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 ...

REASONING- AND DEDUCTIVE EVIDENCE. 423
attention the characters which have passed before them on the
stage of life; or who amuse themselves with marking the trifling and fortuitous circumstances by which the multitude are
decided, in pronouncing their verdicts of foresight or of improvidence.
III. OF 5MATHEIMATICAL DEMONSTRATION. Of the circuanstance on which demonstrative evidence essentially depends. —
The peculiarity of that species of evidence which is called demnonstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other
branches of science, is a fact which must have arrested the
attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet I am doubtful
if a satisfactory account has been hithlerto given of the circumstance from which it arises. MAr. Locke tells us, that " what
constitutes a demonstration is intuitive evidence at every step;"
and I readily grant, that if in a single step such evidence should
fail, the other parts of the demonstration would be of no value.
it does not, however, seem to me that it is on this consideration
that the demonstrative evidence of the conclusion depends, —
not even when we add to it another which is much insisted on
by Dr. Reid, - that, " in demonstrative evidence, our frst prinples must be intuitively certain." The inaccuracy of this remark
I formerly pointed out, when treating of the evidence of axioms;
on which occasion I also observed, that the first principles of
our reasonings in mathematics are not axioms, but definitions.
It is in this last circumstance (I mean the peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found; and I shall accordingly endeavor to
explain it at considerable length, and to state some of the more
important consequences to which it leads.
It was already remarked in the eighth chapter, that whereas,
in all other sciences, the propositions which we attempt to
establish, express facts real or supposed, - in mathematics,
the propositions which we demonstrate only assert a connection
between certain suppositions and certain consequences. Our
reasonings, therefore, in mathematics, are directed to an object