This repetition of that which has been substantially stated before,
brings us to the final consideration of the distinctive nature of
different departments of Thought, as indicated by the Methods of Proof
which respectively prevail in them; and hence as embodying either exact
and definite _Knowledge_, or only varying degrees of _Probability_. We
have already seen that in at least one sphere of intellectual activity
we are able to start from the most basic and fundamental conceptions,
from axiomatic truths so patent and universal that they cannot even be
conceived of as being otherwise than as they are, and to proceed from
them, by equally irresistible Inferences, to conclusions which are, from
the nature of the human mind, inevitable. It is in the Mathematics, in
which the Deductive Method is rightly operative, that this kind of
Proof--Demonstration in the strict sense of the term--prevails. The
various branches of Mathematics have therefore been appropriately
denominated the _Exact_ Sciences, in contradistinction from those
domains of Thought whose Laws or Principles are liable to be somewhat
indefinite or uncertain; hence, called the _Inexact_ Sciences.

Exact Science--in its largest sense, that which extends to all domains
in which the proper Deductive Method has been or may hereafter be
rightly employed--is therefore a _system or series of truths relating to
the whole Universe, or to some department of it, consecutively and
necessarily resulting from, and dependent upon, each other, in a
definite chain or series; and resting primarily upon some fundamental
truth or truths so simple and self-evident, that, when clearly stated,
all men must, by the natural constitution of the human mind, perceive
them and recognize them as true. Demonstration is the pointing out of
the definite links in the chain or series by which we go from
fundamental truths, clearly perceived and irresistible, up to the
particular truth in question_.