Note 3 — Chapter I

With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical analysis, geometers count this last couple of functions among the analytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. 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.