On a Difficulty connected ivitha Demonstration of Euclid. '^^^ 



may be removed in the following simple manner, by making 

 an alteration in Euclid's definitions. It is a singular fact that 

 the difficulty arose solely from his having adopted an obvious 

 but inadequate definition of parallel lines. The fault of his 

 definition is, that it specifies only a negative property of those 

 lines, viz. that being produced ever so far both ways they do 

 not meet; from which it is plain that nothing positive could 

 be deduced. The following definition may be given in its 

 place. 



If a straight line falling on two other straight lines, makes 

 the exterior angle equal to the interior and opposite angle on 

 the same side of it, those lines are said to be parallel. 



From the sixteenth pro- 

 position it may then be 

 readily deduced that such 

 lines wiirnever meet ; for, if 

 possible, let the parallel k 

 lines A B, C B meet to- 

 wards AC in K, then if 

 G E F be the straight line 

 falling upon them, we have 

 by the definition 



ZGEA = zEFC 

 but by prop. 16 



Z G E A is greater than the ^ E F C ; 

 which is absurd, therefore the lines do not meet towards A C. 



In the same manner it may be proved that they do not meet 

 towards B D ; and therefore they never meet. This proposi- 

 tion will supply the place of the old definition, and the twenty- 

 ninth proposition itself is nearly superseded, for the first part 

 of it is contained in the definition, and the second part follows 

 immediately by combining the definition with the thirteenth 

 proposition. 



These few lines may possess some interest for thosie who 

 are concerned in the controversy about axioms and defini- 

 tions. It may be proper to state that the idea was suggested 

 to the author by reading Dugald Stewart's remarks on the use 

 and importance of definitions in abstract reasoning. It struck 

 him that the difficulty which Euclid attempts to remove by 

 the introduction of an axiom, should rather, if Dugald Stew- 

 art's sentiments were correct, be obviated by some change 

 in the definitions. Whether Euclid's error arose from his 

 ignorance of the philosophy of his subject, or whether he was 

 afraid of offending the prejudices of his tyros by a strange de- 

 finition of so simple a thing as parallel lines, the writer will 



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