ELEMENTARY GEOMETRY

INTRODUCTION

THE FOUR FUNDAMENTAL SPACE CONCEPTS

1. Geometry is that branch of mathematical science which treats of the properties of space.

The space in which we live is divisible into parts. Every portion of matter occupies a part of space. The portion of space occupied by a body, considered separately from the matter which it contains, may be regarded as existing unchanged when the body moves into another portion of space.

2. Any portion of space capable of being occupied by a physical solid is called a geometrical solid, or simply a solid.

3. The common boundary of two adjoining solids, or of a solid and the surrounding space, is not a solid; it is a second kind of space element, called a surface.

4. Any surface is likewise divisible into parts; and the common boundary of two adjoining parts of a surface is not a surface; it is a third kind of space element, called a line.

5. Again, any line is divisible into parts; and the common extremity of two adjoining parts of a line is a fourth kind. of space element, called a point.

A point is not divisible into parts; hence, the point is the simplest space element.

6. A fine tracing point, or a dot on a sheet of paper, gives an approximate representation of the ideal geometric point.

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Similarly the lines which we trace on the surface of a sheet of paper give some idea of geometric lines. They are, however, only approximations to ideal geometric lines, no matter how finely they may be traced.

7. It is often convenient to think of a geometric line as traced or generated by a point of a moving body.

The line is then called the path of the point.

In the same way a surface may be imagined as generated by a line that is traced on a moving body.

Again, the surface of a moving body may be imagined as tracing or sweeping out a solid portion of space.

We cannot go on, however, and imagine any motion of a solid that will generate any higher space concept.

Hence, the solid is the most comprehensive space concept we can form.

8. Thus whether we begin with the notion of a solid and proceed downwards to the notion of a point, or whether we begin with the point and build up the solid, there are but three steps in the process: from solid to surface, surface to line, line to point; or else from point to line, line to surface, surface to solid.

Accordingly, the space of our experience is said to have three dimensions.

A point is said to have no extension and no dimensions; a line is said to be extended in one dimension; a surface to be extended in two dimensions; a solid to be extended in three dimensions.

9. Any combination of points, lines, surfaces, or solids, is called a geometric figure.

Two PRIMARY SPACE POSTULATES

10. The postulates of geometry are fundamental agreements or conventions concerning the starting point and scope of the science.

11. Postulate of space-dimensions. It is commonly agreed that ordinary geometry shall treat only of a space of three dimensions.

We cannot, however, assert that a space of four dimensions could not exist under any conditions. We are not able to form a mental picture of such a space, but it does not follow that no one will ever be able to form such a picture.

12. Postulate of figure-transference. It is also commonly agreed that ordinary geometry shall consider only a space in which figures can be transferred in thought from one position to another without further change.

There is a branch of higher geometry which considers the possible existence of a space in which figures are not transferable without change. (See Art. 34.)

Besides the two postulates just stated, other postulates will be introduced in due course.

PRIMARY DEFINITIONS

13. In geometry a definition is a statement of what is to be regarded as the fundamental property of a certain class of figures, sufficient to distinguish the class, and also sufficient to furnish a starting point for deriving other properties by logical inference.

The definitions will be introduced wherever occasion arises. The name to be applied to the class of figures so defined will be italicized when used for the first time in the definition.

14. Superposable figures. Equal figures. If two figures are such that they can, by transference, be so applied to each other that every point of one falls on some point of the other, point for point, the two figures when so applied are said to be coincident, or to be superposed. Figures that are capable of superposition are said to be superposable. Superposable figures are also said to be equal to each other. Thus the phrase "equal figures" will always have the same meaning as "superposable figures."