These two departments, which constitute the immediate object of the calculus of direct functions, are subordinate to a third one, purely speculative, from which both of them borrow their most powerful resources, and which has been very exactly designated by the general name of _Theory of Equations_, although it as yet relates only to _Algebraic_ equations. The numerical resolution of equations, because of its generality, has special need of this rational foundation. This last and important branch of algebra is naturally divided into two orders of questions, viz., those which refer to the _composition_ of equations, and those which concern their _transformation_; these latter having for their object to modify the roots of an equation without knowing them, in accordance with any given law, providing that this law is uniform in relation to all the parts.[9] [Footnote 9: The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical analysis--the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree--is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. The general impossibility of such a decomposition is demonstrated by the author in detail, but more properly belongs to a special treatise.] THE METHOD OF INDETERMINATE COEFFICIENTS. To complete this rapid general enumeration of the different essential parts of the calculus of direct functions, I must, lastly, mention expressly one of the most fruitful and important theories of algebra proper, that relating to the transformation of functions into series by the aid of what is called the _Method of indeterminate Coefficients_. This method, so eminently analytical, and which must be regarded as one of the most remarkable discoveries of Descartes, has undoubtedly lost some of its importance since the invention and the development of the infinitesimal calculus, the place of which it might so happily take in some particular respects. But the increasing extension of the transcendental analysis, although it has rendered this method much less necessary, has, on the other hand, multiplied its applications and enlarged its resources; so that by the useful combination between the two theories, which has finally been effected, the use of the method of indeterminate coefficients has become at present much more extensive than it was even before the formation of the calculus of indirect functions. * * * * * Having thus sketched the general outlines of algebra proper, I have now to offer some considerations on several leading points in the calculus of direct functions, our ideas of which may be advantageously made more clear by a philosophical examination. IMAGINARY QUANTITIES.