I 



DEMONSTRATION, AND NECESSARY TRUTHS. 185 



the doctrine took its place as a universal truth, but as one proved to be 

 such by experience. That the theory itself preceded the proof of its truth 

 — that it had to be conceived before it could be proved, and in order that 

 it miglit be proved — does not imply that it was self-evident, and did not 

 need proof. Otherwise all the true theories in the sciences are necessary 

 and self-evident; for no one knows better than Dr. Whewell that tliey all 

 began by being assumed, for the purpose of connecting them by deduc- 

 tions with those facts of experience on which, as evidence, they now con- 

 fessedly rest.* 



* The Quarterly Review for June, 1841, contained an article of great ability on Dr. "Whe- 

 well'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 supports 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 por- 

 tion 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 : 



"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 mag- 

 nitude in the abstract, which are equally true -of space, time, force, number, and every other 

 magnitude susceptible of aggregation and subdivision. 'Such propositions, where they are 

 not mere definitions, as some of them are, carry their inductive origin on the face of their 



enunciation Those which declare that tw^o straight lines can not inclose a space, and 



tiiat two straigiit lines which cut one another can not both be parallel to a third, are in reality 

 the only ones which express characteristic properties of space, and these it will be w'orth while 

 to consider more nearly. Now the only clear notion we can form of straightness is uniform- 

 ity of direction, for space in its idtimate analysis is nothing but an assemblage of distances 

 and directions. And (not to dwell on the notion of continued contemplation, i. e., mental ex- 

 perience, 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 can not even propose the proposition 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 can not 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 can not attempt mentally to exemplify the conditions of the assertion 

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

 ence, and defacing our mental picture of space as grounded on it. Wliat 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 re- 

 gards the latter axiom, after what has been said it must be clear that the very same course of 

 remarks equallj- 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 simplicity of these primary relations, being called up by the 

 imagination with as much vividness and clearness as could be done by any external impression, 

 which is the only meaning we can attach to the word intuition, as applied to such relations." 



And again, of the axioms of mechanics: "As we admit no such propositions, other than 

 as truths inductively collected from observation, even in geometry itself, it can hardly be ex- 

 pected that, in a science of obviously contingent relations, we should acquiesce in a contraiy 

 view. Let us take one of these axioms and examine its evidence : for instance, that equal 

 forces perpendicularly applied at the opposite ends of equal arms of a straight lever will bal- 

 ance each other. What but experience, we may ask, in the first place, can possibly inform us 

 that a force so applied will have any tendency to turn the lever on its centre at all ? or that 

 force can be so transmitted along a rigid line perpendicular to its direction, as to act elsewhere 

 in space than along its own line of action ? Surely this is so far from being self-evident that 



