SCIENCE ABSOLUTE OF SPACE. 63 



geometry, that the number of assumptions 

 ought not to be increased without necessity ; 

 or as Dedekind has it: " Was beweisbar ist, 

 soil in der Wissenschaft nicht ohne Beuueis 

 geglaubt werden" 



Professor W. B. Smith (Ph. D., Goettingen), 

 has written: " Nothing could be more unfor- 

 tunate than the attempt to lay the notion of 

 Direction at the bottom of Geometry. ' ' 



Was it not this notion which led so good a 

 mathematician as John Casey to give as a 

 demonstration of a triangle's angle-sum the 

 procedure called " a practical demonstration " 

 on page 87 of Richardson's Euclid, and there 

 described as * ' laying a * straight edge ' along 

 one of the sides of the figure, and then turn- 

 ing it round so as to coincide with each side in 

 turn." 



This assumes that a segment of a straight 

 line, a sect, may be translated without rota- 

 tion, which assumption readily comes to view 

 when you try the procedure in two-dimensional 

 spherics. Though this fallacy was exposed by 

 so eminent a geometer as Olaus Henrici in so 

 public a place as the pages of 'Nature,' yet it 

 has just been solemnly reproduced by Pro- 

 fessor G. C. Edwards, of the University of 

 California, in his Elements of Geometry: Mac- 



