258 HERMANN VON HELMHOLTZ 



'I believe that the considerations here adduced are not 

 without weight for the question of the original discovery of 

 the geometrical propositions. For when men were seeking 

 for a mathematical formula which should coincide with their 

 more or less exact observations and measurements, they could 

 find none that they could consistently carry through, save that 

 expressed in the Pythagorean proposition, since as a matter of 

 fact there was no other. And in this, as I believe, lies the 

 foundation of the peculiar sort of conviction that we cherish in 

 regard to the axioms that are unprovable either in theory or in 

 practice. We have indeed no choice but to accept them, unless 

 we mean to forgo all possibility of spatial measurement/ 



Helmholtz dissents altogether from Kant's doctrine of the 

 a priori forms of intuition and of the axioms of geometry, and 

 inquires into the facts that underlie geometry, or the question 

 what geometrical laws express actual facts, and what on the 

 contrary are merely definitions, or conclusions from definitions, 

 and from the special modes of expression selected. The answer 

 to this question, however, presents enormous difficulties, because 

 geometry always has to do with ideal figures, to which the 

 material figures of the actual world can only approximate. 

 The decision whether, e. g., the surfaces of a body are plane, 

 its sides straight, &c., can only be solved by the laws of 

 geometry, the positive accuracy of which has first to be proven. 

 We see without difficulty, that in addition to the Euclidean 

 axioms as usually proposed for geometry, a whole series of 

 other facts are tacitly admitted. It is essential to note in 

 particular that we can only conceive intuitions of such relations 

 of space as can be represented in actual space, and that we 

 must not let ourselves be misled by this power of conception 

 into assuming as a matter of course, what is in reality a par- 

 ticular and by no means self-evident characteristic of the 

 external world that is before us. 



But since analytical geometry only treats of spatial figures 

 as magnitudes, which are determined by other magnitudes, 

 since all the spatial relations known to us are measurable, 

 i.e. can be referred to determinations of magnitude, length of 

 line, angles, surfaces, &c., it has no need of intuition for its 

 proof, and Helmholtz was led by this consideration to the 

 further question what analytical properties of space and spatial 



