(The contents of the next two chapters are based on the excellent biography by Herivel Joseph Fourier The Man and the Physicist, Oxford 1975.)

Joseph Fourier was born in 1768 in Auxerre, the ninth child of a master tailor. Although the death of his father left him an orphan at the age of ten, his intelligence gained him a free place at the local Benedictine school. At the end of a brilliant school career he applied to enter the artillery only to be informed that such a profession was only open to those of noble blood and was closed to him ‘even if he were a second Newton’.

Fourier began to prepare to enter the Benedictine teaching order but, whatever his plans may have been, the course of his life was violently altered by the outbreak of the French Revolution. ‘As the natural ideas of equality developed, it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests and to free from this double yoke the long usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and the most beautiful which any nation has undertaken.’

However, the Revolution was soon threatened by problems of its own making. The collapse of royal authority and the effects of revolutionary zeal and political misjudgement created administrative chaos whilst at the same time driving France into war against most of Europe.

(The following passage is translated from the introduction to Fourier's Théorie Analytique de la Chaleur.)

The equations of heat conduction like those of sound or small oscillations for liquids belong to one of the most recently discovered branches of science which it is most important to extend. Having established these differential equations we must obtain their solution which involves passing from a general to a particular solution obeying all the boundary conditions. This difficult investigation requires a special calculus based on new theorems…. The method which results leaves nothing vague or undetermined in the solutions; it leads us to the point where we can obtain numerical values, a necessary property of any such method, without which we obtain nothing but useless formal transformations.

These same theorems which enable us to solve the heat equation have immediate applications to questions in general analysis and dynamics whose solutions have long been sought.

The profound study of nature is the most fruitful source of mathematical discovery. Not only does this study, by proposing a fixed goal for our research, have the advantage of excluding vague questions and calculations without result. It is also a sure way of shaping Analysis itself, and discovering which parts it is most important to know and must always remain part of the subject. These fundamental principles are those which are found in all natural phenomena.

The discovery of the calculus was followed by a century of brilliant exploitation led by men like Euler and Lagrange. But towards the end of that century it became clear that several problems of physics and pure mathematics were resisting all attacks. In addition the lack of any clear and agreed foundation for the calculus led to paradoxes and obscurities which might be ignored but could not be resolved.

In a much quoted letter to d'Alembert in 1781, Lagrange wrote

These sentiments were echoed by Delambre in his summary for the Academy of Sciences of the state of the mathematical sciences (‘Rapport Historique sur les Progrès des Sciences Mathématiques Depuis 1789’). Speaking of the calculus he described the general view of such discoveries. ‘Prepared by the work of many centuries, when they are sufficiently ripe to be plucked by genius, their first effect is to excite admiration and their second to impose on following generations immense labours whose glory will never equal their difficulty.’