It is obvious from this exhibition of the characteristics of the two leading Scientific Methods, or the two leading Processes of the one Method, in whichever light we may choose to view them, that so far from being the best or the only true Method or Process of intellectual investigation, the Inductive is far inferior to the _true Deductive_ Method or Process, in all the essentials of a Scientific guide. The Inductive can give us only a _high degree_ of precision and definiteness, with only proximate certainty for the future as the result of a slow mode of procedure; while the true Deductive Method gives us perfect precision, exactitude, and complete certainty, as the result of a rapid mode. The true Deductive Method--brought into disrepute 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.