That (as Mr. Russell affirms) 'rigid bodies are unnecessary in geometry' is a proposition which is not likely to be disputed by any one who draws a clear distinction between geometry and mensuration. But if, when he speaks of rigidity of body, the geometer has in mind invariability of figure-and this, I believe, is usually the case-then we have to do merely with a question of phraseology. Substitute invariable figure for rigid body, and the argument is a variant of the one which went before as to the irrelevance of motion; but it is no more convincing. I do not see how we can admit that the notion of invariable figure is derived from that of equal spaces. If any logical priority is to be assigned, I should assign it the other way; because, as I have already urged, if we make abstraction of figure altogether, the notion of spatial quantities disappears altogether.

We come back, then, to the conclusion already more than once expressed; that figures which coincide on superposition, or the same figure in different places, or motion of an invariable figure, are equivalent ways of giving expression to the notion of congruence. The axiom of congruence, using the term axiom in the definite sense of a proposition expressing an immediate and necessary conclusion which follows from the synthesis of two data not themselves conclusions, is obviously contained in Euclid's Axiom 1, where the term 'equal' includes the meaning of 'congruent'.

CHAPTER XVI

SYSTEMS OF PLANE GEOMETRY

Real and Nominal Contradiction.-Conditions of Real Contradiction.— Euclid's and Lobatschewsky's respective hypotheses concerning parallels nominally exclude one another.-If these hypotheses are also real contradictories, we must admit two mutually contradictory planimetries for the surface which Euclid calls plane.-If, on the other hand, the two socalled planimetries are relevant to two different surfaces, both are admissible at the same time, and the contradiction is merely nominal.-The same argument applies to Riemann's planimetry in relation to Euclid's and Lobatschewsky's.-These conclusions are unavoidable unless it is a fact that different kinds of space are conceivable.

It is generally believed that Napoleon died at St. Helena. The assertion, which many people would be ready to make, that Alexander the Great did not die at St. Helena, would not be considered to be in contradiction with the belief that Napoleon died there; the one belief does not exclude the other; we find no difficulty in yielding credence to both assertions. But if A asserts that Napoleon died at St. Helena, and B asserts that Napoleon did not die there, the two assertions are in contradiction. It may not be the case, however, that the beliefs to which A and B thus give expression are irreconcilable: we might on inquiry discover that while A's assertion was intended to be relevant to the great soldier and statesman, B's was intended to refer to Victor Hugo's Napoléon le Petit. Between the two beliefs there would be no contradiction; any one could hold them both; the contradiction would be merely nominal.

It is no different when the predication relates not to indubitable events, but to really uncertain events, or to events to come. If I assert that Tom and Jane will meet here at dusk, this is not contradicted in your assertion that Charles and Mary will not meet here at dusk; nor would there be aught but a merely verbal contradiction, did you assert that Tom and Jane will not meet as aforesaid, you having in mind some Tom and Jane other than those to whom my assertion is relevant.

The case, again, is no different when, instead of events, we are dealing with abstractions. Euclid affirms that two straight

lines in the same plane with, but not identically inclined to, a common transversal, must meet if continually produced. Lobatschewsky, in common with

many other geometers, considers this proposition to be uncertain, or not self-evident. There are, according to him, two alternative and mutually exclusive hypotheses: either, as Euclid affirms, every straight line through A

A

B

C

E

D

which falls within the right

angle CAB will meet BD (which is perpendicular to AB); or else only those straight lines through A which fall within the acute angle EAB (of uncertain magnitude) will meet BD; the straight line AE being, on this side of AB, the limit between those lines which cut BD and those which do not cut it.

In admitting this second hypothesis Lobatschewsky at least verbally contradicts Euclid, or any one who affirms what Euclid affirms. But here, again, it is an underlying condition of the hypotheses being mutually exclusive, that is, of the contradiction being real, that the two assumptions shall be relevant to the same entities. If, for instance, we suppose Lobatschewsky to have meant, by a plane and a straight line, a surface and a line different from the surface and the line which Euclid had in mind -and by different I mean different in shape then the two hypotheses are not mutually exclusive in the sense of standing in logical contradiction each to the other. To admit either of them need not necessarily involve the rejection of the other.

Professor H. J. Stephen Smith says, in his Introduction to Clifford's Mathematical Papers, that Lobatschewsky's assumption 'was in effect to adopt the hypothesis (though it does not appear to have occurred to Lobatschewsky in that light) that a plane has negative curvature'. But evidently no one can assume that a plane has negative curvature if by the term plane' he means a surface of zero curvature, which is what Euclid is understood to have meant, and is what people commonly do mean, by a plane. If it did not occur to Lobatschewsky that his hypothesis is equivalent to assuming that a plane has negative curvature, it is to be presumed that by a plane he did not mean such a surface, but meant the surface shape to which

that name is commonly given; and in that case his assumption is-as he seems to have intended it to be—a real alternative to Euclid's 12th Axiom, and stands in real contradiction to it. But if, on the other hand, Lobatschewsky did consider his hypothesis as equivalent to assuming that a plane has negative curvature, it is to be presumed that when he uses that term in his own development of geometry-Imaginary Geometry, as he calls it—he meant by it a surface of negative curvature; and in that case his hypothesis is not-what he seems to have intended it to be a real alternative to Euclid's, and does not stand in real contradiction to it. No one will find any difficulty in admitting that in a surface of zero curvature there is, or there may be, through a given point, but one parallel to a given straight line, while at the same time admitting that in a surface of constant negative curvature there are, or there may be, through a given point, an infinity of parallels to a given geodesic.

There is, I believe, not a particle of evidence in any of Lobatschewsky's published works which goes to show that by the terms' plane' and 'straight line' he meant anything other than what we suppose Euclid to have meant by them. Indeed, if we cannot be sure of what Lobatschewsky conceived under these terms from the definitions he gives of them and the contexts in which he employs them, neither can we have any assurance, on the same subject, from the like evidence in Euclid's case. And in connexion with this point it may be well to call attention to a remark which Lobatschewsky himself makes in the preface to his small work on the Theory of Parallels (Berlin, 1840): that in opposition to Legendre's opinion, the theory of parallels is wholly independent of such imperfections of principle as that of the definition of the straight line-blemishes, if blemishes they are, which he considers to be irrelevant to the theory in question. Short of explicitly affirming that imperfections in the definition of the straight line are of no moment because this conception, no matter how expressed, is perfectly definite, we could hardly have anything more explicit of Lobatschewsky's attitude than is this remark of his.

For many years after Lobatschewsky first made his researches, mathematicians in general regarded his non-Euclidean geometry as a logical freak, clearly because they supposed this geometry to be relevant to the Euclidean plane and straight line. The

planimetry which he deduces from his rejection of Euclid's 12th Axiom contains many theorems which, as Mr. Poincaré remarks,1 are strange and at first disconcerting. Why do they produce this effect upon us? Simply because we understand them to be predicated of the surface and the line which Euclid calls the plane surface and the straight line. Let it be understood that these theorems relate to the pseudo-spherical surface,2 or surface of constant negative curvature, and we find nothing either strange of disconcerting in such a theorem as, e. g. that two 'straight' lines may be so situated in relation to one another that, being produced on the same side of a common transversal, they first approach one another and then recede from one another;3 or, again, that two straight lines which are perpendicular to a common transversal continually diverge from one another on both sides of the transversal. These and many other theorems involved in Lobatschewsky's planimetry strike us as strange, and even as impossible, because we understand them to be affirmed of the particular shape of surface commonly termed 'plane', and of the particular shape of line commonly called 'straight'.

Bodies exist, and bodies have surfaces and, in general, edges and corners, in the ordinary senses of these terms imposed upon us by the practical affairs of life. But the surfaces, lines, and angles about which we geometrize do not exist in any other sense than as abstract constructions from these concrete experiences. Whosoever admits this, as even roughly and approximatively descriptive of the process of geometrizing, cannot but find the usual mathematical standpoint regarding the modern development of geometry as other than confused and vacillating, while at the same time comprehending how the mathematician has been driven to such a standpoint.

In part this state of things proceeds from drawing the line, but not drawing it firmly and consistently, between geometry and its practical applications. But it is also in part due to the tradition fixed through the unfortunate way in which Euclid 1 La Science et l'Hypothèse, p. 51.

2 Roughly, the shape of a saddle; or of a mountain pass, where the curvatures along and across the pass are opposed in sense.

3 It is the negation of this proposition which Simson takes, as an axiom, in order to demonstrate Euclid's postulate. See his Notes on the Elements, p. 260 of the 25th edition (1841).