"If a figure is composed of straight lines a, b, c ... which all pass through a fixed point or centre S, it can be projected from a straight line or axis s passing through S; the result is a figure composed of planes sa, sb, sc....
"If a figure is composed of straight lines a, b, c ... all lying in a fixed plane, it may be cut by a straight line (transversal) s lying in the same plane; the figure which results is formed by the points sa, sb, sc...."
110. The successive application, to any figure, of two reciprocal operations of projection and section, is regarded as producing a figure protectively indistinguishable from the first, provided only that the dimensions of the original figure were the same as those of the resulting figure, that, for example, if the second operation be section by a plane, the original figure shall have been a plane figure. The figures obtained from a given figure, by projection or section alone, are related to that figure by the principle of duality, of which we shall have to speak later on.
I shall endeavour to show, in what follows, first, in what sense figures obtained from each other by projective transformation are qualitatively alike; secondly, what axioms, or adjectives of space, are involved in the principle of projective transformation; and thirdly, that these adjectives must belong to any form of externality with more than one dimension, and are, therefore, à priori properties of any possible space.
For the sake of simplicity, I shall in general confine myself to two dimensions. In so doing, I shall introduce no important difference of principle, and shall greatly simplify the mathematics involved.
111. The two mathematically fundamental things in projective Geometry are anharmonic ratio, and the quadrilateral construction. Everything else follows mathematically from these two. Now what is meant, in projective Geometry, by anharmonic ratio?
If we start from anharmonic ratio as ordinarily defined, we are met by the difficulty of its quantitative nature[122]. But among the properties deduced from this definition, many, if not most, are purely qualitative. The most fundamental of these is that, if through any four points in a straight line we draw four straight lines which meet in a point, and if we then draw a new straight line meeting these four, the four new points of intersection have the same anharmonic ratio as the four points we started with. Thus, in the figure, abcd, a′b′c′d′, a″b″c″d″, all have the same anharmonic ratio. The reciprocal relation holds for the anharmonic ratio of four straight lines. Here we have, plainly, the required basis for a qualitative definition. The definition must be as follows:
Two sets of four points each are defined as having the same anharmonic ratio, when (1) each set of four lies in one straight line, and (2) corresponding points of different sets lie two by two on four straight lines through a single point, or when both sets have this relation to any third set[123]. And reciprocally: Two sets of four straight lines are defined as having the same anharmonic ratio when (1) each set of four passes through a single point, and (2) corresponding lines of different sets pass, two by two, through four points in one straight line, or when both sets have this relation to any third set.
Two sets of points or of lines, which have the same anharmonic ratio, are treated by projective Geometry as equivalent: this qualitative equivalence replaces the quantitative equality of metrical Geometry, and is obviously included, by its definition, in the above account of projective transformations in general.