hat the hypotheses, which form part of
the premisses of geometry, must, as Dr. Whewell says, not be
arbitrary--that is, that in their positive part they are observed facts,
and only in their negative part hypothetical--happens simply because our
aim in geometry is to deduce conclusions which may be true of real
objects: for, when our object in reasoning is not to investigate, but to
illustrate truths, arbitrary hypotheses (e.g. the operation of British
political principles in Utopia) are quite legitimate.

The ground of our belief in axioms is a disputed point, and one which,
through the belief arising too early to be traced by the believer's own
recollection, or by other persons' observation, cannot be settled by
reference to actual dates. The axioms are really only generalisations
from experience. Dr. Whewell, however, and others think that, though
suggested, they are not proved by experience, and that their truth is
recognised _a priori_ by the constitution of the mind as soon as the
meaning of the proposition is understood. But this assumption of an _a
priori_ recognition is gratuitous. It has never been shown that there is
anything in the facts inconsistent with the view that the recognition of
the truth of the axioms, however exceptionally complete and instant,
originates simply in experience, equally with the recognition of
ordinary physical generalisations. Thus, that we see a property of
geometrical forms to be true, without inspection of the material forms,
is fully explained by the capacity of geometrical forms of being painted
in the imagination with a distinctness equal to reality, and by the fact
that experience has informed us of that capacity; so that a conclusion
on the faith of the imaginary forms is really an induction from
observation. Then, again, there is nothing inconsistent with the theory
that we learn by experience the truth of the axioms, in the fact that
they are conceived by the mind as universally and necessarily true, that
is, that we cannot figure them to ourselves as being false. Our capacity
or incapacity of conceiving depends on our associations. Educated minds
can break up their associations more easily than the uneducated; but
even the former not entirely at will, even when, as is proved later,
they are erroneous. The Greeks, from ignorance of foreign languages,
believed in an inherent connection between names and things. Even Newton
imagined the existence of a subtle ether between th