22. Express 221° in radial measure.

23. Express angle .7854 in degrees, minutes, and seconds.

24. Find the radial measure of an angle whose arc is π feet, the radius of the circle being π yards.

25. What is the radius of a circle the arc of 8".9 on which is 3956 miles long.

26. What is the length of an arc of 16' 2" on a circumference whose radius is 91684792 miles.

27. The circumference of a circle is 300 feet, find its area.

28. The area of a circle is 1000 square meters, find its radius.

29. Find the radius of a circle which is equal in area to the sum of the areas of three circles whose radii are 4, 6, and 6.93 feet, respectively.

GEOMETRY IN SPACE.

BOOK V.

PLANES AND SOLID ANGLES.

I. DETERMINATION OF PLANES IN SPACE, ETC.

DEFINITIONS.

1. We have seen (Def. 6, Book I.) that a plane is a surface in which any two points being taken, the straight line joining them lies wholly in the surface.

NOTE. This surface is indefinite in extent, yet, to represent it we assign limits to it--that is, we represent a plane by a figure traced in it—but the plane must be conceived to extend indefinitely beyond the sides of the figure. This figure is usually a parallelogram in the Theorems of this Book.

2. A straight line is perpendicular to a plane when it is perpendicular to all the straight lines which pass through its foot in the plane. Conversely, the plane is perpendicular to the line. The foot of the perpendicular is the point in which it meets or pierces the plane.

3. A straight line is oblique to a plane when it meets the plane without being perpendicular to it.

4. A straight line is parallel to a plane when it cannot meet that plane, how far soever both be produced. Conversely, this plane is parallel to the line.

5. Two planes are parallel to each other when they cannot meet, how far soever both be produced.

6. The projection of a point, A, on a plane, P, is the foot, a, of the perpendicular let fall from the point on the plane. The perpendicular Aa is called the projecting line of the point, and the plane P is called the plane of projection.

7. The projection of a line on a

A

a

plane is the line which contains the feet of the perpendiculars let fall from all the points of the line on the plane.

PROPOSITION I.
THEOREM.

A straight line cannot be partly in a plane and partly out of it. For, by Definition 1, when a straight line has two points common with a plane, it lies wholly in that plane.

COR. A straight line can meet a plane in one point only.

SCHOLIUM. To discover whether a surface is a plane, we must apply a straight line in different directions to the surface, and observe if it touches the surface throughout its whole length.

PROPOSITION II.

THEOREM.

Two planes, P and Q, which have three points, A, B, and C, not in the same straight line, in common, coincide throughout their whole extent. Join any two of the points, as A and B. Let D be any point of the plane P, on the opposite side of AB from C. Join CD. The lines CD and AB, being in the same plane, P, must meet each other in some point, E. But, since the point C, the straight line AB, and its point, E, are in the plane Q, by hypothesis, the straight line CE, and its point, D, must be in this plane (Prop. I.). Hence, the point D is

C

E

common to both planes. And, as we may make the same contruction for each of the points, A, B, and C, and the line joining the other two, it follows that every point of the plane P is common to the plane Q; therefore the two planes coincide.

COR. 1. Through three points, not in the same straight line, one plane may be made to pass, and but one. In other words, Three points not in the same straight line determine the position of a plane.

COR. 2. A point, C, and a straight line, AB, determine the position of a plane.

COR. 3. Two straight lines, AB and AC, which intersect each other, determine the position of a plane.

For the plane P of the three points A, B, and C contains the lines AB and AC, since each of these lines will have two points in that plane; and conversely, the plane of AB and AC cannot be different from the plane P, since it is the plane of the three points A, B, and C.

B

COR. 5. Hence, through a point we can draw in space only one parallel to a given straight line.

For a line drawn through this point, parallel to the given line, must be in the plane determined by the point and the given line, and in a plane only one parallel can be drawn to a given line through a given point.

SCHOLIUM I. In Geometry in Space, the name Geometric Locus is given to a line or a surface containing all the points in space which fulfil a given condition as to position, or, as it is expressed, possess a particular geometrical property. Thus it follows from this Proposition that The Geometric Locus of a line which passes through a fixed point and meets a fixed line is the plane determined by the point and fixed line.

SCHOLIUM 2. The plane of an angle, BAC, may be generated by a straight line moving along the side AB, and remaining constantly parallel to the side AC; or, more generally, by a straight

B

B

line which moves, resting in any manner whatever on the two lines AB and AC.

PROPOSITION III

THEOREM.

If two planes cut each other, their common intersection will be a straight line.

For, if among the points common to the two planes, there be three which are not in the same straight line, then the planes passing each through these three points must form only one and the same plane, which contradicts the hypothesis.

COR. I. Three planes may meet in a point. This point is where the line common to two of the planes pierces the third plane.

COR. 2. Through three points, A, B, and C, in the same straight line, any number of planes may be drawn. Hence, at each point of a straight line, AB, any number of perpendiculars may be drawn to that line. planes which contains AB.

A

B

For we can draw one in each of the

PROPOSITION IV.
THEOREM.

At any point, O, of a plane, MN, one perpendicular can always be drawn to that plane, and but one.

First. From the point O draw a perpendicular, OC, to any line AB of the plane, and at C erect another perpendicular, CR, to this line. Finally, in the plane OCR, determined by CO and CR, let the perpendicular OD be drawn to the line OC. Then will OD be perpendicular to any line, OP, drawn

M

R

H

D

S

P

B