Abstract
There is as yet no complete history of potential theory covering the 18th century in detail. There are two major works which deal with it — Bacharach and Sologub — and three minor ones — Cohen Paraira, Becker and Brenneke1. Each of these fails: Bacharach is the best but neglects Euler and much pre-1770; Sologub mentions Euler but not anyone else pre-1770; Cohen-Paraira copied too much and analyzed too little besides being incomplete; Becker was a Gaussophile and pedantic but did recognize Euler and Bernoulli; and Brenneke was an Eulerian, with no mention of anyone else. The three minor papers are priority battles, with disastrous consequences for history; the two major sources concentrate on events post-1820 and the 18th century is put to one side.
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Notes
Cf. Max Bacharach. Abriss der Geschichte der Potentialtheorie, Thein’sche Druckerei (Stürtz), Würzburg 1883 it was his Inaugural-Dissertation; G.F. Becker, “Potential” a Bernoullian term, American Journal of Science (3) 45, 1893, p.97–100; Rudolf Brenneke, Die Verdienste Leonhard Eulers um den Potentialbegriff, Zeitschrift für Physik 25, 1924. p.42–45; Mozes Cohen Paraira, Over de methoden ter bepaling van de aantrekking eener ellipsoide op en willekeurig punt, Gebroeders Binger, Amsterdam 1879 Akademisch proefschrift; V. S. Sologub, Razvitie teorii elliptičeskikh uravnenii v XVIII i XIX stoletijakh, Naukova Dumka, Kiev 1975
Speaking about the Bernoulli family is difficult: to avoid repeating Jacob I, Johann I and Daniel, I use the phrase “the Bernoullis” as a generic term, generally meaning the two brothers to 1730 and the latter two thereafter.
Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem, secundum quam volunt a Deo eandem semper quantitatem motus conservari, qua et in re mechanica abutuntur, in: Acta eruditorum, 1686, = p. 117–119 of Mathematische Schriften, VI, Die mathematischen Abhandlungen, originally edited by C. I. Gerhardt in 1860 (with an introduction on p. 3–16) and reprinted by Georg Olms, Hildesheim, New York, 1971. There is a Beilage to the paper on p.119–123, and the text of his long manuscript text on dynamics on p.281–514. Cf. also vols. V and VII, and note 15 below.
Cf. István Szabó, Geschichte der mechanisehen Prinzipien und ihrer wiehtigsten Anwendungen, Birkhäuser, Basel, Stuttgart 1977, 2.Auflage 1979 — a very useful history of the five topics it covers (cf. Truesdell’s review in Centaurus, vol. 23, 1980, p.163–175).
Cf. Der Briefwechsel von Johann Bernoulli, Bd. I, Birkhäuser, Basel 1955, Otto Spiess (ed.), p.323, footnote 3, and almost the whole of his Opera omnia, vol. III, and De vera notione virium vivarum, in: Acta eruditorum, 1735, p. 210 = Opera Omnia, vol. IV. The letter from l’Hôpital indicates that while Leibniz’s paper was known in Paris its message had not been accepted. The influence of Newton on Paris was as yet minimal, which will have severe implications for Clairaut.
Cf. the articles Nos. LXXV and XCIII on p.768–778 and 874–887 in Jacob Bernoulli, Opera, tomus I, originaly published by Cramer & Philibert, Geneva 1744, and reprinted in Bruxelles: Culture et Civilisation, 1967. The Bernoullis’ ideas are discussed in John L. Greenberg’s pair of excellent papers on Fontaine quoted in note 13; the reference to the Bernoullis is Greenberg (1981), p. 259. The best source on these matters is Steven B. Engelsman, Families of Curves and the Origins of Partial Differentiation, Meppel 1982 (Proefschrift).
Jacob Hermann, Phoronomia, sive de viribus et motibus cor porum solidorum et fluidorum libri duo, Amstelodami (Amsterdam) 1716. The terminology of this book differs from that of Leibniz substantially, leading to a confusion between vis and potentia, between the meanings “energy” and “force” for both of them, and this Hermann tradition through Euler leads to potentia in the meaning “force” passing into the French school through Lagrange.
For Newton and his influence on Euler, cf. Truesdell, Essays in the History of Mechanics, Springer, Berlin etc., 1968; for the Bernoullis, see the articles by Truesdell in the Leonhardi Euleri Opera Omnia quoted in note 14, the articles by G. Eneström in Bibliotheca mathematica, series 2 and 3, 1897–1906, and Otto Spiess’s Leonhard Euler. Ein Beitrag zur Geistesgeschichte des XVIII. Jahrhunderts, Frauenfeld, Leipzig 1929.
To save space I have quoted Euler’s works according to the enumeration in Gustaf Enestrom’s Verzeichnis der Schriften Leonhard Eulers (BV Enestrom, 1910–1913). Usually I add a short title as an aide-memoire. The Mechanica was originally published in 1736.
Daniel Bernoulli. Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Argentorati (Strasbourg) 1738.
The corresponding statements in Newton are in the Philosophiae naluralis principia mathematica in the first six definitiones.
The liveliest biography of Maupertuis, checked by several of his contemporaries including Euler, is by Laurent Angliviel de La Beaumelle, Vie de Maupertuis, suivi de lettres inédites de Frédéric le Grand et de Maupertuis, avec des Notes et un Appendice, Paris 1856. The description of Maupertuis’ influence is given on p. 32–33 and probably stems from an eyewitness account by La Condamine who formed part of the group. The manuscript of the biography was completed before 1769.
John L. Greenberg, Alexis Fontaine’s “Fluxio-differential method” and the origins of the calculus of several variables, Annals of Science 38, 1981, p. 251–290, followed by his Alexis Fontaine’s integration of ordinary differential equations and the origins of the calculus of several variables, Annals of Science 39, 1982, p. 1–36.
We have mentioned the main work in note 10 but full descriptions are given in the following works: Clifford Truesdell, O. II, 11, Sectio secunda: The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788. Introduction to O. II, 10, 11, Zürich 1960. We also quote from: Leonard Euler, O. II, 12: Commentationes mechanicae ad theoriam cor porum fluidorum pertinentes edidit Clifford Ambrose Truesdell. Volumen prius Editor’s introduction entitled: Rational fluid mechanics, 1687–1765, Zürich 1964, and from Leonhard Euler, O. II, 13: Commentationes mechanicae ad theoriam cor porum fluidorum pertinentes edidit Clifford Ambrose Truesdell III, Volumen posterius Editor’s introduction; I. The first three sections of Euler’s treatise on fluid mechanics 1766; II. The theory of aerial sound, 1687–1788; III. Rational fluid mechanics, 1765–1788, Zürich 1965.
Cf. De infinitis curvis..., E.44, 45, where the papers of Leibniz and Hermann are quoted in the O. I, 22, p.36–75. Cf. also S. Engelsman, Historia Mathematica, vol. 8, 1981, p. 71.
Cf. Greenberg (1981), p.255–263 (note 13).
A bibliography of Clairaut is given by René Taton, Inventaire chronologique de l’Œuvre d’Alexis-Claude Clairaut (1713–1765), Rev. Hist. Sci. 29, 1976, p. 97–122; and Supplement a I’«Inventaire de l’Œuvre de Clairaut» (I), ibidem 31, 1978, p.269–271. From this list, quoted as T, cf. T1–T8 for Clairaut’s early work on geometry. In addition, for the work of Laplace, we shall quote from Gillispie’s useful list, given in Charles C. Gillispie, Laplace, Pierre-Simon, Marquis de, in: Dictionary of Scientific Biography, vol. XV, Charles Scribner’s Sons, New York 1978, p.387–403, prefixing the dates and numbers by the letter G.
Cf. his Institutiones calculi differentialis (E. 212/O. I, 10), §§ 208–232.
Of the type Pdx + Q dy, but in Euler P and Q are functions of a single variable (x for P and y for Q) and y is a function of x, while in Clairaut P and Q are functions of both x and y where x and y are independent.
Cf. Greenberg (1981), p.283, and Paul Stäckel’s Introduction to the Mechanica in O. II, 1, p. X.
Cf. Greenberg (1981), p.253.
Cf. T 25, p. 425–426, and Greenberg (note 13), passim.
Cf. T 28, p. 293–294, 322, as well as T 28, p.425–426.
Cf. notes 22 and 23 and Greenberg (1982), p.20–34.
Cf. Thomas L. Hankins: Jean d’Alembert: Science and the Enlightenment, Clarendon Press, Oxford 1970, p.30ff.
For the controversy about the principle of least action see J. O. Fleckenstein’s Introduction to O. II, 5; for the Euler-Bernoulli correspondence cf. P. H. Fuss. Correspondance mathématique et physique..., tomes I and II, St. Pétersbourg 1843. reprinted by Johnson Reprint Corp., New York, London 1968 (BV Fuss, P.H., 1843), and G. Enestrom, in: Bibliotheca mathematica, (3),4–8, 1903–1908.
Cf. his Opera omnia, vol. IV. published in 1743, and also Truesdell. note 14. O. II. 11 Sectio 2. Cf. also notes 10, 14 and 28.
The translation of these works given in Hunter Rouse (ed.), Hydrodynamics by Daniel Bernoulli & Hydraulics by Johann Bernoulli. translated by.... Dover. New York 1968, is unfortunate in that it does not comment on this major aspect of the terminology, and Becker (note 1) also fails to comment.
Cf. Jean-le-Rand d’Alembert. Recherches sur la courbe que forme une corde ten due. mise en vibration, Histoire de l’Academie royale des Sciences et Belles-Lettres de Berlin (1747) 1749, p. 214–219, and Suite des..., ibidem. p. 220–249.
O.IV A, 1. Letters 1503 (10.12.1745), 1506 (24.5.1746), and 1532 (8.5.1748) to Maupertuis, and the letters 387 (30.10.1740) and 388 (26.12.1740) between Euler and Clairaut. They are printed in O.IV A, 5.
Cf. Fuss, note 26, p.457 for the French phrase, p.506 for the Latin phrase (20.10.1742); cf. also p. 524 (23.4.1743) and 533 (4.9.1743).
Cf. Becker (note 1), p.98–99.
Cf. E. 65/O. I, 24, p. 231–232, 298. Cf. also note 31.
Cf. notes 29 and 33, Hankins, op. cit. (note 25, p. 46–52), and Truesdell, O.II, 112 and 12.
Cf. Fleckenstein, O.II, 5.
See Hankins, op. cit., p.32–37, and C. B. Waff, Alexis Clairaut and his proposed modification of Newton’s inverse square law of gravitation, in: Avant, avec, aprés Copernic, Paris: Blanchard, 1975, p.281–288.
Cf. Truesdell, O. II, 12, p. L-LVIII, and Hankins, op. cit., p.48–51.
Cf. Ivor Grattan-Guiness, The Development of the Foundations of Mathematical Analysisfrom Euler to Riemann, Cambridge (Mass.): The MIT Press, 1970; Adolphe P. Youschkevitch, The concept offunction up to the middle of the 19th century, Arch. Hist. Exact Sci. 16 (1976–1977), p.37–85.
Cf. note 4.
Neue Grundsätze der Artillerie, 1745 (E.77/O.II, 14).
Scientia navalis, parts I and II, 1749 (E. 110, 11110.11, 18, 19), edited by Clifford Truesdell.
Cf. notes 29 and 34, as well as his Essai d’une nouvelle théorie de la resistance desfluides, Paris: David l’ainé, 1752, reprinted in Bruxelles: Culture et Civilisation, 1967, especially §§ 42–47, 57–62, and Figures 13, 16 and 17.
Cf. notes 29 and 42.
These are heavily quoted in Brenneke (note 1) along with E.289, 375, 396, 409, and 424 which fall outside our period.
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Cross, J. (1983). Euler’s Contributions to Potential Theory 1730–1755. In: Leonhard Euler 1707–1783. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9350-3_16
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