560 GEOMETRY 



groups of transformations which leave those properties unchanged. 

 Thus almost all discussions on algebraic curves are connected with 

 the group of displacements (more properly the so-called principal 

 group), or the group of projective transformations, or the group of 

 birational transformations; and the distinction between such theories 

 is more fundamental than the distinction between the theories of 

 curves, of surfaces, and of complexes. 



Historically, the advance has been, in general, from small to larger 

 groups of transformations. The change thus produced may be likened 

 to the varying appearance of a painting, at first viewed closely in all 

 its details, then at a distance in its significant features. The analogy 

 also suggests the desirability of viewing an object from several stand- 

 points, of studying geometric configurations with respect to various 

 groups. It is indeed true, though in a necessarily somewhat vague 

 sense, that the more essential properties are those invariant under 

 the more extensive groups; and it is to be expected that such groups 

 will play a predominating role in the not far distant future. 



The domain of geometry occupies a position, as indicated in the 

 programme of the Congress, intermediate between the domain of 

 analysis on the one hand and of mathematical physics on the other; 

 and in its development it continually encroaches upon these adjacent 

 fields. The concepts of transformation and invariant, the algebraic 

 curve, the space of n dimensions, owe their origin primarily to the 

 suggestions of analysis; while the null-system, the theory of vector 

 fields, the questions connected with the applicability and deforma- 

 tion of surfaces, have their source in mechanics. It is true that some 

 mathematicians regard the discussion of point sets, for example, 

 as belonging exclusively to the theory of functions, and others look 

 upon the composition of displacements as a part of mechanics. 

 While such considerations show the difficulty, if not impossibility, 

 of drawing strict limits about any science, it is to be observed that 

 the consequent lack of definiteness, deplored though it be by the 

 formalist, is more than compensated by the fact that such overlap- 

 ping is actually the principal means by which the different realms 

 of knowledge are bound together. 



If a mathematician of the past, an Archimedes or even a Descartes, 

 could view the field of geometry in its present condition, the first 

 feature to impress him would be its lack of concreteness. There are 

 whole classes of geometric theories which proceed, not merely with- 

 out models and diagrams, but without the slightest (apparent) use 

 of the spatial intuition. In the main this is due, of course, to the 

 power of the analytic instruments of investigation as compared 

 with the purely geometric. The formulas move in advance of thought, 

 while the intuition often lags behind; in the oft-quoted words of 

 d'Alembert, "L'algebre est ge'ne'reuse, elle donne souvent plus qu'on 



