DEMONSTRATION, AND NECESSARY TRUTHS. 279 



on Dr. Whewell s two great works (since acknowledged and reprinted in Sir 

 John Herschel s Essays) which maintains, on the subject of axioms, the doctrine 

 advanced in the text, that they are generalizations from experience, and sup 

 ports that opinion by a line of argument strikingly coinciding with mine. 

 When I state that the whole of the present chapter (except the last four 

 pages, added in the fifth edition) was written before I had seen the article, 

 (the greater part, indeed, before it was published,) it is not my object to 

 occupy the reader s attention with a matter so unimportant as the degree 

 of originality which may or may not belong to any portion of my own 

 speculations, but to obtain for an opinion which is opposed to reigning doc 

 trines, the recommendation derived from a striking concurrence of sentiment 

 between two inquirers entirely independent of one another. I embrace the 

 opportunity of citing from a writer of the extensive acquirements in physical 

 and metaphysical knowledge and the capacity of systematic thought which the 

 article evinces, passages so remarkably in unison with my own views as the 

 following ; 



&quot; The truths of geometry are summed up and embodied in its definitions 

 and axioms. . . . Let us turn to the axioms, and what do we find ? A string 

 of propositions concerning magnitude in the abstract, which are equally true of 

 space, time, force, number, and every other magnitude susceptible of aggrega 

 tion and subdivision. Such propositions, where they are not mere definitions, 

 as some of them are, carry their inductive origin on the face of their enuncia 

 tion. . . . Those which declare that two straight lines cannot inclose a space, 

 and that two straight lines which cut one another cannot both be parallel to a 

 third, are in reality the only ones which express characteristic properties of space, 

 and these it will be worth while to consider more nearly. Now the only clear 

 notion we can form of straightness is uniformity of direction, for space in its 

 ultimate analysis is nothing but an assemblage of distances and directions. And 

 (not to dwell on the notion of continued contemplation, i.e., mental experience, 

 as included in the very idea of uniformity ; nor on that of transfer of the contem 

 plating being from point to point, and of experience, during such transfer, of 

 the homogeneity of the interval passed over) we cannot even propose the propo 

 sition in an intelligible form to any one whose experience ever since he was born 

 has not assured him of the fact. The unity of direction, or that we cannot march 

 from a given point by more than one path direct to the same object, is matter of 

 practical experience long before it can by possibility become matter of abstract 

 thought. We cannot attempt mentally to exemplify the conditions of the assertion 

 in an imaginary case opposed to it, without violating our habitual recollection of 

 this experience, and defacing our mental picture of space as grounded on it. 

 What but experience, we may ask, can possibly assure us of the homogeneity of 

 the parts of distance, time, force, and measurable aggregates in general, on 

 which the truth of the other axioms depends ? As regards the latter axiom, after 

 what has been said it must be clear that the very same course of remarks equally 

 applies to its case, and that its truth is quite as much forced on the mind as that 

 of the former by daily and hourly experience, . . . including always, be it 

 observed, in our notion of experience, that which is gained by contemplation of 

 the inward picture which the mind forms to itself in any proposed case, or which 

 it arbitrarily selects as an example such picture, in virtue of the extreme sim- 



