376 Prof. Stevelly on the Doctrine of Parallel Lines. 



cation, and because it affords a fair specimen of the defective 

 logic and incorrect reasoning which pervade the whole. 



To sustain his position, tlftit the method of limits resorted to 

 by me, if allowable, was unnecessary, Mr. Ilennessy asserts that 

 he has shown in the third (fifth ?) volume of the present series^' 

 of this Journal, that the entire doctrine of parallel lines may be 

 derived from the following definition : — " Parallel lines are such, 

 that if they meet a third line, the two interior angles on the 

 same side will be (together) equal to two right angles." 



Now it is quite certain that this definition, as such, does not 

 give a solid foundation for the doctrine of parallel lines, neither 

 does it warrant the deductions derived from it in the paper 

 referred to. Let us examine it even partially. 



I meet two lines, wx and , i i, 



yz, so related to "a third" \^ ",i '^jpy 



line, loy, that the two interior — 

 angles on the same side are 



together equal to two right 



angles. Well, the definition 

 fairly warrants me in calling 

 these lines " parallel.'* This is the essential difi'erence which 

 distinguishes them from other pairs of lines, and from which all 

 their properties must be proved to flow. Now the 29th propo- 

 sition of Euclid's Elements calls on me to prove that " any " 

 line, as ivy, which cuts those, also makes the two interior angles 

 on the same side together equal to two right angles. This is a 

 relation of these two lines to all others which cut them which 

 requires to be proved, and which neither the above definition 

 nor any other definition can warrant us to assume as a truth 

 without proof. It is a '^ property " which must be shown to be 

 an essential concomitant of that which has already fixed the rela- 

 tion of those two lines to one another, and to all others which cut 

 them, and which we have by the definition merely agreed shall 

 entitle us to denominate them " parallel lines " in any individual 

 case in which we find it to obtain. And yet this simple assump- 

 tion is all that Mr. Hennessy gives in the paper referred to as 

 proof of the 29th and 27th of Euclid's first book of Elements. 



When I was a boy at school, I thought for a day or two that 

 I had proved the entire doctrine of parallel lines by defining 

 them to be such, that if *' any line " was drawn across them they 

 would make the alternate angles equal. But I am happy to add 

 that I detected my false logic before I had committed myself by 

 its publication. 



I am. Gentlemen, 



Your obedient Servant, 

 «yn;» John Stevelly. 



