GEOMETRY OF SPACE.

BOOK VII.

PLANES AND SOLID ANGLES.

Definitions.

1. A STRAIGHT line is perpendicular to a plane when it is perpendicular to every straight line which it meets in that plane.

Conversely, the plane in this case is perpendicular to the line.

The foot of the perpendicular is the point in which it meets the plane.

2. A straight line is parallel to a plane when it can not meet the plane, though produced ever so far.

Conversely, the plane in this case is parallel to the line.

3. Two planes are parallel to each other when they can not meet, though produced ever so far in every direction.

4. The angle contained by two planes which meet one another is the angle contained by two lines drawn from any point in the line of their common section, at right angles to that line, one in each of the planes.

This angle may be acute, right, or obtuse. If it is a right angle, the two planes are perpendicular to each other.

5. A solid angle is the angular space contained by more than two planes which meet at the same point, and not lying in the same plane.

To represent a plane in a diagram, we are obliged to take a limited portion of it; but the planes treated of in this Book are supposed to be indefinite in extent.

PROPOSITION I. THEOREM.

One part of a straight line can not be in a plane, and another part without it.

For, from the definition of a plane (B. I., Def. 11), when a straight line has two points common with a plane, it lies wholly in that plane.

Scholium. To discover whether a surface is plane, we apply a straight line in different directions to this surface, and see if it touches throughout its whole extent.

PROPOSITION II. THEOREM.

Any two straight lines which cut each other are in one plane, and determine its position.

B

Let the two straight lines AB, BC cut each other in B; then will AB, BC be in the same plane.

Conceive a plane to pass through the straight line BC, and let this plane be turned about BC until it pass through the point A. Then, because the points A and B are situated in this plane, the straight line AB lies in it (B. I., Def. 11). Hence the position of the plane is determined by the condition of its containing the two lines AB, BC; for if it is turned in either direction about BC, it will cease to contain the point A. Therefore, any two straight lines, etc. Cor, 1. A triangle ABC, or three points A, B, C, not in the same straight line, determine the position of a plane.

A

F

C.

-D

Cor. 2. Two parallel lines AB, CD deterB mine the position of a plane. For, if the line EF be drawn, the plane of the two straight lines AE, EF will be the same as that of the parallels AB, CD; and it has already been proved that two straight lines which cut each other determine the position of a plane.

If two planes cut each other, their common section is a straight line.

Let the two planes AB, CD cut each other, and let E, F be two points in their common section. From E to F draw the straight line EF. Then, since te points E and F are in the

plane AB, the straight line EF which joins them must lie wholly in that plane (B. I., Def. 11). For the same reason, EF must lie wholly in the plane CD. Therefore the straight line EF is common to the two planes AB, CD; that is, it is their common section. Hence, if two planes, etc.

If a straight line be perpendicular to each of two straight lines at their point of intersection, it will be perpendicular to the plane in which these lines are.

Let the straight line AB be perpendicular to each of the straight lines CD, EF which intersect at B; AB will also be perpendicular to the plane MN which passes through these lines.

Through B draw any line BG, in the plane MN; let G. be any point of this line, and through G draw DGF, so that DG shall be equal to GF (B. V., Pr. 21). Join AD, AG, and AF.

N

Then, since the base DF of the triangle DBF is bisected in G, we shall have (B. IV., Pr. 14),

BD2+BF2=2BG2+2GF2.

Also, in the triangle DAF,

AD2+AF2-2AG2+2GF2.

Subtracting the first equation from the second, we have

AD2-BD2+AF2-BF2-2AG2-2BG2.

But, because ABD is a right-angled triangle,

AD2-BD2-AB2;

and, because ABF is a right-angled triangle,

AF2-BF2-AB2.

Therefore, substituting these values in the former equation, we

have

whence

or

AB2+AB2=2AG2-2BG2;

AB2-AG2-BG2,

AG2-AB2+BG2.

Wherefore ABG is a right angle (B. IV., Pr. 13, Sch.); that is, AB is perpendicular to the straight line BG. In like manner, it may be proved that AB is perpendicular to any other straight line passing through B in the plane MN; hence it is perpen

dicular to the plane MN (Def. 1). Therefore, if a straight line,

etc.

Scholium. Hence it appears not only that a straight line may be perpendicular to every straight line which passes through its foot in a plane, but that it always must be so whenever it is perpendicular to two lines in the plane, which shows that the first definition involves no impossibility.

M

G

D

Cor. 1. The perpendicular AB is shorter than any oblique line AD; it therefore measures the true distance of the point A from the plane MN.

Cor. 2. Through a given point B in a plane, only one perpendicular can be drawn to this plane. For, if there could be two perpendiculars, suppose a plane to pass N through them, whose intersection with the plane MN is BG; then these two perpendiculars would both be at right angles to the line BG, at the same point and in the same plane, which is impossible (B. I., Pr. 1).

It is also impossible, from a given point without a plane, to let fall two perpendiculars upon the plane. For, suppose AB, AG to be two such perpendiculars; then the triangle ABG will have two right angles, which is impossible (B. I., Pr. 27, Cor. 3).

Oblique lines drawn from a point to a plane, at equal distances from the perpendicular, are equal; and of two oblique lines unequally distant from the perpendicular, the more remote is the longer.

Let the straight line AB be drawn perpendicular to the plane MN; and let AC, AD, AE be oblique lines drawn from the point A, equally distant from the perpendicular; also, let AF be more remote from the perpendicular than AE; then will the lines AC, AD, AE all be equal to each other, and AF be longer than AE.

For, since the angles ABC, ABD, ABE are right angles, and BC, BD, BE are equal, the triangles ABC, ABD, ABE have two sides and the included angle equal; therefore the third sides AC, AD, AE are equal to each other.

So, also, since the distance BF is greater than BE, it is plain that the oblique line AF is longer than AE (B. I., Pr. 17).

Cor. All the equal oblique lines AC, AD, AE, etc., terminate in the circumference CDE, which is described from B, the foot of the perpendicular, as a centre.

If, then, it is required to draw a straight line perpendicular to the plane MN, from a point A without it, take three points in the plane C, D, E, equally distant from A, and find B, the centre of the circle which passes through these points. Join AB, and it will be the perpendicular required.

.

Scholium. The angle AEB is called the inclination of the line AE to the plane MN. All the lines AC, AD, AE, etc., which are equally distant from the perpendicular, have the same inclination to the plane, because all the angles ACB, ADB, AEB, etc., are equal.

PROPOSITION VI. THEOREM.

If a straight line is perpendicular to a plane, every plane which passes through that line is perpendicular to the first-mentioned plane.

Let the straight line AB be perpendicular to the plane MN; then will every plane which passes through AB be perpendicular to the plane MN.

Suppose any plane, as AE, to pass through AB, and let EF be the common section of the planes AE, MN. In the plane MN, through the point B, draw CD perpendicular to the common section EF.

N

Then, since the line AB is perpendicular to the plane MN, it must be perpendicular to each of the two straight lines CD, EF (Def. 1). But the angle ABD, formed by the two perpendiculars BA, BD, to the common section EF, measures the angle of the. two planes AE, MN (Def. 4), and, since this is a right angle, the two planes must be perpendicular to each other. Therefore, if a straight line, etc.

Scholium. When three straight lines, as AB, CD, EF, are perpendicular to each other, each of these lines is perpendicular to the plane of the other two, and the three planes are perpendicular to each other.