758 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



right line." For centuries the greatest efforts have been made to 

 demonstrate that postulate, and then during the last century the 

 Russian Lobatschefski and the Hungarian Bolyai almost simulta- 

 neously showed that such a demonstration is impossible. Yet the 

 Acad6mie des Sciences each year receives a dozen or so pseudo- 

 demonstrations of that postulate. 



Lobatschefski did even better: Supposing that several parallel 

 lines can be drawn through a point parallel to a given right line, 

 retaining the other axioms of geometry, he proved a succession of 

 strange theorems, between which it was impossible to find any incon- 

 sistency, and built a new geometry, no more unmipeachable logically 

 than the ordinary Euclidean geometry. Then Riemann and yet 

 others came who showed that we could construct as many more geom- 

 etries as we wished, each perfectly logical and coherent. The theo- 

 rems of these new geometries are sometimes very odd. For instance, 

 the following one, unagmed by Poincare himself, has been demon- 

 strated: A real right line can be perpendicular to itself. 



I imagine that architects and engineers would scarcely admit such 

 deductions, although they are in no way logically contradictory. 

 That brings us to the kernel of our discussion. If, as results from 

 what has preceded, the axioms of geometry are only conventions, or, 

 as Poincar^ has expressed them, ''definitions in disguise," and if the 

 Euclidean geometry is no more absolutely true than any other, why 

 have men chosen and used it ? Because it is better adapted to our 

 needs, to our daily life, to the exterior world in which we live; because 

 in this world its theorems reduce to the simplest possible form the 

 relationships between things. A measurer could express just as 

 accurately by means of a Lobatschefskian geometry the relation 

 between the vohmie and the sides of a cube of wood. But it happens 

 from the nature of a cube of wood, or rather from the way our senses 

 comprehend it, that those relations would be more complex than with 

 the ordinary Euclidean geometry. It is possible to imagine a world 

 so constructed physically that men having our brains — that is, our 

 kind of logic — would not find Euclidean geometry the simplest. 



Geometry, then, is no longer the inner temple of the absolute. It 

 is an arbitrary creation of our mtellect. It can inform us only rela- 

 tively to the corresponding logical developments. However, in a 

 certain sense geometry depends also on experience, since, as we have 

 just seen, the exterior world appears simplest m the Euclidean 

 aspect. That does not mean that geometrical truths can be proved 

 or invalidated by experiment. Our instruments and our senses are 

 imperfect, whereas a geometrical theorem which is not exactly true 

 is false. If we measure with our instruments the sum of the angles 

 of a triangle drawn upon paper, we shall never fuid them exactly 

 equal to two right angles. Sometimes we will find the sum smaller, by 



