Haughton — Euclid's Doctrine of Parallel Lines. 71 



yill. — Ceitical and Literaet Eemauks on Euclid's Doctetne op 

 Parallel Lines. By Eev. Samuel Haughton, M.D. ; F.E.S. ; 

 S.P.T.C.D. 



[Eead, February, 11, 1884.] 



Pbofessoe Catlet, President of the British Association for the ad- 

 vancement of Science, in his Address delivered at Southport, 1883, 

 ably defended the position that Euclid's Doctrine of parallel lines 

 truly rests upon an axiomatic or self-evident base, which is as fol- 

 lows : — 



" If a right line falling upon two right lines make the internal 

 angles on the same side less than two right angles ; these lines pro- 

 duced to infinity will meet on the side on which the angles are less 

 than two right angles." 



On the day following the President's Address, 1 called his atten- 

 tion to the fact, that he had defended Euclid beyond what he required, 

 because the best MSS. placed the foregoing statement among the 

 Postulates and not the Axioms. Euclid proves absolutely (1. 17), that 

 any two angles of a triangle are less than two right angles ; and he 

 requires the reader to grant him the converse of this proposition, as 

 the necessary base of the doctrine of parallel lines. Some minds will 

 grant the converse as self-evident, others will accept it as an un- 

 proved condition, on which Euclid proceeds to construct his doctrine 

 of parallels. 



I have examined the old editions of Euclid in the Library of 

 Trinity College, and find the following results : — 



The following editions place the proposition among the Pos- 

 tulates : — 



1. 



2. 

 3. 



J. Campanus^ (Yenice), 

 B. Zamberti^ (Paris), ■ 

 Orontius (Paris), 









1482 

 1516 

 1544 



4. 

 5. 



Billingsley (London), 

 Commandinus (Pesara), 









1570 

 1572 



The following editions place 



the proposition 



among 



;he Axioms : — 



1. 

 2. 

 3. 

 4. 



Grrynseus (Basle),^ . 

 Dasypodius (Strasburg), 

 Candalla (Paris), 

 Clavius (Bome), 







1533 

 1564 

 1566 

 1574 



The best edition of Euclid yet published 



is that 



of E. Peyrard, 



1 This is a Latin translation from the Arabic [Editio princeps.] 



2 A Latin translation. 

 ^ [Editio princeps.] 



