Mathematics and Temperature

Abstract

In the previous chapter, we looked at the preliminary discourse of Fourier’s Analytical Theory of Heat. Herein, he provided an overview of the types of (previously insoluble) problems his theory would address, and how his theory would be formulated: in terms of differential equations. These differential equations govern the flow of heat—and hence the temperature distribution—throughout bodies subjected to sources of heat. In order to arrive at a solution to these differential equations for a given body, one must have knowledge of certain specific qualities of the body. In particular, one must know the body’s (i) specific heat (its power to contain heat), (ii) surface conductivity (its power to receive or transmit heat across its surface), and (iii) thermal conductivity (its power to conduct heat through the interior of its mass). Once these qualities are known—along with the thermal conditions existing at the surface of the body—then finding the temperature distribution within the body is reduced to the mathematical process of solving a differential equation given certain boundary conditions. This is not to say it is easy: Fourier would have to develop a new method—the series solution—to solve many such problems. But the technique which Fourier discovered provided a new way of addressing the problem of heat. In the reading selection that follows, Fourier begins to flesh out the mathematical methods outlined in the preliminary discourse. He begins by way of example—describing the distribution of temperature within bodies subjected to various sources of heat.

Mathematical analysis has therefore necessary relations with sensible phenomena; its object is not created by human intelligence; it is a pre-existent element of the universal order, and is not in any way contingent or fortuitous.

—Joseph Fourier