An Essay on the Foundations of Geometry

axioms refer to actual rigid bodies. Accordingly, we find errors in his mathematics, such as the axiom of Monodromy, and the assumption that the measure of curvature must be positive. Nevertheless, the proof of Riemann's arc-formula is extremely able, and has, on the whole, been substantiated by Lie's more thorough investigations. 27. Helmholtz's other writings on Geometry are almost wholly philosophical, and will be discussed at length in Chapter II. For the present, we may pass to the only other important writer of the second period, Beltrami. As his work is purely mathematical, and contains few controverted points, it need not, despite its great importance, detain us long. The "Saggio di Interpretazione della Geometria non-Euclidea[32]," which is principally confined to two dimensions, interprets Lobatchewsky's results by the characteristic method of the second period. It shows, by a development of the work of Gauss and Minding[33], that all the propositions in plane Geometry, which Lobatchewsky had set forth, hold, within ordinary Euclidean space, on surfaces of constant negative curvature. It is strange, as Klein points out[34], that this interpretation, which was known to Riemann and perhaps even to Gauss, should have remained so long without explicit statement. This is the more strange, as Lobatchewsky's "Géométrie Imaginaire" had appeared in Crelle, Vol. XVII.[35], and Minding's article, from which the interpretation follows at once, had appeared in Crelle, Vol. XIX. Minding had shewn that the Geometry of surfaces of constant negative curvature, in particular as regards geodesic triangles, could be deduced from that of the sphere by giving the radius a purely imaginary value ia[36]. This result, as we have seen, had also been obtained by Lobatchewsky for his Geometry, and yet it took thirty years for the connection to be brought to general notice. 28. In Beltrami's Saggio, straight lines are, of course, replaced by geodesics; his coordinates are obtained through a point-by-point correspondence with an auxiliary plane, in which straight lines correspond to geodesics on the surface. Thus geodesics have linear equations, and are always uniquely determined by two points. Distances on the surface, however, are not equal to distances on the plane; thus while the surface is infinite, the corresponding portion of the plane is contained within a certain finite circle. The distance of two points on the surface is a certain function of the coordinates, not the ordinary function of elementary Geometry. These relations of plane and surface are important in connection with Cayley's theory of distance, which we shall have to consider next. If we were to define distance on the plane as that function of the coordinates which gives the corresponding distance on the surface, we should obtain what Klein calls "a plane with a hyperbolic system of measurement (Massbestimmung)" in which Cayley's theory of distance would hold. It is evident, however, that the ordinary notion of distance has been presupposed in setting up the coordinate system, so that we do not really get alternative Geometries on one and the same plane. The bearing of these remarks will appear more fully when we come to consider Cayley and Klein. 29. The value of Beltrami's Saggio, in his own eyes, lies in the intelligible Euclidean sense which it gives to Lobatchewsky's planimetry: the corresponding system of Solid Geometry, since it has no meaning for Euclidean space, is barely mentioned in this work. In a second paper[37], however, almost contemporaneous with the first, he proceeds to consider the general theory of n-dimensional manifolds of constant negative curvature. This paper is greatly influenced by Riemann's dissertation; it begins with the formula for the linear element, and proves from this first, that Congruence holds for such spaces, and next, that they have, according to Riemann's definition, a constant negative measure of curvature. (It is instructive to observe, that both in this and in the former Essay, great stress is laid on the necessity of the Axiom of Congruence.) This work has less philosophical interest than the former, since it does little more than repeat, in a general form, the results which the Saggio had obtained for two dimensions—results which sink, when extended to n dimensions, to the level of mere mathematical constructions. Nevertheless, the paper is important, both as a restoration of negative curvature, which had been overlooked by Helmholtz, and as an analytical treatment of Lobatchewsky's results—a treatment which, together with the Saggio, at last restored to them the prominence they deserved. Third Period. 30. The third period differs radically, alike in its methods and aims, and in the underlying philosophical ideas, from the period which it replaced. Whereas everything, in the second period, turned on measurement, with its apparatus of Congruence, Free Mobility, Rigid Bodies, and the rest, these vanish completely in the third period, which, swinging to the opposite extreme, regards quantity as a perfectly irrelevant category in Geometry, and dispenses with congruence and the method of superposition. The ideas of this period, unfortunately, have found no exponent so philosophical as Riemann or Helmholtz, but have been set forth only by technical mathematicians. Moreover the change of fundamental ideas, which is immense, has not brought about an equally great change in actual procedure; for though spatial quantity is no longer a part of projective Geometry, quantity is still employed, and we still have equations, algebraic transformations, and so on. This is apt to give rise to confusion, especially in the mind of the student, who fails to realise that the quantities used, so far as the propositions are really projective, are mere names for points, and not, as in metrical Geometry, actual spatial magnitudes. Nevertheless, the fundamental difference between this period and the former must strike any one at once. Whereas Riemann and Helmholtz dealt with metrical ideas, and took, as their foundations, the measure of curvature and the formula for the linear element—both purely metrical—the new method is erected on the formulae for transformation of coordinates required to express a given collineation. It begins by reducing all so-called metrical notions—distance, angle, etc.—to projective forms, and obtains, from this reduction, a methodological unity and simplicity before impossible. This reduction depends, however, except where the space-constant is negative, upon imaginary figures—in Euclid, the circular points at infinity; it is moreover purely symbolic and analytical, and must be regarded as philosophically irrelevant. As the question concerning the import of this reduction is of fundamental importance to our theory of Geometry, and as Cayley, in his Presidential Address to the British Association in 1883, formally challenged philosophers to discuss the use of imaginaries, on which it depends, I will treat this question at some length. But first let us see how, as a matter of mathematics, the reduction is effected. 31. We shall find, throughout this period, that almost every important proposition, though misleading in its obvious interpretation, has nevertheless, when rightly interpreted, a wide philosophical bearing. So it is with the work of Cayley, the pioneer of the projective method. The projective formula for angles, in Euclidean Geometry, was first obtained by Laguerre, in 1853. This formula had, however, a perfectly Euclidean character, and it was left for Cayley to generalize it so as to include both angles and distances in Euclidean and non-Euclidean systems alike[38]. Cayley was, to the last, a staunch supporter of Euclidean space, though he believed that non-Euclidean Geometries could be applied, within Euclidean space, by a change in the definition of distance[39]. He has thus, in spite of his Euclidean orthodoxy, provided the believers in the possibility of non-Euclidean spaces with one of their most powerful weapons. In his "Sixth Memoir upon Quantics" (1859), he set himself the task of "establishing the notion of distance upon purely descriptive principles." He showed that, with the ordinary notion of distance, it can be rendered projective by reference to the circular points and the line at infinity, and that the same is true of angles[40]. Not content with this, he suggested a new definition of distance, as the inverse sine or cosine of a certain function of the coordinates; with this definition, the properties usually known as metrical become projective properties, having reference to a certain conic, called by Cayley the Absolute. (The circular points are, analytically, a degenerate conic, so that ordinary Geometry forms a particular case of the above.) He proves that, when the Absolute is an imaginary conic, the Geometry so obtained for two dimensions is spherical Geometry. The correspondence with Lobatchewsky, in the case where the Absolute is real, is not worked out: indeed there is, throughout, no evidence of acquaintance with non-Euclidean systems. The importance of the memoir, to Cayley, lies entirely in its proof that metrical is only a branch of descriptive Geometry. 32. The connection of Cayley's Theory of Distance with Metageometry was first pointed out by Klein[41]. Klein showed in detail that, if the Absolute be real, we get Lobatchewsky's (hyperbolic) system; if it be imaginary, we get either spherical Geometry or a new system, analogous to that of Helmholtz, called by Klein elliptic; if the Absolute be an imaginary point-pair, we get parabolic Geometry, and if, in particular, the point-pair be the circular points, we get ordinary Euclid. In elliptic Geometry, two straight lines in the same plane meet in only one point, not two as in Helmholtz's system. The distinction between the two kinds of Geometry is difficult, and will be discussed later. 33. Since these systems are all obtained from a Euclidean plane, by a mere alteration in the definition of distance, Cayley and Klein tend to regard the whole question as one, not of the nature of space, but of the definition of distance. Since this definition, on their view, is perfectly arbitrary, the philosophical problem vanishes—Euclidean space is left in undisputed possession, and the only problem remaining is one of convention and mathematical convenience[42]. This view has been forcibly expressed by Poincaré: "What ought one to think," he says, "of this question: Is the Euclidean Geometry true? The question is nonsense." Geometrical axioms, according to him, are mere conventions: they are "definitions in disguise[43]." Thus Klein blames Beltrami for regarding his auxiliary plane as merely auxiliary, and remarks that, if he had known Cayley's Memoir, he would have seen the relation between the plane and the pseudosphere to be far more intimate than he supposed[44]. A view which removes the problem entirely from the arena of philosophy demands, plainly, a full discussion. To this discussion we will now proceed. 34. The view in question has arisen, it would seem, from a natural confusion as to the nature of the coordinates employed. Those who hold the view have not adequately realised, I believe, that their coordinates are not spatial quantities, as in metrical Geometry, but mere conventional signs, by which different points can be distinctly designated. There is no reason, therefore, until we already have metrical Geometry, for regarding one function of the coordinates as a better expression of distance than another, so long as the fundamental addition-equation[45] is preserved. Hence, if our coordinates are regarded as adequate for all Geometry, an indeterminateness arises in the expression of distance, which can only be avoided by a convention. But projective coordinates—so our argument will contend—though perfectly adequate for all projective properties, and entirely free from any metrical presupposition, are inadequate to express metrical properties, just because they have no metrical presupposition. Thus where metrical properties are in question, Beltrami remains justified as against Klein; the reduction of metrical to projective properties is only apparent, though the independence of these last, as against metrical Geometry, is perfectly real. 35. But what are projective coordinates, and how are they introduced? This question was not touched upon in Cayley's Memoir, and it seemed, therefore, as if a logical error were involved in using coordinates to define distance. For coordinates, in all previous systems, had been deduced from distance; to use any existing coordinate system in defining distance was, accordingly, to incur a vicious circle. Cayley mentions this difficulty in a note, where he only remarks, however, that he had regarded his coordinates