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 ...

O~U REAiR S 0IN.
these axioms, without ever seeing one jot the more of mathematical truths." But surely, if this be granted, and if, at the
same time, by the first principles of a science be meant those
fundamental propositions from which its remoter truths are
derived, the axioms cannot, with any consistency, be called the
first principles of mathematics. They have not, it will be admitted, the most distant analogy to what are called the first
principles of natural philosophy;- to those general facts, for
example, of the gravity and elasticity of the air, from which
may be deduced, as consequences, the suspension of the mercury
in the Torricellian tube, and its fall when carried up to an
eminence. According to this meaning of the word, the principles of mathematical science are, not the axioms, buit the dejfnitions; which definitions hold, in mathematics, precisely the same
place that is held in natural philosophy by such general fiacts
as have now been referred to.*
* In order to prevent cavil, it may be necessary for me to remark here,
that when I speak of mathematical axioms, I have in view only such as
are of the same description with the first nine of those which are prefixed
to the Elements of Euclid; for, in that list, it is well known, that there
are several which belong to a class of propositions altogether different from
the others. That " all right angles (for example) are equal to one
another;" that " when one straight line falling on two other straight lines
makes the two interior angles on the same side less than two right angles,
these two straight lines, if produced, shall meet on the side, where are the
two angles less than two right angles;" are manifestly principles whictl
bear no analogy to such barren truisms as these, "Things that are equal
to one and the same thing, are equal to one another." "If equals he
added to equals, the wholes are equal." " If equals be taken from equals,
the remainders are equal." Of these propositions, the two former (the
10th and 11th axioms, to wit, in Euclid's list) are evidently theorems
which, in point of strict logical accuracy, ought to be demonstrated; as
may be easily done, with respect to the first, in a single sentence. That
the second has not yet been proved in a simple and satisfactory manner,
has been long considered as a sort of reproach to mathematicians; and
I have little doubt that this reproach will continue to exist, till the basis
of the science be somewhat enlarged, by the introduction of one or two
new definitions, to serve as additional principles of geometrical reasoning.
[Dr. Whewell and Mr. J. S. Mill have engaged in this discussion