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 ...

456 REASONING AND DEDUCTIVE EVIDENCEa
ences, - that it rests wholly on hypotheses and definitions, and
in no respect upon any statement of facts, true or false. The
ideas, indeed, of extension, of a triangle, and of equality, presuppose the exercise of our senses. Nay, the very idea of
superposition involves that of motion, and consequently (as the
parts of space are immovable) of a material triangle. But
where is there any thing analogous, in all this, to those sensible
facts, which are the principles of our reasoning in physics;
and which, according as they have been accurately or inaccurately ascertained, determine the accuracy or inaccuracy of our
conclusions? The material triangle itself, as conceived by the
mathematician, is the object, not of sense, but of intellect. It is
not an actual measure, liable to expansion or contraction, from
the influence of heat or of cold; nor does it require, in the ideal
zuse which is made of it by the student, the slightest address of
hand or nicety of eye. Even in explaining this demonstration
for the first time to a pupil, how slender soever his capacity
might be, I do not believe that any teacher ever thought of
illustrating its meaning by the actual application of the one triangle to the other. No teacher, at least, would do so, who had
formed correct notions of the nature of mathematical science.
If the justness of these remarks be admitted, the demonstration in question must be allowed to be as well entitled to the
name, as any other which the mathematician can produce; for
as our conclusions relative to the properties of the circle, considered in the light of hypothetical theorems, are not the less
rigorously and necessarily true, that no material circle may anywhere exists corresponding exactly to the definition of that
figure, so the proof-given by Euclid of the fourth proposition
would not be the less demonstrative, although our senses were
incomnparably less acute than they are, and although no material
triangle continued of the same magnitude for a single instant.
Indeed, when we have once acquired the ideas of equality and
of a common measure, our mathematical conclusions would not
be in the least affected, if all the bodies in the universe should
vanish into nothing.
IV. OF OUR REASONINGS CONCERNING PROBABLE OR