adding equals to equals, the rectangles BC, AD and AB, DC are together equal to the rectangles BD, CF and bd, af. But since BD is undivided, and ca divided into two parts in F, the rectangle BD, CA is equal (ii. 1) to the rectangles BD, CF and BD, AF; and things that are equal to the same thing are equal to one another (Ax. 1): therefore the rectangle AC, BD is equal to the two rectangles AB, DC and AD, BC. Which was to be proved.

THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS.

I.

A SOLID is that which has length, breadth, and thick

The extremities of a solid are surfaces.

OBS. Hence a solid figure (Bk. i. Def. 14) is enclosed by one or more boundaries (Bk. i. Def. 13), each of which is a surface.

III.

A straight line is defined to be perpendicular or at right angles to a plane, when it is at right angles to every straight line that is drawn to meet it in that plane.

IV.

A plane is defined to be perpendicular or at right angles to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes are perpendicular to the other plane.

OBS. The common section of two surfaces which cut one another is a line; and it will be proved in Bk. xi. Prop. 3, that in the case of two planes cutting one another, this line is a straight line.

V.

When a straight line meets a plane and is neither in the same plane with it nor at right angles to it, it is said to be obliquely inclined to the plane; and its inclination is the acute angle included by the straight line and another straight line, which is drawn from the point where the first line meets the plane to the point in which a perpendicular to the plane, drawn from any point of the first line, meets the plane.

VI.

When a plane cutting another plane is not perpendicular to it, the two planes are said to be obliquely inclined to one another; and their inclination is the acute angle included by two straight lines drawn each from any point in the common section of the two planes at right angles to it, one in one plane and the other in the other plane.

VII.

Two planes are said to have the same or a like inclination to one another which two other planes have, when the angles of inclination are equal to one another.

VIII.

Parallel planes are such as being produced ever so far every way never meet.

IX.

A solid angle is that which is constituted by the meeting together at a point of more than two plane angles, each of which is in a plane different from all the others, and is contiguous to two of them, i. e. each of the two straight lines which include it is one of the straight lines which include two of the other plane angles.

OBS. 1. A solid angle may also be regarded as formed by the meeting together of more than two planes at a point, each of which is dif

ferent from all the others, and is intersected by two of them; for the common sections of each with the two contiguous planes (which are straight lines by Bk. xi. Prop. 3) form one of the plane angles, which in the defn constitute the solid angle.

OBS. 2. If the plane angles which constitute one solid angle be respectively equal to those constituting another, and the inclination of planes of each pair of contiguous angles in the one is equal to that of the planes of the pair of contiguous equal angles in the other: then these two solid angles shall be equal to one another.

For if one solid angle be applied to the other, so that a plane angle of the one may coincide with the equal plane angle of the other, the two solid angles falling on the same side of it; then the two planes contiguous to the plane angle of the one will fall on the two planes contiguous to the equal plane angle of the other, since by hyps the inclinations of each pair of contiguous angles are equal in each; and therefore the two plane angles contiguous to the plane angle of the one will coincide with the two contiguous to that of the other, since by hyps the plane angles are equal. Similarly it may be shewn that each of the other plane angles of the one coincides with each of those of the other. Hence the two solid angles coincide; and magnitudes that coincide are equal (Ax. 7): therefore the two solid angles are equal to one another. Which was to be proved.

X.

When the surfaces which bound a solid figure are all planes, the solid figure is called a polyhedron.

OBS. I. Since the common section of two planes cutting one another is a straight line (Bk. xi. Prop. 3), each of the bounding planes of the polyhedron will be cut by the other bounding planes in straight lines, and will therefore be a plane rectilinear figure or polygon, of which these straight lines are the sides, and at each of whose angular points more than two bounding planes cut one another. Then :

(1) the straight lines, which are the common sections of pairs of the bounding planes, and are sides common to two of the polygons, are called the edges of the polyhedron ;

(2) the bounding polygons are called the faces of the polyhedron ;

(3) the solid angles at the points where more than two of the bounding planes meet, which are constituted by the plane angles of the corresponding polygons, are called the solid angles of the polyhedron.

OBS. 2. When the faces of a polyhedron are all equal regular polygons, it is called a regular polyhedron.

XI.

One polyhedron is defined to be similar to another polyhedron having the same number of solid angles, and the same number of faces constituting each solid angle, when each solid angle of the one is equal to a solid angle of the other, and the faces constituting each solid angle of the one are respectively similar to the faces constituting the equal solid angle of the . other.

XII.

A pyramid is a polyhedron, bounded by a polygon, and a set of triangles which have the sides of the polygon for their bases, and meet in a common angular point without the plane of the polygon.

XIII.

A prism is a polyhedron, two of the faces of which are equal, similar and parallel to one another, and the rest parallelograms.

XIV.

A sphere is the solid figure generated by the revolution of a semicircle about its diameter, which remains fixed.

XV.

The axis of a sphere is a fixed straight line about which the generating semicircle revolves.

XVI.

The centre of a sphere is the centre of the generating semicircle.

XVII.

A diameter of a sphere is a straight line which passes through the centre, and which is terminated both ways by the surface of the sphere.