we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra. The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations. 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 only important case of this class which has thus far been completely treated is the general integration of linear equations of any order whatever, with constant coefficients. Even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation.

Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it "Differentiatio de curva in curvam." But this comparison had no analogy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant. I propose hereafter to develop this new consideration, in a special work upon the Calculus of Variations, intended to present this hyper-transcendental analysis in a new point of view, which I think adapted to extend its general range. Lacroix has justly criticised the expression of solid, commonly used by geometers to designate a volume. It is certain, in fact, that when we wish to consider separately a certain portion of indefinite space, conceived as gaseous, we mentally solidify its exterior envelope, so that a line and a surface are habitually, to our minds, just as solid as a volume. It may also be remarked that most generally, in order that bodies may penetrate one another with more facility, we are obliged to imagine the interior of the volumes to be hollow, which renders still more sensible the impropriety of the word solid.