IW Mr. J. P. Hennessy on some Facts connected 



" Parallel right lines are such as are in the same plane, and 

 which, being produced continually in both directions, would 

 •never meet/' 



Adopting this definition, we are forced to give an indirect 

 proof to the XXVI Ith ; but if we change its form to an affirma- 

 tive, that proposition can be proved directly. 



I also found, that of the twelve axioms one only was negative. 



This led me immediately to discover a trivial, but extraordi- 

 nary error, which the editors of Euclid* have promulgated in 

 asserting that the first instance of indirect proof is to be found 

 in the demonstration of the sixth proposition. The incorrect- 

 ness of this assertion is easily shown in the following manner. 



The so-called axiom in question is the tenth : — 



" Two right lines cannot enclose a space." 



This is used by Euclid in the concluding portion of the proof 

 he gives to the fourth proposition. It is needless to state this 

 proof at length, as I wish to call attention only to the following 

 paragraph : — 



" Since the extremities of the bases BC and EF coincide, these 

 lines themselves must coincide ; for if they did not, they would 

 include a space (Xth ax.). Hence the sides BC and EF are equal." 



By developing this argument, as we generally do in practice, 

 the latent reductio ad absurdum becomes manifest. 



Since the extremities of the bases BC and EF coincide, the 

 lines themselves must coincide ; if not, let EF fall above BC : 

 according to the tenth axiom, two right lines cannot enclose a 

 space, but here two right lines do enclose a space, which is 

 absurd. In the same manner it can be shown that EF could 

 not fall below BC ; and as it cannot fall either above or below, 

 it must fall on it. 



If the hypothetical assumption, that one line lie above the 

 other, was not adopted by Euclid, why does he call in the aid of 

 the Xth axiom ? And if he has made that assumption, his line of 

 reasoning is analytical, his demonstration clearly indirect. 



As in the preceding instance, the substitution of an affirmative 

 premiss will permit us to give a direct proof to the fourth pro- 

 position. Before doing so I wish to refer to a collateral circum- 

 stance, the scholastic controversy about geometrical axioms, 

 which I think has an important bearing on this investigation. 



The most eminent metaphysicians have differed on the ques- 

 tion, whether we should regard the definitions or the axioms as 

 the principles in mathematical science. Mr. Locke was in favour 

 of the former ; he says, referring to the case of an individual 

 who has learnt that the square of the hypothenuse in a right- 



* See any edition. Dr. Lardner's, p. 20; Mr. R. Potts's, p. 48. 



