On the other hand, there is a suggestion of the empirical in the apparent element of prediction involved in the Euclidean (which is the classically original) form of the axiom. Given certain conditions, then something will happen, viz. the lines in question must eventually meet. It is by no means impossible that this mode of phraseology counted for something in forming the original judgement that the proposition is an empirical one. If this was so, the result for the philosophy of geometry was a very unfortunate one. It is easy to get rid of the apparent element of prediction contained in the so-called axiom by breaking it up into its two subordinate propositions, and by substituting for the second one the converse of the first:

(1) Two straight lines not identically inclined to a transversal are mutually inclined.

(2) Two straight lines which are mutually inclined, and are continually produced, will eventually meet.

Instead of (2), which contains the apparent element of prediction, we substitute the converse of (1), i. e. Two straight lines which are mutually inclined are not identically inclined to a transversal. From these two propositions, (1) and the converse of (1), it can be demonstrated that the sum of the angles of any triangle is equal to two right angles, a proposition which is equivalent to the postulate. But it is more convenient and neither more nor less self-evident to take as axioms the two propositions :

a. Straight lines which are identically inclined to the same straight line are not mutually inclined.

b. Straight lines which are not mutually inclined are identically inclined to the same straight line.

The proposition about the sum of the angles is seen at once to follow from these two.

The proposition (1) above and its converse derive respectively, by the reductio ad absurdum argument, from the propositions b and a, just as we have seen to be the case with the axioms of magnitude. In fact these four propositions relating to direction are in exact analogy with the four propositions relating to magnitude. Take, for instance, the first axiom of magnitude and proposition a:

If A and B are identically related in length to C, A and B are equal, or are identical in length.

The necessity is simply a suggestion, contained in the data, which we are unable to resist. To have the length-relation of A to C given as identical with that of B to C, suggests that it is indifferent whether we consider A or B in this relation to C; and it is this suggestion which imposes itself upon us as a necessity of thought: the identity of A and B in length is a necessary conclusion. Now put proposition a in a similar form:

1

If A and B are identically related to C in direction (are identically inclined to C),1 A and B are identical in direction (are not mutually inclined).

Given the direction-relation of A to C as identical with that of B to C, suggests that it is indifferent whether we consider A or B in this relation to C. Those who find that this suggestion is one which they are unable to resist will, in other words, find that the identity of A and B in direction is a necessity of thought.

If we were to take proposition a as a definition of mutual noninclination of two coplanar straight lines, it would follow (from my point of view) that there are no axioms in geometry other than the axioms of magnitude. But although this idea of the proposition as a definition is at first sight plausible enough, it will not stand careful examination. Evidently if we admit proposition a to be simply a definition of mutual non-inclination, we must then also admit the proposition: Two straight lines which are not identically inclined to a transversal are mutually inclined, to be a definition of mutual inclination. Mutual inclination' would thus be merely another name for the notion of non-identical inclination to a transversal. But it cannot be denied that the notion of mutual inclination is involved in that of the angle. Thus if the notion of mutual inclination is that of non-identical inclination to a transversal, the notion of the angle involves that of non-identical inclination of the straight lines which contain it to a transversal; and this is simply not the

case.

It will be understood that the very close analogy evinced between the two sets of propositions respecting magnitude and direction is not urged as a demonstration of the self-evidence of the latter set. If self-evidence is the foundation of ratiocinative demonstration, the ratiocinative demonstration of self

1 Including non-inclination as a special case.

evidence is an absurdity. It is rather for those who already feel the necessity of the geometrical propositions that the precision of the analogy is in this respect of importance, for the very process of analysis which discloses the analogy affords a guarantee that the necessity already admitted is not an illusion due to over-haste in judgement.

CHAPTER XV

THE AXIOM OF FREE MOBILITY OR CONGRUENCE

This proposition is not necessarily implied in Euclid's process of reasoning. -The assumption that bodies can be moved without change of shape or size is relevant to Mensuration.-The proposition that geometrical figures can be thus moved is, as an assumption, meaningless; it is merely one of several ways of defining the notion of Congruence.-Analysis of Clifford's explanation of this so-called geometrical Axiom.-Mr. Bertrand Russell's explanation of its meaning.

SUPPOSE Some one had suggested-possibly some one did suggest-to Euclid, that in his Elements he assumes, but omits to state, that figures may be freely moved without change of shape or size. Might he not quite legitimately have replied that in geometry we are not concerned with the physical properties or qualities of bodies, with the effects of action; such matters belonging to the province of physics, in which the results of geometrical investigation find a place as do those also of arithmetical investigation?

By a legitimate reply I mean one consistent with the process of reasoning in the Elements. It will at once be objected that Euclid does make this assumption in the proof of Prop. 4, Book I, which may be said to be the fundamental proposition of the Elements. This opinion is general, but in order to maintain it effectively it seems that we must hold the so-called Axiom 8 (magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another) to be the expression of an assumption. I see in it nothing but a definition of 'identical' equality or congruence. Professor Henrici says it 'may be taken' as a definition of 'equal'; and the writer of the article' Axiom' (Ency. Brit.) already quoted, unhesitatingly pronounces it to be a mere definition of equals. But if that is so, the assertion that Euclid tacitly makes the said assumption in the proof of Prop. 4 may very well be disputed. In this proposition the premisses, or what is given as data for reasoning, are that two sides and the included angle of one triangle are equal to two sides and the included angle of another

triangle; that is, these entities are given as coinciding on superposition; the conclusion is, that the third sides and the other angles must also coincide on superposition. Thus if we regard Axiom 8 as defining equality, the opinion is untenable that the demonstration (which consists in the deduction of the conclusion from the premisses) involves the assumption in question. Euclid, in this so-called axiom, defines coincidence as the exact filling of the same space (clearly a useless definition from the point of view of mensuration, or applied geometry); had he said equal spaces the term equal would obviously not have been defined in the proposition. Whence, then, do we derive the notion of equal, and hence also of unequal, spaces? Evidently from that of figure; the notion of equal spaces is here that of the same figure in different places: make abstraction of the notion of configuration, of delimitation in space, and the notion of spatial magnitude disappears.

In applied geometry, in mensuration, every precaution which experience suggests is taken to ensure, so far as possible, invariability of the instruments of measure-evidently because the conception of invariable figure is involved in that of metrical comparison. Thus there is here a clear sense in which it may be said that the validity of the results of measurement depends upon the truth of the assumption that the instruments of measure and the things measured have remained invariable during the process. But if the notion of invariable figure is involved in the very conception of metrical comparison, then, in pure geometry, which excludes every consideration subsumable under the conception of cause and effect, the statement that we assume invariability of figure seems to be bereft of any definite meaning. At first sight the statement that this is one of the axioms of metrical geometry seems so obviously true that it hardly occurs to us to question it, to inquire what, precisely, is the meaning of it. Geometers do not, as a rule, perceive that it calls for an explanation. In the few cases where such an explanation has been attempted, a careful analysis of it brings to light inconsistencies and confusions of thought. Let us take, for instance, that which Clifford gives in his book, The Common Sense of the Exact Sciences.

Geometry, according to Clifford, is a physical science, which deals with the sizes, shapes, and distances of things, and which