4'24- Oti the Theori/ of paruUcl lAncs in Geometry. 



as remote at least as the days of Euclid down to the present 

 time : but their efforts, however powerful, have usually been 

 exerted to very little purpose ; for this difficulty, which Euclid 

 left as an exercise for succeeding geometers, appears to have 

 suffered no change during the lapse of twenty centuries, if it 

 is even now demonstrated. 



*' There is scarce any thing," says Mr. Thomas Simpson, 

 " more obvious to sense, and at the same time more difficult to 

 demonstrate, than the first and most simple properties of pa- 

 rallel lines." So true is this observation, that the very dia- 

 grams themselves seem to refuse being so distorted as to suit 

 the conditions of any supposition contrary to Euclid's 12th 

 axiom, or which denies that the angles of a triangle amount 

 to two right angles : and yet all this distortion, although offen- 

 sive to the eye, is quite consistent when we attempt to reason 

 on it, and compare its several parts. The reason why in this 

 case we arrive at a consistent conclusion even when proceeding 

 on an erroneous supposition, seems to be, that we are not as 

 yet in possession of any property of lines or angles which can 

 counteract our supposition and lead to a contradictory con- 

 clusion, or reductio ad absurdum ; our supposition itself being 

 the only condition that the investigation involves. 



In the Philosophical Magazine for March last, the attention 

 of your mathematical readers has been again directed to this 

 very difficult subject, by your distinguished correspondent 

 Mr. Ivory, who has of late furnished you with so many valu- 

 able articles. After a number of inatructive preliminary re- 

 marks, Mr. I. takes occasion to mention the demonstration of 

 Legendre, but not without pohiting out an objection it is liable 

 to, on account of a new principle or axiom which enters into 

 its composition. Mr. Ivory then proposes a demonstration of 

 his own, which, he says, requires no new principles, and is lia- 

 ble to no objection excepting its length. I suppose therefore 

 that every one is perfectly at liberty to slate any reasonable 

 doubts or objections he may have regarding the legitimacy of 

 that demonstration. 



As to the first part, which is intended to prove that the three 

 angles of a triangle cannot exceed two right angles, there can 

 be no doubt thi t it is rigidly demonslrated. But I cannot en- 

 tertain quite so favourable an opinion of the latter part of the 

 performance, because its learned author ajipears to have over- 

 looked a verv important circumstance in the demonstration <jf 

 his third proposition. This, however, he may still be able to 

 put to rights ; but should he fail in doing so, I suj^pose we 

 may reasonably despair of any other person's giving us an un- 

 objectionable demonstration of the theory of parallels. The 



defect 



