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theory of geometrical reasoning, depends on the axiom, that tri- 
angles, having two sides equal each to each, are equal in all respects. 
Such, nevertheless, is the case ; and when one sees this absurdity 
pass unmodified from edition to edition of Mr Mill’s Logic, and 
when even Mansel, Mr Mill’s watchful enemy, tells us that “against 
the form of the geometrical syllogism, as exhibited by Mr Mill, the 
logician will have no objections to allege ” (Mansel’s Aldrich, 3d ed., 
p. 255), one cannot but think that logic would make more progress 
if logicians would give a little more attention to the processes they 
profess to explain. 
It may perhaps be worth while to show how Mr Mill was led 
into this extraordinary mistake. We shall find that Mr Mill 
chooses rather to sacrifice geometry to his philosophy, than to 
modify his philosophy in accordance with the facts of geometry. 
Mr Mill holds that all general knowledge is derived from experi- 
ence ; meaning by experience the comparison of at least two distinct 
experimental facts. In other words, all knowledge is ultimately 
gained by induction from a series of observed facts. That any general 
truth can be got at intuitively, by merely looking at one case, Mr Mill 
emphatically denies. The fact that two straight lines cannot enclose 
a space, is not self-evident as soon as we know what straight lines are 
[Le., can mentally construct such lines] ; but is got at only by experi- 
ments on “ real” or “ imaginary ” lines (Logic, I. pp. 259, 262). Now 
it is certain, that in the demonstrations of Euclid, we are satisfied of 
the truth of the general proposition enunciated, as soon as we have 
read the proof for the special figure laid down. There is no need 
for an induction from the comparison of several figures. Since 
then one figure is as good as half a dozen, Mr Mill is forced to the 
conclusion that the figure is no essential part of the proof, or that 
“ by dropping the use of diagrams, and substituting, in the demon- 
strations, general phrases for the letters of the alphabet, we might 
prove the general theorem directly ” from “ the axioms and defini- 
tions in their general form ” (p. 213). 
We may just mention, in passing, that this view, combined with 
the doctrine that the definitions of geometry are purely hypo- 
thetical, leads Mr Mill to the curious opinion that we might make 
any number of imaginary sciences as complicated as geometry, by 
applying real axioms to imaginary definitions. We mention this 
