ry may be reduced to the one problem of the
measurement of magnitudes; that is, to the finding the equalities
between them.

Mathematical principles can be applied to other sciences. All causes
operate according to mathematical laws; an effect being ever dependent
on the quantity or a function of the agent, and generally on its
position too. Mathematical principles cannot, indeed, as M. Comte has
well explained, be usefully applied to physical questions, whenever the
causes are either too inaccessible for their numerical laws to be
ascertained, or are too complex for _us_ to compute the effect, or are
ever fluctuating. And, in proportion as physical questions cease to be
abstract and hypothetical, mathematical solutions of them become
imperfect. But the great value of mathematical training is, that we
learn to use its _method_ (which is the most perfect type of the
Deductive Method), that is, we learn to employ the laws of simpler
phenomena to explain and predict those of the more complex.

CHAPTER XXV.

THE GROUNDS OF DISBELIEF.

The result of examining evidence is not always belief, or even
suspension of judgment, but is sometimes positive disbelief. This can
ensue only when the affirmative evidence does not amount to full proof,
but is based on some approximate generalisation. In such cases, if the
negative evidence consist of a stronger, though still only an
approximate, generalisation, we think the fact improbable, and
disbelieve it provisionally; but if of a complete generalisation based
on a rigorous induction, it is disbelieved by us totally, and thought
impossible. Hence, Hume declared miracles incredible, as being, he
considered, contrary to a complete induction. Now, it is true that _in
the absence of any adequate counteracting cause_, a fact contrary to a
complete induction is incredible, whatever evidence it may be grounded
on; unless, indeed, the evidence go to prove the supposed law
inconsistent with some better established one. But when a miracle is
asserted, the presence of an adequate counteracting cause _is_ asserted,
viz. a direct interposition of an act of the will of a Being having
power over nature. Therefore, all that Hume proved is, that we cannot
believe in a miracle unless we believe in the power, and _the will_, of
the Deity to interfere with existing causes by introducing new ones; and
that, in default of such belief, not the most satisfactory evidence of
our senses or of testim