ories and
doubtful generalizations, incapable of Mathematical Proof, have mingled
with their _Demonstrated_ Laws and Phenomena, as a component part of
the Science itself. It has consequently exhibited an ambiguous or
problematical aspect, incompatible with the rigorous requirements of
Exact Science. Even in Professor Silliman's admirable work, _formulae_
are given as Laws, which, however correct, have yet no foundation in
axiomatic truth; while Inferences are drawn from them which are by no
means capable of _Demonstration_. Strictly speaking, however, only those
Laws which _do_ rest upon a Demonstrable basis and the Phenomena derived
from them come within the scope of the _Science_ of Physics. So far as
these prevail, this department of investigation is entitled to the
Mathematical character accorded to it by Professor Silliman, and ranks
as an Exact Science.

Astronomy and Physics, viewed in the light in which they are here
presented, are rather special branches of Mathematics, than distinct
Sciences. But as we often speak of Geometry as a separate Science,
although it is in reality only a division of the Mathematical domain,
and is so classed by Comte; so there is a sense in which both Astronomy
and Physics, as herein defined, may be regarded as individual Sciences,
and in that character they will be considered in this paper.

We have, then, three domains in which the true Deductive Method is
active; in which we can start from universally recognized Truths and
proceed, by irresistible Inferences, to ulterior Principles and Facts.
In three Sciences, in Mathematics as commonly defined and understood, in
Astronomy and Physics as herein circumscribed, we are able to establish
starting points of thought with Mathematical certainty, and to deduce
from them all the Phenomena of their respective realms.

Within the scope of these three Sciences, therefore, our information is
clearly defined, positive, and indisputable. The conclusions to which we
are led by their Principles can no more be gainsayed than human
existence can be doubted. While time shall last, while mankind shall
endure, while the human Mind is constructed on its present basis; while,
in fine, there is a possibility for the exercise of Thought in any way
conceivable to the existing Mentality of the universe, the Laws of
Mathematics, of Astronomy, and of Physics can be apprehended in no way
different from that in which they are now apprehended. There is _no
co