654 SCIENCE AND HYPOTHESIS 



gree is simpler than a polynomial of the second degree; 2nd, because 

 it sufficiently agrees with the properties of natural solids, those bodies 

 which we can compare and measure by means of our senses. 



Space and Geometry 



Let us begin with a little paradox. Beings whose minds were made 

 as ours, and with senses like ours, but without any preliminary edu- 

 cation, might receive from a suitably-chosen external world impres- 

 sions which would lead them to construct a geometry other than that 

 of Euclid, and to localize the phenomena of this external world in a 

 non-Euclidean space, or even in space of four dimensions. As for us, 

 whose education has been made by our actual world, if we were sud- 

 denly transported into this new world, we should have no difficulty in 

 referring phenomena to our Euclidean space. Perhaps somebody may 

 appear on the scene some day who will devote his life to it, and be 

 able to represent to himself the fourth dimension. 



Geometrical Space and Representative Space. It is often said 

 that the images we form of external objects are localized in space, 

 and even that they can only be formed on this condition. It is also 

 said that this space, which thus serves as a kind of framework ready 

 prepared for our sensations and representations, is identical with 

 the space of the geometers, having all the properties of that space. 

 To all clear-headed men who think in this way, the preceding state- 

 ment might well appear extraordinary ; but it is as well to see if they 

 are not the victims of some illusion which closer analysis may be able 

 to dissipate. In the first place, what are the properties of space pro- 

 perly so called? I mean of that space which is the object of geo- 

 metry, and which I shall call geometrical space. The following are 

 some of the more essential : 



1st, it is continuous; 2nd, it is infinite; 3rd, it is of three dimen- 

 sions; 4th, it is homogeneous that is to say, all its points are 

 identical one with another; 5th, it is isotropic. Compare this now 

 with the framework of our representations and sensations, which I 

 may call representative space. 



Visual Space. First of all let us consider a purely visual impres- 

 sion, due to an image formed on the back of the retina. A cursory 

 analysis shows us this image as continuous, but as possessing only 

 two dimensions, which already distinguishes purely visual from what 

 may be called geometrical space. On the Bother hand, the image is 

 enclosed within a limited framework; and there is a no less important 

 difference: this pure visual space is not homogeneous. All the points 

 on the retina, apart from the images which may be formed, do not 

 play the same role. The yellow spot can in no way be regarded as 

 identical with a point on the edge of the retina. Not only does the 



