66 SCIENCE ABSOLUTE OF SPACE. 



* * It is now again made to slide along CA till 

 the point B comes to A, and is turned about A 

 till it lies in the line AB. Hence it follows, 

 since rotation is independent of translation, 

 that the line has performed a whole revolution, 

 that is, it has been turned through four right 

 angles. But it has also described in succession 

 the three exterior angles of the triangle, hence 

 these are together equal to four right angles, 

 and from this follows at once that the interior 

 angles are equal to two right angles. 



' ' To show how erroneous this reasoning is 

 in spite of Sir William Hamilton and in spite 

 of quaternions I need only point out that it 

 holds exactly in the same manner for a triangle 

 on the surface of the sphere, from which it 

 would follow that the sum of the angles in a 

 spherical triangle equals two right angles, 

 whilst this sum is known to be always greater 

 than two right angles. The proof depends 

 only on the fact, that any line can be made to 

 coincide with any other line, that two lines do 

 so coincide when they have two points in com- 

 mon, and further, that a line may be turned 

 about any point in it without leaving the sur- 

 face. But if instead of the plane we take a 

 spherical surface, and instead of a line a great 



