62 Transactions. — Miscellaneous. 



Similiarly, no line through P and passing outside Q P A 

 can be parallel to A B, for the continuation of it would fall 

 within the angle Q' P A. Hence only one straight line can 

 be drawn through P parallel to A B, viz : P Q. Q.E.D. 



I have thus shown that if Euclid's axiom is true, then the 

 modern substitute is true. To establish the exact logical 

 equivalence of the two axioms, I should have to prove the con- 

 verse formally, viz. : that if the modern substitute is true, then 

 Euclid's axiom is true. But I assume it will be conceded 

 that the above reasoning can quite well be put in the converse 

 form. I now pass to the third equivalent, which is alleged by 

 Mr. Skey not to be a real equivalent of the other two. If it be 

 borne in miud that the word parallel in the second equivalent 

 means not equidistance along the whole length of two lines ; but 

 lying in the same plane, plus non-intersection however far produced 

 (see Euclid's definition) — if it be borne in mind that I define 

 paraUelism in this way, I think it will be recognised at once 

 that the second and third forms of the axiom are merely two 

 different ways of saying the same thing. 



However, as truth and falsehood in nature can never be 

 dependent on the signification of words, I may as well say how 

 the axiom would be worded if we define two straight lines to be 

 parallel when they are equidistant along their whole length. (I 

 vastly prefer this definition, though it is not the usual one.) 

 Taking this as the definition of parallelism, Euclid's axiom may 

 be stated thus : — " Two straight lines lying in the same plane, 

 and not being parallel, (i.e., not equidistant along their whole 

 length,) must ultimately intersect if sufficiently produced in both 

 directions." 



In Lobatchewsky's geometry, on the other hand, straight 

 lines in a plane need not intersect though they are not equi- 

 distant along their whole length. They may approach each 

 other for awhile, reach a minimum mutual distance, and then 

 recede more and more continually. Also in Lobatchewsky's 

 geometry no two straight lines can be parallel, in the sense of 

 being equidistant along their whole length. If two lines are 

 parallel (i.e., equidistant along their whole length), they cannot 

 both be straight. One, at least, must be a curved line, i.e., a 

 longer line than some other which could be drawn through any 

 two of its points. 



9. " Nothing is said as to the distance away from this line 

 at which the point is to be placed" (page 103). (This quota- 

 tion refers to the point outside the first line through which the 

 second line is drawn.) The distance of the point from the line 

 may be as short as possible, and still (if Euclid's 12th axiom is 

 untrue) there will be a finite angle tlirough which the rotating 

 line can be turned without ever intersecting the fixed line : the 

 magnitude of this angle depending partly on the distance of the 



