I 

 4 LECTURES AND ESSAYS [1869- 



It may perhaps be worth while to show how Mr. Mill 

 was led into this extraordinary mistake. We shall find 

 that Mr. Mill chooses rather to sacrifice geometry to his 

 philosophy, than to modify his philosophy in accordance 

 with the facts of geometry. 



Mr. Mill holds that all general knowledge is derived 

 from experience ; meaning by experience the com 

 parison of at least two distinct experimental facts. In 

 other words, all knowledge is ultimately gained by 

 induction from a series of observed facts. That any 

 general truth can be got at intuitively, by merely looking 

 at one case, Mr. Mill emphatically denies. The fact that 

 two straight lines cannot enclose a space is not self- 

 evident as soon as we know what straight lines are 

 [i.e. can mentally construct such lines] ; but is got at 

 only by experiments on &quot; real &quot; or &quot; imaginary &quot; lines 

 (Logic, i. pp. 259, 262). Now it is certain that, in the 

 demonstrations of Euclid, we are satisfied of the truth 

 of the general proposition enunciated, as soon as we have 

 read the proof for the special figure laid down. There is 

 no need for an induction from the comparison of several 

 figures. Since then one figure is as good as half a dozen, 

 Mr. Mill is forced to the conclusion that the figure is no 

 essential part of the proof, or that &quot; by dropping the use 

 of diagrams, and substituting, in the demonstrations, 

 general phrases for the letters of the alphabet, we might 

 prove the general theorem directly &quot; from &quot; the axioms 

 and definitions in their general form &quot; (p. 213). 



We may just mention, in passing, that this view, 

 combined with the doctrine that the definitions of geo 

 metry are purely hypothetical, leads Mr. Mill to the 

 curious opinion that we might make any number of 

 imaginary sciences as complicated as geometry, by 

 applying real axioms to imaginary definitions. We 

 mention this merely to illustrate Mr. Mill s position our 

 present business is to see how these views of geometry 

 work in practice. 



