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. 387
Fronm what principle are the various properties of the circle
derived, but from the definition of a circle? From what prinrespecting the nature of axioms and the first principles of mathemamical
reasoning, the former controverting, and the latter supporting, the opinions
of Mr. Stewart. Yet the admission made by Stewart in the former palrt
of this note seems to take away much of the ground of controversy
between him and Dr. Whewell. It seems to be admitted on all hands, by
Mr. Mill as well as by Mr. Stewart, that what Dr. Whewell calls "the
peculiar geometrical axioms," such as the 10th and 11th of Euclid, are
among the first principles of geometry, - that, far from being barren
trt dsms, like the first nine, proper inferences can be deduced. fiom them,
and the whole structure of geometry could not be built up without their
aid. The only doubt is, whether they are properly called axioms, whether
they should not be considered rather as theorems, or propositions which
ought to be demonstrated, though geometricians have not as yet succeeded
in proving' them. On the other hand, Dr. Whellwell does not assert that
geometrical reasoning rests exclusi:ely upon axioms, but allows that definitions must be classed with them, both together constituting the first principles of the science. He urges, " that no one has yet been able to constrect a system of mathematical truths by the aid of definitions alone;
that a definition would not be admissible or applicable, except it agreed
with a distinct conception in the mind; that the definitions which we
employ in mathematics are not arbitrary or hypothetical, but necessary
definitions; and that the real foundation of the truths of mathematics is
the idea of space, which may be expressed, for purposes of demonstration,
partly by definitions and partly by axioms."
Mr. Mill answers, " Those who say that the premises of geometry are
hypotheses, are not bound to maintain them to be hypotheses which have
no relation whatever to fact. Since an hypothesis framed for the purpose of scientific inquiry must relate to something which has real existence, (for there can be no science respecting nonentities,) it follows, that
any hypothesis which we make respecting an object, to facilitate our
study of it, must not involve any thing which is distinctly false, and repugnant to its real nature; we must not ascribe to the thing any propl
erty, which it has not; our liberty extends only to suppressing some of
those which it has, under the indispensable obligation of restoring thcn
whenever, and as far as, their presence or absence would make any
material difference in the truth of our conclusion. Of this nature, accordingly, are the first principles involved in the definitions of geometry.
In their positive part, they are observed facts; it is only in their negactive
part that they are hypothetical."
It had been previously remarked by Mr. Mill, that " there exist no real