DEMONSTRATIOV, AND NECESSARY TRUTHS. 163 



character of necessity in any sense beyond this, as implying an evidence 

 independent of and snporior to observation and experience, must depend 

 upon the previous estabHshment of such a claim in favor of the defini- 

 tions and axioms themselves. With regard to axioms, we found that, 

 considered as experimental truths, they I'est upon superabundant and 

 obvious evidence. We inquired, whether, since this is the case, it be 

 necessaiy to suppose any other evidence of those truths than experi- 

 mental evidence, any other origin for our belief of them than an experi- 

 mental origin. We decided, that the burden of proof lies with those 

 who maintain the affirmative, and we examined, at considerable length, 

 such arguments as they have produced. The examination having led 

 to the rejection of those arguments, we have thought ourselves war- 

 ranted in concluding that axioms are but a class, the highest class, of 



by the counterpressure which supports the fulcrum. But would not this destruction 

 equally arise, and by the same amount of counteracting force, if each force simply pressed 

 its own half of the lever against the fulcrum ? And what can assure us that it is not so, 

 except removal of one or other force, and consequent tilting of the lever ? The other fun- 

 damental axiom of statics, that the pressure on the point of support is the sum of the 

 weights ... is merely a scientific transformation and more refined mode of stating a coarse 

 and obvious result of universal experience, viz., that tire weight of a rigid body is the same, 

 handle it or suspend it in what position or by what point we will, and that whatever sus- 

 tains it sustains its total weight. Assuredly, as Mr. Whewell justly remarks, ' No one 

 probably ever made a trial for the purpose of showing that the pressure on the support is 

 equal to the sum of the weights.' . . . But it is precisely because in every action of his life 

 from earliest infancy he has been continually making the trial, and seeing it made by every 

 other living being about him, that he never dreams of staking its result on one additional 

 attempt made with scientific accuracy. This would be as if a man should resolve to de- 

 cide by experiment whether his eyes were useful for the purpose of seeing, by hermetically 

 sealing himself up for half an hour in a metal case." 



On the " paradox of universal propositions obtained by experience," the same writer 

 says : " If there be necessary and universal truths expressible in propositions of axiomatic 

 simplicity and obviousness, and having for their subject-matter the elements of all our ex- 

 perience and all our knowledge, surely these are the truths which, if experience suggests to 

 us any truths at all, it ought to suggest most readily, clearly, and unceasingly. If it were 

 a truth, universal and necessary, that a net is spread over the whole surface of every plan- 

 etary globe, we should not travel far on our own without getting entangled in its meshes, 

 and making the necessity of some means of extrication an axiom of locomotion. . . .There 

 is, therefore, nothing paradoxical, but the reverse, in our being led by observation to a re- 

 cognition of such truths, as general propositions, coextensive at least with all human expe- 

 rience. That they pervade all the objects of experience, must ensure their continual sug- 

 gestion by experience ; that they are true, must ensure that consistency of suggestion, that 

 iteration of uncontradicted assertion, which commands implicit assent, and removes all oc- 

 casion of exception ; that they are simple, and admit of no misunderstanding, must secure 

 their admission by every mind." 



" A truth, necessary and universal, relative to any object of our knowledge, must verify 

 itself in every instance where that object is before our contemplation, and if at the same 

 time it be simple and intelligible, its verification must be obvious. The setitiment of such a 

 truth cannot, therefore, but be present to our minds whenever that object is contemplated, and must 

 therefore make a part of the menial picture or idea of that object which we may on any occasion stimmon 

 before our imagination. . . . All propositions, therefore, become not only untrue but inconceivable, if 

 . . . axioms be violated in therr enunciation." 



Another high authority (if indeed it be another authority) may be cited in favor of the 

 doctrine that axioms rest upon the evidence of induction. " The axioms of geometry them- 

 selves may be regarded as in some sort an appeal to experience, not corporeal, but mental. 

 When we say, the whole is greater than its part, we announce a general fact, which rests, 

 it is true, on our ideas of whole and part ; but, in abstracting these notions, we begin by 

 considering them as subsisting in space, and tune, and body, and again, in linear, and su- 

 perficial, and solid space. Again, when we say, the equals of equals are equal, we men- 

 tally make comparisons, in equal spaces, equal times, &c., so that these axioms, however self- 

 evident, are still general propositions so far of the inductive kind, that, independently of ex- 

 perience, they would not present themselves to the mind. The only diilerence between 

 these and axioms obtained from extensive induction is this, that, in raising the axioms of 

 geometry, the instances oifer themselves spontaneously, and without the trouble of search, 

 and are few and simple ; in raismg those of nature, they are infinitely numerous, compli- 

 cated, and remote, so that the most diligent research and the utmost acuteness are required 

 to unravel their web and place their meaning in evidence."— Sir J. Herschel's Discourse 

 on the Htudy of Natural Philosophy, pp. 95, 96. 



