566 GEOMETRY 



where there is an ordinary discontinuity, this is indicated by a break 

 in the graph; iff is continuous, but the derivative f has such a dis- 

 continuity, this shows itself by a sharp turn in the curve; if the 

 discontinuity is only in the second derivative, there is a sudden 

 change in the radius of curvature, which is, however, relatively 

 difficult to observe from the figure; finally, if the third derivative 

 is discontinuous, the effect upon the curve is no longer apparent. 

 Does this mean that it is impossible to picture it? Does it not rather 

 indicate a limitation in the usual geometric training which goes 

 only as far as relations expressible in terms of tangency and curva- 

 ture? For the interpretation of the third derivative it is necessary 

 to consider say the osculating parabola at each point of the curve: 

 in the case referred to, as we pass over the critical point, the 

 tangent line and osculating circle change continuously, but there is 

 a sudden change in the osculating parabola. If in fact our intuition 

 were trained to picture osculating algebraic curves of all orders, it 

 would detect a discontinuity in a derivative of any order. A partial 

 equivalent would be the ability to picture the successive evolutes 

 of a given curve; a complete equivalent would be the picturing of 

 the successive slope curves y=f'(x), y=f"(x), etc. All this requires, 

 evidently, only an increase in the intensity of our intuition, not a 

 change in its nature. 



This, however, would not apply to all questions. There are func- 

 tions which, while possessing derivatives of all orders (then neces- 

 sarily continuous), are not analytic (that is, not expressible by power 

 series). What is it that distinguishes the analytic curves among this 

 larger class? Is it possible to put the distinction in a form capable 

 of assimilation by an idealized intuition? In short, what is the 

 really geometric definition of an analytic curve ? 1 



Much recent work in fu action theory has had for its point of de- 

 parture a more general basis than the theory of curves, namely, the 

 theory of sets or assemblages of points, with special reference to 

 the notions of derived set and the various contents or areas. The 

 geometry of point sets must indeed be regarded as one of the most 

 important and promising in the whole field of mathematics. It 

 receives its distinctive character, as compared with the general 

 abstract theory of assemblages (Mengenlehre}, from the fact that it 

 operates not with all one-to-one correspondences, but with the 

 group of analysis situs, the group of continuous one-to-one corre- 

 spondences. From the point of view of the larger group, there is no 

 distinction between a one-dimensional and a two- or many-dimen- 

 sional continuum (Cantor). This is still the case if the correspondence 



1 One method of attack would be the interpretation of Pringsheim's condi- 

 tions; this requires not merely the individual derivative curves, but the limit of 

 the system. 



