e sun and bodies on
which it acts, because, like his rivals the Cartesians, he could not
conceive a body acting where it is not. Indeed, inconceivableness
depends so completely on the accident of our mental habits, that it is
the essence of scientific triumphs to make the contraries of once
inconceivable views themselves appear inconceivable. For instance,
suppositions opposed even to laws so recently discovered as those of
chemical composition appear to Dr. Whewell himself to be inconceivable.
What wonder, then, that an acquired incapacity should be mistaken for a
natural one, when not merely (as in the attempt to conceive space or
time as finite) does experience afford no model on which to shape an
opposed conception, but when, as in geometry, we are unable even to call
up the geometrical ideas (which, being impressions of form, exactly
resemble, as has been already remarked, their prototypes), e.g. of two
straight lines, in order to try to conceive them inclosing a space,
without, by the very act, repeating the scientific experiment which
establishes the contrary.

Since, then, the axioms and the misnamed definitions are but inductions
from experience, and since the definitions are only hypothetically true,
the deductive or demonstrative sciences--of which these axioms and
definitions form together the first principles--must really be
themselves inductive and hypothetical. Indeed, it is to the fact that
the results are thus only conditionally true, that the necessity and
certainty ascribed to demonstration are due.

It is so even with the Science of Number, i.e. arithmetic and algebra.
But here the truth has been hidden through the errors of two opposite
schools; for while many held the truths in this science to be _a
priori_, others paradoxically considered them to be merely verbal, and
every process to be simply a succession of changes in terminology, by
which equivalent expressions are substituted one for another. The excuse
for such a theory as this latter was, that in arithmetic and algebra we
carry no ideas with us (not even, as in a geometrical demonstration, a
mental diagram) from the beginning, when the premisses are translated
into signs, till the end, when the conclusion is translated back into
things. But, though this is so, yet in every step of the calculation,
there is a real inference of facts from facts: but it is disguised by
the comprehensive nature of the induction, and the consequent generality