Fkaxki.wk- //'■ Xon-lHuiiidian Ireometry Vindicated. (Jl 



only to demonstrate that no one else can demonstrate its falsity. Iu 

 other words, he has attempted to demonstrate (and that he has 

 completely succeeded all modern mathematicians allow) that the 

 truth of Euclid's 12th axiom can by no possible succession of 

 syllogisms be deduced from the other axioms and the definitions 

 of the straight hue, plane, parallels, &c. Innumerable attempts 

 had been made to do this — i.e., to put the 12th axiom on the 

 same logical footing as, for instance, the 5th proposition of the 

 First Book. All the attempts had failed. Lobatchewsky 

 proved, once for all, that they must necessarily fail, by con- 

 structing an unimaginable but perfectly self-consistent scheme 

 of geometry, in which all the other axioms were assumed to be 

 true, and ail the definitions remain the same, but in which this 

 one axiom (the 12th) was assumed to be false. The equivalents 

 of Euclid's axiom which I have mentioned are really exact 

 logical equivalents. If one is true, all are true. If one is false, 

 all are false. In Euclid's space all are true : in Lobatchewsky's, 

 all are false. 



8. I propose now to establish the exact logical equivalence 



of the three forms of the parallel-axiom mentioned in my paper. 



Form (a), (Euclid's) is: — "If a straight hue meets two 



straight lines, so as to make the two interior angles on the same 



side of it taken together less than two 

 right angles, these straight lines being 

 continually produced shall at length 

 meet upon that side on which are the 

 angles which are less than two l'ight 

 angles." In other words, if the angle 

 C A I! + the angle A B D < 180°, 

 then A C and B I) will at length meet. 



This is Euclid's axiom, and it is to my mind just as good as 

 any of its modern substitutes. 



I now propose to deduce from this axiom the usual modern 

 substitute: — " It is impossible to draw more than one straight 

 line parallel to a given straight line (i.e., lying in the same 



plane with it, but not intersecting it) 

 through a given point outside it." Let 

 Q V A + PAB = 180°. Then, by a 

 proposition of Euclid .which does not, 

 directly. or indirectly, rest on the 12th 

 axiom, P Q can never intersect A B. 

 Draw any straight line P R within 

 Q P A. Then, 



Since QPA-\-PAB = two right angles 

 .-. R P A + P A B < two right angles. 

 .'. P R will eventually meet A B (Euclid's 12th 

 axiom), i.e., P R cannot be parallel to A B. Hence no line 

 within Q P A and passing through P can be parallel to A B. 



