**Jean-Marie Duhamel**studied at the Lycée in Rennes before becoming a student at the École Polytechnique in 1814. In order to understand the events which totally disrupted his studies there, we need to look at some of the extraordinary events which took place in France around this time.

In 1804 Napoleon had made the École Polytechnique into a military school and the decision was taken to move it to the Montagne Sainte Geneviève. Monge, who was the director of the École Polytechnique, was a staunch supporter of Napoleon and the École was greatly in favour and flourished. However, the allies advanced on France after Napoleon's Russian disaster and crossed the Rhine after the Battle of Leipzig in October 1813. Despite Napoleon's skill in using the troops at his disposal, the allies reached Paris in 1814. Even though the École Polytechnique's students defended Paris fiercely, the city fell to the advancing allies. Napoleon abdicated on 6 April 1814 and was banished to Elba. Monge remained as director of the École Polytechnique and it was as this time that Duhamel began his university studies there.

After Napoleon escaped from Elba and returned to Paris, Monge immediately rallied to him and gave him his full support. Even after Napoleon was defeated at Waterloo, Monge continued to see him until he was put on board a ship on 15 July 1815. By October Monge feared for his life and fled from France but returned to Paris in March 1816. King Louis XVIII had been returned to power by the allies but there was severe friction between supporters of the King and the new Chamber of Deputies which had been elected in August 1815. Monge was dismissed from the directorship of the École Polytechnique immediately following his return, and the students took violent action to support him. There had been other measures imposed on the École which had also angered the students. The King moved against the students sending them all down, and all courses were cancelled until 1817. Duhamel returned to Rennes, but did not go back to Paris after the École Polytechnique was reorganised and reopened in 1817, preferring to remain in Rennes where he studied law.

Duhamel did return to Paris after taking his law degree and taught mathematics and physics both at the Institution Massin and at the Lycée Louis-le-Grand. He then decided to open his own school which was later called the École Sainte-Barbe. Despite having a heavy work laod, Duhamel found time to continue his mathematical studies and in 1823 he presented his first paper *Problèmes et développements sur diverses parties des mathématiques* written jointly with Antoine-André-Louis Reynaud. He began teaching at at the École Polytechnique in 1830, becoming professor there in 1834. He was highly thought of as a teacher of mathematics and was reported to have given very fine lectures. He published articles such as *Sur les équations générales de la propagation de la chaleur dans les corps solides dont la conductibilité n'est pas la même dans tous les sens* (1832) and *Sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans des milieux dont la température varie avec le temps* (1833) in the *Journal of the École Polytechnique*. He submitted his work on the mathematical theory of heat, written up as a doctoral thesis *Théorie mathématique de la chaleur* , to the Faculty of Science and he was awarded his doctorate in 1934. Duhamel was elected to the Academy of Sciences in 1840.

Appointed as entrance examiner at the École Polytechnique in 1835, Duhamel was named professor of analysis and mechanics in 1836. He continued to occupy posts at the École Polytechnique, being made permanent examiner in 1840 and then, during the period 1848 until 1851, was Director of Studies. A commission removed him from his posts in 1850 since he had resisted proposed changes. However, from 1851, he again filled the analysis chair at the École Polytechnique after Liouville was appointed to the vacant chair at the Collège de France. Also from 1851 Duhamel was professor at the Faculté des Sciences in Paris.

Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics, and to acoustics. His acoustical studies involved vibrating strings and the vibration of air in cylindrical and conical pipes. His techniques in the theory of heat were mathematically similar to Fresnel's work in optics with his theory of the transmission of heat in crystal structures based on earlier work by Fourier and Poisson. 'Duhamel's principle' in partial differential equations arose from his contributions to the distribution of heat in a solid with a variable boundary temperature. In fact his theoretical predictions relating to the propogation of heat in non-isotropic solids was later verified experimentally by the physicist Henri de Sénarmont. However, Duhamel did some experimental work of his own, particularly in the area of vibrating strings. He invented a recording instrument which consisted of a pen attached to a vibrating string which left a recond on a moving plate behind it. He suggested that the different sounds which one perceives from musical instruments was due to the ear receiving a complex number of harmonics which were heard as a single sound. This understanding of sound was made independently by G S Ohm.

Duhamel published *Cours d'analyse de l'école polytechnique* in two volumes, the first in 1840 and the second in the following year. Another two volume work *Cours de mécanique* appeared in 1845 and 1846, then both volumes of *Eléments de calcul infinitésimal* were published in 1856. *Des Méthodes dans les sciences de raisonnement* appeared in five volumes between 1866 and 1872.

Paul Tannery gives this interesting description of Duhamel's lectures:-

The pupils of the École Polytechnique have given his name to a glass of sweetened water which, at the beginning of each lesson, he was accustomed to prepare while summarizing, in a voice initially hardly audible, but which rose little by little, the contents of the preceding lesson.

**Article by:** *J J O'Connor* and *E F Robertson*