162 REASONING. 



CHAPTER VL 



THE SAME SUBJECT CONTINUED. 



§ 1. In the examination which formed the subject of the last chapter, 

 into the nature of the evidence of those deductive sciences which are 

 commonly represented to be systems of necessary truth, we have been 

 led to the following conclusions. The results of those sciences are 

 indeed necessary, in the sense of necessarily following from certain first 

 principles, commonly called axioms and definitions; of being certainly 

 true if those axioms and definitions are so. But their claim to the 



pacity 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 

 magnitude 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 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 straight- 

 ness is uniformity of direction, for space in its ultimate analysis is nothing but an assem- 

 blage of distances and directions. And (not to dwell on the notion of continued contem- 

 plation, i. e., mental experience, as included in the very idea of uniformity ; nor on that of 

 transfer of the contemplating 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, ivithout vi- 

 olating 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 homo- 

 geneity 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, i/i virtue of the extreme simplicity of these 

 primary relations, being called up by the imagination with as 7nuch 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 

 expected that, in a science of obviously contingent relations, we should acquiesce in acon- 

 trary 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 balance 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 be- 

 ing self-evident that it has even a paradoxical appearance, which is only to be removed by 

 giving our lever thickness, material composition, -and molecular powers. Again we con- 

 clude, that the two forces, being equal and applied under precisely similar circumstances, 

 must, if they exert any effort at all to turn the lever, exert equal and opposite efforts : but 

 what ^ priori reasoning can possibly assure us that they do act under precisely similar cir- 

 cumstances? that points which differ in place, are similarly circumstanced as regards the 

 exertion of force ? that universal space may not have relations to universal force — or, at all 

 events, that the organization of the material universe may not be such as to place that por- 

 tion of space occupied by it in such relations to the forces exerted in it, as may invalidate 

 the absolute similarity of circumstances assumed ? Or we may argue, what have we to do 

 with the notion of angular movement in the lever at all ? The case is one of rest, and of 

 quiescent destruction of force by force. Now how is this destruction effected ? Assuredly 



