isrepute by being
confounded with the Anticipative or Hypothetical, which differs from it
only in this, that the Principles from which the latter reasons are
_true_, while those of the former are _doubtful_--has thus far prevailed
in Mathematics alone, and _Mathematics_ is, up to our day, _the only
recognized Exact Science_, the only Science in which _Demonstration_, in
the strict sense of that term, is now possible,--the fruits of the
Inductive Method being known as the _Inexact_ Sciences, in which only
Probable Reasoning prevails.

It is necessary to say, in the _strict sense of the term_, because the
same laxity exists in the use of the word _Demonstration_, as in that of
Science, and hence it has lost the distinctive meaning which attaches to
it, in its legitimate use, as signifying a mode of reasoning in which
the _self-evident truths or axioms_, with which we start, and every step
in the deduction, 'are not only perfectly definite, but incapable of
being apprehended differently--if really apprehended, they must be
apprehended alike by all and at all times.' It is because this Method of
proof exists only in Mathematics, that this alone is denominated the
_Exact_ Science, or its branches, the Exact Sciences; Sciences whose
Laws or Principles, and the Facts connected with or deduced from them,
are irresistible conclusions of thought, in all minds, which conclusions
rest upon universally recognized axioms; while the _Inexact Sciences_,
including all except Mathematics, the Sciences in which the Inductive
Method prevails, are systems of Laws or Principles, with their related
Facts, of the truth of which there is great probability, but of which
there is, nevertheless, no complete certainty; whose conclusions are not
_based_ upon universally undeniable axioms, or are not _themselves_
irresistible to the human mind.

The superiority of the Deductive Method, both in its mode of advancing
to the discovery of new truth and in the precision, clearness, and
certainty which accompany its findings, must now easily become apparent.
Whether we regard Induction and Deduction as correlative Processes
belonging to one Method, each of which has been disproportionately in
vogue at different epochs, or as distinctive Methods, having each their
own Deductive and Inductive Processes, in either aspect, Induction is
only a preparative labor, leading in the more important work of the
application of the Law or Principle derived. It is only,