SCIENCE ABSOLUTE OF SPACE. 65 



Rivals," to give us one more equivalent form: 

 "In any circle, the inscribed equilateral tetra- 

 gon is greater than any one of the segments 

 which lie outside it." (A New Theory of 

 Parallels by C. L. Dodgson, 3d. Ed., 1890.) 



Any attempt to define a straight line by 

 means of "direction" is simply a case of "ar- 

 gumentum in circulo." In all such attempts 

 the loose word "direction" is used in a sense 

 which presupposes the straight line. The 

 directions from a point in Euclidean space are 

 only the oc 2 rays from that point. 



Rays not costraight can be said to have the 

 same direction only after a theory of parallels 

 is presupposed, assumed. 



Three of the exposures of Professor G. C. 

 Edwards' fallacy are here reproduced. The 

 first, already referred to, is from Nature, Vol. 

 XXIX, p. 453, March 13, 1884. 



"I select for discussion the 'quaternion 

 proof " given by Sir William Hamilton. . . . 

 Hamilton's proof consists in the following: 



"One side AB of the triangle ABC is turned 

 about the point B till it lies in the continuation 

 of BC; next, the line BC is made to slide along 

 BC till B comes to C, and is then turned about 

 C till it comes to lie in the continuation of AC. 



