THE PRESENT PROBLEMS OF GEOMETRY 



BY DR. EDWARD KASNER 



[Edward Kasner, Instructor in Mathematics, Columbia University, b. New York 

 City, 1877. B.S. College of the City of New York, 1896; A.M. Columbia 

 University, 1899; Ph.D. ibid. 1899. Post-graduate, Fellow in Mathematics, 

 Columbia University, 1897-99 ; Student, University of Gottingen, 1899- 

 1900; Tutor in Mathematics, Columbia University, 1900-05; Instructor, 

 1905; Member American Mathematical Society; Fellow American Associa- 

 tion for Advancement of Science. Associate editor, Transactions American 

 Mathematical Society.] 



IN spite of the richness and power of recent geometry, it is notice- 

 able that the geometer himself has become more modest. It was the 

 ambition of Descartes and Leibnitz to discover universal methods, 

 applicable to all conceivable questions; later, the Ausdehnungslehre 

 of Grassmann and the quaternion theory of Hamilton were believed 

 by their devotees to be ultimate geometric analyses; and Chasles 

 attributed to the principles of duality and homography the same 

 role in the domain of pure space as that of the law of gravitation 

 in celestial mechanics. To-day, the mathematician admits the ex- 

 istence and the necessity of many theories, many geometries, each 

 appealing to certain interests, each to be developed by the most 

 appropriate methods; and he realizes that, no matter how large his 

 conceptions and how powerful his methods, they will be replaced 

 before long by others larger and more powerful. 



Aside from the conceivability of other spaces with just as self- 

 consistent properties as those of the so-called ordinary space, such 

 diverse theories arise, in the first place, on account of the variety 

 of objects demanding consideration, curves, surfaces, congruences 

 and complexes, correspondences, fields of differential elements, and 

 so on in endless profusion. The totality of configurations is indeed 

 not thinkable in the sense of an ordinary assemblage, since the total- 

 ity itself would have to be admitted as a configuration, that is, an 

 element of the assemblage. 



However, more essential in most respects than the diversity in 

 the material treated is the diversity in the points of view from which 

 it may be regarded. Even the simplest figure, a triangle or a circle, 

 has an infinity of properties indeed, recalling the unity of the 

 physical world, the complete study of a single figure would involve 

 its relations to all other figures and thus not be distinguishable from 

 the whole of geometry. For the past three decades the ruling thought 

 in this connection has been the principle (associated with the names 

 of Klein and Lie) that the properties which are deemed of interest 

 in the various geometric theories may be classified according to the 



