Even the scientifically instructed, who possess, in the form of general
propositions, a systematic record of the results of the experience of
mankind, need not always revert to those general propositions in order to
apply that experience to a new case. It is justly remarked by Dugald
Stewart, that though the reasonings in mathematics depend entirely on the
axioms, it is by no means necessary to our seeing the conclusiveness of
the proof, that the axioms should be expressly adverted to. When it is
inferred that AB is equal to CD because each of them is equal to EF, the
most uncultivated understanding, as soon as the propositions were
understood, would assent to the inference, without having ever heard of
the general truth that “things which are equal to the same thing are equal
to one another.” This remark of Stewart, consistently followed out, goes
to the root, as I conceive, of the philosophy of ratiocination; and it is
to be regretted that he himself stopped short at a much more limited
application of it. He saw that the general propositions on which a
reasoning is said to depend, may, in certain cases, be altogether omitted,
without impairing its probative force. But he imagined this to be a
peculiarity belonging to axioms; and argued from it, that axioms are not
the foundations or first principles of geometry, from which all the other
truths of the science are synthetically deduced (as the laws of motion and
of the composition of forces in dynamics, the equal mobility of fluids in
hydrostatics, the laws of reflection and refraction in optics, are the
first principles of those sciences); but are merely necessary assumptions,
self-evident indeed, and the denial of which would annihilate all
demonstration, but from which, as premises, nothing can be demonstrated.
In the present, as in many other instances, this thoughtful and elegant
writer has perceived an important truth, but only by halves. Finding, in
the case of geometrical axioms, that general names have not any talismanic
virtue for conjuring new truths out of the well where they lie hid, and
not seeing that this is equally true in every other case of
generalization, he contended that axioms are in their nature barren of
consequences, and that the really fruitful truths, the real first
principles of geometry, are the definitions; that the definition, for
example, of the circle is to the properties of the circle, what the laws
of equilibrium and of the pressure of the atmosphere are to the rise of
the mercury in the Torricellian tube. Yet all that he had asserted
respecting the function to which the axioms are confined in the
demonstrations of geometry, holds equally true of the definitions. Every
demonstration in Euclid might be carried on without them. This is apparent
from the ordinary process of proving a proposition of geometry by means of
a diagram. What assumption, in fact, do we set out from, to demonstrate by