet the circle whose radius is CA; conceive a right prism having H for its

altitude, and the polygon GHIP for its base. The convex surface of this prism will be equal (2. VIII.) to the contour of the polygon GHIP multiplied by the altitude H: this contour is less than the circumference whose radius is CA; hence the convex surface of the prism is less than circ. CA× H. But, by hypothesis, circ. CAXH is the convex surface of the cylinder whose base has CD for its radius; which cylinder is inscribed in the prism: hence the convex surface of the prism must be less than that of the inscribed cylinder. On the other hand (3. VIII.) it is greater; hence our hypothesis was false: hence, in the first place, the circumference of a cylinder's base multiplied by its altitude cannot be the measure of a smaller cylinder.

We We are next to shew that this product cannot be the measure of a greater cylinder. For, retaining the present figure, let CD be the radius of the given cylinder's base; and, if possible, let circ. CDXH be the convex surface of a cylinder, which with the same altitude has for its base a greater circle, the circle, for instance, whose radius is CA. The same construction being performed as above, the convex surface of the prism will again be equal to the contour of the polygon GHIP multiplied by the altitude H. But this contour is greater than circ. CD; hence the surface of the prism must be greater than circ. CDx H, which, by hypothesis, is the surface of cylinder having the same altitude, and CA for the radius of its base. Hence the surface of the prism must be greater than that of the prism. But even though this prism were inscribed in the cylinder, its surface (3. VIII.) would be less than the cylinder's; still farther is it less when the prism does not reach so far as to touch the cylinder. Hence our last hypothesis also was false; hence, in the second place, the circumference of a cylinder's base multiplied by the altitude cannot measure the surface of a greater cylinder.

Hence, finally, the convex surface of a cylinder is equal to the circumference of its base, multiplied by the altitude.