In his scientific research, Laplace exploited an elementary set-theoretic property of functions to give an unexpected proof of a result such as the fundamental theorem of algebra (see the box at the end of the article "The educational value of passing on knowledge"). Yet he is not generally remembered for this kind of functional reasoning. Such functions would remain central to mathematical practice. Laplace therefore did not create mathematics for its own sake, but always with a specific purpose, whether in celestial mechanics, probability or physics.
The potential function ------------------------
Lagrange was, admittedly, the one who introduced the potential function to study the attraction between masses. But Laplace used the concept to establish that, outside the masses, this function of three spatial variables has a vanishing Laplacian. In other words, the sum of its second-order partial derivatives with respect to the three spatial variables is zero.
Laplace was interested in the non-spherical shape of the planets, the subject of his first book, published in 1784, Théorie du mouvement et de la figure elliptique des planètes (see box). In it, he defines a function V at a point M with coordinates x, y and z in an orthonormal coordinate system. He defines it as the triple integral, over all space, of the mass element at M’—whose coordinates are x’, y’ and z’—divided by the distance r from M’ to M. This function V captures the essence of the attraction: at M, the three components of the attractive force in the orthonormal coordinate system are, up to sign, its partial derivatives with respect to the three variables.
Representing the force analytically by a single function has many advantages, including a vanishing Laplacian, simply because this is a property of the function that maps r > 0 to 1/r. Laplace immediately wrote the Laplacian not in Cartesian coordinates, but in spherical coordinates. He could then turn to solving the Laplace equation, introducing polynomials indexed by their degree that proved to have an orthogonality property, because a certain integral of the product of two polynomials of different degrees is zero. This approach, a distant precursor to Hilbert's geometrization of analysis, had been initiated by Legendre, though without the same generality, and the polynomials are indeed called Legendre polynomials today.