formulated his theory of parallels. His successors were not satisfied with the theory-and no doubt he himself must have been unsatisfied with it; but they never dared break loose from his authority, and recast the theory in such a way as to remedy its principal defect, viz. the complex and artificial character of the 12th Axiom, a proposition which expresses a process of thought so utterly different in kind from those to which the other fundamental propositions give expression. They admitted it as necessary, and yet not as a necessity of thought; as needing demonstration, and yet as apparently indemonstrable. Now if we admit, with Lobatschewsky and others, that Euclid's 12th Axiom is not a necessity of thought, is neither a real axiom nor a proposition derivable from real axioms; and if we understand Lobatschewsky's hypothesis as he apparently intended it to be understood, viz. as a real alternative to, and as in real contradiction with, Euclid's 12th Axiom; how does the case then stand? We are none the less obliged to admit that the results which Lobatschewsky deduces from his assumption are logical results, and this (remembering that we take Lobatschewsky to be dealing with the same geometrical entities as Euclid) is to admit that there are two possible but mutually exclusive planimetries of the surface which Euclid calls plane. Moreover, this conclusion is entirely independent of whether Lobatschewsky did or did not intend that which he appears to have intendedhis work supplies us with the data for the conclusion, and we cannot but draw it whether he intended it or not. Now set against this the received view, as expressed by Stephen Smith, that Lobatschewsky's hypothesis is in effect to assume that the plane has negative curvature. It was reserved for an Italian mathematician, Beltrami-as Professor Smith goes on to sayto show that the plane geometry of Lobatschewsky is identical with the geometry of a pseudo-spherical surface, i. e. of a surface of constant negative curvature. That is to say, the metrical relations of the sides and angles of a rectilinear triangle, strictly so called, derived by Lobatschewsky from his anti-Euclidean assumption, are identical with the metrical relations of the sides and angles of a rectilinear triangle, loosely so called, on a surface of constant negative curvature.

Now if we cannot admit that the relations of the sides and angles of a geodesic triangle on a surface of zero curvature are

identical with the relations of the sides and angles of a geodesic triangle on a surface of constant negative curvature, we cannot admit the hypothesis from which this identity flows. Euclid's 12th Axiom is thus a necessity of thought; that is, it is either what Cayley, in Playfair's version of it, found it to be a selfevident proposition, not needing demonstration; or, as I have ventured to suggest, a proposition which derives from the real axioms of direction which Euclid should have given us, but did not. And, on this view, Lobatschewsky's hypothesis should not be described as in effect equivalent to the assumption that a plane has negative curvature, which is an ambiguous piece of phraseology, but as in effect equivalent to giving an extension of meaning to the term ' plane', so that it shall mean the surface of constant negative curvature as well as that of zero curvature. And such an extension of the term, even though it may tend more to confusion than to clearness of thought, cannot be condemned as purely arbitrary, for the assigned community of name does at least correspond to a property common to the two surfaces, viz. that uniformity which is defined in mathematical language as the constancy of the measure of curvature. But this extension of meaning once admitted, analogy in conception drives us a step further on the road of metaphor, and we are committed to calling the surface of constant positive measure of curvature, as well as that of constant negative measure of curvature, a plane.

If we thus extend the meaning of the term plane, we can no doubt go on to speak of Euclid's planimetry, of Lobatschewsky's planimetry, of Riemann's planimetry; but the conceptions thus denoted do not involve that of different 'systems' of geometry, the admission of any one of which excludes the others. These planimetries' are merely different branches of geometry. Riemann's geometry, in two dimensions, as Mr. Poincaré remarks,1 does not differ from spherical geometry, which is a branch of 'ordinary' geometry. And here evidently the path we have been following comes to an abrupt end: we are left in contemplation of three branches of geometry, but without the faintest indication of a departure from, or modification of, the ordinary notion of space, or of an attribution of meaning to the expression systems' of geometry. According to the metageometer, however, we have to distinguish between (1) surfaces of constant 1 La Science et l'Hypothèse, p. 55.

positive, and of constant negative, measure of curvature in 'Euclidean' space; and (2) the plane in 'elliptic' and in' hyperbolic' space.1 The condition of our ability to make this distinction can of course only lie in our ability to conceive these different kinds of space. It is to the explanations of this development of the notion of space that we must now turn.

1 See 'Geometry, non-Euclidean 'in the Ency. Brit., vol. xxviii, p. 670.

CHAPTER XVII

POPULAR EXPOSITIONS OF METAGEOMETRY

Helmholtz's explanation of a means by which we may attain variety in the conception of Space.-The way suggested is through analogy with the differing space-conceptions of logical' two-dimensional beings' inhabiting different kinds of surface-worlds.-The space-conceptions of these figurative beings are, however, nothing but geometrical abstractions from our own spatial experience, clothed in allegorical language; and the ground of the analogy is thus itself an illusion.-Lotze's ineffective attack on Metageometry.-Clifford's attempt to carry the conception of Elementary Flatness from the surface to space. The false analogy involved in Clifford's reasoning.

LET us suppose that the phrase 'species of space and the systems of geometry relevant to these several species' does briefly indicate, for some few human beings, a certain real process of conception, of which the great mass of human beings have no cognizance. For this great mass or-let us go so far-even for the educated and intelligent man, the expression species of space' is simply devoid of meaning. In saying that there are different kinds of space, metageometers of course do not intend to assert that different kinds of space 'exist'; but that we are able to conceive different kinds of space, and that we do not know which of these differently conceived spaces is 'our' space, or the space which 'exists'.

There are two ways in which the non-Euclidean geometer or, as we might perhaps call him, the more-than-Euclidean geometer, explains how we can attain to a generalized conception of space and a correspondingly generalized conception of geometry. There is the academical and mathematical way taken by Riemann in his essay Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (1854),1 and there is the simpler and less technical way, better suited than Riemann's to the mind not highly trained in mathematical methods of expression. The latter mode of explanation is employed by Helmholtz in his discussion of the Origin and Significance of Geometrical Axioms.2 I do not know 1 Not published till 1867, after Riemann's death.

2 Popular Lectures on Scientific Subjects, translated by E. Atkinson, Ph.D., F.C.S., 1881.

whether he was the original inventor of it, but it has certainly met with approval, for it has been repeated with insignificant variations by almost every mathematician who has sought, in current phraseology, to make clear to an intelligent inquirer the non-Euclidean doctrine. And, as will I think be seen further on, the purely analytical (in the mathematical sense of the term) part of Riemann's exposition really depends for its geometrical relevance upon whether it is or is not a fact that the ordinary notion of space is susceptible of generalization or transformation, upon whether we can or cannot conceive space as having other attributes or 'properties' than those we commonly assign to it.

Helmholtz's explanation is rather long, but as it has obtained wide currency, though not altogether without criticism, it will be as well to quote it in extenso :

'Let us, as we logically may, suppose reasoning beings of only two dimensions to live and move on the surface of some solid body. We will assume that they have not the power of perceiving anything outside this surface, but that upon it they have perceptions similar to ours. If such beings worked out a geometry, they would of course assign only two dimensions to their space. They would ascertain that a point in moving describes a line, and that a line in moving describes a surface. But they could as little represent to themselves what further spatial construction would be generated by a surface moving out of itself, as we can represent what would be generated by a solid moving out of the space we know. By the much abused expression "to represent or "to be able to think how something happens" I understand and I do not see how anything else can be understood by it without loss of all meaningthe power of imagining the whole series of sensible impressions that would be had in such a case. Now as no sensible impression is known relating to such an unheard-of event, as the movement to a fourth dimension would be to us, or as a movement to our third dimension would be to the inhabitants of a surface, such a "representation " is as impossible as the "representation" of colours would be to one born blind, if a description of them in general terms could be given to him.

Our surface-beings would also be able to draw shortest lines in their superficial space. These would not necessarily be straight lines in our sense, but what are technically called geodetic lines of the surface on which they live; lines such as are described by a tense thread laid along the surface, and which can slide on it freely. I will henceforth speak of such lines as the straightest lines of any particular surface or given space, so as to bring out