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GEOMETRY.

PAR T I.

SECTION I.

DEFINITIONS AND REMARKS.

1. A Line is length, without breadth or thickness.

2. The Extremities of a Line are called points; and any place between the extremities, is also called a point.

3. A Straight Line, is the shortest distance from one point to another.

A

-B Thus, AB is a straight line, and the shortest distance from A to B.

4. A Curve Line, is one which changes its direction at every point. Thus, ABC is a curve line.

B

A

The word Line, when used by itself means a straight line; and the word Curve, means a curve line.

QUEST.-1. What is a line? 2. What are the extremities of a line called? What is any place between the extremities called ? 3. What is a straight line? Make it on the black board. 4. What is a curve line ? Make one.

When the word line is used by itself, what does it mean?

Definitions and Remarks.

5. A Surface is that which has length and breadth, without height or thickness.

6. A Plane Surface is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide.

7. A Curved Surface has length and breadth without thickness, and like a curve line is constantly changing its direction.

8. A Solid or Body is that which has length, breadth, and thickness. Length, breadth, and thickness, are called Dimensions. Hence, a solid has three dimensions, a surface two, and a line one. A point has no dimensions, but position only.

9. Geometry treats of lines, surfaces, and solids.

10. A Demonstration is a course of reasoning which establishes a truth.

QUEST.-5. What is a surface? Has a surface thickness? 6. What is a plane surface ? If you lay a straight line on a plane surface will it touch it in its whole length ? Is the surface of a looking-glass a plane surface ? Is the surface of a sheet of paper a plane? 7. What is a curved surface ? Is the surface of a ball plane or curved ? Is the surface of the earth plane or curved? Will a straight line coincide with a curved surface ? 8. What is a solid ? What are length, breadth, and thickness called? How many dimensions has a solid ? What are they? How many dimensions has a surface? What are they? How many dimensions has a line? What is it? Has a point dimensions? What has it? 9. What does Geometry treat of ? 10. What is a demonstration ?

Definitions and Remarks-Axioms.

11. A Theorem is something to be proved by demonstration.

12. A Problem is something proposed to be done.

13. A Proposition is something proposed either to be done or demonstrated-and may be either a problem or a theorem.

14. A Corollary is an obvious consequence, deduced from something that has gone before.

15. An Hypothesis is a supposition on which a system of reasoning may be founded.

16. A Scholium is a remark on one or more preceding propositions.

17. An Axiom is a self evident truth.

AXIOMS.

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

2. If equals be added to equals the wholes will be equal.

3. If equals be taken from equals the remainders will be equal.

QUEST.–11. What is a theorem ? 12. What is a problem? 13. What 18 a proposition ? May it be a problem? May it be a theorem? 14. What is a corollary? 15. What is an hypothesis ? 16. What is a scholium? 17. What is an axiom? What is the first axiom? What the second ? What is the third axiom ?

Axioms.—Of Angles.

4. Things which are double of the same thing are equal to each other.

5. Things which are halves of the same thing are equal to each other.

6. A whole is greater than any of its parts.
7. A whole is equal to the sum of all its parts.

8. Things which being applied to each other, coincide throughout their whole extent, are equal.

SECTION II.

OF ANGLES.

A

1. An Angle is the opening or inclination of two lines which meet each other in a point. Thus the lines AC,

B AB, form an angle at the point A. The lines AC, AB, are called the sides of the angle; and the point A, at which they meet, is called the vertex of the angle. An angle is generally read by placing the letter at

QUEST.-What is the fourth axiom? What the fifth ? What the sixth ? What the seventh? What the eighth ? 1. What is an angle? What are the lines called which form the angle? What is the point of intersection called ? How is the angle generally read ? How else may it be read ?

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