An Essay on the Foundations of Geometry

Section A. THE AXIOMS OF PROJECTIVE GEOMETRY. 102. Projective Geometry proper, as we saw in Chapter I., does not employ the conception of magnitude, and does not, therefore, require those axioms which, in the systems of the second or metrical period, were required solely to render possible the application of magnitude to space. But we saw, also, that Cayley's reduction of metrical to projective properties was purely technical and philosophically irrelevant. Now it is in metrical properties alone—apart from the exception to the axiom of the straight line, which itself, however, presupposes metrical properties[116]—that non-Euclidean and Euclidean spaces differ. The properties dealt with by projective Geometry, therefore, in so far as these are obtained without the use of imaginaries, are properties common to all spaces. Finally, the differences which appear between the Geometries of different spaces of the same curvature—e.g. between the Geometries of the plane and the cylinder—are differences in projective properties[117]. Thus the necessity which arises, in metrical Geometry, for further qualifications besides those of constant curvature, disappears when our general space is defined by purely projective properties. 103. We have good ground for expecting, therefore, that the axioms of projective Geometry will be the simplest and most complete expression of the indispensable requisites of any geometrical reasoning: and this expectation, I hope, will not be disappointed. Projective Geometry, in so far as it deals only with the properties common to all spaces, will be found, if I am not mistaken, to be wholly à priori, to take nothing from experience, and to have, like Arithmetic, a creature of the pure intellect for its object. If this be so, it is that branch of pure mathematics which Grassmann, in his Ausdehnungslehre of 1844, felt to be possible, and endeavoured, in a brilliant failure, to construct without any appeal to the space of intuition. 104. But unfortunately, the task of discovering the axioms of projective Geometry is far from easy. They have, as yet, found no Riemann or Helmholtz to formulate them philosophically. Many geometers have constructed systems, which they intended to be, and which, with sufficient care in interpretation, really are, free from metrical presuppositions. But these presuppositions are so rooted in all the very elements of Geometry, that the task of eliminating them demands a reconstruction of the whole geometrical edifice. Thus Euclid, for example, deals, from the start, with spatial equality—he employs the circle, which is necessarily defined by means of equality, and he bases all his later propositions on the congruence of triangles as discussed in Book I.[118] Before we can use any elementary proposition of Euclid, therefore, even if this expresses a projective property, we have to prove that the property in question can be deduced by projective methods. This has not, in general, been done by projective geometers, who have too often assumed, for example, that the quadrilateral construction—by which, as we saw in Chap. I., they introduce projective coordinates—or anharmonic ratio, which is primâ facie metrical, could be satisfactorily established on their principles. Both these assumptions, however, can be justified, and we may admit, therefore, that the claims of projective Geometry to logical independence of measurement or congruence are valid. Let us see, then, how it proceeds. 105. In the first place, it is important to realize that when coordinates are used, in projective Geometry, they are not coordinates in the ordinary metrical sense, i.e. the numerical measures of certain spatial magnitudes. On the contrary, they are a set of numbers, arbitrarily but systematically assigned to different points, like the numbers of houses in a street, and serving only, from a philosophical standpoint, as convenient designations for points which the investigation wishes to distinguish. But for the brevity of the alphabet, in fact, they might, as in Euclid, be replaced by letters. How they are introduced, and what they mean, has been discussed in Chapter I. Here we have only to repeat a caution, whose neglect has led to much misunderstanding. 106. The distinction between various points, then, is not a result, but a condition, of the projective coordinate system. The coordinate system is a wholly extraneous, and merely convenient, set of marks, which in no way touches the essence of projective Geometry. What we must begin with, in this domain, is the possibility of distinguishing various points from one another. This may be designated, with Veronese, as the first axiom of Geometry[119]. How we are to define a point, and how we distinguish it from other points, is for the moment irrelevant; for here we only wish to discover the nature of projective Geometry, and the kind of properties which it uses and demonstrates. How, and with what justification, it uses and demonstrates them, we will discuss later. 107. Now it is obvious that a mere collection of points, distinguished one from another, cannot found a Geometry: we must have some idea of the manner in which the points are interrelated, in order to have an adequate subject-matter for discussion. But since all ideas of quantity are excluded, the relations of points cannot be relations of distance in the ordinary sense, nor even, in the sense of ordinary Geometry, anharmonic ratios, for anharmonic ratios are usually defined as the ratios of four distances, or of four sines, and are thus quantitative. But since all quantitative comparison presupposes an identity of quality, we may expect to find, in projective Geometry, the qualitative substrata of the metrical superstructure. And this, we shall see, is actually the case. We have not distance, but we have the straight line; we have not quantitative anharmonic ratio, but we have the property, in any four points on a line, of being the intersections with the rays of a given pencil. And from this basis, we can build up a qualitative science of abstract externality, which is projective Geometry. How this happens, I shall now proceed to show. 108. All geometrical reasoning is, in the last resort, circular: if we start by assuming points, they can only be defined by the lines or planes which relate them; and if we start by assuming lines or planes, they can only be defined by the points through which they pass. This is an inevitable circle, whose ground of necessity will appear as we proceed. It is, therefore, somewhat arbitrary to start either with points or with lines, as the eminently projective principle of duality mathematically illustrates; nevertheless we will elect, with most geometers, to start with points[120]. We suppose, therefore, as our datum, a set of discrete points, for the moment without regard to their interconnections. But since connections are essential to any reasoning about them as a system, we introduce, to begin with, the axiom of the straight line. Any two of our points, we say, lie on a line which those two points completely define. This line, being determined by the two points, may be regarded as a relation of the two points, or an adjective of the system formed by both together. This is the only purely qualitative adjective—as will be proved later—of a system of two points. Now projective Geometry can only take account of qualitative adjectives, and can distinguish between different points only by their relations to other points, since all points, per se, are qualitatively similar. Hence it comes that, for projective Geometry, when two points only are given, they are qualitatively indistinguishable from any two other points on the same straight line, since any two such other points have the same qualitative relation. Reciprocally, since one straight line is a figure determined by any two of its points, and all points are qualitatively similar, it follows that all straight lines are qualitatively similar. We may regard a point, therefore, as determined by two straight lines which meet in it, and the point, on this view, becomes the only qualitative relation between the two straight lines. Hence, if the point only be regarded as given, the two straight lines are qualitatively indistinguishable from any other pair through the point. 109. The extension of these two reciprocal principles is the essence of all projective transformations, and indeed of all projective Geometry. The fundamental operations, by which figures are projectively transformed, are called projection and section. The various forms of projection and section are defined in Cremona's "Projective Geometry," Chapter I., from which I quote the following account. "To project from a fixed point S (the centre of projection) a figure (ABCD ... abcd ...) composed of points and straight lines, is to construct the straight lines or projecting rays SA, SB, SC, SD, ... and the planes (projecting planes) Sa, Sb, Sc, Sd, ... We thus obtain a new figure composed of straight lines and planes which all pass through the centre S. "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? 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,