Denote its circumference by circ. CA, its radius by R, and its diameter by D. From the last Proposition, we have, area CA = «R2;

and, from Proposition XIV., we have,

circ. CA 2πR, or, circ. CA = «D.

That is, the circumference of any circle is equal to 3.1416 times its diameter.

Scholium 1. The abstract number, equal to 3.1416, denotes the number of times that the diameter of a circle is contained in the circumference, and also the number of times that the square constructed on the radius is contained in the area of the circle (P. XV.). Now, it has been proved by the methods of Higher Mathematics, that the value of T is incommensurable with 1; hence, it is impossible to express, by means of numbers, the exact length of a circumference in terms of the radius, or the exact area in terms of the square described on the radius. We may also infer that it is impossible to square the circle; that is, to construct a square whose area shall be exactly equal to that of the cir

cle.

Scholium 2. Besides the approximate value of, 3.1416, usually employed, the fractions 22 and 355 are also used, when great accuracy is not required.

113

BOOK VI.

PLANES AND POLYEDRAL

ANGLES.

DEFINITIONS.

1. A straight line is PERPENDICULAR TO A PLANE, when it is perpendicular to every line of the plane which passes through its Foor; that is, through the point in which it meets the plane.

In this case, the plane is also perpendicular to the line.

2. A straight line is PARALLEL TO A PLANE, when it cannot meet the plane, how far soever both may be produced. In this case, the plane is also parallel to the line.

3. TWO PLANES ARE PARALLEL, when they cannot meet, how far soever both may be produced.

4. A DIEDRAL ANGLE is the amount of divergence of two planes.

The line in which the planes meet, is called the edge of the angle, and the planes themselves are called faces of the angle.

A

The measure of a diedral angle is the same as that of a plane angle formed by two lines, one drawn in each face, and both perpendicular to the edge at the same point. A diedral angle may be acute, obtuse, or a right angle. In the latter case, the faces are perpendicular to each other.

5. A POLYEDRAL ANGLE is the amount of dir gence of several planes meeting at a common point.

This point is called the vertex of the angle ; which the planes meet are called edges of the the portions of the planes lying between the called faces of the angle. Thus, S is the vertex of the polyedral angle, whose edges are SA, SB, SC, SD; and whose faces are ASB, BSC, CSD, DSA.

A polyedral angle which has but three faces, is called a. triedral

angle.

he lines in

angle, and edges are

POSTULATE.

A line may be drawn perpendicular to a p ne from any point of the plane, or from any point without the plane.

PROPOSITION I. THEOREM.

If a straight line has two of its points in a plane, it will lie wholly in that plane.

For, by definition, a plane is a surface sun, that if any two of its points be joined by a straight line, that line will lie wholly in the surface (B. I., D. 8).

Cor. Through any point of a plane, an infinite number of straight lines may be drawn which will lie in the plane. For, if a line be drawn from the given point to any other point of the plane, that line will lie wholly in the plane.

Scholium. If any two points of a plane be joined by a straight line, the plane may be turned about that line as an

axis, so as to take an infinite number of positions. Hence, we infer that an infinite number of planes may be passed through a given line.

Through three points, not in the same straight line, one plane can be passed, and only one.

Let A, B, and C be the three points: then can one plane be passed through them, and only one.

Join two of the points, as A and B, by the line AB. Through AB let a plane be passed, and let this plane be turned around AB until it contains the point C; in this position it will pass through the three points A,

B,

and C. If now, the plane be turned

B

about AB, in either direction, it will no longer contain the point Chence, one plane can always be passed through three points, and only one; which was to be proved.

Cor. 1. Three points, not in a straight line, determine the position of a plane, because only one plane can be passed through them.

Cor. 2. A straight line and a point without that line, determine the position of a plane, because only one plane can be passed through them.

Cor. 3. Two straight lines which intersect, determine the position of a plane. For, let AB and AC intersect at A then will either line, as AB, and one point of the other, as C, determine the position of a plane.

Cor. 4. Two parallel lines determine the position of a

plane.

For, let AB and CD be parallel. By definition (B. I., D. 16) two parallel lines always lie in the same plane. But either line, as AB, and any point of the other, as F, determine the position of a plane hence, two parallels determine the position of a plane.

PROPOSITION III. THEOREM.

A.

B

-D

F

The intersection of two planes is a straight line.

Let AB and CD be two planes: then will their inter

will lie wholly in that plane: hence, every point of EF is common to both planes. Furthermore, the planes can have no common point lying without EF, otherwise there would be two planes passing through a straight line and a point lying without it, which is impossible (P. II., C. 2); hence, the intersection of the two planes is a straight line; which was to be proved.

If a straight line is perpendicular to two straight lines at their point of intersection, it is perpendicular to the plane of those lines.

Let MN be the plane of the two lines BB, CC, and

.

let AP be perpendicular to these lines at P: then will