528 POPULAR SCIENCE MONTHLY 



Since the enunciation of our laws may vary with the conventions 

 that we adopt, since these conventions may modify even the natural 

 relations of these laws, is there in the manifold of these laws some- 

 thing independent of these conventions and which may, so to speak, 

 play the role of universal invariant? For instance, the fiction has 

 been introduced of beings who, having been educated in a world dif- 

 ferent from ours, would have been led to create a non-Euclidean 

 geometry. If these beings were afterward suddenly transported into 

 our world, they would observe the same laws as we, but they would 

 enunciate them in an entirely different way. In truth there would 

 still be something in common between the two enunciations, but this 

 is because these beings do not yet differ enough from us. Beings still 

 more strange may be imagined, and the part common to the two sys- 

 tems of enunciations will shrink more and more. Will it thus shrink 

 in convergence toward zero, or will there remain an irreducible residue 

 which will then be the universal invariant sought? 



The question calls for precise statement. Is it desired that this 

 common part of the enunciations be expressible in words? It is clear 

 then that there are not words common to all languages, and we can 

 not pretend to construct I know not what universal invariant which 

 should be understood both by us and by the fictitious non-Euclidean 

 geometers of whom I have just spoken; no more than we can construct 

 a phrase which can be understood both by Germans who do not under- 

 stand French and by French who do not understand German. But 

 we have fixed rules which permit us to translate the French enuncia- 

 tions into German, and inversely. It is for that that grammars and 

 dictionaries have been made. There are also fixed rules for translating 

 the Euclidean language into the non-Euclidean language, or, if there 

 are not, they could be made. 



And even if there were neither interpreter nor dictionary, if the 

 Germans and the French, after having lived centuries in separate 

 worlds, found themselves all at once in contact, do you think there 

 would be nothing in common between the science of the German books 

 and that of the French books? The French and the Germans would 

 certainly end by understanding each other, as the American Indians 

 ended by understanding the language of their conquerors after the 

 arrival of the Spanish. 



But, it will be said, doubtless the French would be capable of 

 understanding the Germans even without having learned German, 

 but this is because there remains between the French and the Germans 

 something in common, since both are men. We should still attain 

 to an understanding with our hypothetical non-Euclideans, though 

 they be not men, because they would still retain something human. 

 But in any case a minimum of humanity is necessary. 



