tion in a plane, is called the Angle of Inclination of the line and the plane.

PARALLEL LINES AND PLANE.

538. Theorem.—If a straight line in a plane is parallel to a straight line not in the plane, then the second line and the plane can not have a common point.

For if any line is parallel to a given line in a plane, and passes through any point of the plane, it will lie wholly in the plane (121). But, by hypothesis, the second line does not lie wholly in the plane. Therefore, it can not pass through any point of the plane, to whatever extent the two may be produced.

539. Such a line and plane, having the same direction, are called parallel.

540. Corollary. If one of two parallel lines is parallel to a plane, the other is also.

541. Corollary.-A line which is parallel to a plane is parallel to its projection on that plane.

542. Corollary.-A line parallel to a plane is everywhere equally distant from it.

APPLICATIONS.

543. Three points, however placed, must always be in the same plane. It is on this principle that stability is more readily obtained by three supports than by a greater number. A threelegged stool must be steady, but if there be four legs, their ends should be in one plane, and the floor should be level. Many surveying and astronomical instruments are made with three legs.

544. The use of lines perpendicular to planes is very frequent in the mechanic arts. A ready way of constructing a line perpendicular to a plane is by the use of two squares (114). Place the angle of each at the foot of the desired perpendicular, one side of

each square resting on the plane surface. Bring their perpendieular sides together. Their position must then be that of a jerpendicular to the plane, for it is perpendicular to two lines in the plane.

545. When a circle revolves round its axis, the figure undergoes no real change of position, each point of the circumference taking successively the position deserted by another point.

On this principle is founded the operation of millstones. Two circular stones are placed so as to have the same axis, to which their faces are perpendicular, being, therefore, parallel to each other. The lower stone is fixed, while the upper one is made to revolve. The relative position of the faces of the stones undergoes no change during the revolution, and their distance being properly regulated, all the grain which passes between them will be ground with the same degree of fineness.

546. In the turning lathe, the axis round which the body to be turned is made to revolve, is the axis of the circles formed by the cutting tool, which removes the matter projecting beyond a proper distance from the axis. The cross section of every part of the thing turned is a circle, all the circles having the same axis.

DIEDRAL ANGLES.

547. A DIEDRAL ANGLE is formed by two planes meeting at a common line. This figure is also called simply a diedral. The planes are its faces, and the intersection is its edge.

In naming a diedral, four letters are used, one in each face, and two on the edge, the letters on the edge being between the other two.

This figure is called a diedral angle, because it is similar in many respects to an angle formed by two lines.

MEASURE OF DIEDRALS.

548. The quantity of a diedral, as is the case with a linear angle, depends on the difference in the directions

Geom.-16

of the faces from the edge, without regard to the extent of the planes. Hence, two diedrals are equal when they can be so placed that their planes will coincide.

519. Problem.-One diedral may be added to another. In the diagram, AB, AC, and AD

represent three planes having the common intersection AE.

Evidently the sum of BEAC and CEAD is equal to BEAD.

550. Corollary. Diedrals

may
A

be subtracted one from another. liedral may be bisected or divided in any required ratio by a plane passing through its edge.

E

B

551. But there are in each of these planes any number of directions. Hence, it is necessary to determine which of these is properly the direction of the face from the edge. For this purpose, let us first establish the following principle:

552. Theorem.-One diedral is to another as the plane angle, formed in the first by a line in each face perpendicular to the edge, is to the similarly formed angle in the other.

553. Corollary-A diedral is said to be measured by the plane angle formed by a line in each of its faces perpendicular to the edge.

554. Corollary.-Accordingly, a diedral angle may be acute, obtuse, or right. In the last case, the planes are perpendicular to each other.

555. Many of the principles of plane angles may be applied to diedrals, without further demonstration.

All right diedral angles are equal (90).

When the sum of several diedrals is measured by two right angles, the outer faces form one plane (100). When two planes cut each other, the opposite or vertical diedrals are equal (99).

PERPENDICULAR PLANES.

556. Theorem.—If a line is perpendicular to a plane, then any plane passing through this line is perpendicular to the other plane.

If AB in the plane PQ is perpendicular to the plane MN, then AB must be perpendicular to every line in MN which passes through the

point B (519); that is, to RQ, the intersection of the two planes, and to BC, which is made perpendicular to the intersection RQ. Then, the angle ABC measures the inclina

M

P.

A

R

C

B

N

tion of the two planes (553), and is a right angle. Therefore, the planes are perpendicular.

557. Corollary.-Conversely, if a plane is perpendicular to another, a straight line, which is perpendicu

lar to one of them, at some point of their intersection, must lie wholly in the other plane (524).

553. Corollary.-If two planes are perpendicular to a third, then the intersection of the first two is a line perpendicular to the third plane.

OBLIQUE PLANES.

559. Theorem.-If from a point within a diedral, perpendicular lines be made to the two faces, the angle of these lines is supplementary to the angle which measures the diedral.

Let M and N be two planes whose intersection is AB, and CF and CE perpendicu

lars let fall upon them from the point C; and DF and DE the intersections of the plane FCE with the two planes M and N. Then the plane FCE must be perpendicular to each of the planes M and N (556).

M

E

B

Hence, the line AB is perpendicular to the plane FCE (558), and the angles ADF and ADE are right angles. Then the angle FDE measures the diedral. But in the quadrilateral FDEC, the two angles F and E are right angles. Therefore, the other two angles at C and D are supplementary.

560. Theorem. Every point of a plane which bisects a diedral is equally distant from its two faces.

Let the plane FC bisect the diedral DBCE. Then it is to be proved that every point of this plane, as A, for example, is equally distant from the planes DC and EC.

From A let the perpendiculars AH and AI fall upon the faces DC and EC, and let IO, AO, and HO be the