Prof. Stevelly on the Doctrine of Parallel Lines. 375 



ri(It was in the attempt to establish this theorem that M. Le- 

 gendre deviated from the method of EucUd by employing the 

 doctrine of limits. Setting out frAn this theorem, the demon- 

 strations of the various properties of parallels are simple and 

 obvious. To give these final propositions would be only to repeat 

 what M. Legendre has done ; for to that distinguished geometer 

 must belong the merit of having shown that the whole question 

 depended upon a proof of the 32nd proposition. His proof of 

 that proposition is inadmissible ; but his subsequent demonstra- 

 tions are in accordance with the principles of plane geometry. 



Although the proof just given of this theorem will probably 

 be acknowledged to be consistent both with the definitions and 

 with the method of Euclid, nevertheless I cannot regard it as 

 the most satisfactory solution of the difficulty. It may be in- 

 teresting as the first successful attempt of the class to which it 

 belongs. But the real solution I still believe to consist in treat- 

 ing the conception of parallel lines, as we treat any other geo- 

 metrical conception, with a clear and unequivocal definition. 



119 Jermyn Street, 

 October 8. 1856. 



XL VI, On the Doctrine of Parallel Lines, 

 By Professor Stevelly. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, Holywood, Oct. 13, 1856. 



FOR the personally kind terms in which Mr. Hennessy has 

 stated his objections to the method of treating the doctrine 

 of parallel lines which I lately communicated to you, I beg to ex- 

 press my acknowledgements. I also feel much gratified that he 

 does not profess to have detected any defect in the chain of rea- 

 soning on which the proofs are founded. 



I trust, however, Mr. Hennessy will not consider me deficient 

 in courtesy when I add, that regard for scientific truth compels 

 me to say, that there is scarcely an assertion which he has made 

 in that paper, or in the one to which he refers in (the fifth not) 

 the third volume of the present series of this Journal, in which I 

 can concur. 



Since a detailed examination of the defective logic and incor- 

 rect statements contained in those papers would far exceed any 

 reasonable limits, and since an examination of even two or three 

 of the most obvious would wear too much the appearance of con- 

 troversy, and seem to invest the entire subject with a very undue 

 share of importance, I shall simply select one as an example, 

 because of its close connexion with the subject of my communi- 



