"_To cut by a fixed plane σ (transversal plane_) a figure (_αβγδ_ ... _abcd_ ...) made up of planes and straight lines, is to construct the straight lines or _traces σα, σβ, σγ_ ... and the points or _traces σa, σb, σc_....[121] By this means we obtain a new figure composed of straight lines and points lying in the plane _σ_. "_To project from a fixed straight line s_ (the _axis_) a figure _ABCD_ composed of points, is to construct the planes _sA_, _sB_, _sC_.... The figure thus obtained is composed of planes which all pass through the axis _s_. "_To cut by a fixed straight line s_ (a _transversal_) a figure _αβγδ_ ... composed of planes, is to construct the points _sα_, _sβ_, _sγ_.... In this way a new figure is obtained, composed of points all lying on the fixed transversal _s_. "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? [Illustration]