PROPOSITION XXI. THEOREM.

The surface of a spherical polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the quadrantal triangle.

E

D

Let ABCDE be any spherical polygon. From the vertex B draw the arcs BD, BE to the opposite angles; the polygon will be divided into as many triangles as it has sides, minus two. But the surface of each triangle is measured by the sum of its angles minus two right angles, multiplied by the quadrantal triangle. Also, the sum of all the angles of the triangles, is equal to the sum of all the angles of the polygon; hence the surface of the polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the quadrantal triangle.

A

B

Cor. If the polygon has five sides, and the sum of its an gles is equal to seven right angles, its surface will be equal to the quadrantal triangle; if the sum is equal to eight right angles, its surface will be equal to two quadrantal triangles; if the sum is equal to nine right angles, the surface will be equal to three quadrantal triangles, etc.

BOOK X.

THE THREE ROUND BODIES.

Definitions.

1. A cylinder is a solid described by the revolution of a rectangle about one of its sides, which remains fixed. The bases of the cylinder are the circles described by the two revolving opposite sides of the rectangle.

2. The axis of a cylinder is the fixed straight line about which the rectangle revolves. The opposite side of the rectangle describes the convex surface.

3. A cone is a solid described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. The base of the cone is the circle described by that side containing the right angle, which revolves.

4. The axis of a cone is the fixed straight line about which the triangle revolves. The

hypothenuse of the triangle describes the convex surface. The side of the cone is the distance from the vertex to the circumference of the base.

5. A frustum of a cone is the part of a cone next the base, cut off by a plane parallel to the base.

6. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

PROPOSITION I. THEOREM.

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

Let ACE-G be a cylinder whose base is the circle ACE and altitude AG; then will its convex surface be equal to the product of AG by the circumference ACE.

H

E

B

C

In the circle ACE inscribe the regular polygon ABCDEF; and upon this polygon G Cet a right prism be constructed of the same altitude with the cylinder. The edges AG, BH, CK, &c., of the prism, being perpendicular to the plane of the base, will be contained in the convex surface of the cylinder. The convex surface of this prism is equal to the product of its altitude by the perimeter of its base (Prop. I., B. VIII.). Let, now, the arcs subtended by the sides AB, BC, &c., be bisected, and the number of sides of the polygon be indefinitely increased; its perimeter will approach the circumference of the circle, and will be ultimately equal to it (Prop. XI., B. VI.); and the convex surface of the prism will become equal to the convex surface of the cylinder. But whatever be the number of sides of the prism, its convex surface is equal to the product of its altitude by the perimeter of its base; hence the convex surface of the cylinder is equal to the product of its altitude by the circumference of its base.

Cor. If A represent the altitude of a cylinder, and R the radius of its base, the circumference of the base will be represented by 2 R (Prop. XIII., Cor. 2, B. VI.); and the convex surface of the cylinder by 2πRA.

PROPOSITION II. THEOREM.

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

Let ACE-G be a cylinder whose base is the circle ACE and altitude AG; its solidity G is equal to the product of its base by its altitude.

B

E

C

In the circle ACE inscribe the regular polygon ABCDEF; and upon this polygon let a right prism be constructed of the same altitude with the cylinder. The solidity of this prism is equal to the product of its base by its altitude (Prop. XI., B. VIII.). Let, now, the number of sides of the polygon be indefinitely in creased; its area will become equal to that of the circle, and the solidity of the prism becomes equal to that of the cylinder. But whatever be the number of sides of the prism, its solidity is equal to the product of its base by its altitude; hence the solidity of a cylinder is equal to the product of its base by its altitude

Cor. 1. If A represent the altitude of a cylinder, and R the radius of its base, the area of the base will be represented by TR (Prop. XIII., Cor. 3, B. VI.); and the solidity of the cylinder will be πR'A.

Cor. 2. 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. 3. 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 since the cylinders are similar, the diameters of the bases are as their altitudes (Def. 6). Therefore the bases are as the squares of the altitudes; and hence the products of the bases by the altitudes, or the cylinders themselves, will be as the cubes of the altitudes.

PROPOSITION III. THEOREM.

The convex surface of a cone is equal to the product of half its side, by the circumference of its base.

Let A-BCDEFG be a cone whose base is the circle BDEG, and its side AB; then will its convex surface be equal to the product of half its side by the circumference of the circle BDF.

C

A

E

In the circle BDF inscribe the regular polygon BCDEFG; and upon this polygon let a regular pyramid be constructed having B A for its vertex. The edges of this pyramid will lie in the convex surface of the cone. From A draw AH perpendicular to CD, one of the sides of the polygon. The convex surface of the pyramid is equal to the product of half the slant height AH by the perimeter of its base (Prop. XIV., B. VIII.). Let, now, the arcs subtended by the sides BC, CD, &c., be bisected, and the number of sides of the polygon be indefinitely increased, its perimeter will become equal to the circumference of the circle, the slant height AH becomes equal to the side of the cone AB, and the convex surface of the pyramid becomes equal to the convex surface of the cone. But, whatever be the number of faces of the pyramid, its convex surface is equal to the prodact of half its slant height by the perimeter of its base; hence the convex surface of the cone, is equal to the product of half its side by the circumference of its base.

Cor. If S represent the side of a cone, and R the radius

of its base, then the circumference of the base will be represented by 2πR, and the convex surface of the cone by 2πR XIS, or πRS.

PROPOSITION IV. THEOREM.

The convex surface of a frustum of a cone is equal to the product of its side, by half the sum of the circumferences of its two bases.

Let BDF-bdf be a frustum of a cone whose bases are BDF, bdf, and Bb its side; its convex surface is equal to the product of Bb by half the sum of the circumferences BDF, bdf.

C

D

Complete the cone A-BDF to which the frustum belongs, and in the circle BDF inscribe the regular polygon BCDEFG; and upon this polygon let a regular pyramid be constructed having A for its B vertex. Then will BDF-bdf be a frustum of a regular pyramid, whose convex surface is equal to the product of its slant height by half the sum of the perimeters of its two bases (Prop. XIV., Cor. 1, B. VIII.). Let, now, the number of sides of the polygon be indefinitely increased, its perimeter will become equal to the circumference of the circle, and the convex surface of the pyramid will become equal to the convex surface of the cone. But, whatever be the number of faces of the pyramid, the convex surface of its frustum is equal to the product of its slant neight, by half the sum of the perimeters of its two bases. Hence the convex surface of a frustum of a cone is equal to the product of its side by half the sum of the circumferences of its two bases.

Cor. It was proved (Prop. XIV., Cor. 2, B. VIII.), that the convex surface of a frustum of a pyramid is equal to the product of its slant height, by the perimeter of a section at equal distances between its two bases; hence the convex surface of a frustum of a cone is equal to the product of its side, by the circumference of a section at equal distances between the

two bases.

H