DEMONSTRATION AND NECESSARY TRUTHS. 



i6s 



the word necessity, even in this ac- 

 ceptation of it, means no more than 

 certainty. But their claim to the 

 character of necessity in any sense 

 beyond this, as implying an evidence 



us that a force so applied will have any 

 tendency to turn the lever on its centre at 

 all ? or that force c;in be so transmitted 

 along a rigid line perpendicular to its di- 

 rection as to act elsewhere in space tlian 

 along its own line of action ? Surely this 

 is so far from being self-evident that it 

 has even a paradoxical appearance, which 

 is only to be removed by giving our lever 

 thickness, material composition, and mole- 

 cular powers. Again, we conclude 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 d priori reasoning can possibly 

 assure us that they do act under precisely 

 similar circumstances ? that points which 

 differ in place are similarly circumstanced 

 as regards the exertion of force ? that uni- 

 versal space may not have relations to 

 universal force— or, at all events, that tiie 

 organisation of the material universe may 

 not be such as to place that portion of 

 space occupied by it in sue a 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 move- 

 ment 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 by the counter-pres- 

 sure 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 aurainst the fulcrum ? And 

 what can assure us that it is not so, except 

 removal of one or other force, and conse- 

 quent tilting of the lever ? The other fun- 

 dament al axiom of statics, that the pres 

 sure on tiie 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 uni- 

 versal experience, viz. that the weight of 

 a rigid body is the same, handle it or sus- 

 pend it in what position or by what point 

 we will, and that whatever sustains it 

 sustains its total weight. Assuredly, as 

 Mr. Wliewell justly romarks, ' No one pro- 

 bably 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 



independent of and superior to obser- 

 vation and experience, must depend 

 on the previous establishment of such 

 a claim in favour of the definitions 

 and axioms themselves. With regard 



as if a man should resolve to decide by 

 experiment whether liis eyes were usetul 

 for the purpose of seeintr by hermetically 

 sealing himself up for half an hour in a 

 metal case. " 



On the "paradox of universal proposi- 

 tions obtained by experience," the same 

 writer says: "If there be necessary and 

 universal truths expressible in pro[)Osition8 

 of axiomatic simplicity and obviousness, 

 and having for their siibject-matter the 

 elements of all our experience 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 erf 

 every planetary globe, weshotdd not travel 

 far in 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 bemg led by observation to a recogni- 

 tion of such truths as general propositions, 

 co-extensive at least with all human ex- 

 perieiice. That they pervade all the objects 

 of experience must ensure their continual 

 suggestion by experience ; tliat they are 

 true, must ensure that consistency of sug- 

 gestion, that iteration of uncontradicted 

 assertion, which commandsimplicit assent, 

 and removes all occasion of exception ; that 

 they are simple, and admit of no misunder- 

 standing, must secure their admission by 

 every mind." 



"A truth, necessary and universal, re- 

 lative to any object of our knowledge, 

 must verity itself in every instance where 

 til at object is before our contemplation, 

 and if at the same time it be simple and 

 intelligible, its verification must be ob- 

 vious. The sentiment of such a truth cannot, 

 therefore, but be present to our minds when- 

 ever 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 eminent mathematician had 

 previously sanctioned by his authority the 

 doctrine of the origin of geometrical axioms 

 in experience " Giometry is thus founded 

 likewise on observation ; but of a kind so 

 familiar and obvious that the pri-nary 

 notions which it furnishes might seem 

 intuitive."— 6'ir John Leslie, quoied by 

 Sir Wilham Hamilton, Discourses, die, p. 

 272. 



