9. If one line meets another so as to make the adjacent

angles unequal, each of these angles is called Oblique, in distinction from the right angle; as, A CD and D C B.

Oblique angles are of two classes: A

acute and obtuse.

10. An Acute Angle is one which is less than a right angle, as the angle A.

D

B

D

13. The Complement of an angle is the difference between

a right angle and the given angle.

14. The Supplement of an angle is the difference between two right angles and the given angle.

GENERAL TERMS.

1. A Proposition is something proposed or stated; as, 'Youth is thoughtless.'

2. A Demonstration is a course of reasoning by which the truth of a statement becomes evident. Thus:

All men are mortal;

Socrates was a man;

Therefore Socrates was mortal.

3. A Theorem is a truth requiring demonstration. Something is asserted, and the truth of it required to be proved.

4. A Corollary is a truth which follows, as a necessary consequence, from one or more preceding propositions.

5. A Hypothesis is a supposition assumed to be true in order to argue from it the truth or falsity of a proposition; and it is made either in the statement of the point in question, or in the course of the reasoning.

6. A Scholium is a remark upon some feature of a proposition.

AXIOMS.

An Axiom is a self-evident truth:

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

2. If the same operation be performed upon equals, the results will be equal. Thus, if the same quantity be added to each of two equal quantities, the sums will be equal.

3. If the same operation be performed upon unequals, the results will be unequal. For instance, if the same quantity be added to each of two unequal quantities, it is plain that the results must be unequal.

4. The whole is greater than any of its parts.

5. The whole is equal to the sum of all its parts.

6. All right angles are equal.

7. Only one straight line can be drawn from one point to another.

8. A straight line is the shortest distance between any two points.

POSTULATES.

A Postulate is a self-evident possibility:

1. A straight line can be drawn from any one point to any other.

2. A straight line may be prolonged to any distance.

3. Afinite straight line can be bisected; that is, divided into

two equal parts.

4. An angle may be bisected.

5. An angle may be constructed equal to a given angle.

6. A straight line may be drawn perpendicular to a given straight line either from a point without the line or from a point in it.

Definitions, axioms, and postulates form the basis of geometrical science.

EXERCISES.

1. Take a square' piece of paper, and suspending it in the position A B C D, move it to the position E F G H; then the form of the entire A figure will represent the space through which the paper has passed, or the path which the paper has described. What kind of a solid is this path? B What kind of a solid is the paper itself?

One edge of the paper is A B, and A B F E represents its path in passing from the first position to the second. What is this path called? How, then, may a surface be defined?

What represents the path of A D? of D C? of BC?

The corners of the paper are A, B, C, and D. The path of A is A E; what is it called? The path of B is what? of C? of D?

Hence a line may be considered as the path of what?

2. Does a line in motion necessarily generate a surface? 3. Why are two letters sufficient to read a straight line? 4. Which is the greater angle, a or

7. How many straight lines can be drawn through the north and south poles of the earth?

8. How many straight lines can be drawn through 15 points, each line connecting two points?

CHAPTER II.

RECTILINEAR FIGURES.

THEOREM I.

If one straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles.

A

E

D

B

For, let CE be drawn perpendicular to A B at C (Post. 6), and

let R denote a right angle. By

definition, each of the angles ACE and ECB is a right angle; and therefore their sum equals two right angles; that is,

ACE ECB = 2 R.

Now it is plain that A CD and BCD together fill exactly the same space as ACE and E CB; hence,

ACD+BCD=ACE+ECB

= 2 R.

Therefore, if one straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles.

Cor. The sum of all the angles formed at a point on the same side of a straight line is equal to two right angles.

(2) ACD+DCB 2 R, as just proved;

.. (3) a+b+c+d+e+f= 2 R.

QUERIES.

1. On what two axioms is (1) founded?

2. By what axiom is (3) deduced from (1) and (2)? What is the thing'? What are the 'equal things' ?

same

3. In the proof of the theorem, what axiom is involved in the second

equation?

EXERCISES.

1. If an angle is a right angle,

what is its complement?

2. If an angle is a right angle,

what is its supplement?

C

90 degrees (expressed 90°), and c =

4. If an angle is % of a right angle, what is its supplement? 5. If a right angle

what?

6. If c 50°, and a = 45°, b = what?

= 60°, ABC =

Note.-The student must not think he is bound to use the precise words, or follow in every particular, the precise arrangement of the author. Let him cultivate self-reliance and self-assertion, the habit of original thought and original expression. To this end he is advised, in his beginning, after having studied a demonstration, to draw the figure a second time, using different letters; or a third time, using no letters. Let his sole aim be to discover the ideas, regardless of the expression; to distinguish, each by itself, the different steps in a demonstration; to ascertain and accurately to note the connection of one step with another, and how each succeeding step leads to the desired result, namely, the proof of his theorem. Lastly, these different steps should be separately fixed in the memory.

DEFINITIONS.

If two straight lines cut, or intersect, each other, four angles are formed about the point of intersection, which are distinguished as follows:

1. Those which lie on the same side of one of the lines and on opposite sides of the other are called Adjacent Angles.