February 25, 1892] 



NATURE 



405 



It is impossible to construct a figure similar to a given figure, 

 but of different dimensions. 



If a circle be divided into n equal parts, and tangents 

 be drawn to the points of division, these n tangents will meet 

 and form a polygon, provided that the radius of the circle 

 be small enough ; but if this radius is sufficiently large, they 

 will not meet. It is useless to multiply these examples ; the 

 propositions of Lowatchewski have no longer any connection 

 with those of Euclid, but they are not less logically connected 

 together. 



The Geometry of Riemann. — Let us imagine a world peopled 

 only with beings deprived of thickness ; and let us suppose that 

 these animals, "infinitely flat," are all in one plane, and are not 

 able to get out of it. Let us admit, further, that this world is 

 removed sufficiently from others to be free from their influence. 

 As we are making these assumptions, we may as well endow 

 these beings both with reasoning powers and the capacity of 

 founding a geometry. In this case they would certainly attribute 

 to space only two dimensions. 



But let us suppose, however, that these imaginary animals, all 

 still devoid of thickness, have the form of a portion of a spherical 

 figure, and not of a v>lane one, and are all on one and the same 

 sphere without being able to leave it. What geometry would they 

 construct ? It is clear at once that they would only attribute to 

 space two dimensions : that which will play for them the part of 

 the straight line will be the shortest distance between two 

 points on the sphere— that is to say, an arc of a great circle ; in 

 a word, their geometry would be spherical geometry. 



What they will call space will be this sphere which they 

 cannot leave, and on which occur all the phenomena of which 

 they can have any knowledge. Their space then will be without 

 limits, since on a sphere one can always go forward, without 

 ever coming to an end, and nevertheless it will be finite — one 

 will never find the limit, but one can make the circuit of it. 



In fact, the geometry of Riemann is spherical geometry 

 extended to three dimensions. To construct it, the German j 

 mathematician had to throw overboard not only the postulates i 

 of Euclid, but even the first axiom :• Only one straight line can \ 

 be draivn between tico points. 



On a sphere only one great circle in general can be drawn 

 through two given points (which, as we have just seen, would 

 play the part of the straight line to our imaginary beings) ; but | 

 to this there is an exception ; for, if the two given points are I 

 diametrically opposed, an infinite number of great circles can | 

 be made to pass through them. I 



In the same way, in the geometry of Riemann, only one I 

 straight line in general can be drawn between two points ; but j 

 there are exceptional cases where an infinite number of straight j 

 lines can be drawn between them. | 



There is a kind of opposition between the geometry of 

 Riemann and that of Lowatchewski. 



Thus, the sum of the angles of a triangle is — 

 Equal to two right angles in Euclid's geometry. 

 Less than two right angles in that of Lowatchewski. 

 Greater than two right angles in that of Riemann. 



The number of parallels that can be drawn to a given straight 

 line through a given point is equal — 

 To one in the geometry of Euclid. 

 To zero in that of Riemann. 

 To infinity in that of Lowatchewski. 



Let us add that the space of Riemann is finite although without 

 limit, in the sense already given to these two words. 



Surfaces of Constant Curvature. — There was, however, one 

 possible objection. The theorems of Lowatchewski and of 

 Riemann present no contradiction, but, however numerous the 

 consequences which these two geometers have drawn from their 

 hypotheses, they were compelled to stop before they had ex- 

 hausted all of them, for the number would be infinite : who can 

 say, therefore, that, if they had carried their deductions further, 

 they would not finally have found such contradictions ? 



This difficulty does not exist for the geometry of Riemann, 

 provided that it is limited to two dimensions ; for, in fact, the 

 geometry of Riemann for two dimensions does not differ, as we 

 have seen, from spherical geometry, which is only a branch of 

 ordinary geometry, and consequently outside all discussion. 



M. Beltrami, in considering in the same way the two-dimen- 

 sional geometry of Lowatchewski to be only a branch of 

 ordinary geometry, has equally refuted the objection in this 

 case. 



This he has done this in the following manner : — Consider on 



a surface any figure. Imagine this figure, traced on a flexible 

 and inextensible cloth, to be laid on this surface, in such a way 

 that when the cloth is moved and changes its shape, the various 

 lines of this figure can change form without altering their length. 

 In general this flexible and inextensible figure cannot leave its 

 place without quitting the surface ; but there are certain par- 

 ticular surfaces for which a similar movement would be possible ; 

 these are the surfaces with constant curvature. 



If we resume the comparison that we previously made, and 

 imagine beings without thickness living on one of these surfaces, 

 they will regard the movement of a figure all of whose lines 

 preserve a constant length as possible. A like movement, on 

 the other hand, would appear absurd to animals without thick- 

 ness living on a surface whose curvature was variable. 



These surfaces of constant curvature are of two kinds : — 



Some are of positive curvature, and can be so deformed as to 

 be laid on a sphere. The geometry of these surfaces becomes 

 then spherical geometry, which is that of Riemann. 



Others are o{ negative curvature. M. Beltrami has shown that 

 the geometry of these surfaces is none other than that of 

 Lowatchewski. The two-dimensional geometries of Riemann 

 and Lowatchewski are thus found to be re-attached to Euclidian 

 geometry. 



Interpretation of Non- Euclidian Geometries. — Thus the ob- 

 jection disappears as regards geometries of two dimensions. 



It would be easy to extend M. Beltrami's reasoning to 

 geometries of three dimensions. The minds which space of 

 four dimensions does not repel will see here no difficulty ; but 

 they are few. I prefer, then , to proceed otherwise. 



Let us consider a particular plane that we will call funda- 

 mental, and construct a kind of dictionary, making a double 

 series of words, written in the two columns, correspond each to 

 each, in the same way that the words of two languages, having 

 the same signification correspond in ordinary dictionaries : — 



Space Portion of space situated above the funda- 

 mental plane. 



Plane Sphere cutting orthogonally the funda- 

 mental plane. 



Right line Circle cutting orthogonally the fundamental 



plane. 



Sphere Sphere. 



Circle Circle. 



Angle Angle. 



{Logarithm of the anharmonic ratio of 

 these two points and the intersections 

 of the fundamental plane with a circle 

 passing through these two points and 

 cutting it orthogonally. 

 &c., &c. 



Let us take, then, the theorems of Lowatchewski, and translate 

 them by means of this dictionary, as we should translate a 

 German text with the aid of a German-French dictionary. We 

 shall obtain then the theorems of ordinary geometry. 



For example, this theorem of Lowatchewski — " The sum of 

 the angles of a triangle is less than two right angles " — is trans- 

 lated thus : " If a curvilinear triangle has for its sides the arcs 

 of a circle which if prolonged would cut orthogonally the 

 fundamental plane, the sum of the angles of this curvilinear 

 triangle will be less than two right angles." Thus, however 

 far one pushes the results of the hypotheses of Lowatchewski, 

 one will never be led to a contradiction. Indeed, if two of 

 Lowatchewski's theorems were contradictory, the translations of 

 these two theorems, made with the help of our dictionary, would 

 also be contradictory ; but these translations are theorems of 

 ordinary geometry, and everyone agrees that ordinary geometry 

 is free from contradictions. Whence comes this certainty, 

 and is it justified? This is a question that I cannot treat here, 

 but which is very interesting, and, as I believe, soluble. The 

 objection that I have formulated above no longer then exists. 



But this is not all. The geometry of Lowatchewski, sus- 

 ceptible of a concrete interpretation, ceases to be a frivolous 

 logical exercise, and is capable of application : I have not the 

 time to mention here either these applications or the use that 

 M. Klein and myself had made of them for the integration of 

 linear equations. 



This interpretation, moreover, is not unique, and one could 

 construct several dictionaries analogous to that given above, and 

 by which we could by a simple "translation" transform the 

 theorems of Lowatchewski into theorems of ordinary geometry 



NO. I 165, VOL. 45] 



