39§ POPULAR SCIENCE MONTHLY 



THE VALUE OF SCIENCE 

 BY m. h. poincare 



MEMBER OF THE INSTITUTE OF FRANCE 



Chapter III. The Notion of Space 



1. Introduction 



[~N the articles I have heretofore devoted to space I have above all 

 -*- emphasized the problems raised by non-Euclidean geometry, while 

 leaving almost completely aside other questions more difficult of ap- 

 proach, such as those which pertain to the number of dimensions. All 

 the geometries I considered had thus a common basis, that tridimen- 

 sional continuum which was the same for all and which differentiated 

 itself only by the figures one drew in it or when one aspired to 

 measure it. 



In this continuum, primitively amorphous, we may imagine a net- 

 work of lines and surfaces, we may then convene to regard the meshes 

 of this net as equal to one another, and it is only after this convention 

 that this continuum, become measurable, becomes Euclidean or non- 

 Euclidean space. From this amorphous continuum can therefore arise 

 indifferently one or the other of the two spaces, just as on a blank sheet 

 of paper may be traced indifferently a straight or a circle. 



In space we know rectilinear triangles the sum of whose angles is 

 equal to two right angles; but equally we know curvilinear triangles 

 the sum of whose angles is less than two right angles. The existence 

 of the one sort is not more doubtful than that of the other. To give 

 the name of straights to the sides of the first is to adopt Euclidean 

 geometry; to give the name of straights to the sides of the latter is to 

 adopt the non-Euclidean geometry. So that to ask what geometry it 

 is proper to adopt is to ask, to what line is it proper to give the name 

 straight ? 



It is evident that experiment can not settle such a question; one 

 would not ask, for instance, experiment to decide whether I should 

 call AB or CD a straight. On the other hand, neither can I say that 

 I have not the right to give the name of straights to the sides of non- 

 Euclidean triangles because they are not in conformity with the eternal 

 idea of straight which I have by intuition. I grant, indeed, that I 

 have the intuitive idea of the side of the Euclidean triangle, but I have 



