MODERN GEOMETRY.

CHAPTER I.

LINES AND PLANES.

Definition. The object of the science of Geometry, is to investigate those properties of bodies which have relation to space.

The foundation of the science rests on a few simple results of common observation. Very early in life the eye begins to distinguish differences in the form, size, distance, and colour of objects. It is by these differences that we learn to recognise things about us, such as articles of furniture, houses, trees, or animals.

All bodies occupy certain portions of space, and possess, therefore, certain properties having relation to the space occupied; as, for instance, position, form, and extent. The investigation of these properties is the province of Geometry.

Bodies have other properties, such as colour, hardness, weight, &c.; but as these have no relation to space, they are not considered in Geometry.

Surfaces. Every object marks out two regions of space, namely, a limited one within, and an unlimited one outside itself. These regions are separated from one another by the surface or surfaces of the object; a ball, for example, is enclosed by one surface, and a table or box by several surfaces.

Lines. The meeting of two surfaces forms a line, hence a line may be considered as the boundary of a surface.

Lines may be either bent, curved, or straight.

In a block of wood or stone, prepared so that its surfaces are bounded by lines of different kinds, it is an easy matter to distinguish the straight lines from those which are curved or bent. It is not so easy, however, to say exactly what is a straight, or a curved, or a bent line. With the view of forming definitions of these terms, let us examine further the properties of lines.

B

FIG. I.

FIG. 2.

If a very fine flexible thread be stretched between two points (Fig. 1), the figure presented by the thread at once suggests a straight line. If the thread be not fully stretched, as in Fig. 2, it will hang in a curve. But in neither case does the string represent what is in Geometry called a line; for it has a certain thickness, and, consequently, may have other lines drawn on any of its different sides.

So also is the line drawn in the figure to represent the thread something more than a geome

trical line. It is, in fact, a surface, although a very narrow one, with lines on each side of it.

It is the same, too, with all the marks made on paper to represent lines. They are really small surfaces indicative simply of the positions of the lines they represent, and the finer they are drawn the more exactly do they indicate those positions.

Geometrically considered, a line has length but no thickness.
Points. The extremities of a line are termed points.

Points, like lines, have no thickness, and only mark position; but, unlike lines, they have no length either, and, having neither length nor breadth, they possess no magnitude

FIG. 3.

whatever.

The intersection of two straight lines is a point. The best way to mark the position of a point is to draw two short intersecting lines (Fig. 3).

Straight Lines.A straight line is distinguished from all other lines by having the same direction throughout its entire length. This property alone, therefore, serves to define it. It may, however, be otherwise expressed; as, for example:

A straight line is the line which lies evenly between its extreme points. A straight line is such, that if one portion were moved forward upon the other, then would the first lie throughout its entire length upon the second.

When the direction changes the line is not straight.