664 SCIENCE AND HYPOTHESIS 



geometry. It will be noticed that my description of these fantastic 

 worlds has required no language other than that of ordinary geo- 

 metry. Then, were we transported to those worlds, there would be 

 no need to change that language. Beings educated there would no 

 doubt find it more convenient to create a geometry different from ours, 

 and better adapted to their impressions ; but as for us, in the presence 

 of the same impressions, it is certain that we should not find it more 

 convenient to make a change. 



Experiment and Geometry 



1. I have on several occasions in the preceding pages tried to show 

 how the principles of geometry are not experimental facts, and that in 

 particular Euclid's postulate cannot be proved by experiment. How- 

 ever convincing the reasons already given may appear to me, I feel I 

 must dwell upon them, because there is a profoundly false conception 

 deeply rooted in many minds. 



2. Think of a material circle, measure its radius and circumference, 

 and see if the ratio of the two lengths is equal to IT. What have we 

 done? We have made an experiment on the properties of the matter 

 with which this roundness has been realized, and of which the measure 

 we used is made. 



3. Geometry and Astronomy. The same question may also be 

 asked in another way. If Lobatschewsky's geometry is true, the paral- 

 lax of a very distant star will be finite. If Kiemann's is true, it will 

 be negative. These are the results which seem within the reach of 

 experiment, and it is hoped that astronomical observations may enable 

 us to decide between the two geometries. But what we call a straight 

 line in astronomy is simply the path of a ray of light. If, therefore, x 

 we were to discover negative parallaxes, or to prove that all parallaxes 

 are higher than a certain limit, we should have a choice between two 

 conclusions: we could give up Euclidean geometry, or modify the 

 laws of optics, and suppose that light is not rigorously propagated in 

 a straight line. It is needless to add that every one would look upon 

 this solution as the more advantageous. Euclidean geometry, there- 

 fore, has nothing to fear from fresh experiments. 



4. Can we maintain that certain phenomena which are possible 

 in Euclidean space would be impossible in non-Euclidean space, so 

 that experiment in establishing these phenomena would directly con- 

 tradict the non-Euclidean hypothesis? I think that such a question 

 cannot be seriously asked. To me it is exactly equivalent to the fol- 

 lowing, the absurdity of which is obvious : There are lengths which 

 can be expressed in metres and centimetres, but cannot be measured in 

 toises, feet, and inches; so that experiment, by ascertaining the exist- 

 ence of these lengths, would directly contradict this hypothesis, that 



