372 Mr. J. P. Hennessy on a Solution oftlte Theory of Parallels, 



in the introduction of a method of reasoning different from that 

 employed by Euclid. This is an extensive class. It includes 

 the attempts of Professor Franceschini*, of M. Legendrc, of 

 MM. Bertrand and Lacroixf, of J)r. Oliver^, of Mr. Playfair§, 

 of Colonel Thompson ||, and of many other writers. (2) Those 

 which, preserving Euclid^s method of reasoning, were based on 

 a change in his definitions. To this class belong the attempts 

 of many ancient geometers, as well as of many modern English 

 geometers. (3) Lastly, those attempts which were made to solve 

 the difficulty by taking Euclid's definitions as they stand, exclu- 

 ding the 12th axiom, and employing the ordinary method of 

 geometrical demonstration . 



Of the two first of these classes it is unnecessary for me to say 

 anything. Of the third class it is only necessary to observe, that 

 no successful attempt has hitherto been published. 



Those who believe that the foundation of a system of geometry 

 should be laid as narrow as possible, and that nothing should be 

 taken for granted in the commencement which might possibly 

 be made the subject of subsequent proof, will probably assert 

 that Euclid's definition of a square is defective, and should be 

 altered. I allude to this for the purpose of observing that such 

 a question is foreign to this paper. It is not with the propriety 

 or impropriety of Euclid's definition that I am now dealing. I 

 am dealing simply with that definition as we find it, and with 

 the assumption on which its possibility is based. 



That assumption, as stated in formal language, is as follows : — 

 "A quadrilateral, the sides of which are respectively equal to 

 each other, and the angles of which are also respectively equal 

 to each other, contains four, and only four right angles.'' 



This assumption of Euclid's can be employed in either of two 

 ways : either in the shape of a problem, on the well-known prin- 

 ciple that the theorems of Euclid may be proved independently 

 of the problems, the construction of which can be assumed ; or, 

 which seems to be the more satisfactory course, by using it as 

 one of the regular premises of a theorem. It is in the latter way 

 that it is here employed. 



Proposition I. 



The three interior angles of a right-angled isosceles triangle are 

 together equal to two right angles. 



Produce the sides AB and AC, which include the right angle, 



* La Teoria delle parallele rigorosamente dimostrata : OpuscoU Mate- 

 matichi, 1787. 



t Ellens de G4om4trie, p. 23. 



X De rectarum linearum parallelismo, ^c. 1604. 



$ Geometry, p. 409d^iir >v || Geometry without Axioms, p. 84-98. 



