DEMONSTRATION, AND NECESSARY TRUTHS. 173 



pion of the contrary opinion as Dr. Whewell has found occasion for a most 

 elaborate treatment of the whole theory of axioms, in attempting to con- 

 struct 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 represented. 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 elsewhere 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 can not 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 />roues the axiom ; but that its truth is per- 

 ceived a priori, by the constitution of the mind itself, from the first mo- 

 ment 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 can not, however, but allow that the truth of the axiom. Two 

 straight lines can not inclose a space, even if evident independently of ex- 

 perience, is also evident from experience. Whether the axiom needs con- 

 firmation or not, it receives confirmation in almost every instant of our 

 lives ; since we can not look at any two straight lines which intersect one 

 another, without seeing that from that point they continue to diverge more 

 and more. Experimental proof crowds in upon us in such endless j^rofu- 

 sion, 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 be- 

 lieving 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 be- 

 lieve it with an intensity of conviction far greater than we accord to any 

 ordinary physical truth : and this too at a time of life much earlier than 

 that from which we date almost any part of our acquired knowledge, and 

 much too early to admit of our retaining any recollection of the history of 

 our intellectual operations at that period. Where then is the necessity for 

 assuming that our recognition of these truths has a different origin from 

 the rest of our knowledge, when its existence is perfectly accounted for by 

 supposing its origin to be the same? when the causes which produce be- 

 lief in all other instances, exist in this instance, and in a degree of strength 

 as much superior to what exists in other cases, as the intensity of the be- 

 lief itself is superior? The burden of proof lies on the advocates of the 

 contrary opinion : it is for them to point out some fact, inconsistent with 

 the supposition that this part of our knowledge of nature is derived from 

 the same sources as every other part.* 



* Some persons find themselves prevented from believing that the axiom, Two straight lines 

 can not inclose a space, could ever become known to us through experience, by a difficulty 

 which may be stated as follows : If the straight lines spoken of are those contemplated in the 

 definition — lines absolutely without breadth and absolutely straight — that such are incapable 

 of inclosing a space is not proved by experience, for lines such as these do not present them- 

 selves in our experience. If, on the other hand, the lines meant are such straight lines as we 

 do meet with in experience, lines straight enough for practical purposes, but in reality slightly 

 zigzag, and with some, however trifling, breadth ; as applied to these lines the axiom is not 



