Notas

There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously. Some other concrete functions may be usefully introduced into the number of analytical elements, certain conditions being fulfilled. It is thus, for example, that the labours of M. Legendre and of M. Jacobi on elliptical functions have truly enlarged the field of analysis; and the same is true of some definite integrals obtained by M. Fourier in the theory of heat. Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes, a and b, x³ + 3ax = 2b, as is the case in the problem of the trisection of an angle. We see at once that the dependence between x on the one side, and ab on the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution.

When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the form x = ∛(b + √(b² + a³)) + ∛(b - √(b² + a³)), the work of algebra is finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work of arithmetic will now consist in taking that formula for its starting point, and finding the number x when the values of the numbers a and b are given. I have thought that I ought to specially notice this definition, because it serves as the basis of the opinion which many intelligent persons, unacquainted with mathematical science, form of its abstract part, without considering that at the time of this definition mathematical analysis was not sufficiently developed to enable the general character of each of its principal parts to be properly apprehended, which explains why Newton could at that time propose a definition which at the present day he would certainly reject. This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.

Simple as may seem, for example, the equation a^x + b^x = c^x, 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.