TENDENCIES IN MATHEMATICAL SCIENCE 337 



national societies are formed, and international congresses held. 

 These tendencies are naturally not confined to the mathemat- 

 ical sciences. In some respects mathematics has merely en- 

 joyed its share in the prosperity of a more scientific age, in some 

 it has perhaps suffered, at any rate relatively, from the powerful 

 stimulus given the natural sciences by the working out of evolu- 

 tionary theories. From a position of acknowledged primacy 

 among a small number of recognized sciences, it has come to be 

 regarded as but one of many. 



It is impossible here even to enumerate the different branches 

 of mathematical science developed during this period. Certain 

 typical features may however be touched upon. 



NON-EUCLIDEAN GEOMETRY. Each century takes over as a 

 heritage from its predecessors a number of problems whose solution 

 previous generations of mathematicians have arduously but vainly 

 sought. It is a signal achievement of the nineteenth century to have 

 triumphed over some of the most celebrated of these problems. 



The most ancient of them is the quadrature of the circle, which 

 already appears in our oldest mathematical document, the Papyrus 

 Rhind, B.C. 2000. Its impossibility was finally shown by Lindemann, 

 1882. 



But of all problems which have come down from the past, by far 

 the most celebrated and important relates to Euclid's parallel axiom. 

 Its solution has profoundly affected our views of space and given rise to 

 questions even deeper and more far-reaching, which embrace the entire 

 foundation of geometry and our space conception. Pierpont (1904). 



I am convinced more and more that the necessary truth of our 

 geometry cannot be demonstrated, at least not by the human intellect 

 to the human understanding. Perhaps in another world we may gain 

 other insights into the nature of space which at present are unattain- 

 able to us. Until then we must consider geometry as of equal rank 

 not with arithmetic, which is purely a priori, but with mechanics. 



- Gauss (1817). 



There is no doubt that it can be rigorously established that the 

 sum of the angles of a rectilinear triangle cannot exceed 180. But 

 it is otherwise with the statement that the sum of the angles cannot 

 be less than 180 ; this is the real Gordian knot, the rocks which cause 



