PRESENT PROBLEMS OF GEOMETRY 565 



The work on logical foundations has been confined almost entirely 

 to the Euclidean and projective geometries. It is desirable, however, 

 that other geometric theories should be treated in a similar deductive 

 fashion. In particular, it is to be hoped that we shall soon have 

 a really systematic foundation for the so-called inversion geometry, 

 dealing with properties invariant under circular transformations. 

 This theory is of interest, not only for its own sake and for its appli- 

 cations in function theory, but also because its study serves to free 

 the mind from what is apt to become, without some check, slavery to 

 the projective point of view. 



The Curve Concept Analysis Situs 



Although curves and surfaces have constituted the almost exclu- 

 sive material of the geometric investigation of the thirty centu- 

 ries of which we have record, it can hardly be claimed that the con- 

 cepts themselves have received their final analysis. Certain vague 

 notions are suggested by the naive intuition. It is the duty of mathe- 

 maticians to create perfectly precise concepts which agree more or 

 less closely with such intuitions, and at the same time, by the reac- 

 tion of the concepts, to refine the intuition. The problem, evidently, is 

 not at all determinate. It would be of interest to trace the evolution 

 which has actually produced several distinct curve concepts defining 

 more or less extensive classes of curves, agreeing in little beyond the 

 possession of an infinite number of points. 



The more familiar special concepts or classes of curves are defined 

 in terms of the corresponding equation y=f(x] or function f(x). 

 Such are, for example: (1) algebraic curves; (2) analytic curves; 

 (3) graphs of functions possessing derivatives of all orders; (4) the 

 curves considered in the usual discussions of infinitesimal geometry, 

 in which the existence of first and second derivatives is assumed; 

 (5) the so-called regular curves with a continuously turning tangent 

 (except for a finite number of corners); (6) the so-called ordinary 

 curves possessing a tangent and having only a finite number of 

 oscillations (maxima and minima) in any finite interval; (7) curves 

 with tangents; (8) the graphs of continuous functions. 



How far are such distinctions accessible to the intuition? Of 

 course there are limitations. For over two centuries, from Descartes 

 to the publication of Weierstrass's classic example, the intuition of 

 mathematicians declared the classes (7) and (8) to be identical. Still 

 later it 'was found that such extraordinary (pathological or crinkly) 

 curves may present themselves in class (7). However, even here 

 partially successful attempts to connect with intuition have been 

 made by Wiener, Hilbert, Schoenflies, Moore, and others. 



Let us consider a simpler extension in the field of ordinary curves. 

 If the function f(x) is continuous except for a certain value of x 



