onstration, each of the others will be equivalent to this third parallelopiped; they will, therefore, be equivalent to each other.

Cor. The volume of any parallelopiped on a rectangular base, is equal to the area of its base multiplied by its altitude: for it is equivalent to a right parallelopiped of the same base and altitude.

DEFINITIONS.

11. A prism is a polyedron bounded by plane faces, of which two are equal and parallel polygons, and the others parallelograms. It includes the parallelopiped as one species.

The equal and parallel polygons are called the bases. The other faces together form the convex or lateral surface. The edges joining the corresponding angles of the two bases are called the principal edges.

A prism is called triangular, quadrangular, pentagonal, etc., according as its base is a triangle, a quadrilateral, a pentagon, etc.

12. A right prism is one whose principal edges are perpendicular to the bases. Any other prism is called oblique.

13. A cylinder is a solid described by the revolution of a rectangle about one side, which remains fixed.

The fixed side is called the axis of the cylinder. The opposite side describes the convex surface. The circles described by two other sides are called the bases.

14. A prism is said to be inscribed in a

E. G.-12.

cylinder when its bases are inscribed in the bases of the cylinder, and its principal edges lie in the convex surface of the cylinder.

15. The height, or altitude, of a prism or a cylinder, is the perpendicular distance between the planes of its bases.

THEOREM XX.

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

Let ABE be a right prism. Since its principal edges are, by definition, perpendicular to the bases, any one of them may be taken as the height of the prism. Now, AD and BC, being both perpendicular to the lower base, are also perpendicular to AB, which they meet in the plane of that base (?); hence, the parallelogram ABCD is a rectangle, and its area is equal to its base AB multiplied by its altitude BC (Cor. 1, Theo. XXIV, Book I). In the same manner it may be shown that the area of each of the other faces composing the convex surface is equal to its base multiplied by the altitude of the prism. Therefore, the sum of their areas is equal to the sum of their bases multiplied by the common altitude. But the sum of their bases constitutes the perimeter of the base of the prism.

Hence, the convex surface, etc.

THEOREM XXI.

The volume of any prism is equal to the area of its base multiplied by its altitude.

D

Let ABCD be any prism. Now, whatever may be the form of its base, AB, it is evident that it may be divided into an indefinitely large number of small rectangles, so that the remainder, if there be any, shall be less than any assignable quantity; and the base DC may be divided into the same number of equal rectangles having their sides respectively parallel to the sides of the others. Now, each rectangle in the lower base, and its corresponding rectangle in the upper base, may be considered as the opposite bases. of a small parallelopiped; and the prism will be made up of such parallelopipeds. But the volume of each of these will be equal to the area of its base multiplied by its altitude (Cor., Theo. XIX), which is the same as the altitude of the prism; hence, the sum of their volumes will be equal to the sum of their bases multiplied by their common altitude.

That is, the volume of any prism is equal, etc.

THEOREM XXII.

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

Let ABCD be a cylinder, having a prism inscribed in it whose base is a regular polygon. Now, if the number of sides of this polygon be indefinitely increased, its perimeter will ultimately coincide with the circumference of the base of the cylinder. Then, also, the convex surface

D

tude (Theo. XXI).

of the prism will coincide with the convex surface of the cylinder, and the volume of the prism with the solidity of the cylinder. But the convex surface of the prism is equal to the perimeter of the base multiplied by the altitude (Theo. XX), and its volume is equal to the area of the base multiplied by the alti

Therefore, the convex surface of a cylinder, etc.

SECTION XVII.-PYRAMIDS AND CONES.

DEFINITIONS.

1. A pyramid is a polyedron bounded by plane faces, of which one is any polygon, and the others triangles having a common vertex. The polygon is called the base. The triangles together form the convex or lateral surface.

Pyramids are called triangular, quadrangular, pentagonal, etc., according as their bases are triangles, quadrilaterals, pentagons, etc. 2. A regular pyramid is one whose base is a regular polygon, and the triangular faces are equal and isosceles.

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 fixed side is called the axis of the cone. The hypotenuse describes the convex surface. The circle described by the other revolving side is called the base.

4. The altitude of a pyramid or cone is the perpendicular distance from the vertex to the plane of the base.

5. The slant height of a regular pyramid is the perpendicular let fall from the vertex upon the base of any one of its triangular faces. The side or slant height of a cone is the straight line drawn from the vertex to any point in the circumference of the base.

6. A frustum of a pyramid or cone is the portion next the base cut off by a plane parallel to the base. The slant height of a frustum is that part of the slant height of the whole solid which lies on the frustum.

7. A section of any solid is the surface in which it is divided by a plane which passes through it.

THEOREM XXIII.

The convex surface of a regular pyramid is equal to half the product of the perimeter of the base by the slant height.

Let A-BCDE be a regular pyramid, of which AF is the slant height. The area of the triangle ACD is equal to half the product of its base CD into its altitude, which is the slant height. Consequently, the areas of all the equal triangles composing the convex surface are together equal to half the product of the sum of their bases by the slant height. But the sum of their bases constitutes the perimeter of the base of the pyramid.

B

E

Therefore, the convex surface of a regular pyramid, etc.