# D'Alembert, Lagrange, and Reduction of Order - The Correspondence Between D'Alembert and Lagrange

Author(s):
Sarah Cummings (Wittenberg University) and Adam E. Parker (Wittenberg University)

Lagrange and d'Alembert started their correspondence in 1759.  At this time, d'Alembert was 41 and one of the preeminent living mathematicians.  Lagrange was 22 and already establishing himself as a first class mathematician in his own right.  A year earlier, Lagrange and his students started the Turin Academy of Sciences and published the first issue of the Mémoires of their new society.  This journal went by many names over the years, but today is known as Miscellanea Taurinensia.  On September 27, 1759, d'Alembert wrote to Lagrange from Paris and began by complimenting him on the first issue (see Figure 2). This letter started a correspondence that lasted for 24 years.

 Monsieur, I've received the first volume of your Memoires that you have done the honor of sending me with much gratitude and read with much satisfaction. Your essay on sound is full of the most scholarly and ingenious research.

Figure 2. The beginning of years of correspondence between d'Alembert and Lagrange [12, p. 1].  In each of Figures 2 through 6, D'Alembert's or Lagrange's French is followed by the authors' translation into English.  (Source: Excerpts from the correspondence of d'Alembert and Lagrange reproduced in Figures 2 through 6 are from a copy of Lagrange's Oeuvres [12] held by Bibliothèque nationale de France, département Littérature et art, V-15599 (BIS).)

The first mention of reduction of order occurred in a letter from Lagrange to d'Alembert on January 26, 1765 (see Figure 3).  In this letter, Lagrange told d'Alembert he had a technique of reducing the order of a linear differential equation and moreover he planned to publish his method in the upcoming third volume of Miscellanea Taurinensia.  Notice Lagrange did not provide details of his method here.

 $X$ being some function of $x$, and $y=A+Bx$; and this solution in and of itself is only a specific case for an integration method of which I calculate the full value of $y$ in this equation of $m$th degree: $Py+Q\frac{dy}{dx}+R\frac{dy^2}{dx^2}+\dots = X$ $(P, Q, \dots, X$ being arbitrary functions of $x)$ when assuming that I know $m$, or least $m-1$, specific values of $y$ in the equation $Py+Q\frac{dy}{dx}+R\frac{dy^2}{dx^2}+\dots = 0.$ This will be the subject of a memoire that I will insert in the third volume of our Mélanges.

Figure 3. Lagrange alluded to a method to reduce the order, but gave few details [12, pp. 30-31].

Lagrange continued by complimenting d'Alembert's methods on the famous three body problem, then asked if d'Alembert would be willing to publish them in the upcoming issue as well (see Figure 4).

 Your method of integration for the three bodies problem is extremely simple and convenient; I will send you mine, which is completely different, as soon as you wish; allow me to assure you in advance, it will not displease. Our society is preparing to publish a new volume: would you like to do it the honor of putting your name on it? This would certainly have a great effect, and it would speed up its acceptance in the community. Send us some of your papers; I will put them in order and have them printed with all due care.

Figure 4. Lagrange asked d'Alembert to publish his studies on the three body problem in Miscellanea Taurinensia [12, p. 31].

D'Alembert responded in March of 1765.  He described (in more detail than Lagrange shared) his own method for reduction of order.  It's clear this is our modern process (see Figure 5).

 I've provided, in the first edition of my Traité de Dynamique, a method for integrating the equation $ddy+Mydz^2+Pdz^2 = 0$ which allows me very simply to integrate the equation $Py+Q \frac{dy}{dx}+R\frac{d^2y}{dx^2} +\dots = X$ when one has $m-1$ values of $y$ in $x$ in the case of $X=0$. So $y=\nu z$, $z$ being one of the values and $\nu$ an undetermined variable; I have $Pz+Q\frac{dz}{dx} + R\frac{d^2 z}{dx^2} + \dots = 0$ and by substitution, it gives me a degree $m$ equation which will only have $\nu$ as an unknown  and no term at all with $V$; so, having $\frac{dv}{dz}=q$ I will have an equation of $m-1$ degree where I will have $m-2$ values of $q$ in the case of $X=0$, since $\nu$ or $\frac{y}{z}$ has $m-2$ known values (hyp.) and as a result, $\frac{dv}{dz}$ or $q$; therefore by continuing in this way, I will arrive at an equation of the form $dr +Z r dz+\zeta dz = 0$ which is evidently integrable.

Figure 5.  D'Alembert responded with his own method for reduction of order [12, pp. 33-34].

He also addressed Lagrange's request for a paper (see Figure 6), and mentioned conflicts with the Academies of Paris and Berlin which made him hesitant to publish with Miscellanea Taurinensia.  This may not be a surprise considering historian of science Thomas Hankins’ statement [8, p. 28] that “In pure contentiousness d'Alembert came in a close second to the Bernoullis; whatever he lacked in viciousness he made up in the dogged determination with which he pursued his pet theories and claims of priority."  The story of these conflicts is interesting on its own, though it is not highly relevant to Lagrange’s role in developing the reduction of order technique.  Please see the Appendix for details on these controversies.

 In regard to your proposal to me my dear and illustrious friend, to insert a piece about my method in the Memoirs, it’s an honor which I appreciate very much and which I very much desire to be able to respond, but since I want to avoid any difficulties with the Academy, to whom I no longer send any memoires for the reasons of which you are already aware, and the same with the Academy of Berlin, where for a long time I have no longer sent anything either. Here’s what I could to do: Write you a long letter in which I would summarily treat different matters and in which (which is more important and dear to me) I would have the opportunity to do you, without it seeming to flatter, the justice you deserve. You could give this writing the title “Extrait de différentes lettres de M. d'Alembert à M. de la Grange.” This would be like a kind of analysis of the main things that I must treat in the fourth volume of my Opuscules. If it suits you, tell me in what time to have it ready. You can count on my word, provided I have the time before me, because I neither want to nor am I able to hurry. My health does not allow me to complete extensive work on the same material.

Figure 6.  D'Alembert proposed a compromise concerning contributing to Miscellanea Taurinensia [12, pp. 34-35].

D'Alembert did not completely reject Lagrange's request to contribute to Miscellanea Taurinensia, but rather proposed a compromise.  He agreed to send some notes to Lagrange, which Lagrange could organize and publish under the title “Extrait de différentes lettres de M. d'Alembert à M. de la Grange.”  In March 1765, Lagrange accepted the offer, referring to d'Alembert's reduction of order technique as being “très-belle” and noted that his method is different [12, p. 37].  In June, d'Alembert asked for a concrete deadline [12, p. 40] and in July, Lagrange said he needed it by the end of the year [12, p. 42].  On September 6, he reminded d'Alembert again, and on September 28, d'Alembert promised the material by the end of December [12, p. 45].  D'Alembert was true to his word and sent the notes with plenty of time to spare ... on December 28.

It appears that the relationship between mathematicians, editors, and deadlines hasn't changed much in 250 years.

Sarah Cummings (Wittenberg University) and Adam E. Parker (Wittenberg University), "D'Alembert, Lagrange, and Reduction of Order - The Correspondence Between D'Alembert and Lagrange," Convergence (September 2015)