i8 7 3] GEOMETRICAL REASONING 9 



is^, The axioms are more numerous than Mr. Mill 

 thinks, for his proof of Euc. I. 5 is lost for want of more 

 axioms. 



2nd, The indefinite extension of geometry depends on 

 the power of indefinitely extended construction [but 

 where there is construction there is intuition nay, mental 

 intuition is mental construction] . 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 con 

 struction 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 repre 

 sentation 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. 



$rd, Obviously it is not by logic that we can satis 

 factorily determine how far geometry contains synthetic 

 elements peculiar to itself. We have, however, in analyti 

 cal geometry a ready criterion how far geometry can be 

 developed without the addition of new geometrical con 

 siderations. 



Now we find that we cannot begin analytical geometry 

 from the mere axioms and definitions. We must by 

 synthetic geometry, by actual seeing, learn the qualities 

 of lines and angles before we can begin to use analysis. 

 Then, given so many synthetic propositions, we can 

 deduce others by algebra ; but only by a use of actual 

 intuition, first, in translating the geometrical enunciation 

 into algebraic formulae ; and, second, in translating the 

 algebraic result (if that result is not merely quantitative) 

 into its geometric meaning. The answer to a proposition 

 in analytical geometry is simply a rule to guide us in 

 actually constructing, by a new use of our eyes or imagina 

 tion, the new lines which we must have to interpret the 

 result. Analysis does not enable us to dispense with 



