GEOMETRICAL AXIOMS. 67 



every part of space if no external force affects them, 

 is a single case falling under the general principle. 



In conclusion, I do not, of course, maintain that man- 

 kind first arrived at space-intuitions, in agreement with 

 the axioms of Euclid, by any carefully executed systems 

 of exact measurement. It was rather a succession of 

 everyday experiences, especially the perception of the 

 geometrical similarity of great and small bodies, only 

 possible in flat space, that led to the rejection, as im- 

 possible, of every geometrical representation at variance 

 with this fact. For this no knowledge of the neces- 

 sary logical connection between the observed fact of 

 geometrical similarity and the axioms was needed ; but 

 only an intuitive apprehension of the typical relations 

 between lines, planes, angles, &c., obtained by nume- 

 rous and attentive observations an intuition of the 

 kind the. artist possesses of the objects he is to repre- 

 sent, and by means of which he decides with certainty 

 and accuracy whether a new combination, which he tries, 

 will correspond or not with their nature. It is true 

 that we have no word but intuition to mark this ; but 

 it is knowledge empirically gained by the aggregation 

 and reinforcement of similar recurrent impressions in 

 memory, and not a transcendental form given before 

 experience. That other such empirical intuitions of 

 fixed typical relations, when not clearly comprehended, 

 have frequently enough been taken by metaphysicians 



