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 DEDUCTIVEl EVIDENCE. 4{55
That the reasoning employed by Euclid in proof of the fourth
proposition of his first boolk is completely demonstrative, will
be readily granted by those who compare its different steps
with the conclusions to which we were formerly led, when treating of the nature of mathematical demonstration. In none of
these steps is any appeal made to facts resting on the evidence
of sense, nor, indeed, to any facts whatever. The constant appeal is to the definition of equality*' Let the triangle A B
C," says Euclid, " be applied to the triangle D E F; the point
A to the point D, and the straight line A B to the straight line
D E; the point B will coincide with the point E, because A B
is equal to D E. And A B coinciding with D E, A C will
coincide with D F, because the angle B A C is equal to the
angle E D F." A similar remark will be found to apply to
every remaining step of the reasoning; and, therefore, this
reasoning possesses the peculiar characteristic which distinguisiLes mathematical evidence from that of all the other sciposed, an inexact and purely mechanical mode of demonstration. Superposition in mathematics does not consist in applying one figure to the
other, in order to judge by the eye whether they differ or coincide, just as
a workman applies his foot-rule to a line in order to measure it; it consists in imagining one figure placed over the other, and concluding, from
the supposed equality of certain parts of the two figures, the coincidence
of these parts with each other, and from their coincidence inferring the
coincidence of the other parts; whence results the perfect equality and
similitude of the whole figures."]
About a century before the time when D'Alembert wrote these observations, a similar view of the subject was taken by Dr. Barrow; a writer
who, like D'Alembert, added to the skill and originality of an inventive
mathematician, the most refined, and, at the same time, the justest ideas
concerning the theory of those intellectual processes which are subservient
to mathematical reasoning.
It was before observed, that Euclid's eighth axiom (magnitultdes which
coincide with each other are equal) ought, in point of logical rigor, to
have been stated in the form of a definition. In our present arilment,
however, It is not of material consequence whether this criticism be
adopted or not. Whether we consider the proposition in question in the
light of an axiom or of a definition, it is equally evident, that it does not
express a fact ascertained by observation or by experiment.