GEOMETRICAL AXIOMS. 33 



tial part in the system of geometry. At first sight, 

 these appear to be practical operations, introduced for 

 the training of learners; but in reality they estab- 

 lish the existence of definite figures. They show that 

 points, straight lines, or circles such as the problem re- 

 quires to be constructed are possible under all con- 

 ditions, or they determine any exceptions that there 

 may be. The point on which the investigations turn, 

 that we are about to consider, is essentially of this 

 nature. The foundation of all proof by Euclid's 

 method consists in establishing the congruence of 

 lines, angles, plane figures, solids, &c. To make the 

 congruence evident, the geometrical figures are sup- 

 posed to be applied to one another, of course without 

 changing their form and dimensions. That this is 

 in fact possible we have all experienced from our 

 earliest youth. But, if we proceed to build necessities 

 of thought upon this assumption of the free trans- 

 lation of fixed figures, with unchanged form, to every 

 part of space, we must see whether the assumption 

 does not involve some presupposition of which no 

 logical proof is given. We shall see later on that it 

 does indeed contain one of the most serious import. 

 But if so, every proof by congruence rests upon a fact 

 which is obtained from experience only. % 



I offer these remarks, at first only to show what 

 difficulties attend the complete analysis of the pre- 

 ii. p 



