DEMONSTRATION, AND NECESSARY TRUTHS. 327 



which commands implicit assent, and removes all occasion of excep- 

 tion ; 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 sentiment 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 

 mental picture or idea of that object which we may on any occasion 

 summon before our imagination. . . . All propositions, therefore, 

 become not only untrue but inconceivable, if ... axioms be violated 

 in their enunciation." 



Another high authority (if indeed it be another authority) may 

 be cited in favour of the doctrine that axioms rest upon the evidence 

 of induction. "The axioms of geometry themselves 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 time, and body, and 

 again, in linear, and superficial, and solid space. Again, when we 

 say, the equals of equals are equal, we mentally 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 experience, they would not present 

 themselves to the mind. The only difference between these and 

 axioms obtained from extensive induction is this, that, in raising 

 the axioms of geometry, the instances offer themselves spontaneously, 

 and without the trouble of search, and are few and simple; in 

 raising those of nature, they are infinitely numerous, complicated, 

 and remote, so that the most diligent research and the utmost acute- 

 ness are required to unravel their web and place their meaning in 

 evidence." SIR J. HERSCHEL'S Discourse on the Study of Natural 

 Philosophy, pp. 95, 96. 



