258 REASONING. 



definitions, in this, that they are true without any mixture of 

 hypothesis. That things which are equal to the same thing 

 are equal to one another, is as true of the lines and figures in 

 nature, as it would be of the imaginary ones assumed in the 

 definitions. In this respect, however, mathematics are only 

 on a par with most other sciences. In almost all sciences 

 there are some general propositions which are exactly true, 

 while the greater part are only more or less distant approxi- 

 mations to the truth. Thus in mechanics, the first law of 

 motion (the continuance of a movement once impressed, until 

 stopped or slackened by some resisting force) is true without 

 qualification or error. 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 diminu- 

 tion 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, though they really want something 

 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 ob- 

 servation. 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 

 induction from the evidence of our senses. 



This opinion runs counter to a scientific prejudice of long 

 standing and great strength, and there is probably no pro- 

 position 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. I consider it very fortunate that so emi- 

 nent a champion of the contrary opinion as Dr. Whewell, has 



