"DEMONSTRATION, AND NECESSARY TRUTHS. 259 



found occasion for a most elaborate treatment of the whole 

 theory of axioms, in attempting to construct the philosophy 

 of the mathematical and physical sciences on the basis of the 

 doctrine against which I now contend. Whoever is anxious 

 that a discussion should go to the bottom of the subject, must 

 rejoice to see the opposite side of the question worthily re- 

 presented. If what is said by Dr. Whewell, in support of an 

 opinion which he has made the foundation of a systematic 

 work, can be shown not to be conclusive, enough will have 

 been done, without going further in quest of stronger argu- 

 ments and a more powerful adversary. 



It is not necessary to show that the truths which we call 

 axioms are originally suggested by observation, and that we 

 should never have known that two straight lines cannot inclose 

 a space if we had never seen a straight line : thus much being 

 admitted by Dr. Whewell, and by all, in recent times, who 

 have taken his view of the subject. But they contend, that it 

 is not experience which proves the axiom ; but that its truth 

 is perceived a priori, by the constitution of the mind itself, 

 from the first moment when the meaning of the proposition is 

 apprehended ; and without any necessity for verifying it by 

 repeated trials, as is requisite in the case of truths really 

 ascertained by observation. 



They cannot, however, but allow that the truth of the 

 axiom, Two straight lines cannot inclose a space, even if 

 evident independently of experience, is also evident from 

 experience. Whether the axiom needs confirmation or not, 

 it receives confirmation in almost every instant of our lives ; 

 since we cannot look at any two straight lines which intersect 

 one another, without seeing that from that point they con- 

 tinue to diverge more and more. Experimental proof crowds 

 in upon us in such endless profusion, and without one instance 

 in which there can be even a suspicion of an exception to the 

 rule, that we should soon have stronger ground for believing 

 the axiom, even as an experimental truth, than we have for 

 almost any of the general truths which we confessedly learn 

 from the evidence of our senses. Independently of a priori 

 evidence, we should certainly believe it with an intensity of 



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