§ 1. Axioms.—The object of geometry is to investigate the properties of space. The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning. They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete. They must, in fact, completely “define” space.
§ 2. Definitions.—The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,—hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz. solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This has shape, size, position, and may be moved. Its boundary or boundaries are called surfaces. They separate one part of space from another, and are said to have no thickness. Their boundaries are curves or lines, and these have length only. Their boundaries, again, are points, which have no magnitude but only position. We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension. Hence we say a solid has three dimensions, a surface two, a line one, and a point none. Space itself, of which a solid forms only a part, is also said to be of three dimensions. The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.
Euclid gives the essence of these statements as definitions:—
Def. 1, I. A point is that which has no parts, or which has no magnitude.
Def. 2, I. A line is length without breadth.
Def. 5, I. A superficies is that which has only length and breadth.
Def. 1, XI. A solid is that which has length, breadth and thickness.
It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of “surface,” “line,” and “point” from the experimental realization of a “solid” does not find a place in his system, although possessing more advantages.
If we allow motion in geometry, we may generate these entities by moving a point, a line, or a surface, thus:—
| The path of a moving point is a line. The path of a moving line is, in general, a surface. The path of a moving surface is, in general, a solid. |