ELEMENTARY GEOMETRY.

INTRODUCTION.

DEFINITIONS.

1. SPACE is indefinite extension in every direction. All material bodies occupy limited portions of space, and have length, breadth, thickness, form, and position. The material body occupying any portion of space is called a physical solid. The part of space which is or may be occupied by a material body is called a geometric solid. A physical solid is therefore a real body, while a geometric solid is only the form of a physical solid, and is the one treated of in Geometry. The term solid will be used for brevity to denote a geometric solid.

2. A solid is a limited portion of space, and has length, breadth, and thickness. Length, breadth, and thickness are

called the three dimensions of the solid.

3. A surface is the limit or boundary of a solid, and has only two dimensions, length and breadth.

A surface has no thickness, for if it had any, however small, it would form part of the solid, and would be space of three dimensions.

4. A line is the limit or boundary of a surface, and has only one dimension, namely, length.

A line has no breadth, for if it had any, however small, it would form part of the surface, and would be space of two dimensions; and if in addition it had any thickness, it would be space of three dimensions; hence a line has neither breadth nor thickness.

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5. A point is the limit or extremity of a line, and has position, but neither length, breadth, nor thickness.

A point has no length, for if it had any, however small, it would form part of the line of which it is the extremity; and it can have neither breadth nor thickness because the line has none.

6. If we suppose a solid to be divided into two parts which touch each other, the division between the two parts is a surface. This surface can have no thickness, for if it had a thickness, however small, it would be a part either of the one solid or the other, and would therefore be a solid and not a surface.

Again, if we suppose a surface cut into two parts which touch each other, the division between the two parts is a line. This line can have no thickness, because the surface has none, and it can have no breadth, for it forms no part of either surface.

If we suppose a line cut into two parts which touch each other, the division between the two parts is a point. This point can have neither breadth nor thickness, because the line has none, and it can have no length, for it forms no part of either line.

Euclid regarded a point merely as a mark of position, and he attached to it no idea of size and shape.

Similarly, he considered that the properties of a line arise only from its length and position, without reference to that minute breadth which every line must really have if actually drawn, even though the most perfect instruments are used.

We cannot make the points, lines, and surfaces of Geometry. A dot, made on paper or on the blackboard, will have length, breadth, and thickness, and hence will not be a real point. Yet the dot may

be taken as an imperfect representation of the real point. So also a line, drawn on paper or on the blackboard, will have breadth and thickness, and hence will not be a real line. Yet the line which we draw may be taken to represent the real line.

7. We have considered a surface as the boundary of a solid, a line as the boundary of a surface, and a point as the limit of a line. On the other hand, inversely, we may re

gard a line as generated by the motion of a point, a surface as generated by the motion of a line, and a solid as generated by the motion of a surface. Again, each of these may be regarded in a purely abstract manner, distinct from each other.

Thus, we may suppose a surface to exist in space separately from the solid whose boundary it forms, and to be of unlimited extent.

Similarly, we may suppose a line to exist in space separately from the surface whose boundary it forms, and to be of unlimited length.

Likewise we may suppose a point to exist in space separately from the line, and to have only position.

The points, lines, surfaces, and solids of Geometry are called geometric points, lines, surfaces, and solids.

8. A straight line, or right line, is one which has the same direction at every point, as the line AB.

9. A curved line is one no part of which is straight, but changes its direction at every point, as the line CD.

10. A broken line is a line made up of different successive straight

lines, as the line EF.

A

B

F

Fig. 1

The word line, used alone, signifies a straight line; and the word curve, a curved line.

11. A plane surface, or, simply, a plane, is a surface in which the right line joining any two points in it lies wholly in the surface.

12. A curved surface is one no part of which is plane. 13. A figure is any definite combination of points, lines, surfaces, or solids.

A plane figure is one formed of points and lines in a plane. If the figure is formed of right lines only, it is called a rectilinear, or right-lined, figure.

The figure of a solid depends upon the relative position