By axiom the sum of the first members of the three equations equals the sum of the second members; but the sum of the first members equals V; therefore,

V = [(DFB) + (FBL) + (BLH)] (a); and since DFB + FBL + BLH = DFLHB, or B,

V = (B) (a).

Q. E. D.

91. COROLLARY 1. The volume of any cylinder is equal to the product of its base and altitude.

For the cylinder is the limiting case of the inscribed prism; and accordingly the theorem of (90) must be true alike for the prism and for the cylinder (88).

92. COROLLARY 2. Prisms are to each other as the products of their bases and altitudes. When the altitudes are equal, the prisms are to each other as their bases. When the bases are equal, the prisms are to each other as their altitudes. This COROLLARY is also true of the cylinder.

Use V, B, and a to represent the volume, base, and altitude, respectively, of a prism; and correspondingly for any other prism use V', B' and a'; then,

93.

COROLLARY 3. Using V and a, as in (92), and R to

denote the radius of the base, then for any circular cylinder

V = (TR2) (a).

PROPOSITION XVIII.

94. THEOREM. The lateral surface of a prism is equal to the product of the perimeter of a right section by a lateral edge.

Let DE + EF + FD be the perimeter of a right section of any prism ABC-K. Let / represent the length of any lateral edge (59). Let S represent the lateral surface; then will

95. COROLLARY 1. The lateral surface of a cylinder is equal to the product of the perimeter of a right section by an element of the surface.

For the cylinder is the limiting case of the inscribed prism (88); and, accordingly, theorem (94) must be true alike for the prism and for the cylinder.

96. COROLLARY 2. The lateral surface of a right prism or cylinder is equal to the product of the altitude and the perimeter of the base.

97. COROLLARY 3. Lateral surfaces of right prisms or cylinders, if the perimeters of the bases are equal, are to each other as the altitudes of the solids; if the altitudes are equal, the lateral surfaces are to each other as the perimeters of

the bases.

EXERCISES.

98. THEOREM. The volumes of twc similar cylinders of revolution are to each other as the cube. of their altitudes, or as the cubes of the radii of their bases.

99. THEOREM. The surfaces, lateral or total, of two similar cylinders of revolution are to each other as the squares of their altitudes or as the squares of the radii of their bases. Use diagram of (98) for (99).

100. THEOREM. The opposite faces of a parallelopiped are equal.

101. THEOREM. The diagonals of any parallelopiped

bisect each other.

102. DEFINITION. Similar prisms are such as have the same number of faces, each face of the one being similar to a corresponding face of the other, and similarly placed with respect to adjoining faces.

103. THEOREM. With respect to their triedral angles, two similar prisms are mutually equiangular.

Use diagram of (104) for (103).

104. THEOREM. Two similar prisms are to each other as the cubes of their corresponding edges.

Place the prisms, as in the diagram, with their corresponding bases on the plane MN. Draw the perpendiculars AB and ab. Then AB, ab, AD, and DB lie in the same plane; DB being in the plane MN.

AB and ab are the altitudes of the respective prisms.

E

D

SECTION III.

PYRAMIDS AND CONES.

DEFINITIONS.

105. A pyramid is a solid enclosed by the faces of a polyedral angle and a plane cutting all these faces (36). The cutting plane bounded by its intersections with the faces is called the base. The line from the vertex (36) to the centre of the base is called the axis. The perpendicular distance from the vertex to the base is termed the altitude. The lateral faces are all the faces except the base. The lateral surface is the sum of the lateral faces. The lateral edges are the edges of the polyedral angle (36).

B

106. A right pyramid is one whose axis is perpendicular to the base; if the base is a regular polygon the right pyramid is termed a regular pyramid.

107. A truncated pyramid is a portion. of a pyramid included between the base

and any section cutting all the lateral edges. If the section is parallel to the base, the truncated pyramid is called a frustum, and the section is its

the lateral faces.

upper base.

The altitude of the frustum

is the distance between its bases. The lateral faces are all the faces except the bases, and the lateral surface is the sum of