116 ROYAL SOCIETY OF CANADA 



The meaning of (1) is that two[point& 



determine a straight line completely, 



and determine only one straight line. 



Yet (2) is necessary. For while A, B 



^ 3 might uniquely determine A B C D, 



still C D might uniquely determine C D B F. The same consideration 



shows the need of both (4) and (6) . 



The assumption " any two points on a straight line determine it " 

 ic the co-relative in this system of geometry of the familiar axiom 

 '' Two straight lines cannot enclose a space." Yet it is evidently more 

 elementary than the latter; it makes no such assumption as is implied 

 in the word "enclose." 



The conception of a line as having an infinite number of points 

 en it, and the conception of a plane as having an infinite number of 

 lines and points on it, are not implied in the preceding assumptions. 

 They will appear later as deductions from the assumptions respecting 

 '' betweenness." 



Consideration will show that it is not necessary to physically con- 

 ceive the elements point, line and plane, difficult though it may be to 

 avoid doing so. Hilbert has not discussed this matter in the published 

 reproduction of his lectures, whatever he may have done in the lectures 

 themselves. I conjecture that he felt it proper that each student 

 should impart to the subject tlie degree of abstraction he felt possible 

 by reason of his mental make-up. It is to be borne in mind, how- 

 ever, that the object of this geometry is to get away from the evidence 

 of our senses by reason of the doubt which the mind casts on the reli- 

 ability of such, evidence. Strictly speaking, the subject is a purely 

 logical one, though, I think, throughout its study we are expected to 

 note the complete correspondence between the conclusions we reach and 

 the facts of the physical universe. 



We have no difficulty in making deductions from the preceding 

 assumptions : — 



Theorem 1. — Two straight lines cannot have two points in com- 

 mon. For since the two points determine a straight line nniquely 

 [(1), (2)], there can be only one straight line through these two 

 points. 



Theorem 2. — Two planes have no point in common, or they have 

 a straight line in common. For if they have one point in common 

 they have a second point in common, [(71 ; and, therefore, each con- 

 tains thte straight line which is determined bv these two points [ (8)^ 

 (2)1. 



