all the other fundamental propositions are definitions of geometrical abstractions.

Euclid's 12th Axiom, the axiom of parallels, is a proposition which is complex not only in expression but also from a logical point of view. It can be broken up logically into two propositions, the second one of which is geometrically equivalent to the converse of the first. These converse propositions I hold to be axioms in the sense already defined. The process of thought which they formulate is identical with the process of thought which the axioms of magnitude formulate. They are axioms of direction which Euclid ought to have given us in place of his 12th Axiom, which is, indeed, as Cayley says, self-evident-like those propositions the proof of which the Epicureans deridedbut is not axiomatic.

Euclid, according to the modern view, makes use in the Elements of an assumption which he omits to state, viz. that figures can be freely moved without change of shape or size. It was contended, however, that Euclid with perfect consistency might have repudiated this interpretation of his procedure in demonstration, provided the authorities on Euclid's text are right in their opinion that the so-called Axiom 8 was added by later editors. However this may be, whatever Euclid might have had to say on the subject, an examination of the analyses of this supposed assumption, given respectively by the late Professor Clifford and by Mr. Bertrand Russell, leads to the conclusion that this assumption is relevant only to problems of mensuration. In geometry, where we make abstraction of the conceptions of cause and effect, physical interaction, 'non-geometrical circumstances '—to use Clifford's expression-the statement that figures can be moved freely without change of shape or size is, as an assumption, meaningless; it merely defines a constituent of the notion of metrical relation, i. e. 'identical' equality or congruence.

There are thus, in geometry, four axioms (or eight, if the point of view taken in the note to chap. xiii is admitted); one pair relevant to the notion of magnitude, the other to the notion of direction-the two most fundamental notions of geometry. In each case the pair is a pair of converse propositions. The other premisses of geometry are propositions which define the fundamental geometrical abstractions, or are these abstractions pre

sented as 'indefinables'. Assumption, in the ordinary sense of the word, has no relevance to geometry

With chap. xvi we pass from what may be called the positive side of the argument respecting Euclid to the negative side. No matter how convincing a case has been, or might be, made for the apodeictic certainty of ordinary geometry, it must necessarily fall to pieces if, as is asserted to be the case, other systems of geometry are really conceivable, provided of course that the term 'geometry' is throughout employed in the same sense. The question is, then, are other systems of geometry really conceivable ?

Comparing what are called the mutually exclusive planimetries of Euclid and of Lobatschewsky, it was pointed out that the latter only contradicts the conclusion affirmed in Euclid's 12th Axiom provided his own affirmation is relevant to the data of Euclid's proposition, that is, relevant to the surface and line which Euclid calls respectively plane and straight, and that this appears to have been Lobatschewsky's intention. In that case, admitting the premiss substituted by Lobatschewsky for Euclid's —which, save as a mere exercise in logic, we can only do if we consider Euclid's 12th Axiom to be not apodeictic-Lobatschewsky establishes an alternative planimetrical system to Euclid's, where' plane' has the same meaning in the two systems. But if it cannot be admitted as possible that we can have two mutually exclusive sets of metrical relations for Euclid's plane, this is equivalent to admitting that Euclid's 12th Axiom is apodeictic; and in that case Lobatschewsky establishes a selfconsistent metrical system which rests upon a false premiss. On the other hand, if it is held that Lobatschewsky's premiss is relevant to some other surface than that which Euclid calls the plane, then Lobatschewsky does not really contradict Euclid, he does not establish a different system of geometry, but he gives us the geometry of a surface of constant negative measure of curvature. We can escape from this argument only by falling back upon the meaningless statement that Euclid assumes the existence' of the surface of zero curvature, and that Lobatschewsky rejects this assumption'. But the question assumes a totally different aspect when it is alleged that these two geometries, and subsequently the third geometry of Riemann, arise respectively from three different conceptions of space, and that

we have thus to distinguish between the two notions: (1) the sphere and the pseudo-sphere in 'Euclidean' space, (2) the plane in each of the two 'non-Euclidean' spaces.

The significance of the metageometrical doctrine thus depends upon whether we can in fact conceive different kinds of space. But neither in the popular expositions of Helmholtz and Clifford, nor in the academical dissertation of Riemann, is there any evidence that this is really possible, while on the other hand there is in all of them evidence of real confusions of thought. Helmholtz's explanation confounds what may briefly be described as the logical treatment of an allegory, with a development of analogy in conception; and Clifford's rests upon what appears to be simply a false analogy. Riemann, as we saw, and as seems to be admitted by at least some metageometers, fails to construct a general conception of externality which includes as a particular case of it the ordinary notion of space. The conception which he calls that of an n-fold extended magnitude, and to which his purely mathematical argument is relevant, is the conception of a general system of purely abstract quantitative relations, or of a series of particular systems of such relations. He calls these systems of quantitative relations possible systems of measure-relations of space. But this is permissible only if a general conception of externality has been constructed, otherwise the question is begged, since this general conception is the necessary condition of the systems of abstract quantitative relations being conceived as systems of spatial measure relations. We do not escape this conclusion by merely talking about analytical conceptions of space, or by referring to non-Euclidean conceptions of space as being analytical constructions, for until that in which Riemann failed is accomplished, these are and remain question-begging phrases.

In Cayley's Theory of Distance we have a doctrine which is also metageometrical or non-Euclidean, though in a completely new sense. Here there is no longer any question of conceiving different kinds of space; but, nominally, of conceiving different systems of metrical relations in space. I say 'nominally' because the systems of relations are metrical in name, while in conception they are descriptive. We do not in fact conceive different systems of metrical relations, but, according to the choice of the fundamental figure which Cayley calls the Absolute,

we have different systems of descriptive equivalence. The supposition that we conceive different kinds of distance is an illusion engendered by an unrealized reaction of symbolic forms, verbal or algebraic, on the thinker's judgement of his own process of thought. It leads him to say, e. g., that the ordinary notion of distance is included, as a special case, in the generalized notion of distance, while the real fact is that this generalized notion of distance is generalized, not from the ordinary notion of distance, but from a notion of projective relation.

Finally, in discussing the relation of mensuration to metrical geometry, in the literal sense of the term 'metrical', it was contended that the view of this relation commonly entertained by metageometers is indefensible. For if Mr. Russell's criticism of Lotze's argument for the apodeictic certainty of ordinary geometry is valid, it is no less so when turned against the belief that the objective validity of ordinary geometry might be disproved by greater precision or larger amplitude of astronomical observation. The further discussion of this relation, from the points of view respectively of the Euclidean and the Metageometer, led to the question of the relevance of the notion of direction to that of non-Euclidean spaces, and to the conclusion that, were such spaces really conceived, the ordinary notion of direction could not find a place in these conceptions of space, and would logically be irrelevant to them; their construction would involve non-Euclidean notions of direction. Thus the fact that the ordinary notion of direction is employed in the construction of these alleged non-Euclidean conceptions of space would alone be sufficient to warrant the rejection of the doctrine that different kinds of space are conceivable.

INDEX

(See also Table of Contents)