GEOMETRICAL AXIOMS. 29 



It appears thereby to postulate, for this a priori 

 form, not only the character of a purely formal scheme 

 of itself quite unsubstantial, in which any given result 

 experience would fit ; but also to include certain pe- 

 culiarities of the scheme, which bring it about that 

 only a certain content, and one which, as it were, is 

 strictly defined, could occupy it and be apprehended 

 by us. 1 



It is precisely this relation of geometry to the theory 

 of cognition which emboldens me to speak to you on 

 geometrical subjects in an assembly of those who for 

 the most part have limited their mathematical studies 

 to the ordinary instruction in schools. Fortunately, 

 the amount of geometry taught in our gymnasia will 

 enable you to follow, at any rate the tendency, of the 

 principles I am about to discuss. 



I intend to give you an account of a series of 

 recent and closely connected mathematical researches 

 which are concerned with the geometrical axioms, their 



1 In his book, On the Limits of Philosophy, Mr. W. Tobias main- 

 tains that axioms of a kind which I formerly enunciated are a 

 misunderstanding of Kant's opinion. But Kant specially adduces 

 the axioms, that the straight line is the shortest (Kritik der reinen 

 Vernunft, Introduction, v. 2nd ed. p. 16) ; that space has three di- 

 mensions (Ibid, part i. sect. i. 3, p. 41) ; that only one straight line 

 is possible between two points (Ilrid. part ii. sect. i. * On the Axioms 

 of Intuition '), as axioms which express a priori the conditions of 

 intuition by the senses. It is not here the question, whether these 

 axioms were originally given as intuition of space, or whether they 

 are only the starting-points from which the understanding can 

 develop such axioms a priori on which my critic insists. 



