Nineteenth-Century Science and Religion i ^j 



their relativity as against rationalistic a priori reasoning. Thus 

 was shown the subjective character of those sensorial elements 

 which the mechanical theory had raised to the position of ultimate 

 reality. Were not resistance, space, and time perhaps representa- 

 tions which, no less than sound, colour, scent, and taste, depended 

 on the particular physiological structure of the organisms ? What 

 right have we to consider the former as objective and primary 

 qualities, and the latter as subjective and secondary qualities ? 

 According to Helmholtz, the only legitimate distinction between 

 these elements is of a practical character, in that some of them 

 serve better than the others to make us turn towards* facts, by 

 arousing in us expectations which habitually come true. The 

 possibility of discovering the psychological origin of these com- 

 plexities and of breaking them up into their component parts 

 seemed, in the first enthusiasm aroused by research, a clear proof 

 of the empirical nature of geometrical truths, and, with them, of 

 mechanics. 



4. A further contribution was made in the same direction by 

 the new non-Euclidian geometry, which aroused a belief in the 

 incontrovertible certainty of mathematical truths and led to a 

 logical re-elaboration of them. The principles on which thev 

 were founded showed themselves to be freely created by the mind, 

 and, losing the character of intuitive evidence which rationalism 

 attributed to them, were reduced merely to " useful fictions " 

 (Poincare). Mathematical theories assumed the structure of 

 " hypothetical deductive systems " (Fieri), based on a certain 

 number of indefinable factors and undemonstrable propositions 

 freely chosen by convention. Mathematical truths have there- 

 fore no character of objective necessity, but are dependent upon 

 those initial conventions. Euclid's postulate, " If two straight 

 lines on the same plane are intersected by a third they will be inter- 

 sected by that part in which the sum of the two interior angles is 

 less than two right angles," is, in fact, one of these conventions ; 

 while the theorem that " The sum of the interior angles of a 

 triangle is equal to two right angles " is only true if that postulate is 

 admitted. 



5. The most serious blow to the mechanical theory came 

 from Carnot's principle. Mechanics, as ordinarily understood, 

 is the study of reversible phenomena. If the parameter, which 

 represents time, and which has taken increasing values during the 



