requires the collected and compared experiences of many centuries. In the
first place, repeated confirmation in experience brings the induction,
upon which the hypothesis rests, so near completeness that in practice it
takes the place of certainty, and is regarded as diminishing the value of
the hypothesis, its source, just as little as the incommensurability of
straight and curved lines diminishes the value of the application of
geometry, or that perfect exactness of the logarithm, which is not
attainable, diminishes the value of arithmetic. For as the logarithm, or
the squaring of the circle, approaches infinitely near to correctness
through infinite fractions, so, through manifold experience, the
induction, _i.e._, the knowledge of the cause from the effects,
approaches, not infinitely indeed, but yet so near mathematical evidence,
_i.e._, knowledge of the effects from the cause, that the possibility of
mistake is small enough to be neglected, but yet the possibility exists;
for example, a conclusion from an indefinite number of cases to all cases,
_i.e._, to the unknown ground on which all depend, is an induction. What
conclusion of this kind seems more certain than that all men have the
heart on the left side? Yet there are extremely rare and quite isolated
exceptions of men who have the heart upon the right side. Sense-perception
and empirical science have, therefore, the same kind of evidence. The
advantage which mathematics, pure natural science, and logic have over
them, as _a priori_ knowledge, rests merely upon this, that the formal
element in knowledge upon which all that is _a priori_ is based, is given
as a whole and at once, and therefore in it we can always proceed from the
cause to the effect, while in the former kind of knowledge we are
generally obliged to proceed from the effect to the cause. In other
respects, the law of causality, or the principle of sufficient reason of
change, which guides empirical knowledge, is in itself just as certain as
the other forms of the principle of sufficient reason which are followed
by the _a priori_ sciences referred to above. Logical demonstrations from
concepts or syllogisms have the advantage of proceeding from the reason to
the consequent, just as much as knowledge through perception _a priori_,
and therefore in themselves, _i.e._, according to their form, they are
infallible. This has greatly assisted to bring demonstration in general
into such esteem. But this infallibility is merely relative; the
demonstration merely subsumes under the first principles of the science,
and it is these which contain the whole material truth of science, and
they must not themselves be demonstrated, but must be founded on
perception. In the few _a priori_ sciences we have named above, this
perception is pure, but everywhere else it is empirical, and is only
raised to universality through induction. If, then, in the empirical
sciences also, the particular is proved from the general, yet the general,