An Essay on the Foundations of Geometry

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. 112. We have next to consider the quadrilateral construction[124]. This has a double purpose: first, to define the important special case known as a harmonic range; and secondly, to afford an unambiguous and exhaustive method of assigning different numbers to different points. This last method has, again, a double purpose: first, the purpose of giving a convenient symbolism for describing and distinguishing different points, and of thus affording a means for the introduction of analysis; and secondly, of so assigning these numbers that, if they had the ordinary metrical significance, as distances from some point on the numbered straight line, they would yield –1 as the anharmonic ratio of a harmonic range, and that, if four points have the same anharmonic ratio as four others, so have the corresponding numbers. This last purpose is due to purely technical motives: it avoids the confusion with our preconceptions which would result from any other value for a harmonic range; it allows us, when metrical interpretations of projective results are desired, to make these interpretations without tedious numerical transformations, and it enables us to perform projective transformations by algebraical methods. At the same time, from the strictly projective point of view, as observed above, the numbers introduced have a purely conventional meaning; and until we pass to metrical Geometry, no reason can be shown for assigning the value –1 to a harmonic range. With this preliminary, let us see in what the quadrilateral construction consists. 113. A harmonic range, in elementary Geometry, is one whose anharmonic ratio is –1, or one in which the three segments formed by the four points are in harmonic progression, or again, one in which the ratio of the two internal segments is equal to the ratio of the two external segments. If a, b, c, d be the four points, it is easily seen that these definitions are equivalent to one another: they give respectively: ab bc / ad dc = – 1 , 1 ab – 1 ac = 1 ac – 1 ad , and ab bc = ad cd . But as they are all quantitative, they cannot be used for our present purpose. Nor are any definitions which involve bisection of lines or angles available. We must have a definition which proceeds entirely by the help of straight lines and points, without measurement of distances or angles. Now from the above definitions of a harmonic range, we see that a, b, c, d have the same anharmonic ratio as c, b, a, d. This gives us the property we require for our definition. For it shows that, in a harmonic range, we can find a projective transformation which will interchange a and c. This is a necessary and sufficient condition for a harmonic range, and the quadrilateral construction is the general method for giving effect to it. Given any three points A, B, D in one straight line, the quadrilateral construction finds the point C harmonic to A with respect to B, D by the following method: Take any point O outside the straight line ABD, and join it to B and D. Through A draw any straight line cutting OD, OB in P and Q. Join DQ, BP, and let them intersect in R. Join OR, and let OR meet ABD in C. Then C is the point required. To prove this, let DRQ meet OA in T, and draw AR, meeting OD in S. Then a projective transformation of A, B, C, D from R on to OD gives the points S, P, O, D, which, projected from A on to DQ, give R, Q, T, D. But these again, projected from O on to ABD, give C, B, A, D. Hence A, B, C, D can be projectively transformed into C, B, A, D, and therefore form a harmonic range. From this point, the proof that the construction is unique and general follows simply[125]. The introduction of numbers, by this construction, offers no difficulties of principle—except, indeed, those which always attend the application of number to continua—and may be studied satisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle of it is, to assign the numbers 0, 1, ∞ to A, B, D and therefore the number 2 to C, in order that the differences AB, AC, AD may be in harmonic progression. By taking B, C, D as a new triad corresponding to A, B, D, we find a point harmonic to B with respect to C, D and assign to it the number 3, and so on. In this way, we can obtain any number of points, and we are sure of having no number and no point twice over, so that our coordinates have the essential property of a unique correspondence with the points they denote, and vice versa. 114. The point of importance in the above construction, however, and the reason why I have reproduced it in detail, is that it proceeds entirely by means of the general principles of transformation enunciated above. From this stage onwards, everything is effected by means of the two fundamental ideas we have just discussed, and everything, therefore, depends on our general principle of projective equivalence. This principle, as regards two dimensions, may be stated more simply than in the passage quoted from Cremona. It starts, in two dimensions, from the following definitions: To project the points A, B, C, D ... from a centre O, is to construct the straight lines OA, OB, OC, OD.... To cut a number of straight lines a, b, c, d ... by a transversal s, is to construct the points sa, sb, sc, sd....[126] The successive application of these two operations, provided the original figure consisted of points on one straight line or of straight lines through one point, gives a figure projectively indistinguishable from the former figure; and hence, by extension, if any points in one straight line in the original figure lie in one straight line in the derived figure, and reciprocally for straight lines through points, the two operations have given projectively similar figures. This general principle may be regarded as consisting of two parts, according to the order of the operations: if we begin with projection and end with section, we transform a figure of points into another figure of points; by the converse order, we transform a figure of lines into another figure of lines. 115. Before we can be clear as to the meaning of our principle, we must have some notion as to our definition of points and straight lines. But this definition, in projective Geometry, cannot be given without some discussion of the principle of duality, the mathematical form of the philosophical circle involved in geometrical definitions. Confining ourselves for the moment to two dimensions, the principle asserts, roughly speaking, that any theorem, dealing with lines through a point and points on a line, remains true if these two terms, wherever they occur, are interchanged. Thus: two points lie on one straight line which they completely determine; and two straight lines meet in one point, which they completely determine. The four points of intersection of a transversal with four lines through a point have an anharmonic ratio independent of the particular transversal; and the four lines joining four points on one straight line to a fifth point have an anharmonic ratio independent of that fifth point. So also our general principle of projective transformation has two sides: one in which points move along fixed lines, and one in which lines turn about fixed points. This duality suggests that any definition of points must be effected by means of the straight line, and any definition of the straight line must be effected by means of points. When we take the third dimension into account, it is true, the duality is no longer so simple; we have now to take account also of the plane, but this only introduces a circle of three terms, which is scarcely preferable to a circle of two terms. We now say: Three points, or a line and a point, determine a plane: but conversely, three planes, or a line and plane, determine a point. We may regard the straight line as a relation between two of its points, but we may also regard the point as a relation between two straight lines through it. We may regard the plane as a relation between three points, or between a point and a line, but we may also regard the point as a relation between three planes, or between a line and a plane, which meet in it. 116. How are we to get outside this circle? The fact is that, in pure Geometry, we cannot get outside it. For space, as we shall see more fully hereafter, is nothing but relations; if, therefore, we take any spatial figure, and seek for the terms between which it is a relation, we are compelled, in Geometry, to seek these terms within space, since we have nowhere else to seek them, but we are doomed, since anything purely spatial is a mere relation, to find our terms melting away as we grasp them. Thus the relativity of space, while it is the essence of the principle of duality, at the same time renders impossible the expression of that principle, or of any other principle of pure Geometry, in a manner which shall be free from contradictions. Nevertheless, if we are to advance at all with our analysis of geometrical reasoning and with our definitions of lines and points, we must, for a while, ignore this contradiction; we must argue as though it did not exist, so as to free our science from any contradictions which are not inevitable. 117. In accordance with this procedure, then, let us define our points as the terms of spatial relations, regarding whatever is not a point as a relation between points. What, on this view, must our points be taken to be? Obviously, if extension is mere relativity, they must be taken to contain no extension; but if they are to supply the terms for spatial relations, e.g. for straight lines, these relations must exhibit them as the terms of the figures they relate. In other words, since what can really be taken, without contradiction, as the term of a spatial relation, is unextended, we must take, as the term to be used in Geometry, where we cannot go outside space, the least spatial thing which Geometry can deal with, the thing which, though in space, contains no space; and this thing we define as the point[127]. Neglecting, then, the fundamental contradiction in this definition, the rest of our definitions follow without difficulty. The straight line is the relation between two points, and the plane is the relation between three. These definitions will be argued and defended at length in section B of this Chapter[128], where we can discuss at the same time the alternative metrical definitions;