562 GEOMETRY 



highly probable, but do not constitute a complete proof; while the 

 exact meaning of the term general can be determined only from 

 the context. 



The reaction against such loose methods is represented by Study 1 

 in algebraic geometry, and Hilbert in differential geometry. The 

 tendency of a considerable portion of recent work is towards the 

 exhaustive treatment of definite questions, including the considera- 

 tion of the special or degenerate cases ordinarily passed over as 

 unimportant. Another aspect of the same tendency is the discussion 

 of converses of familiar problems, with the object of obtaining con- 

 ditions at once necessary and sufficient, that is, completely character- 

 istic results. 2 



Another set of problems is suggested by the relation of geometry 

 to physics. It is the duty of the geometer to abstract from the physical 

 sciences those domains which may be expressed in terms of pure 

 space, to study the geometric foundations (or, as some would put it, 

 the skeletons) of the various branches of mechanics and physics. 

 Most of the actual advance, it is true, has hitherto come from the 

 physicists themselves, but undoubtedly the time has arrived for 

 more systematic discussions by the mathematicians. In addition to 

 the importance which is due to possible applications of such work, 

 it is to be noticed that we meet, in this way, configurations as inter- 

 esting and remarkable as those created by the geometer's imagina- 

 tion. Even in this field, one is tempted to remark, truth is stranger 

 than fiction. 



We have now considered, briefly and inadequately, some of the 

 leading ideals and influences which are at work towards both the 

 widening and the deepening of geometry in general ; and turn to our 

 proper topic, a survey of the leading problems or groups of problems 

 in certain selected (but it is hoped representative) fields of contem- 

 poraneous investigation. 



Foundations 



The most striking development of geometry during the past decade 

 relates to the critical revision of its foundations, more precisely, its 

 logical foundations. There are, of course, other points of view, for 



1 " [Es 1st eine] tief eingewurzelte Gewohnheit yieler Geometer, Satze zu formu- 

 lieren, die 'im allgemeinen ' gelten sollen, d. h. einen klaren Sinn iiberhaupt nicht 

 haben, zudem noch haufig als allgemein giiltig hingestellt oder mangelhaft be- 

 griindet werden. [Dies Verfahren wird], trotz etwanigen Verweisungen auf Trager 

 sehr beriihmter Namen, spateren Geschlechtern sicher als ganz unzulassig erschei- 

 nen, scheint aber in unserem ' kritischen ' Zeitalter von vielen als eine berechtigte 

 Eigentumlichkeit der Geometric betrachtet zu werden . . ." Jahr. Deut. Math.- 

 Ver., vol. xi (1902), p. 100. 



2 As an example may be mentioned* the theorem of Malus and Dupin, known 

 for almost a century, that the rays emanating from a point are converted, by any 

 refraction, into a normal congruence. Quite recently, Levi-Civitta succeeded in 

 showing that this property is characteristic; that is, any normal congruence may 

 be refracted into a bundle. 



