April 12, 18771 



NATURE 



515 



O^ 



ON THE SIMPLEST CONTINUOUS MANI- 

 FOLDNESS OF TWO DIMENSIONS AND 

 OF FINITE EXTENT' 



,NE of the most remarkable speculitions of the pre- 

 sent century is the speculation that the axioms of 

 geometry may be only approximately true, and that the 

 actual properties of space may be somewhat different 

 from those which we are in the habit of ascribing to it. 

 It was Lobatchewsky who first worked out the con- 

 ception of a space in which some of the ordinary laws of 

 geometry should no longer hold good. Among the 

 axioms which lie at the foundation of the Euclidian 

 scheme, he assumed all to be true except the one which 

 relates to parallel straight lines. An equivalent form of 

 this axiom, and the one now generally employed in works 

 on geometry, is the statement that it is impossible to 

 draw more than one straight line parallel to a given 

 straight line through a given point outside it. In other 



words, if we take a fixed straight line, A B, prolonged in- 

 finitely in both directions, and a fixed point, p, outside it ; 

 then, if a second straight line, also infinitely prolonged in 

 both directions, be made to rotate about P, there is only 

 one position in which it will not intersect AB. Now 

 Lobatchewsky made the supposition that this axiom 

 should be untrue, and that there should be a finite angle 

 through which the rotating line might be turned, without 

 ever intersecting the fixed straight line, A B. And in fol- 

 lowing out the consequences of this assumption he was 

 never brought into collision with any of the other axioms, 

 but was able to construct a perfectly self-consistent 

 scheme of propositions, all of them valid as analytical 

 conceptions, but all of them perfectly incapable of being 

 realised in thought. 



Many of the results he arrived at were very curious ; 

 such as, for instance, that the three angles of a triangle 

 would not be together equal to two right angles, but would 

 be together less than two right angles by a quantity propor- 

 tional to the area of the triangle. If we were to increase 

 the sides of such a triangle, keeping them always in the 

 same proportion, the angles would become continually 

 smaller and smaller, until at last the three sides would 

 cease to form a triangle, because they would never meet 

 at all. 



There are many other assumptions, at variance with 

 the axioms of Euclid, which may be made respecting dis- 

 I tance-relations, and which yield self-consistent schemes 

 i of propositions differing widely from the propositions of 

 geometry. We see, therefore, that geometry is only a 

 particular branch of a more general science, and that the 

 conception of space is a particular variety of a wider and 

 more general conception. This wider conception, of 

 which time and space are particular varieties, it has been 

 proposed to denote by the term manifoldness. Whenever 

 a general notion is susceptible of a variety of specialisa- 

 tions, the aggregate of all such specialisations is called a 

 tmanifoldness. Thus space is the aggregate of all points, 

 land each point is a specialisation of the general notion of 

 bositioti. In the same way time is the aggregate of all 



' Read before the London Mathematical Society, December 14, 1876 



instants, and each instant is a specialisation of the gene- 

 ral notion of position in time. Space and time are, in 

 fact, of all manifoldnesses, the ones with which we are by 

 far the most frequently concerned. 



Now there is an important feature in which these two 

 manifoldnesses agree. They are both of them of such a 

 nature that no limit can be conceived to their divisibility. 

 However near together two points in space may be, we 

 can always conceive the existence of intermediate points. 

 And the same thing holds in regard to time. Mathe- 

 maticians express this fact by saying that space and time 

 are continuous manifoldnesses. But there is another fea- 

 ture, equally important with the foregoing, in regard to 

 which space and time are strikingly contrasted. If we 

 wish to travel away from any particular instant in time, 

 there are only two directions in which we can set out. 

 We mubt either ascend or descend the stream. But from 

 a point in space we can set out in an infinite number of 

 directions. This difference is expressed by saying that 

 time is a manifoldness of 07ie dimension, and that space 

 is a manifoldness of more than one dimension. An a^^gre- 

 gate of points in which we could only travel backwards 

 or forwards would be, not solid space, but a line. A line, 

 therefore, is a manifoldness of one dimension. A surface, 

 again, may be regarded as an aggregate of lines ; and it 

 is an aggregate of such a nature, that if we wish to travel 

 away from a particular line, there are only two directions 

 in which we can set out. It is therefore a line-aggregate 

 of one dimension. Considered as a point-aggregate it 

 has two dimensions, and accordingly it is a manifoldness 

 of two dimensions. In the same way it will be seen that 

 solid space is a manifoldness of three dimensions. 



I have endeavoured by these remarks to explain what 

 is meant when we speak of a continuous manifoldness of 

 two dimensions. It is the object of this paper to commu- 

 nicate some results I have arrived at respecting the pro- 

 perties of the simplest of such manifoldnesses which has 

 a finite extent. The existence of the particular manifold- 

 ness I shall endeavour to describe has been referred to in 

 a remarkable lecture by Prof. Clifford on " The Postu- 

 lates of the Science of Space," but, so far as I am aware, 

 its properties have not hitherto been worked out in detail. 

 The simplest of all doubly-extended continuous mani- 

 foldnesses is the plane. But it is not a manifoldness of 

 finite extent. It reaches to infinity in every direction, 

 and its area is greater than any assignable area. It is 

 therefore not the manifoldness of which we are in search. 

 Now the circumstance in which the plane differs from 

 those doubly-extended manifoldnesses which are next to 

 it in order of simplicity, is the possibility that figures 

 constructed in it may be magnified or diminished to any 

 extent without alteration of shape ; in o.her words, that 

 figures which can be constructed in it at all can be con- 

 st'ructed to any scale. That this property is not possessed 

 by curved surfaces, may be seen by considering the case 

 of a spherical triangle. If the sides of a triangle con- 

 structed on a given sphere be all of them mcreased or 

 diminished in the same proportion, the shape of the 

 triangle will not remain the same. Now it has been found 

 by Prof. Riemann that this property of the plane is equi- 

 valent to the following two axioms :— (i) That two geodesic 

 lines which diverge from a point will never intersect agam, 

 or, as Euclid puts it, that two straight lines cannot in- 

 close a space ; and (2) that two geodesic lines which do 

 not intersect will make equal angles with every other 

 geodesic line. The second is precisely equivalent to 

 Euclid's twelfth axiom. Deny the first of these axioms, 

 and you have a manifoldness of uniform positive curva- 

 ture ; deny the second, and you have one of uniform 

 negative curvature. The plane lies midway between the 

 two, and its curvature is zero at every point. 



Let us consider, then, the case of a doubly-extended 

 manifoldness, of which the curvature is uniform and 

 positive. The first of the before-mentioned two axioms 



