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ROYAL SOCIETY OF CANADA 



it is an assumption from which important results must follow. The 

 existence of such a foundation stone as (4) may seem to be one of 

 the reasons for preferring "' assumption ' to ' axiom/ since in point of 

 elementarinessl (1), {2) and (3) seem to be in a class different from 

 (4). The real reason, however, for preferring "assumption" is that 

 " axiomatic " and " self-evident " have come to be regarded as synony- 

 mous, and in the latter word, in such a connection, there seems to be 

 a reference to our experiential knowledge. The demand, also, that 

 the ' axioms ' of Euclid be conceded is certainly an appeal to our 

 experiential laiowledge. Thus it is that the word ' axiom ' does not 

 suggest the ground on which these foundation stones are introduced 

 into this system of geometry, or their relation to the system. The 

 word " assumption " on the other hand, is not suggestive of self- 

 evidentness or of any appeal to previous knowledge. It is important 

 'to keep this point clearly in mind, otherwise the whole spirit of this 

 system of geometrj^ may be lost sight of. 



From the preceding assumptions we see that there is an unlimited 

 number of points on a straight line, of straight lines in a plane, and 

 of planes in space. 



For from {%) we see there is an unlimited number of points on 

 a straight line. 



Also A, B, C being [I, (6) ] three points on a plane, not co- 

 straight, then A, B determine one straight 

 line and A, C another. On each of these 

 lines there is an unlimited number of 

 points, and every different combination of a 

 point on one with a point on another de- 

 termines a different straight line. For the 

 line determined by B", C" could not be the 

 line determined by B', C. If it were, then 

 each would be both of the lines A B and A C 

 [I. (2)], and the points A, B, C would be co- 

 straight, which is contrary to hypothesis . 



Similarly A, B, C, D being four non-co 

 planar points we reach an unlimited number of 

 planes in space. 



Theorem 6. — A, B, C are three non-co-straighi 

 points. Then a straight line cannot have 

 points within all three sects B C, C A, A B 

 For let D, E, F be points of such a straight line. Of the three points 

 D, E, F, one of them, say E, must be between the other two [II. (3)]. 



