114 ROYAL SOCIETY OF CANADA 



Before I enter with any detail into an explanation of this rational 

 geometry, that you may understand at once how much more fimda- 

 mental it is than the system of Euclid, and how much more rigourously 

 it seelcs to exclude our experimental knowledge of the physical world, 

 let me mention some of the assumptions and deductions of the system: — 



Assumption. — "If A, B, C are points on a 

 straight line, and B lies between A and C, 

 then B lies also between C and A." 



Assumption. — " A, B, C are three points 

 not co-straight. If a line, a, cuts the sect 

 A C, then it also cuts the sect A B or the 

 sect C B." 



Theorem,. — " Every straight line a which lies in a plane separates 

 the plane into two regions such that every point A of one region 

 with every point B of the other region determines a sect A B within 

 which lies a point of the straight line a ; and any two points A, A' of the 

 same region determine a sect A A' which contains no point of a." 



Theorem. — " No straight line can lie wholly within a triangle." 



jS'ow, with a view to placing my hearers in the position of isolation 

 Ticcessary for an appreciation of the assumptions at the base of Hilbert's 

 geometry, with the purpose of suggesting to them the degree of abstrac- 

 tion with which the subject must be approached, let me briefly outline 

 a purely abstract geometry: — 



A point may be defined to be that which is determined by two 

 numbers, x and y. (I am not in the least sugesting the Cartesian 

 method with its co-ordinate axes). We may suppose a straight line 

 to be defined as that which is determined by two ratios, u: v: w, still 

 without thie suggestion of physical representation. Further, we may 

 say that such a point {x, y) is said to lie on such a line u:v: w when 

 the equation ux-\-vy-\-w^^^o is satisfied. But with such a basis, 

 when thre^e poinjts (x^, y^,), ^{x^, y 2), (^3, y 3) lie on such a line, how 

 can we say that one point lies ' between ' the other two ? Clearly 

 some convention must be adopted, possibly with respect to the mag- 

 nitudes of the numbers, the x's or the y's, without which there is no 

 such thing necessarily as ' betweenness '. And with such a basis how 

 can we speak of the ' sides ' of such a line ? Clearly some convention 

 must be adopted, possibly that all points which make ux -\- v y -\- w 

 positive shall be said to lie on one side of the line, and all points that 

 make it negative shall be said to lie on the other side. I do not say 



