Of still another domain, that of Physics, Professor Silliman says, 'all its phenomena are dependent on a limited number of general laws ... which may be represented by numbers and algebraic symbols; and these condensed _formulae_ enable us to conduct investigations with the certainty and precision of pure Mathematics.' The various branches of Physics have not hitherto been ranked as Exact Sciences, because, as in Astronomy, unsubstantiated theories 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.