fSO "Mr. J. P. Hennessy on some Facts connected 



parallel lines maybe deduced, is certainlyas obvious as that towhich 

 he improperly gave the title of axiom. The numerous attempts 

 which have been made to supersede his theory have generally 

 been met by two objections, — (1), that the subsequent demon- 

 strations are much more complex, (2) and full of embaiTassing 

 reductio ad absurdum. These are put forward by Dr. Lardner 

 very clearly ; he says, " When you have once admitted Euclid's 

 axiom, all his theorems flow from that and his definition, as the 

 most simple and obvious inferences. In other theories, after 

 conceding an axiom much further removed from self-evidence 

 than Euclid's, a labyrinth of complicated and indirect demon- 

 stration remains to be threaded, requiring much subtlety and 

 attention to be assured that error and fallacy do not lurk in its 

 mazes.'' 



It will presently be seen that the demonstrations I propose to 

 give are not open to either of those objections ; whereas Euclid's, 

 a fact which Dr. Lardner appears to have overlooked, are most 

 decided specimens of indirect proof. 



The definitions given by Euclid of a right line and plane sur- 

 face are precisely the same, and have been objected to for many 

 reasons, of which, I conceive, the most serious, that they cannot 

 be used in his geometry, is equally applicable to those given by 

 Archimedes and Plato. Now M. Legendre has adopted the 

 definition of a straight line given by the former, that it is the 

 shortest distance between any two points ; but he has not taken 

 Archimedes' definition of a plane surface, which is exactly 

 similar : in this case he has substituted for Euclid's the following, 

 attributed to Hero : — "A plane surface is such that the right line 

 joining every two points which can be assumed upon it lies 

 entirely in the surface." This definition has received the unani- 

 mous approbation of modern geometers, even those most conser- 

 vative of Euclid's arrangement. It is not this, however, which 

 induced me to make a definition of right lines precisely similar ; 

 but the more important fact, that every other definition I have 

 seen was either of no utility in geometric reasoning or of a nega- 

 tive form. I cannot avoid citing the tacit acknowledgement 

 that such a definition is better than Euclid's Xth axiom, which is 

 given by Mr. Mill*, in tracing deductive truths to their original 

 inductive foundation. He proves the fifth proposition from first 

 principles by six formulae, the third of which is " straight lines 

 having their extremities coincident coincide." It is obvious that 

 the simplicity and elegance of the proof in this case would have 

 been interfered with if this definition has not been assumed. 



♦ System of Logic, 2nd edit. vol. i. p. 286. 



