INTRODUCTION.

GEOMETRY had its origin, as the name indicates, in the need of a method for the accurate description of limited portions of the Earth's surface.

At the present time the development of the science is such as to include a vast amount of material which is beyond the scope of a book devoted to the consideration of elementary forms in such space as that in which we live and exercise our senses.

Like any other science, Geometry is built upon definitions and axioms. Definitions describe, in as simple a manner as possible, the objects with which we have to deal. Axioms are self-evident truths that relate to the objects described, and the operations to be performed. An axiom is a truth self-evident to one who understands the terms in which it is stated.

Upon the definitions and axioms of Geometry is built up our knowledge of fundamental relations. Upon these more complex relations are based. Upon these new relations, still more advanced relations are built; and thus the science is developed. As is true of every other

science, the extent to which Geometry may be pursued is without limit.

While the beginnings are based upon definitions and axioms, it does not follow that all of the definitions and axioms are to be brought forward at the opening of the subject. As the information of the student grows, new definitions and new axioms are appropriate.

Advance in Geometry is made through an orderly arrangement of theorems and problems.

A theorem is a general statement of relations. These relations are established by a course of reasoning.

A problem is a demand that a construction be made; that certain relations be established; or that relations. between certain things be established.

ELEMENTS OF GEOMETRY.

CHAPTER I.

1. Definitions. An enclosed portion of space, no matter what its form or material, or if it exist in the imagination only, is called a volume.

That which separates a volume from excluded space is called surface, and does not partake of the character of the material, if there be any used. The surface that

FIG. 1.

separates the hull of a ship from the water is neither water nor the material of which the ship may be constructed.

2. Definitions. A position in space that is without magnitude is called a point.

If a point move it will generate a line; it may move at a snail's pace, or it may move with the rapidity of thought.

The position of one point with respect to another determines direction. If we represent the position of one point by the letter (4), placed near it, and the position of another point by the letter (B), placed near it, the direction (AB) is established. The direction of (B) from (A) and the direction of (4) from (B) are opposites.

Relations expressed by the words same and opposite, whether they be of interpretation or of operation, are expressed symbolically by the vertical cross (+), and by the horizontal dash (-). If (+4) represents a number of miles in the direction of (B) from (4), (-4) will represent the same distance in the direction of (4) from (B). If (+6) represents six years to come, (-4) will represent four years that have passed.

Anything which is so large that it is beyond our comprehension, is said to be infinitely large. Space is infi nitely large; the number of points in space is infinite; time is infinite.

DIRECTION AXIOM. From any point in space there will be an infinite number of directions; and each direction will have its opposite direction.

There will also be an infinite number of points in any assumed direction from a given point, which points will each be at different distances from the given point. In each opposite direction there will also be an infinite number of points. The rays of light from an arc-light or from a fixed star are fairly good illustrations.

FIG. 2.