THE VALVE OF SCIENCE 405 



Thus the definition I have just given does not differ essentially 

 from the usual definitions; I have only endeavored to give it a form 

 applicable not to the mathematical continuum, but to the physical con- 

 tinuum, which alone is susceptible of representation, and yet to retain 

 all its precision. Moreover, we see that this definition applies not 

 alone to space; that in all which falls under our senses we find the 

 characteristics of the physical continuum, which would allow of the 

 same classification; that it would be easy to find there examples of 

 continua of four, of five, dimensions, in the sense of the preceding 

 definition; such examples occur of themselves to the mind. 



I should explain finally, if I had the time, that this science, of which 

 I spoke above and to which Eiemann gave the name of analysis situs, 

 teaches us to make distinctions among continua of the same number 

 of dimensions and that the classification of these continua rests also 

 on the consideration of cuts. 



From this notion has arisen that of the mathematical continuum 

 of several dimensions in the same way that the physical continuum of 

 one dimension engendered the mathematical continuum of one dimen- 

 sion. The formula 



A > C, A=B, B = C, 



which summed up the data of crude experience, implied an intolerable 

 contradiction. To get free from it it was necessary to introduce a new 

 notion while still respecting the essential characteristics of the physical 

 continuum of several dimensions. The mathematical continuum of 

 one dimension admitted of a scale whose divisions, infinite in number, 

 corresponded to the different values, commensurable or not, of one 

 same magnitude. To have the mathematical continuum of n dimen- 

 sions, it will suffice to take n like scales whose divisions correspond to 

 different values of n independent magnitudes called coordinates. We 

 thus shall have an image of the physical continuum of n dimensions, 

 and this image will be as faithful as it can be after the determination 

 not to allow the contradiction of which I spoke above. 



4. The Notion of Point 



It seems now that the question we put to ourselves at the start is 

 answered. When we say that space has three dimensions, it will be 

 said, we mean that the manifold of points of space satisfies the defini- 

 tion we have just given of the physical continuum of three dimensions. 

 To be content with that would be to suppose that we know what is the 

 manifold of points of space, or even one point of space. 



Now that is not as simple as one might think. Every one believes 

 he knows what a point is, and it is just because we know it too well 

 that we think there is no need of defining it. Surely we can not be 



