Jean-le-Rond D'Alembert (1717 - 1783)

From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.

Jean-le-Rond D'Alembert was born at Paris on November 16, 1717, and died there on October 29, 1783. He was the illegitimate child of the chevalier Destouches. Being abandoned by his mother on the steps of the little church of St. Jean-le-Rond, which then nestled under the great porch of Notre-Dame, he was taken to the parish commissary, who, following the usual practice in such cases, gave him the Christian name of Jean-le-Rond; I do not know by what authority he subsequently assumed the right to prefix de to his name. He was boarded out by the parish with the wife of a glazier in a small way of business who lived near the cathedral, and here he found a real home, though a humble one. His father appears to have looked after him, and paid for his going to a school where he obtained a fair mathematical education.

An essay written by him in 1738 on the integral calculus, and another in 1740 on ``ducks and drakes'' or ricochets, attracted some attention, and in the same year he was elected a member of the French Academy; this was probably due to the influence of his father, It is to his credit that he absolutely refused to leave his adopted mother, with whom he continued to live until her death in 1757. It cannot be said that she sympathised with his success, for at the height of his fame she remonstrated with him for wasting his talents on such work: ``Vous ne serez jamais qu'un philosophe,'' said she, ``et qu'est-ce qu'un philosophe? c'est un fou que se tourmente pendant sa vie, pour qu'on parle de lui lorsqu'il n'y sera plus.''

Nearly all his mathematical works were produced during the years 1743 to 1754. The first of these was his Traité de dynamique, published in 1743, in which he enunciates the principle known by his name, namely, that the ``internal forces of inertia'' (that is, forces which resist acceleration) must be equal and opposite to the forces which produce the acceleration. This may be inferred from Newton's second reading of his third law of motion, but the full consequences had not been realized previously. The application of this principle enables us to obtain the differential equations of motion of any rigid system.

In 1744 D'Alembert published his Traité de l'équilibre et du mouvement des fluides, in which he applies his principle to fluids; this led to partial differential equations which he was then unable to solve. In 1745 he developed that part of the subject which dealt with the motion of air in his Theorie générale des vents, and this again led him to partial differential equations. A second edition of this in 1746 was dedicated to Frederick the Great of Prussia, and procured him an invitation to Berlin and the offer of a pension; he declined the former, but subsequently, after some pressing, pocketed his pride and the latter. In 1747 he applied the differential calculus to the problem of a vibrating string, and again arrived at a partial differential equation.

His analysis had three times brought him to an equation of the form

partial^2 u / partial t^2 =  partial^2 u / partial x^2,
and now he succeeded in shewing that it was satisfied by
u = \phi(x + t) + \psi(x - t).
where \phi   and \psi   are arbitrary functions. It may be interesting to give his solution which was published in the transactions of the Berlin Academy for 1747. He begins by saying that, if partial u / partial x   be denoted by p and partial u / partial t   by q, then
du = p dx + q dt.
But, by the given equation, partial q / partial t = partial p / partial x, and therefore p dt + q dx is also an exact differential: denote it by dv. Therefore
dv = p dt + q dx.
du + dv = (p dx + q dt) + (p dt + q dx) = (p + q)(dx + dt),
du - dv = (p dx + q dt) - (p dt + q dx) = (p - q)(dx - dt).
Thus u + v must be a function of x + t and u - v must be a function of x - t. We may therefore put
u + v = 2\phi(x + t)
u - v = 2\psi(x - t)
u = \phi(x + t) + \psi(x - t).

D'Alembert added that the conditions of the physical problem of a vibrating string demand that, when x = 0, u should vanish for all values of t. Hence identically

\phi(t) + \psi(-t) = 0.
Assuming that both functions can be expanded in integral powers of t, this requires that they should contain only odd powers. Hence
\psi(-t) = - \phi(t) = \phi(-t)
u = \phi(x + t) + \phi(x - t).

Euler now took the matter up and shewed that the equation of the form of the string was

partial^2 u / partial t^2 =  a^2 partial^2 u / partial x^2,
and that the general integral was
u = \phi(x + at) + \psi(x - at).
where \phi   and \psi   are arbitrary functions.

The chief remaining contributions of D'Alembert to mathematics were on physical astronomy, especially on the precession of the equinoxes, and on variations in the obliquity of the ecliptic. These were collected in his Système du monde, published in three volumes in 1754.

During the latter part of his life he was mainly occupied with the great French encyclopaedia. For this he wrote the introduction, and numerous philosophical and mathematical articles; the best are those on geometry and on probabilities. His style is brilliant, but not polished, and faithfully reflects his character, which was bold, honest, and frank. He defended a severe criticism which he had offered on some mediocre work by the remark, ``j'aime mieux être incivil qu'ennuyé''; and with his dislike of sycophants and bores it is not surprising that during his life he had more enemies than friends.

This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).

Transcribed by

D.R. Wilkins
School of Mathematics
Trinity College, Dublin