124 ROYAL SOCIETY OF CANADA 



It is then possible to show that through a given point (A) without 

 a given straight line (BC), there is always one straight line parallel 

 to the given one (B C). 



For let < D A B be E < A D C, ^ 



and suppose that A B meets C B. 

 Let D C be E A B. Then the 

 triangles B A D, C D A are con- 



-c:!. 



gruent, and the angles similarly B 3 ^ 



marked are equal. Hence the angles at A are 'adjacent angles,' and 

 we should have the straight Une BAC meeting B C in two points. 



IV. Assumption of parallels. 

 It is then assumed that, — 



Through a given point there is not more than one parallel to a 

 given straight line. 



The geometry is thus made Euclidean. 



V. Assumption of continuity. 



Finally there is the assumption of continuity, frequently spoken 

 of as the axiom of Archimedes : — 



If Aj be any point on a straight line between any given points 

 A and B ; and the points A2, A3 .... be taken on the line such that 

 Ai lies between A and Ag, Ag between A^ and A3, etc.; and such that 

 the sects A A^, A^ A2, .... are all congruent; then in the series of 

 points A,, A3, ... . there is always a point An such that B lies between 

 A and An. 



This makes possible the introduction into geometry of the idea 

 of continuity, and is the expression of the idea of continuity in terms 

 of a sect calculus. It claims that the magnitudes with which we deal 

 in geometryi are continuous. 



The statement of this principle by Archimedes constitutes his fifth 

 assumption and is as follows : " Further, of unequal lines, unequal 

 surfaces, and unequal solids, the greater exceeds the less by such a 

 magnitude as, when added to itself, can be made to exceed any assigned 

 magnitude among those which are comparable with one another." 

 It will be remembered that Euclid in his definitions (Def. 4, Bk. V) ; 

 says, ''' Magnitudes are said to have a ratio to one another when the 

 less can be multiplied so as to exceed the greater." Proposition 1 of 

 Book X, Euclid, which constitutes Lemma 1 of Book XII, is as fol- 

 lows : " If from the greater of two unequal magnitudes there be taken 

 more than its half, and from the remainder more than its half, and 

 so on; there shall at length remain a magnitude less than the least 



