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 ...

REASON. 385
that peculiar evidence which is aliowed to accompany mathemlatical demonstration. "All the sciences," it has been said,
"rest ultimately on first principles, which we must take for
granted without proof; and whose evidence determines, both in
kind and degree, the evidence which it is possible to attain in
our conclusions. In some of the sciences, our first principles
are intuitively certain; in others, they are intuitively probable;
and such as the evidence of these principles is, such must that
of our conclusions be. If our first principles are intuitively
certain, and if we reason from them consequentially, our conclusions will be demonstratively certain; but if our principles
be only intuitively probable, our conclusions will be only demonstratively probable. In mathematics, the first principles froml
which we reason are a set of axioms, which are not only intuitively certain, but of which we find it impossible to conceive the
contraries to be true; and hence the peculiar evidence which
belongs to all the conclusions that follow from these principles
as necessary consequences."
Deftnitions, not axioms, are tlhe first princples of mcathematics. - That there is something fundamentally erroneous in
these very strong statements with respect to the relation which
Euclid's axioms bear to the geometrical theorems which follow,
appears sufficiently fiom a consideration which was long ago
mentioned by Locke, that from these axioms it is not possible
for human ingenuity to deduce a single inference. "It was not,"
says Locke, " the influence of those maxims which are taken for
principles in mathematics, that hath led the masters of that
science to those wonderful discoveries they have made. Let a
man of good parts know all the maxims generally made use of
in mathematics never so perfectly, and contemplate their extent
and consequences as much as he pleases, he will, by their assistance, I suppose, scarce ever come to know, that'the sq.tare
of the hypothenuse in a right angled triangle, is equal to the
squares of the two other sides.' The knowledge that' the whole
is equal to all its parts,' and,'if you take equals from equals,
the remainders will be equal,' helped him not, I presume, to this
demonstration; and a man may, I think, pore long enough on
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