SPECIAL TERMS

An axiom is a truth assumed as self-evident.

A theorem is a truth which becomes evident by a train of reasoning called a demonstration.

A theorem consists of two parts, the hypothesis, that which is given, and the conclusion, that which is to be proved.

A problem is a question proposed which requires a solution.

A proposition is a general term for either a theorem or problem. One theorem is the converse of another when the conclusion of the first is made the hypothesis of the second, and the hypothesis of the first is made the conclusion of the second.

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The converse of a truth is not always true. Thus, "If a man is in New York City he is in New York State," is true; but the converse, If a man is in New York State he is in New York City," is not necessarily true. When one theorem is easily deduced from another the first is sometimes called a corollary of the second.

A theorem used merely to prepare the way for another theorem is sometimes called a lemma.

GEOMETRY

INTRODUCTION

FUNDAMENTAL CONCEPTIONS

1. Def.-Geometry is the science of space.

2. Every one has a notion of space extending indefinitely in all directions. Every material body, as a rock, a tree, or a house, occupies a limited portion of space. The portion of space which a body occupies, considered separately from the matter of which it is composed, is a geometrical solid. The material body is a physical solid. Only geometrical solids are here considered, and they are called simply solids. Def.-A solid is, then, a limited portion of space.

3. Def.-The boundaries of a solid are surfaces (that is, the surfaces separate it from the surrounding space). A surface is no part of a solid.

4. Def.-The boundaries of a surface are lines. A line is no part of a surface.

5. Def.-The boundaries (or ends) of a line are points. A point is no part of a line.

6. The solid, surface, line, and point are the four fundamental conceptions of geometry. They may also be considered in the reverse order, thus:

(1.) A point has position but no magnitude.

(2.) If a point moves, it generates (traces) a line.

This motion gives to the line its only magnitude, length.
(3.) If a line moves (not along itself), it generates a surface.
This motion gives to the surface, besides length, breadth.

(4.) If a surface moves (not along itself), it generates a solid. This motion gives to the solid, besides length and breadth, thickness Def.-A figure is any combination of points, lines, surfaces, or solids.

7. Def.-A straight line is a line which is the shortest path between any two of its points.

8. Def.-A plane surface (or simply a plane) is a surface such that, if any two points in it are taken, the straight line passing through them lies wholly in the surface.

9. Def. Two straight lines are parallel which lie in the same plane and never meet, however far produced.

GEOMETRIC AXIOMS

10. All the truths of geometry rest upon three fundamental axioms, viz.:

(a.) Straight line axiom.-Through every two points in space there is one and only one straight line.

This is sometimes expressed as follows: Two points determine a straight line.

(b.) Parallel axiom.-Through a given point there is one and only one straight line parallel to a given straight line.

(c.) Superposition axiom.—Any figure in a plane may be freely moved about in that plane without change of size or shape. Likewise, any figure in space may be freely moved about in space without change of size or shape.

GENERAL AXIOMS

11. In reasoning from one geometric truth to another the following general axioms are also employed, viz. :

(1.) Things equal to the same thing are equal to each other.
(2.) If equals be added to equals, the wholes are equal.

(3.) If equals be taken from equals, the remainders are equal. (4.) If equals be added to unequals, the wholes are unequal in the same order.

(5.) If equals be taken from unequals, the remainders are unequal in the same order.

(6.) If unequals be taken from equals, the remainders are unequal in the opposite order.

(7.) If equals be multiplied by equals, the products are equal; and if unequals be multiplied by equals, the products are unequal in the same order.

(8.) If equals be divided by equals, the quotients are equal; and if unequals be divided by equals, the quotients are unequal in the same order.

(9.) If unequals be added to unequals, the lesser to the lesser and the greater to the greater, the wholes will be unequal in the same order.

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

(11.) The whole is equal to the sum of all its parts.

(12.) If of two unequal quantities the lesser increases (continuously and indefinitely) while the greater decreases; they must become equal once and but once.

(13.) If of three quantities the first is greater than the second and the second greater than the third, then the first is greater than the third.

12. Def.-Plane Geometry treats of figures in the same plane.

13. Def.-Solid Geometry, or the geometry of space, treats of figures not wholly in the same plane.