Elements of the philosophy of the human mind. By Dugald Stewart. Rev. and abridged, with critical and explanatory notes, for the use of colleges and schools. By Francis Bowen ...

nRESOTING AND )DEIUCTIVE EVTIDENCNE. 433
the solution is as mathematically certain, as the truth of any
property of the triangle or of the circle. The three postulates
of Euclid are, indeed, nothing more than the definition of a
circle and a straight line thrown into a form somewhat different;
and a similar remark may be extended to the corresponding
distribution of propositions into theorems and problems. Notwithstanding the many conveniences with which this distribution
is attended, it was evidently a matter of choice rather than
of necessity; all.the truths of geometry easily admitting of
being moulded into either shape, according to the fancy of the
nmathematician. As to the axioms, there cannot be a doubt,
whatever opinion may be entertained of their utility or of their
insignificance, that they stand precisely in the same relation to
both classes of propositions.
iow far it is true that all mathematical evidence is resolvable
into identical propositions. -I had occasion to take notice, in
the first section of the preceding chapter, of a theory with respect to the nature of mathematical evidence, very different
from that which I have been now attempting to explain. According to this theory (originally, I believe, proposed by Leibnitz) we are taught, that all mathematical evidence ultimately
resolves into the perception of identity; the innumerable variety
of propositions which have been discovered, or which remain
to be discovered in the science, being only diversified expressions of the simple formula, a — a.
As this account of mathematical evidence is quite irreconcilable with the scope of the foregoing observations, it is necessary,
before proceeding further, to examine its real import and
amount; and what the circumstances are from which it derives
that plausibility which it has been so generally supposed to
possess.
Ideal superposition is the only way of proving that one space
is equal to another. - That all mathematical evidence resolves
ultimately into the perception of identity, has been considered
by some as a consequence of the commonly received doctrine,
which represents the axioms of Euclid as the first principles of
all our subsequent reasonings in geometry. Upon this view of
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