IBAKER] THE FOUNDATIONS OF GEOMETRY 117 



Theorem 3. — Two planes cannot have two straight lines in com- 

 mon. Por if they have, on each line there are two points [(3)] ; and 

 three points determine a plane [(5]. Therefore, there is but one 

 plane. 



Theorem ^. — A straight line and a point not on it determine a 

 plane. For on the line are two points [(3)]; and these with the 

 civen point determine a plane [(5)] on which every point on the line 

 lies [(8)]. 



Theorem 5. — If a plane contains a straight line and a point in 

 another straight line, but not such other straight line, then no plane 

 can contain both lines. For a plane containing both lines would con- 

 tain the first line and the point on the second, and, therefore, would 

 be identical with the first plane (Thm. 4), which by hypothesis does 

 not contain the second line. 



II. The assumptions of betweenness are: — 



(1). If A, B, C be points on a straight line, and B lies between 

 A and C, then it also lies between C and A. 



(2). If A and C be points on a straight line, then thene is at 

 least one point B on the line between A and C, and also one point 

 D on the line such that C lies between A and D. 



(3). Of three points on a straight line one and only one lies 

 between the other two. 



(4). Pasch's assumption. — Let A, B, C be three points not co- 

 straight, and a a straight line in the plane ABC but going through 



none of the points A, B, C. Then if a goes 

 through a point within the sect A B, it must 

 also go through a point within the sect A C, 

 or a point within the sect B C. 



The first three assumptions are intended to 

 fix the fact that points exist on a straight 

 ^ line in a certain order. 

 It is important to observe that no such idea has been introduced 

 as that the points on a line are continuous; or that we cannot get 

 from one ' side ' of a line to the other without ' passing through ' 

 or ' cutting ' the line. If we are to reach such a fact evidently it 

 must come to us from assumption (4). Indeed, it will be noted that, 

 compared with assumptions (1), (2) and (3), there is a startling com- 

 plexity in assumption (4) ; and we receive it in much the same way 

 a? we recollect receiving the eleventh axiom of Euclid. One feels that 

 the founder of this geometry must have introduced it with reluctance, 

 and only after a struggle to find a substitute; and yet one feels that 



