We will return now to the consideration of the assumptions which, according to Professor Henrici, Euclid actually makes. Euclid assumes that 'Right angles, as defined in Def. 10, are possible, and all right angles are equal; . . .' (Axiom 11).

This means that Euclid assumes that two straight lines can have the angular relation defined, viz. that one of them can stand on the other so as to make the adjacent angles equal. But since Euclid proves, in Props. II and 12, Book I, that this relation is possible, I do not see how he can be held to have assumed the possibility.1 That the proof rests upon assumptions' does not warrant our counting this as a fresh one. The fact is the writer seems to be somewhat inconsistent in his treatment of Euclid. For instance, the list of assumptions said to be made by Euclid does not include parallels as defined in Def. 35. Why not? Evidently because in Prop. 27 Euclid 'proves the existence of parallel lines' (p. 378), i. e. proves this relation between straight lines to be possible. But if this is a good reason for not including Def. 35 among the assumptions, why not apply it in the case of Def. 10? The conception of the ' plane rectilineal' angle is general; it is the conception of angular magnitude or difference of direction between two straight lines in general. In Def. 10 (the right angle) Euclid defines a 'particular and precise' idea, i.e. that of a particular angular magnitude, just as in Def. 4 he defines a particular linear shape. Thus when in Axiom II he

1 I put the case from what I conceive to be Professor Henrici's own standpoint, not from mine. I do not regard the notion of perpendicularity as something which we must either assume or demonstrate. I do not perceive the relevance either of assumption or of demonstration to the notion. The notion of a straight line which rotates in a plane about a point fixed in another straight line in this plane, involves the notion of perpendicularity, or equality of adjacent angles, as a particular case in the continuous series of cases; just as it also involves, as another particular case of the same series, that of coincidence, or disappearance of the adjacent angles (which we somewhat paradoxically express in saying that these become respectively equal to o° and 180°). But although the notion of perpendicularity is necessarily involved as a particular case of the general conception of (linear) angular magnitude, it is quite a different matter to demonstrate that this geometrical relation (i. e. perpendicularity) is necessarily connected with other geometrical relations; and this is what, from a purely geometrical standpoint, Euclid does in the 11th and 12th propositions. It is not always easy, indeed, to analyse Euclid's procedure as purely geometrical, because he draws no systematic distinction between purely geometrical questions and questions which involve, in a greater or less degree, the notion of application or mensuration.

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'assumes' that all right angles are equal, he assumes' that all angles of a particular magnitude are equal. The axiom is mere surplusage.

Let me return now to the consideration of Axiom 10-two straight lines cannot enclose a space, or one straight line only can lie between two points-which seems to me to be merely an alternative to Def. 4. This proposition

has been regarded by some writers as either a mere definition of straight lines, or as contained by direct implication in the definition; but incorrectly. If it is held to be a definition, nothing is too complex to be so called, and the very meaning of a definition as a principle of science is abandoned; while, if it is said to be a logical implication of the definition, the whole science of geometry may as well be pronounced a congeries of analytic propositions. When straight line is strictly defined, the assertion is clearly seen to be synthetic.'1

This is not a little dogmatic. To say that if this proposition is held to be a definition, nothing is too complex to be so called, is mere rhetorical exaggeration; beating the big drum is not argument, but only drowns it. What proposition could well be less complex than the one in question? And the addendum, that the very meaning of a definition as a principle of science is abandoned, is but a begging of the question at issue, since it is precisely the difference between definition and axiom which is in dispute.

Lobatschewsky, in his Geometrical Researches on the Theory of Parallels (Berlin, 1840), gives a number of simple propositions (all in accord with Euclid) as premisses of his geometrical development. The first two of these propositions is as follows:

'A straight line is, in every position, superposable upon itself. I mean by this, that if about two points of a straight line we rotate the surface which contains it, this line does not change its place.'

Lobatschewsky merely gives this as a proposition. Is it a definition of the straight line, or is it an axiom? The two statements contained in it are evidently intended to be supplementary to one another, so that we may get at the exact meaning of the author. Evidently, also, a line which, in every position, is

1 Art. ' Axiom', Ency. Brit., vol. iii, p. 160.

superposable upon itself, is merely another way of saying, lines which, in every position, are superposable upon one another. The second statement is meant to give the precise sense of the ' in every position' of the first. It seems thus impossible to take this proposition as anything but a mere verbal variant of Euclid's Axiom 10. Yet the very next proposition which Lobatschewsky lays down is: Two straight lines cannot intersect twice. We cannot suppose that he intended to give the proposition twice over in different words. I conclude that he intended the first as a definition of the straight line, the second as an axiom about straight lines. Yet what difference, other than mere verbal difference, is there between them? Euclid's Def. 4 is equivalent to either of them; and so is also his Axiom 10.

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pass over Axiom 12, which will be considered further on under the theory of parallels; I also leave aside the assumption which Euclid is said to make sub silentio, that figures may be freely moved in space without change of shape or size. The discussion is rather long; it will form the subject of a separate chapter. There remain the three Postulates, about which I have as yet said nothing.

Some writers have seen in Euclid's Postulates merely statements which limit the use of instruments in geometrical construction to the ruler and compasses. But this is a side issue which has nothing to do with the foundations of geometrical reasoning. Whether the figures in which we embody or represent our geometrical conceptions are actually drawn or only imagined, and, if drawn, are free-hand or engineered no matter by what auxiliary instruments, has clearly no bearing on the kind of questions we have been discussing.

It is known that Euclid himself grouped the 10th, 11th, and 12th ' axioms', together with the three postulates, into one class under the name of airýμara, and that the arrangement of the preliminary propositions actually found in the modern editions of his Elements was the work of his successors; the ground of the alteration being 'the distinction between postulates and axioms which has become the familiar one, that they are indemonstrable principles of construction and demonstration respectively '1 This distinction seems to be no longer accepted, at least by the quite modern school of geometers; and it is on 1 Art. ' Axiom', Ency. Brit., vol. iii, p. 159.

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the whole strange that it should ever have been generally admitted; for, accidents of phraseology apart, it seems to be a distinction without a difference. Thus 'Axiom' 10, if expressed as it very commonly is expressed, viz. one straight line only can be drawn between two points, might just as well be called a principle of construction as a principle of demonstration. Suppose now that we say: A line which lies evenly between any two of its points will lie evenly between any two other points, this would be to state the so-called axiom of congruence for straight lines only, but is it to say anything in essence different from what Euclid says in the first postulate, that a straight line may be drawn from any one point to any other point? Personally I find no essential difference between the two; but this is no doubt due to the unusual interpretation I attach to the so-called axiom of congruence or assumption that figures may be freely moved, &c., which has yet to be discussed.

In the second postulate Euclid is said to assume that straight lines may be indefinitely produced; and, indeed, in this as in the other postulates, Euclid asks that something may be granted. But the particular form in which these propositions are cast may very possibly be due to the lack of a clear and consistent distinction between geometry and mensuration. It is at all events easy to see that if, in Euclid's day, that distinction was not observed, and it were simply asserted, e. g. that a circle can be described from any centre and at any distance from that centre, it would have been too easy for the critic to object that this is not possible. Euclid asks that this may be granted as a premiss: whoever will not grant it must go elsewhere for his geometry. But if we admit that in geometry we are concerned with what Hume called ' relations of ideas', while in mensuration our object is the application of these relations in matter of fact', which is the distinction in question, a distinction now very generally admitted by mathematicians, then to say that we assume the straight line to be indefinitely producible will be found to mean in substance this: that we assume the straight line not to be a circle. It is more than likely the reader may think that to call this an assumption is absurd or is an abuse of language, and that mathematicians cannot possibly intend anything of the kind. It may be they do not; but let us consider what we can make of the following passage, which is taken from Felix Klein's

Lectures on Non-Euclidean Geometry. Speaking of Riemann's conception of a constant 'measure of curvature' of space, he says:

Riemann was here confronted with the question: what is the value of this constant measure of curvature? If it is equal to o, we have the premisses of the ordinary Euclidean geometry. If it is negative we get Hyperbolic geometry, that is, the geometry developed by Gauss, Lobatschewsky, and Bolyai. But how stands the case if this value is positive? This possibility had so far been overlooked, or rather it had been put on one side because space had always and very naturally been taken as infinitely extended. But now Riemann observes that this third case can also quite well be admitted. For if it should indeed be that space is finite, that is, returns into itself, and that the length of the straight lines in which it returns into itself is merely very great, yet we can in no way become aware of the first of these properties, and we are therefore not warranted in neglecting this possibility. Thus while the earlier investigators naturally regarded space as infinite, Riemann says: space is indeed necessarily unbounded, but because space is unbounded it does not follow that it is infinite.'

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The Euclidean assumption is, then, that a straight line does not return into itself, and since it is a condition of metrical geometry that the unit of measurement is conceived as invariable both in shape and size, the Euclidean assumption is in essence that the straight line is not a circle. The conception of a line which returns into itself is simple; the phrase, space is finite, or returns into itself, is also simple; but does it give expression to any conception? Mr. Poincaré, like other eminent geometers, conceives two-dimensional beings living on a sphere. Their space will be unbounded, since on a sphere one can always go forward without being stopped, and yet it will be finite; you can find no end to it but can go round it'. Subsequently he tells us that 'Riemann's space is finite although unbounded, in the sense assigned above to those two words '.2 Only what is this sense when we pass from a figure (which is a boundary in space) to space? Beings as 'twodimensional' and a surface as a 'space' are merely misleading metaphors unless they express real analogies in thought. Space

1 Nicht-Euklidische Geometrie, i und ii, von F. Klein. Zweiter Abdruck. Göttingen, 1893.

2 La Science et l'Hypothèse, pp. 53, 54.