DEMONSTRATION, AND NECESSARY TRUTHS. 305 



error ; it is not affected by the frictions, rigidities, and 

 miscellaneous disturbing causes, which qualify, for 

 example, the theories of the lever and of the pulley. 

 The rotation of the earth in twenty-four hours, of the 

 same length as in our time, has gone on since the 

 first accurate observations, without the increase or 

 diminution of one second in all that period. These are 

 inductions which require no fiction to make them be 

 received as accurately true : but along with them there 

 are others, as for instance the propositions respecting 

 the figure of the earth, which are but approximations 

 to the truth ; and in order to use them for the further 

 advancement of our knowledge, we must feign that 

 they are exactly true, although they really want some- 

 thing of being so. 



$ 4. It remains to inquire, what is the ground of 

 our belief in axioms what is the evidence on which 

 they rest ? I answer, they are experimental truths ; 

 generalizations from observation. The proposition, 

 Two straight lines cannot inclose a space or in other 

 words, Two straight lines which have once met, do 

 not meet again, but continue to diverge is an induc- 

 tion from the evidence of our senses. 



This opinion runs counter to a philosophic pre- 

 judice of long standing and great strength, and there is 

 probably no one proposition enunciated in this work for 

 which a more unfavourable reception is to be expected. 

 It is, however, no new opinion ; and even if it were 

 so, would be entitled to be judged, not by its novelty, 

 but by the strength of the arguments by which it can 

 be supported. 1 consider it very fortunate that so 

 eminent a champion of the contrary opinion as Mr. 

 Whewell, has recently found occasion for a most 

 elaborate treatment of the whole theory of axioms, in 

 VOL. i. x 



