of Edinburgh, Session 1868 - 69 . 479 
merely to illustrate Mr Mill’s position — our present business is to 
see how these views of geometry work in practice. 
Mr Mill’s example, as we have said, is Euclid I 5 , which he under- 
takes to deduce from the original deductive foundations. We have 
first (p. 241) some preliminary remarks, 
which afford a remarkably happy instance 
of the way in which Mr Mill is accustomed 
to keep himself safe from all opponents, 
by alternately supporting each of two con- 
trary views of a subject. “ First,” says 
he, speaking of the angles ABE, CBE ; 
ACD, BCD, “ it could be perceived in- 
tuitively that their differences were the 
angles at the base.” If this intuition is 
really a step in the proof, then, since in- 
tuition is just actual looking at the figure, what becomes of the 
doctrine that the figure is not essential, or of the still more funda- 
mental ductrine, that no general truth can flow from a single 
intuition?* In this, however, Mi’ Mill only falls foul of himself. 
A more serious matter is, that when he sets about his regular demon- 
stration, he falls foul of the truths of geometry. 
Having shown that AD = AE, Mr Mill proceeds thus: — “Both 
these pairs of straight lines ” [AC, AB : AD, AE] “ have the pro- 
perty of equality; which is a mark that, if applied to each other 
they will coincide. Coinciding altogether, means coinciding in 
every part, and of course at their extremities, D, E, and B, C.” 
How, “ straight lines, having their extremities coincident, coincide. 
BE and CD have been brought within this formula by the pre- 
ceding induction ; they will, therefore, coincide.” [!] If Mr Mill 
generalises this conclusion, I think he will find it to be that two 
triangles, having two sides of each equal, are equal in all respects ; 
and from this theorem he may at once conclude, by his own fourth 
formula [“ angles having their sides coincident, coincide ”], that 
* This is no mere slip on Mr Mill’s part. To show that the angles at the 
base are the differences of the angles in question, without appealing to the 
figure, we must have a new axiom [proved, of course, by induction !] viz., that 
if a side of a triangle be produced to any point, the line joining that point 
with the opposite angle falls wholly without the triangle. 
