482 Proceedings of the Royal Society 
(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 disputed in two 
interests : — 
fa) 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]. 
(/3) 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 analytically), 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 : — 
1st, The axioms are more numerous than Mr Mill thinks, for his 
proof of Euc. I 5 is lost for want of more axioms. 
2d, The indefinite extension of geometry depends on the power 
of indefinitely extended construction [but where there is construc- 
tion there is intuition — nay, mental intuition is mental construc- 
tion]. Now here our opponents may suppose [A] that the general 
conclusion really flows from the particular construction, which, in 
the language of logic, supplies the middle term. But since the 
construction is particular, we should thus be involved in the fallacy 
of the undistributed middle. Again, [B] it may be said that the 
construction is only the sensible representation of a general axiom . 
But as the construction is new and indispensable, the general axiom 
must be so also. Therefore, if geometry is proved from axioms, 
these axioms must be unlimited in number. 
3 d, Obviously.it is not by logic that we can satisfactorily deter- 
mine how far geometry contains synthetic elements peculiar to 
itself. We have, however, in analytical geometry a ready criterion 
how far geometry can be developed without the addition of new 
geometrical considerations. 
Now we find that we cannot begin analytical geometry from the 
