8 LECTURES AND ESSAYS [1869- 



processes, since they are the longest.&quot; If now we do 

 call in new axioms whenever we construct an essentially 

 new figure, must not Mr. Mill admit, on his own showing, 

 that every advance in geometry involves an advance in 

 uncertainty ; that the geometry of the circle is less 

 certain than that of the straight line, solid geometry 

 than plane, conic sections than Euclid, etc. ? Surely 

 this is a reductio ad absurdum of the whole theory. 



The principles of geometry involved in the question 

 are so important that we may profitably separate them 

 from Mr. Mill s blunders in a special case. 



I. The proofs of geometry are clearly not inductive. 

 There is no mental comparison of various figures needed 

 during the proof. The inductions involved (if any) 

 must have been previously formed. 



II. The proofs then must be reduced either to actual 

 perception (intuition), or to deduction from axioms. 

 But since the proof is general, the former assumption 

 involves the reality of general intuition, i.e. of a general 

 judgment from a single perception. 



III. The theory of intuition is sufficient, but is dis 

 puted in two interests : 



(a) In the interest of syllogism, which claims to give 

 indefinitely extensive conclusions from limited premises 

 [but many, as Whewell, hold that these premises are 

 intuitive axioms]. 



(ft) In the interests of empiricism, which makes all 

 arguments be ultimately from particulars to particulars. 



Mr. Mill combines the two objections. 



Now we have seen that if objection (a) falls (i.e. if 

 the premises of geometry are not reducible to a limited 

 number of axioms from which everything follows ana 

 lytically), the security of geometric reasoning can be 

 established only if each premise has apodictic certainty. 

 To overthrow Mill s whole theory, it is therefore enough 

 to show the fallacy of the limited -number -of -axioms 

 hypothesis. On this we observe : 



