2. A straight line may be drawn from one point to another.

This requires the use of a straight-edge or ruler. The straight-edge and the compasses are the only instruments recognized in elementary geometry.

3. A straight line may be produced through either extremity to any length. 4. All straight angles are equal to one another.

This follows directly from the definition of a straight angle.

There are also a number of simple statements, of a general nature, so obvious that the truth of them may be taken for granted. These are called axioms.

The following are the axioms most frequently used in geometry:

1. Things which are equal to the same thing, or to equal things, are equal to each other.

That is, (1) if A = B, and C = B, then A = C. Or, (2) if A = B, and B = C, and C = D, then A = D.

2. If equals are added to equals, the sums are equal.

That is, if A = B, and if C = D, then A + C = B + D.

3. If equals are subtracted from equals, the remainders are equal. That is, if A = B, and if C = D, A being greater than C, then A - C =B-D

4. If equals are added to unequals, the sums are unequal in the same

sense.

That is, if A = B, and if C is greater than D, then A+ C is greater than B +D.

5. If equals are subtracted from unequals, the remainders are unequal in the same sense.

That is, if A = B, and if C is greater than D, then, if D is greater than B, CA is greater than D-B.

6. If equals are multiplied by equal numbers the products are equal. That is, if A B, and m be any number, then mA = mB.

=

7. If equals are divided by equal numbers the quotients are equal.

8. The whole is greater than any of its parts, and equals the sum of all

its parts.

The latter part of the axiom is merely the definition of whole.

Illustrate each of the axioms, (1) using only numbers, (2) using lines, (3) using angles.

SYMBOLS AND ABBREVIATIONS.

The following are used in this work, and are inserted here

References to preceding propositions are made by book and theorem (or problem) thus, I, th. 4; if the Roman numeral is omitted, the proposition is in the current book.

Other simple abbreviations are occasionally used, but they will be easily understood.

Section 3.- Preliminary Theorems.

THE following theorems are designed to show to the beginner the nature of a geometric proof, and to lead him by easy steps to appreciate the logic of geometry. Some of them might properly have been incorporated in Book I, and others might have been omitted altogether; but they form a group of simple propositions that lead the student up to the more difficult work of geometry, and for that reason they are inserted here.

Theorem 1. All right angles are equal.

[SUGGESTION. The only angles of whose equality we are thus far assured are straight angles. Hence in some way we must base our proof of this theorem on postulate 4, which asserts this fact. We then consider how a right angle is related to a straight angle, and the proof is at once suggested.]

3... all right angles, and hence r and r', are equal.

Ax. 7

(If equals are divided by equal numbers the quotients are equal.)

NOTE. In th. 1 we have proved directly from the definition of straight angle that all right angles are equal. In th. 2, on p. 12, a different method of proof will be followed. We shall there suppose that the theorem is false and show that this supposition is absurd. Such proofs have long been known by the name "reductio ad absurdum," a reduction to an absurdity. They are also called indirect proofs.

Theorem 2. At a given point in a given line not more than one perpendicular can be drawn to that line in the same plane.

(For it is given that YY' XX', and the def. of a

is given in step 2.)

(The whole is greater than any of its parts, etc.)

6... the supposition of step 1 is absurd, and a second perpendicular is impossible.

Theorem 3. The complements of equal angles are equal.

[SUGGESTION. Three lines of proof may present themselves. We may base our proof on the equality of straight angles, as we did in th. 1, or we may take an indirect proof as in th. 2, beginning by supposing the theorem false and showing the absurdity of this supposition, or we may base the proof on th. 1. Since the complements suggest right angles, which of the three methods would it probably be best to follow ?] two equal, AOB, A'O'B',

Given

and their complements, BOC,

B'O'C', respectively.

To prove

that BOC=/ B'O'C'.

A

Α'

Proof. 1.

AOC and A'O'C' are rt. .

(Two are said to be complements if their sum is a rt. .)

Def. compl.

Th. 1

Given

Ax. 3

(If equals are subtracted from equals, the remainders are equal.)

Theorem 4. The supplements of equal angles are equal. [Let the student draw the figure and give the proof after the manner of th. 3. Use only four steps in the proof.]

Given

To prove Proof.

Theorem 5. The conjugates of equal angles are equal.

[Any one of the three lines of proof suggested under th. 3 may be followed, or we may base the proof upon the definition of equal angles. The last plan is here adopted, leaving the others to be considered, if desired, by the student.]

be so placed that a lies along a', and b along b'.

Def. equal

(Two Д, ab, a'b', are said to be equal when ab can be placed so that a lies along a', and b along b'.)

2. But then

ba must equal ▲ b'a'.

Def. equal

EXERCISE 19. Illustrate ths. 3, 4, 5, by cutting from paper two equal angles, and showing that their complements, supplements, and conjugates are respectively equal.