481 
of Edinburgh, Session 1868 - 69 . 
the first time does not at once conclude the general truth of this 
formula from the one figure before him, but either brings the 
formula with him to the proof as a result of previous induction, or 
requires to pause in the proof, and satisfy himself of the truth of 
the formula by a comparison of a series of figures. 
It is easily shown, by the same species of analysis as we have 
adopted here, that wherever a real step is made in geometry we 
must either use the figure or introduce a new general axiom [not 
of course in mere converses, as Euc. I 19 , I 25 ]. All geometrical con- 
struction is in the last resort a means of making clear to the eye 
complicated relations of figures. 
Now, if we can at once and with certainty conclude from the one 
case figured in the diagram to the general case — if, that is, axioms 
are proved not by induction, but by intuition, and are necessarily 
true — there is no difficulty about geometrical reasoning ; but if each 
new axiom is gained by a new induction (and that on Mr Mill’s 
showing an u inductio per enumerationem simplicemf) we get a 
difficulty which Mr Mill curiously enables us to state in his own 
words (I. p. 301) — “If it were necessary,” in adding a second step to 
an argument, u to assume some other axiom, the argument would 
no doubt be weakened.” But, says Mr Mill, it is the same axiom 
which is repeated at each step. If this were not so, u the deductions 
of pure mathematics could hardly fail to be among the most uncertain 
of argumentative processes, since they are the longest.” 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, &c.? Surely 
this is a reductio ad absurdum of the whole theory. 
The principles of geometry involved in the question are so im- 
portant 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 
