with Geometrical Reasoning. 131 



The following are the direct demonstrations of the twenty- 

 seventh and twenty-ninth propositions,, which flow from the 

 definition of parallel lines I have given. 



XXVII. If a line intersect two right lines 

 and make the alternate angles equal to each 

 other, these right lines are parallel. DCB 

 and ECB are equal to two right angles, but 

 the former is equal to ABC, therefore ABC 

 and ECB are equal to two right angles; 

 therefore (def.) AB and CD are parallel. 



XXIX. If a right line intersect two parallel right lines it makes 

 the alternate angles equal. (In the same figure) ABC and ECB 

 are equal to DCB and ECB ; therefore ABC and DCB are equal. 



IV. Having assumed the definition which I suggest, a direct 

 demonstration immediately occurs : this is so obvious that a 

 proof is unnecessary. 



The fact that this proposition is proved indirectly, and that 

 such a form of proof only can be given to it, unless an alteration 

 similar to what I have made be adopted, may point out a method 

 of testing the accuracy of the very interesting conjecture which 

 Mr. Sylvester has published on this subject. For example, if it 

 is indispensable for the proof of the following theorem, I think 

 we should assume such proof to be fundamentally indirect : the 

 theorem is taken from Mr. Sylvester^s paper which appeared in 

 the Philosophical Magazine for November 1852, but according 

 to his criterion should admit of the direct form : — " To prove 

 that if from the middle of a circular arc two chords be drawn, 

 and the nearer segments of these cut off by the line joining the 

 end of the arc be equal, the remoter segments will also be equal .^' 

 It is difficult to conceive any proof of this theorem which would 

 not ultimately rest on the fourth proposition, and consequently rest 

 on reductio ad absurdum. Such a proof, however, may be found 

 to exist, although the writer of this paper has been unable to dis- 

 cover it. An opinion differing from Mr. Sylvester's, in a mathe- 

 matical inquiry where certainty has not been arrived at, should 

 be put forward with diffidence ; still more so when that gentle- 

 man's conjecture has received, to a certain extent, the approval 

 of Professor De Morgan *. 



Queen's College, Cork, 

 December 28, 1852. 



* See Phil. Mag. December 1862. 



