a postulate necessary to the metageometer; without it the doctrine vanishes in a mist of words and algebraic symbols. Only there is here a difference of opinion among metageometers themselves. Some of them (the late well-known astronomer, Simon Newcomb, was one) suppose that a three-dimensional 'curved' space is curved in a four-dimensional 'plane' space. Whence does such a supposition arise? To all appearance from the impossibility, for them, of attaching any meaning to curvature of space save 'by analogy' with curvature of surface. Thus easily do we conceive four-dimensional space. On the other hand there are metageometers who will by no means admit the necessity of this supposition. Indeed, it is the very absence of reference to a higher Euclidean space which is chiefly interesting about the non-Euclidean spaces.'1 Space, according to the latter authorities, can very well be conceived to have a measure of curvature which involves no reference to any higher dimensions. Mr. Russell gives a very ingenious explanation of how this is to be done; 2 only he proceeds on the supposition, which may be allowed to stand for the moment,' that space is a particular case of Riemann's manifold; he does not consider what is to happen when, after having allowed the supposition to stand for a moment, we have to reject it as invalid.3 But apart from this, the explanation appears to me to involve a fallacy which touches the foundation of the metageometrical doctrine, and it will therefore be time well spent to consider it as briefly as may be consistent with clearness. The argument is that Gauss's researches on the curvature of surfaces puts us in possession of a conception of measure of surface curvature which involves no reference to anything outside the surface: measure of curvature from within, as Mr. Russell calls it; that we thus see how an analogous conception for space might be formed; and that Riemann's dissertation actually constructs this conception for us. 1 Ency. Brit., vol. xxviii, p. 670. • Foundations of Geometry, pp. 19–21. • The supposition is admitted on the ground that the 'analytical conception of space' with which Riemann deals is a particular case of the continuous manifold. But if the 'analytical conception of space' is not a question-begging phrase, then the conception of a continuous manifold really does include space as a particular case, and there ought to be no need for the reservation which Mr. Russell makes. The simplest expression for the measure of curvature of a surface at any point of the surface involves reference to the radii of curvature of the surface at the point. But, as already mentioned, Gauss obtained a formula, for the infinitesimal distance at any point of a surface, which contains no terms implying measure of anything not in the surface; and from this formula alone he constructs an expression for the measure of curvature of the surface at the point. The measure of curvature of a surface thus appears as an inherent or intrinsic property of the surface or-as Klein puts it-of the length of the lineelement, and no reference to space 'outside' the surface is involved in it. This, indeed, is a marvellous result when we reflect that Gauss demonstrates that this expression for the measure of curvature is equivalent to the previous expression 1/R,R2; a demonstration which necessarily implies identity of the unit of length on and off the surface. The truth is that on a surface of variable curvature there can be no unit of length intrinsic to the surface. Mr. Russell himself remarks that it is logically impossible to set up a precise co-ordinate system, in which the co-ordinates represent spatial magnitudes, without the axiom of Free Mobility', and this involves constancy of the measure of surface curvature. Thus Gauss's expression for the measure of curvature will only be really free from reference to the third dimension when we are dealing with a surface of constant measure of curvature'. But we cannot deal with ', i. e. conceive, merely one surface of invariable curvature: this would be to deprive the term 'measure' itself of any meaning in relation to surface curvature. Nor is measure of curvature conceivable as constant unless it is also conceived as variable. And, so far as the argument is concerned, it makes not the slightest difference whether we have in mind variability of curvature from point to point of the same surface, or variety of surfaces of different, but each of constant, curvature. Gauss's formula for the measure of curvature at any point of a surface whose curvature varies continuously from point to point, is the formula for the measure of curvature of any surface whose curvature is constant; and just as in the former case there is no unit of length intrinsic to the surface, so in the latter is there no unit of length common to all the surfaces, other than a unit not intrinsic to any one of them. It is therefore as impossible in the one case as it is in the other to be really rid of implicit reference to the third dimension. Thus when 'by analogy' we come to consider ' measure of curvature of space'-a phrase which has somehow to be accepted as really significant if the whole theory is not to lapse into the inane-it is no wonder that some metageometers are driven to suppose a space of higher dimensions in which 'our' space may be curved. It is interesting to remark that this fallacy concerning measure of surface curvature which Mr. Russell appears to have overlooked in his work on the Foundations of Geometry confronts him under a wider aspect in the article on Non-Euclidean Geometry in the Encyclopaedia Britannica. In that article he sees with perfect clearness that if the measure of curvature or constant of a space is an inherent property of the space, there can be no quantitative relation between the constants of different spaces, the question thus arising, What is the relation between different spaceconstants? It is a tight corner. Mr. Russell admits the question to be 'somewhat perplexing'. To find an issue from a blind alley without retracing the steps that have led us into it certainly is perplexing. The tentative answer which Mr. Russell gives to this question appears to me to be no less perplexing than the question itself. Yet the answer is in a sense easy, a foregone conclusion. Since space-constants can have no quantitative relation, they can evidently only differ qualitatively—like the shades of red, as Mr. Russell suggests. It is a simple answer, but a solution of the question it is not; for to accept it is to make chaos of the relation of the general to the particular. The general notion of a space-constant is, according to this view, purely a qualitative notion. The particulars subsumed under it, i.e. the different space-constants, cannot therefore be other than purely qualitative. Yet they are also necessarily quantitative notions, since the space-constant of any non-Euclidean space has a particular length, that is, has some quantitative relation to a selected unit of length in that space. Thus these particular notions are at once subsumable and not subsumable under the general notion. When we can consider any particular thing as qualitatively or quantitatively differing from any others, the general notion which comprises the particulars must also have a qualitative and a quantitative aspect. Take, for instance, the case of red and the shades of red. The relation of general to particular is here perfectly clear, no matter whether we regard red as a common or general quality and its shades as particular cases of the quality, or whether we regard the shades as the wave-lengths x1, X2, X, xn, and red as the series x of wave-lengths, where x may have any of these particular values. The explanation, in short, involves our forming a conception under conditions which do not admit of our forming one. Would it not be more rational to retrace our steps? Thus if we had to choose between the metageometers who see in the measure of curvature of space an inherent property of space and those who see in it a relation to a space of higher dimensions, it would be simpler to take our stand with the latter, even though we should thus, according to Mr. Russell, lose the chief interest of non-Euclidean spaces. A four-dimensional space has at least the great advantage that it needs no explanation. You must simply accept the phrase either as expressing a real conception or as evidence of a mystical delusion. CHAPTER XIX CAYLEY'S THEORY OF DISTANCE. GEOMETRY AND MENSURATION Cayley's Theory, though non-Euclidean, does not imply non-Euclidean spaces.—It is in expression a Theory of Distance, but in conception it is a theory of different systems of descriptive equivalence, depending upon the choice of the fundamental figure which Cayley calls the Absolute. Astronomical observations as a test of geometrical theory.-Illusiveness of this test.-Meaning of the rectilinear propagation of light-rays.-Interpretation, from the Euclidean and from the non-Euclidean standpoints, of an assumed non-Euclidean result of measurement.-The ordinary notion of Direction has no logical foothold in non-Euclidean space-conceptions; it is nevertheless employed in the alleged construction of these spaceconceptions. CAYLEY'S views on fundamental questions connected with geometry have been cited on several occasions in the course of this work. We may not always agree with them, but, with one exception, they are at least readily comprehensible. The exception is the standpoint from which he regards the axiom of parallels. When one becomes acquainted with the nature of his work on non-Euclidean systems of geometry however, this standpoint, though perhaps not as perfectly clear as one could wish it to be, at least grows more intelligible. Cayley may be regarded as the founder of what Klein has called the 'third period' in the development of non-Euclidean geometry; Gauss, Lobatschewsky, and Bolyai being the principal figures in the first period, Riemann and Helmholtz in the second. In the first period the foundation of the debate is plainly the alleged, and generally acknowledged, incertitude of Euclid's axiom of parallels; this and nothing more: are there many parallels, through a given point, to a given straight line, or is there one only? The alternative system is developed simply and unambiguously as a logical consequence of admitting the first of the two hypotheses; and the possibility of this development is held to show the indemonstrability of the socalled 12th Axiom from the other indubitable premisses of the elements. There is no question, there is at least no overt ques |