281 MpyJ^'P. Hennessy on the Theory of Parallels: 



and complete. No mathematician can for a moment entertain a 

 doubt as to its correctness. It is, however, open to one fatal objec- 

 tion, an objection which applies with equal force to Mr. Stevelly's 

 demonstration, — that it introduces amongst the propositions of 

 Euclid a method of proof which is totally foreign to the pure 

 and simple reasoning of plane geometry. This is, the doctrine 

 of limits. That Mr. Stevelly employs this method is placed 

 beyond all doubt when he speaks of " making m! approach un- 

 liniitedly close to E by taking b^B = Bb unlimitedly small," &c. 

 But there seems to me to be another objection to Mr. Stevelly's 

 demonstration. The introduction of the method of limits, even 

 if allowable, was quite unnecessary. It is only necessary where 

 we adhere to Euclid^s definition of parallel lines. Mr. Stevelly 

 very properly, I think, put aside that definition and substituted 

 one of his own. In the third volume of the present series of 

 the Philosophical Magazine, it will be seen that the question is 

 treated, as far as a change of definition is concerned, in a similar 

 manner. Instead of Euclid's* definition, I made use of the fol- 

 lowing : — 



" Parallel lines are such, that if they meet a third right line, 

 the two interior angles on the same side will be equal to two 

 right angles." 



From this, as I stated at the time, all the properties of parallel 

 lines can be deduced. The researches in question were but 

 slightly connected with this subject, and I did not think it ne- 

 cessary to give many reasons for choosing such a definition. I 

 was contented with showing that it obviated in certain cases the 

 necessity of employing reductio ad absurdum. 



It has, however, always appeared to me that the difficulty 

 which seems to surround the theory of parallel lines is owing, 

 not so much to any inherent peculiarity in the theory itself, as 

 to the manner in which it has been discussed by geometers. 

 There appears to have been an unwillingness to do for parallel 

 lines what has been done for every other geometncal conception. 

 Before investigating the various theorems connected with the 

 square or with the circle, Euclid took a distinct property of each, 

 in logical phraseology a differentia^ and on that he built his defi- 

 nition. But he did not adopt the same course with respect to 

 parallel lines. In reality he gave no definition whatsoever of 

 them. Attention has been called to the fact that his so-called 

 definition is negative f; and it is hardly necessary to say, that a 

 negative proposition cannot possibly be considered a definition. 



* An interesting driticism on Euclid's definition, which supports the view 

 I ventured to take as to its inadequacy, has lately been published. See 

 "Wedgwood's * Geometry/ p. 5. 



t Philosophical Magazine, February 1863. 



