Jean le Rond d'Alembert

Historically, Jean d’Alembert precedes Augustin-Louis Cauchy. However, in the context of functional equations, it seems more natural to consider his contributions after Cauchy.

Jean d’Alembert was a man of many names. The illegitimate son of an army officer, Louis-Camus Destouches, and a writer, Claudine Guérin de Tencin, he was born in Paris in 1717, while his father was abroad. Shortly after his birth, his mother abandoned him at the church of Saint-Jean-le-Rond. Following tradition, he was named Jean le Rond after the church, and placed in an orphanage. Upon the return of his father, he was removed from the orphanage, and placed with Mme. Rousseau, the wife of a glazier. Although Destouches continued to support his son financially, he chose not to publicly acknowledge his son. In 1738, Jean le Rond entered law school, where he was registered under the name Daremberg. He later changed this name to d’Alembert.

I am sorry not to have known the mathematician who first made use of this method because I would have cited him. Regarding the researches of d'Alembert and Euler could one not add that if they knew this expansion they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not seem that anyone had developed a constant in cosines of multiple arcs, the first problem which I had to solve in the theory of heat.

The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the fundamental theorem of algebra states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why?

Clairaut and Dalembert in their Lunar Theories... introduce... several, now commonly known, Trigonometrical formulæ.

In... Thomas Simpson... the Author evidently intended the one... at p. 76, as preparatory to the... Theory of the Moon; and Euler... states as a reason for cultivating the algorithm of sines, its great utility in the mixed Mathematics.

The first considerable extension of Trigonometry, beyond its original object, was made about twenty years after the death of Newton. It was then, on the ground-work laid down by that great man, that... Clairaut, Dalembert and Euler, and Thomas Simpson... began to establish a system of Physical Astronomy more perfect...

[T]hey laid aside the Geometrical method which Newton had used... and adopted the Analytical. ...[T]hey perceived the formulæ of Trigonometry to be of continual use and recurrence, and the language, by which the process of demonstration was conducted... in a great degree, of symbols and phrases borrowed from that science. ...[T]he advancement of Trigonometry, the pure and subsidiary science, was contemporaneous with that of Astronomy, the mixed and principal one.

D'Alembert was always surrounded by controversy. … he was the lightning rod which drew sparks from all the foes of the philosophes. … Unfortunately he carried this... pugnacity into his scientific research and once he had entered a controversy, he argued his cause with vigour and stubbornness. He closed his mind to the possibility that he might be wrong...

d'Alembert, who wrote the introduction to the Encyclopédie, resigned his editorship with the scathing remark that the work was like a harlequin's coat: some good stuff, but mostly rags.

The evolution of scientific thought is inseparable from the history of man's efforts to resolve the perplexities of his own existence.