# Who did first use the Method of Characteristics?

Which mathematician did introduce the Method of Characteristics for solving linear Partial differential equations?

I some papers I saw that Saint Venant, Lagrange, Cauchy or Riemann were attributed with this method. But who was the first mathematician who introduced the method, which is taught in universities?

[EDIT: check the (*) for a potential earlier reference in Lagrange] The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt (?-1784). According to I. Grattan-Guinness et S. Engelsman, The manuscripts of Paul Charpit, Historia Mathematica, vol. 9, 1982, p. 65-75:

He presented his paper on partial differential equations to the Académie on June 30, 1784, but he died in Paris on December 28 of that year.

The memoir was lost, then found again. Lagrange learnt about it in 1793 only (* see below for a comment). There is a series of papers on that topic by N. Saltykow, such as Méthodes classiques d'intégration des équations aux dérivées partielles du premier ordre à une fonction inconnue, 1931. Page 7:

Charpit a eu, d'abord, la chance de formuler, le premier, les équations différentielles ordinaires des caractéristiques, que l'on attribue fréquemment à Lagrange

with the poor translation:

Charpit was lucky enough to be the first to express the ordinary differential equations of characteristics, which are often attributed to Lagrange.

Full proof (in a modern sense) was apparently given by Jacobi.

(*) For a comment by Gerald, on page 198 of Mathematik für Physiker (Band 2: Gewöhnliche und partielle Differentialgleichungen, mathematische Grundlagen der Quantenmechanik), 2014 by Helmut Fischer and Helmut Kaul, it is said that:

Die Charakteristikenmethode geht auf Lagrange zuruck. Nachdem er 1772 die nichtlineare DG $$\frac{∂u}{∂y} = f(x, y, u, \frac{∂u}{∂x})$$ auf eine quasilineare DG zurückgeführt und 1774 eine Theorie für die allgemeine Lösung aufgestellt hatte, gab er 1779 Differentialgleichungen für die Charakteristiken quasilinearer Probleme an. Eine geometrische Begründung der Charakteristikenmethode fand 1784 Gaspard Monge. Pfaff 1815 und Cauchy 1819 erweiterten diese für $$n > 2$$.

• +1: Thank you for your efforts :) Jun 18 '16 at 21:18
• Fun to do. Nothing interesting on TV. Jun 18 '16 at 21:26
• According to a german textbook by Fischer and Kaul doi.org/10.1007/978-3-658-00477-4 (see p198) it was first used by Lagrange in 1779. Nov 5 '20 at 9:46
• I have added a mention from your reference, thank you. There seems to be a debate on whether Lagrange did it first Nov 5 '20 at 13:47

As a graphical method for the integration of hyperbolic PDEs the method of characteristics was proposed by the Belgian engineer Massau in the late 19th century.
https://archive.org/stream/mmoiresurlintgr05massgoog#page/n4/mode/2up

• I don’t know what this could be good for but still worth knowing something like this exists. Feb 4 '18 at 8:09

Sophus Lie's 1895 paper "Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung [General Theory of Partial Differential Equations of Arbitrary Order]" begins with a historical summary in which he seems to think Monge was the first (p. 4 of the English translation):

…theory of characteristics [Theorie der Charakteristiken] developed by Monge, Laplace, Ampére and Darboux…

and

…the notion of characteristics [Begriff Charakteristik] introduced by Monge…

• Is there any other source supporting this claim? Feb 4 '18 at 8:07

It is worth noting that Lagrange was born in 1736 (died in 1813), Cauchy in 1789, St. Venant in 1797, and Riemann in 1826.

So between those four, it would "have" to be Lagrange, who published his work on this subject in 1786, before the others were born.

It is also worth noting that the method of characteristics is tied to the Euler equations, and that Euler's lifespan was 1707-1783 (he died before Lagrange published his relevant work in 1786).

• Thank you but I need a proper source that is stating who first proved/used the modern method of characteristics Jun 18 '16 at 12:35