PUNISHING A SENIOR WRANGLER, 149 



answering, he gives me his opinion to the effect that the laws of mo- 

 tion are proved true by the truth of the " Principia " deduced from them. 

 Of this hereafter. My present purpose is to show that Newton did 

 not say this, and gave every indication of thinking the contrary. 

 He does not call the laws of motion " hypotheses ; " he calls them 

 " axioms." He does not say that he assumes them to be true pro-cision- 

 ally, and that the warrant for accepting them as actually true will 

 be found in the astronomically-proved truth of the deductions. He 

 lays them down just as mathematical axioms are laid down — posits 

 them as truths to be accepted a priori^ from which follow consequences 

 which must therefore be accepted. And, though the reviewer thinks 

 this an untenable position, I am quite content to range myself with 

 Newton in thinking it a tenable one — if, indeed, I may say so without 

 undervaluing the reviewer's judgment. But now, having shown that the 

 reviewer evaded the issue 1 raised, which it was inconvenient for him 

 to meet, I pass to the issue he substitutes for it. I will first deal with 

 it after the methods of ordinary logic, before dealing with it after the 

 methods of what may be called transcendental logic. 



To establish the truth of a proposition postulated, by showing that 

 the deductions from it are true, requires that the truth of the deduc- 

 tions shall be shown in some way that does not directly or indirectly 

 assume the truth of the proposition postulated. If, setting out with 

 the axioms of Euclid, we deduce the truths that "the angle in a semi- 

 circle is a right angle," and that " the opposite angles of any quadri- 

 lateral figure described in a circle are together equal to two right 

 angles," and so forth ; and, if, because these propositions are true, we 

 say that the axioms are true, w^e are gmltj of sl petitio principii, I do 

 not mean simply that, if these various propositions are taken as true on 

 the strength of the demonstrations given, the reasoning is circular, 

 because the demonstrations assume the axioms, but I mean more — I 

 mean that any supposed experimental proof of these propositions, by 

 mea&urement, itself assumes the axioms to be justified. For, even 

 when the supposed experimental proof consists in showing that some 

 two lines, demonstrated by reason to be equal, are equal when tested 

 in perception, the axiom, that things which are equal to the same thing 

 are equal to one another, is taken for granted. The equality of the 

 two lines can be ascertained only by carrying from the one to the 

 other some measure (either a movable marked line or the space be- 

 tween the points of compasses), and by assuming that the two lines 

 are equal to one another, because they are severally equal to this 

 measure. The ultimate truths of mathematics, then, cannot be estab- 

 lished by any experimental prooffthat the deductions from them are 

 true ; since the supposed experimental proof takes them for granted. 

 The same thing holds of ultimate physical truths. For the alleged 

 a 2^osteriori proof of these truths has a vice exactly analogous to the 

 vice I have just indicated. Every evidence yielded by astronomy, 



