MATHEMATICS IN THE NINETEENTH CENTURY 493 



As a result of all these investigations, both in the protective as 

 also in the metric differential direction, we are led irresistibly to the 

 same conclusion, namely: The facts of experience can be explained 

 by all three geometries when the constant k is taken small enough. 

 It is, therefore) merely a question of convenience whether we adopt 

 the parabolic, hyperbolic, or elliptic geometry. 



The critical synthetic direction represents a return to the old syn- 

 thetic methods of Euclid, Lobatschewski, and Bolyai, with the added 

 feature of a refined and exacting logic. Its principal object is no 

 longer a study of non-Euclidean but of Euclidean geometry. Its 

 aim is to establish a system of axioms for our ordinary space which 

 is complete, compatible, and irreducible. The fundamental terms 

 point, line, plane, between, congruent, etc., are introduced as ab- 

 stract marks whose properties are determined by inter-relations in 

 the form of axioms. Geometric intuition has no place in this order 

 of ideas which regards geometry as a mere division of pure logic. 

 The efforts of this school have already been crowned with eminent 

 success, and much may be expected from it in the future. Its leaders 

 are Peano, Veronese, Fieri, Padoa, Burali-Forti, and Levi-Civitta, in 

 Italy, Pasch and Hilbert in Germany, and Moore in America. 



Closing at this point our hasty and imperfect survey of mathe- 

 matics in the last century, let us endeavor to sum up its main charac- 

 teristics. What strikes us at once is its colossal proportions and rapid 

 growth in nearly all directions, the great variety of its branches, the 

 generality and complexity of its methods; an inexhaustible creative 

 imagination, the fearless introduction and employment of ideal 

 elements, and an appreciation for a refined and logical development 

 of all its parts. 



We who stand on the threshold of a new century can look back on 

 an era of unparalleled progress. Looking into the future, an equally 

 bright prospect greets our eyes; on all sides fruitful fields of re- 

 search invite our labor and promise easy and rich returns. 



Surely this is the golden age of mathematics. 



