by multiplying its base ABE, BCE or CDE by the third part of its altitude SO : hence the sum of these triangular pyramids, or the polygonal pyramid SABCDE, will be measured by the sum of the triangles ABE, BCE, CDE, or the polygon ABCDE, multiplied by SO. Hence every pyramid is measured by a third part of the product of its base by its altitude.

Cor. 1. Every pyramid is the third part of the prism which has the same base and the same altitude.

Cor. 2. Two prisms having the same altitude, are to each other as their bases.

Scholium. The solidity of any polyedral body may be computed, by dividing the body into pyramids; and this division may be accomplished in various ways. One of the simplest is to make all the planes of division pass through the vertex of one solid angle; in that case, there will be formed as many partial pyramids as the polyedron has faces, minus those faces which form the solid angle whence the planes of division proceed,

PROPOSITION XVIII.

THEOREM. Two similar pyramids are to each other as the cubes of their homologous sides.

For, two pyramids being similar, the smaller may be placed within the greater, so that the solid anglè S shall be common to both. In that position the bases ABCDE, abcde will be parallel; because, since the homologous faces are similar, the angle Sab is equal to SAB, and Sbc. to SBC hence the plane ABC is parallel A to the plane abc. This granted, let SO be the perpendicular drawn from the vertex

E

B

S to the plane ABC, and o the point where this perpendicular meets the plane abc from what has already been shown (B. VI, Prop. xv), we shall have

So So :: SA: Sa :: AB: ab; and,

consequently, SO So :: AB

ab.

But the bases ABCDE, abcde being similar figures, we ABCDE abcde: AB2 : ab2.

have

Multiply the corresponding terms of these two proportions; there results the proportion,

ABCDE × 1 SO : abcde × 1 So :: AB3 : ab3.

Now ABCDE X SO is the solidity of the pyramid SABCDE, and abcde × ¦ So is that of the pyramid Sabcde (B. VII, Prop. XVII): hence two similar pyramids are to each other as the cubes of their homologous sides.

T2

2

BOOK EIGHTH.

DEFINITIONS.

1. A cylinder is a solid, which may be produced by the revolution of a rectangle D ABCD, conceived to turn about the immovable side AB.

B

H

N

M

G

B

E

In this rotation, the sides AD, BC, continuing always perpendicular to AB, describe equal circular planes DHP, CGQ, c which are called the bases of the cylinder; the side CD at the same describing the convex surface. The immovable line AB is called the axis of the cylinder. Every section KLM made in the cylinder, at right angles to the axis, is a circle equal to either of the bases; for, while the rectangle ABCD revolves about AB, the line KI, perpendicular to AB, describes a circular plane, equal to the base, which is a section made perpendicular to the axis at the point I.

Every section PQGH passing through the axis, is a rectangle, and is double of the generating rectangle ABCD. 2. A cone is a solid, which may be produced by the revolution of a rightangled triangle SAB, conceived to turn about the immovable side SA.

In this rotation, the side AB describes a circular plane BDCE, named the base of the cone; and the hypothenuse SB, its convex surface.

B

H

K

E

A

D

The point S is named the vertex of the cone; SA, its axis or altitude.

Every section HKFI formed at right angles to the axis, is a circle. Every section SDE passing through the axis, is an isosceles triangle double of the generating triangle SAB.

3. If, from the cone SCDB, the cone SFKH be cut off by a section parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone. We may conceive it to be described by the revolution of a trapezium ABHG, whose angles A and C are right, about the side AG. The immovable line AG is called the axis or altitude of the frustum; the circles BDC, HFK are its bases, and BH is its side.

4. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases.

5. If, in the circle ACD which forms the base of a cylinder, a polygon ABCDE is inscribed, a right prism, constructed on this base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism.

D

C

B

N

The edges AF, BG, CH, &c. of the prism, being perpendicular to the plane of the base, are evidently included in the convex surface of the cylinder: hence the prism and the cylinder touch one another along these edges.

6. In like manner, if ABCD is a polygon circumscribed about the base of a cylinder, a right prism, constructed on this base ABCD, and equal in altitude to the cylinder, is said to be cir cumscribed about the cylinder, or the cylinder to be inscribed in the prism.

Let M, N, &c. be the points of contact in the sides AB, BC, &c.; and through the points M, N, &c. let MX,

NY, &c. be drawn perpendicular to the plane of the base: those perpendiculars will evidently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of contact.

7. A sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre.

The sphere may be conceived to be generated by the revolution of a semicircle DAE about its diameter DE; for the surface described in B this movement, by the curve DAE, will have all its points equally distant from the centre C.

I

G

NX

H

MM

8. The radius of a sphere, is a straight line drawn from the centre to any point in the surface; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface.