April 28, 1916] 



SCIENCE 



591 



lowing passage in the epilogue of his 

 famous book: 



The figures of geometry wHeli once appeared 

 rigid and motionless — as one might say, lifeless, 

 acquired from the theory of transformations an 

 unlooked for vitality, by virtue of which they were 

 changed one into another, disclosing thus kinships 

 before unknown and establishing relations which 

 had been previously not even suspected. 



This expresses well the esthetic feeling of 

 a scientist who ponders upon the meaning of 

 his work, and it contains a hint of the mys- 

 tery of the fleeting fact and the truth 

 which endures. 



Within the century we see geometry com- 

 ing to definite ideal statements of her 

 foundations and her aspirations. Hilbert 

 has described the one, Klein the other. No 

 longer are we to see interminable debates 

 concerning empirical warrant or intuitive 

 warrant for the truths of this exalted sci- 

 ence; though these debates may be profita- 

 ble, they are not geometry. Mathematics 

 begins when we are agreed upon premises. 

 No longer is there to be the illusion of com- 

 pleteness, as if the problems could all be. 

 finished, their invariants determined and 

 interpreted. The question for an investi- 

 gator who considers a problem is now tha/t 

 of the miner who is prospecting for pre- 

 cious metals; he must ask himself: Have 

 questions like this proved simple enough 

 to be solved, and have the results proved 

 interesting or useful ? As for the range of 

 choice, there are appallingly long lists of 

 classes of geometries in the new ' ' Bncyklo- 

 padie der Mathematischen Wissensehaf ten. ' ' 

 Some I have named: differential geometry 

 is full of attractions; the study of twisted 

 curves and of the systems of curves on 

 surfaces is practically still in its beginnings, 

 with vigorous workers calling for recruits; 

 finite point-systems have claim to early con- 

 sideration ; manifolds in space of more than 

 three dimensions will later assume increas- 

 ing importance ; recent procedures in anal- 



ysis must some time be subjected to geo- 

 metrical statement in the hope of simplifi- 

 cation ; and nothing could be imagined more 

 exciting than the geometrical and kinemat- 

 ical speculations upon the configurations or 

 constellations that the physicists caU atoms 

 and molecules. 



My choice of topics has been apparently 

 capricious, for there is matter of impor- 

 tance and intense interest in all directions. 

 It is worth mentioning that no fewer than 

 five living Americans have produced books 

 on the theory of functions, or some great 

 division of that subject. Elliptic functions 

 and the vast subject of hyperelliptic and 

 Abelian functions are temporarily less 

 active, while differential equations, and 

 their successors, integral equations, are in 

 the forefront of progress. The theory of 

 numbers in its modern form dates back 

 only to 1800, and teems with marvels un- 

 foreseen. It is within fifty years that 

 Lindemann succeeded in proving the num- 

 ber TT transcendental — the ratio of circum- 

 ference to diameter, and it was almost 

 twenty years later that the base of Napier- 

 ian logarithms, the number e, was put into 

 the same category. The classification and 

 discovery of transcendental numbers is still 

 going on. Concerning the combinatory 

 analysis with its store of theories yet incom- 

 plete, one does not need to bring news to 

 Syracuse, nor coals to Newcastle. But I 

 may mention the researches on point-sets, 

 set in motion by Kantor at Halle, and refer 

 you to the summary views and keen com- 

 mentaries of Van Vleck, professor in the 

 University of "Wisconsin, in his recent ad- 

 dress as retiring president of the American 

 Mathematical Society.^ 



The close of the past century saw the 



extraordinary growth of scientific societies, 



and in particular of mathematical societies. 



Three I may instance, all less than thirty 



2 See Science, Vol. 39 (new series), pp. 113-124. 



