BOOK FIRST.

SOLID GEOMETRY.

SECTION FIRST.

Planes.

PROPOSITION I.

Three points not in the same straight line determine the (460) position of a plane.

For, let the plane, P, pass through two of the points, as A, B; then, revolving upon these points, it will become fixed or determined in position, when it shall contain the third point, C.

P

Fig. 108.

Cor. 1. Two intersecting lines, as AC, BC, determine (461) the position of a plane.

Cor. 2. A triangle, as ABC, is always in the same plane (462) and determines its position.

Cor. 3. A plane is determined in position by two paral- (463) lel lines, as AP, BQ.

Cor. 4. Two planes cannot intersect each other in more (464) lines than one.

For if A and B be any points common to the planes P and P„ it is obvious from the definition of a plane that the straight line AB will lie wholly in both planes, and will therefore be a line of intersection. Now the planes, P, P2, cannot have a second line of intersection, as ACB; since this hypothesis would reduce the planes to coincidence (460), the three points, A, C, B, not in the same straight line, becoming common to P and P2.

Cor. 5. The intersections of planes are straight lines.

(465)

Cor. 6. Planes coinciding in three points, not in the same straight line, coincide throughout.

(466)

PROPOSITION II.

A line drawn through the intersection of two other lines (467) and perpendicular to both of them, will be perpendicular to their plane.

b

For let p be the perpendicular, a, a, equal portions of the intersecting lines, b, b, the hypothenuses to p, a, -- p, a, which will therefore be equal, and let d be any line drawn through the angle (a, a) and terminating in the line m+n joining the extremities of a, a, and divided by d into the parts m, n; it only remains to show that e, joining the extremities of p, d, is a hypothenuse. We have (137)

Fig. 109.

Cor. 1. If one side of a right angle be made a fixed axis (468) of revolution, the other side, in revolving, will describe a plane. For, if the right angle (a, p) revolve about p as an axis, a will be found constantly in the plane of (a, a).

Cor. 2. Of oblique lines drawn from any point in a per- (469) pendicular to a plane and terminating in this plane, the more distant will be the greater, those equally distant will be equal and terminate in the circumference of the same circle having the foot of the perpendicular for centre.

Cor. 3. Planes which are perpendicular to the same (470) straight line are parallel to each other; and, conversely, if a line be perpendicular to one of two parallel planes, it will be perpendicular to the other also.

Fig. 1092. Cor. 4. If from any point without a plane, a perpendic- (471) ular be dropped upon the plane, and from the foot of this perpendicular a second perpendicular be let fall upon any line in the plane the line joining the first and last-mentioned points, will be perpendicular to the line drawn in the plane.

For if m=n, d and e will both be perpendicular to m+n.

Cor. 5. Through the same point, either without or within (472) a plane, but a single perpendicular can be drawn.

For let e be any line intersecting the perpendicular, p; e is obviously inclined to d and therefore to the plane (a, a).

Cor. 6. A plane passing through a perpendicular to a (473) second plane, is perpendicular to the same.

For let d revolve to take up a position perpendicular to a; then (p, d) being a right angle, the plane (p, a) is said to be perpendicular to the plane (a, a, d).

Cor. 7. The intersection of two planes perpendicular to (474) a third, is perpendicular to the same plane.

Cor. 8. Lines perpendicular to the same plane are paral- (475) lel to each other.

For let P, P2, P3, ..., be lines perpendicular to the plane, P, and a, b, ..., the lines joining the points in which the perpendiculars intersect the plane; it follows from (473), (474), that the planes (p, a), (P3, b), will intersect in P2; whence p, P29 being perpendicular to a, P2, P3, to b, ..., P, P2, P3, are parallel to each other.

Fig. 1093.

Cor. 9. A line parallel to a perpendicular to a plane is (476) itself a perpendicular to the same plane.

Cor. 10. Lines parallel to the same line situated any (477) way in space, are parallel to each other.

For they will be perpendicular to the same plane.

PROPOSITION III.

If a plane cut parallel planes, the lines of intersection (478) will be parallel.

For, if the intersections m, n, of the plane, P, with the parallel planes, M, N, were not parallel, but met on being produced, then would M, N, cut each other in the same point, which is contrary to the hypothesis.

m

Fig. 110.

Cor. 1. The segments, as r, s, t, of parallel lines inter- (479) cepted by parallel planes, are equal.

Cor. 2. Conversely, two planes intercepting equal seg- (480)

ments of three parallel lines not situated in the same plane, are parallel.

Cor. 3. Parallel planes are everywhere equally distant. (481) [Let r, s, t, be perpendicular to M, N.]

B

Cor. 4. Two angles, having their sides parallel and open- (482) ing in the same direction, are equal, and their planes are parallel. For, let the sides AB, AC, of the angle A, be parallel respectively to the sides ab, ac, of the angle a, and open in the same direction; draw Bb, Cc, parallel to Aa, then will the quadrilaterals, Ab, Ac, Bc, be parallelograms, and the sides of the triangles BAC, bac, Fig. 1102. severally equal,— ... ▲ A = a; but Aa Bb = Cc, .. (480) the plane BAC will be parallel to the plane bac.

=

Cor. 5. A Dihedral angle, or the angle which one plane (483) makes with another, is measured by the inclination of two perpendiculars drawn through the same point in its edge, one in each side.

For, make A a perpendicular to the planes BAC, bac, then will Aa be perpendicular to AB, AC, ab, ac, and the dihedral angle BAac will be measured by the plane angle BAC = bac; from which it follows that the point A, through which the perpendiculars AB, AC, are drawn, may be taken anywhere in the edge, Aa, of the dihedral angle.

Cor. 6. The segments of lines intercepted by parallel (484) planes are proportional.

For, let ABC, abc, be any lines whatever, piercing the parallel planes, P, P, P, in the points A, B, C, a, b, c; and through B draw mBn parallel to abc, piercing the planes in m, n. Since A, B, C, m, n, are in the same plane, and mA parallel to Cn, we have

AB: BC= mB: Bn = ab : bc.

A

B

n C

Fig. 1103.

PROPOSITION IV.

If a line passing through a fixed point revolve in any (485) manner so as constantly to intersect two parallel planes, the figures thus described will be similar.