Petite histoire de la force de Coriolis
En 1835, Gustave Coriolis dériva de ses équations les forces qui agissent sur les corps en déplacement dans un système en rotation. Son travail trouva l’inspiration dans l’étude des machines tournantes – nous étions alors en pleine révolution industrielle. Pourtant, le premier système en rotation qui s’impose à nous naturellement, et de la même façon à travers les époques, est bien la Terre elle-même. Il n’est donc pas étonnant que la question de l’influence de la rotation de la Terre sur les mouvements des corps ait été abordée par d’autres scientifiques, bien avant Coriolis. Nous proposons ici un récapitulatif de ces travaux précurseurs. Les premières idées concernant l’influence de la rotation de la Terre sur les mouvements
Link to the Journal website
In this online article, the readers will find useful bibliographical references on the subject that complete the publication in Reflets de la Physique.
Historical sources on the ’Coriolis’ force
Much of the early literature can be downloaded from the website of the Bibliothèque Nationale de France. For the convenience of the reader, we have gathered the available PDFs here.
Eastward deflection of falling bodies
The first detailed study on a manifestation of the "Coriolis" force was made by Giovanni Borelli in the 1660s, when he considered the problem of falling bodies on a rotating Earth. In a theoretical analysis, he found that they will undergo a small eastward deflection during their fall.
Much of the source material was reproduced by Koyré:
This work was later translated in French and published as a book:
The sources reproduced in Koyré’s 1955 paper were used by Burstyn, and extended with more material on the debate between Hooke and Newton:
Borelli’s reasoning led to the correct direction of the deflection (i.e. eastward) as well as to the right order of magnitude, but the reasoning itself is inconsistent at some points. Burstyn attempted to disentangle the contradictions, but his interpretation of Borelli’s underlying assumptions is not always convincing. For more on this, see:
T.Gerkema (2009): On Borelli’s analysis concerning the deflection of falling bodies, (Unpublished note) |
Early experiments to test the idea of eastward deflection were inconclusive, for the effect is very small and easily dwarfed by other effects, such as wind (in open-air experiments, e.g. when dropping balls from a tower), or by a minute horizontal velocity accidentally imparted to the object at the start of its fall.
Only at the beginning of the 19th century were experiments done in a sufficiently careful manner to detect the deflection. For example,
F. Reich (1832): Fallversuche über die Umdrehung der Erde, Engelhardt, Freiberg. |
Benzenberg’s book also contains a detailed overview of earlier attempts and studies, and discusses the contemporary theoretical progress by Gauss and Laplace, who, independently, derived the expression for the eastward deflection of a falling object. Gauss’ work is cited in Benzenberg, and Laplace’s was published as
P.S. Laplace (1803): Mémoire sur le mouvement d’un corps qui tombe d’une grande hauteur. Bull. Soc. Philomatique, 3, 109-115. |
This paper was reproduced in his Oeuvres Complètes, Tome 14, from which the following pdf is taken: laplace-1803.pdf
An extensive historical overview on the subject can be found in the first volume of the 3-part series:
The appendix by Stein offers a summary of the thesis by Kamerlingh Onnes, dealing with theory and experiments on Foucault’s pendulum and related devices.
Deflection of projectiles
One special case of movement, akin to the previous one, is that of an object thrown vertically upward. During the upward motion, the deflection will be to the west; returning downward, it will be to the east. It was argued by Poisson that the two do not cancel: the upward motion starts purely vertically, but when the body reaches its highest position, it has acquired a westward velocity, and so the eastward deflection during its fall will not be as large as it would be when falling from rest. As a result, the object lands to the west of the position from which it was launched. (In fact, this had already been established by Laplace in 1805, see below.) This argument is presented on page 7 of the following document :
S.D. Poisson (1838): Sur le mouvement des projectiles dans l’air, en ayant égard à la rotation de la Terre, Journal de l’École Polytechnique, XXVI Cahier, Tome XVI, 1-68. poisson-1838.pdf |
A summary of the paper was published a year earlier:
S.D. Poisson (1837): Extrait de la première partie d’un Mémoire sur le Mouvement des projectiles dans l’air, en ayant égard à leur rotation et à l’influence du mouvement diurne de la Terre, C.R. Acad. Sci. Paris, t. 5, 660-667. |
Poisson treats the general problem of projectiles fired in an arbitrary direction, and the deflections they undergo as a result of the Earth’s rotation; he also included effects of friction.
Poisson refers to Laplace’s 1803 paper (see above), and asserts that Laplace ignored friction. This is correct so far as the 1803 paper is concerned, but Laplace later added friction in his more general treatise of the problem in the Mécanique Céleste:
P.S. Laplace (An XIII = 1805), Traité de Mécanique Céleste, t. IV, Courcier, Paris. |
Here Laplace also derived the equations, with the deflecting force included, for a body launched in an arbitrary direction, but now without friction (p. 303ff). Indeed, in his equations presented at the top of p. 304, the modern-day reader will easily recognize the four terms representing the deflecting "Coriolis" force, two of them being proportional to the cosine of latitude, the other two, to the sine. (NB: the angle theta denotes co-latitude.)
In particular, Laplace showed that a body launched vertically upward would land slightly westward.
That problem has an amusing history. In the 17th century, speculation on the fate of vertically launched objects elicited a lively debate, and attempts were made to settle the issue experimentally: balls were shot upward. Amazingly, many balls were never found again ! (see Benzenberg, p. 260-261 ). This didn’t surprise Descartes : "... [une expérience] que je voudrais que quelques curieux, qui en pourraient avoir la commodité, entreprissent de faire exactement, avec une grosse pièce de canon pointée tout droit vers le zénith, au milieu de quelque plaine. Car l’auteur [Jean Leurechon] dit que cela a déjà été expérimenté plusieurs fois, sans que la balle soit retombée en terre; ce qui peut sembler fort incroyable à plusieurs, mais je ne le juge pas impossible, et je crois que c’est une chose très digne d’être examinée." (Lettre à Mersenne, avril 1634; Oeuvres et Lettres, Pléiade, Gallimard, p.952). Mersenne then did the experiment, but the balls disappeared once again – a result applauded by Descartes: "Je vous remercie aussi de celle de la balle tirée vers le zénith, qui ne retombe point, ce qui est fort admirable." (Lettre à Mersenne, mars 1636, op. cit, p. 957). Elsewhere he argues why the balls never did return: "... on doit juger que la force du coup les portent fort haut, les eloigne si fort du centre de la Terre, que cela fait perdre leur pesanteur" (letter cited by Benzenberg, p. 261) – a principle with which a modern-day physicist would find it hard to disagree !
Laplace’s tidal theory and the "Traditional Approximation"
A full account of the effects of Earth’s rotation was offered by Laplace in his dynamical theory of tides, which found its definitive form in the Mécanique Céleste. Adopting a geographical coordinate system, he finds four terms describing a deflecting force, proportional and perpendicular to the velocity of the moving object. This derivation is presented in sections 35 and 36 of Livre I of
P.S. Laplace (An VII = 1798), Traité de Mécanique Céleste, t. I, Crapelet, Paris. |
He also argues that two of these four terms can be considered negligible. (The argument is not entirely clear – and, it would seem, not bulletproof, either – but a central element of his reasoning is the notion that the oceans form a thin layer compared to the Earth’s radius.) With this, he introduced what is now known as the "Traditional Approximation", i.e. neglecting the two Coriolis terms that are proportional to the cosine of latitude.
The Mécanique Céleste doesn’t make easy reading, but a detailed and clarifying summary of Laplace’s tidal theory was written by Edm. Dubois, in a now almost forgotten work:
Edm. Dubois (1885): Résumé analytique de la théorie des marées telle qu’elle est établie dans la Mécanique Céleste de Laplace, Revue Maritime et Coloniale, 85, 116-154 | |
Edm. Dubois (1885): Résumé analytique de la théorie des marées telle qu’elle est établie dans la Mécanique Céleste de Laplace, Revue Maritime et Coloniale, 85, 419-460. | |
Edm. Dubois (1885): Résumé analytique de la théorie des marées telle qu’elle est établie dans la Mécanique Céleste de Laplace, Revue Maritime et Coloniale, 85, 687-710. |
The derivation of the four deflecting terms, as well as the introduction of the "Traditional Approximation", are dealt with in the first part.
Eötvös effect
One of the two terms neglected under the "Traditional Approximation" is responsible for the eastward deflection of falling objects.
The other term represents un upward force on eastward moving objects, reducing their weight – the so-called Eötvös effect. This phenomenon had been observed in the early 20th century on research ships on which gravity was measured; they noticed that measurements of g yielded smaller values when the ship went eastward, and larger ones when they went westward. These observations are mentioned by Eötvös in a paper in which he provides the explanation of the effect:
R. Eötvös (1919): Experimenteller Nachweis der Schwereänderung, die ein auf normal geformter Erdoberfläche in östlicher oder westlicker Richtung bewegter Körper durch diese Bewegung erleidet, Annalen der Physik, 364 (16), 743-752. |
In the paper, he also describes a device with which the effect can be demonstrated experimentally.
Foucault’s pendulum
Foucault’s famous experiment is reported in
L. Foucault (1851): Démonstration physique du mouvement de rotation de la terre au moyen du pendule, C.R. Acad. Sci. Paris, t. 32, 135-138. |
At the end of the paper, he offers a qualitative explanation of his experiment in terms of Poisson’s work on the deflection of projectiles.
When hearing of the experiments, the director of the Museum for Natural History in Florence remembered having seen something similar in old notes from members of the Accademia del Cimento, an institution devoted to experimental physics in the 1660s. They didn’t realize that its cause could lie in the Earth’s rotation, and later fixed their pendulum with two ropes, in order to avoid this undesired rotation ! Antinori relates the story in
V. Antinori (1851): Anciennes observations faites par les Membres de l’Académie del Cimento sur la marche du pendule, C.R. Acad. Sci. Paris, t. 32, 635-636. |
Still in the same year, Binet offered a theoretical explanation, inspired by the treatment of the effects of Earth’s rotation by Laplace in his fourth volume of the Mécanique Céleste and Poisson’s 1837 paper, to which he refers in the second part, on p. 197:
J. Binet (1851): Note sur le mouvement du pendule simple en ayant égard à l’influence de la rotation diurne de la terre, C.R. Acad. Sci. Paris, t. 32, 157-160. |
J. Binet (1851): Suite de la note sur le mouvement du pendule simple en ayant égard à la révolution diurne de la terre, C.R. Acad. Sci. Paris, t. 32, 197-205. |
A full treatment of the problem was given only decades later, by Kamerlingh Onnes in his doctoral thesis:
A summary of this work was provided by Stein (see above).
G. Coriolis
His paper on the force centrifuge composée, now known as the Coriolis force, was published in 1835:
G. Coriolis (1835): Sur les équations du mouvement relatif des systèmes de corps, Journal de l’École Polytechnique, XXIV Cahier, Tome XV, 142-154. |
This paper contains an abstract analysis of the forces acting in a rotating system. It was inspired by a desire to understand the mechanics of rotating machines, waterwheels (roues hydrauliques) in particular, as is clear from an earlier paper:
G. Coriolis (1832): Sur le principe des forces vives dans les mouvements relatifs des machines, Journal de l’École Polytechnique, XXI Cahier, Tome XIII, 142-154. |
Notes on Coriolis’ first names
Coriolis signed his papers as "G. Coriolis", giving no clue as to what the "G" actually stands for...
Ever since, his first name has been clouded by confusion. Dictionaries, encyclopedias and other references offer conflicting information. There is an avenue Gustave Coriolis in Toulouse, but Le petit Larousse (2002) calls him Gaspard, while Le petit Robert (2003) combines both names: Gustave Gaspard. Rare visitors to his grave at Montparnasse will find a degraded stone from which his name can be deciphered as Gaspard Gustave.
In an early biographical sketch he is called "Gustave-Gaspard de Coriolis" or, for short, "Gustave de Coriolis":
N.A. Renard (1861): Notice historique sur la vie et les travaux de Gustave de Coriolis. Mémoires de l’Académie de Stanislas (Nancy), xii-xxxix. |
Renard also mentions that Coriolis himself refrained from using the aristocratic "de" (p. xxxvi); he signed his papers modestly as "G. Coriolis". Renard has corresponded with Coriolis’ sister, so he can be considered a reliable source when he implies that Coriolis was called "Gustave" in daily life. But was he correct in stating his full official first names as "Gustave-Gaspard"?
In a later biographical sketch by Lapparent, published in 1895 (and reprinted in the 1990 Gabay edition of some of Coriolis’ papers) he is called "Gaspard-Gustave Coriolis":
A. de Lapparent (1895): Coriolis (1792-1843), in: Livre du centenaire de l’École Polytechnique 1794-1894, Gauthier-Villars. |
As a matter of fact, Lapparent was correct about the official first names, for Coriolis’ death certificate states:
ACTE DE DÉCÈS: l’an mil huit-cent quarante-trois, le dix-neuf septembre, est décédé, à Paris, rue Descartes, 21, douzième arrondissement, Gaspard Gustave Coriolis, directeur des études à l’école Polytechnique, agé de cinquante-un ans, trois mois, né à Paris, célibataire. coriolis-acte-deces.jpg
(NB: rue Descartes is now in the 5th arrondissement, but before 1860 it was in the 12th.)
In summary, three things seem now certain: 1)his common first name was "Gustave"; 2)he dropped the "de"; 3)his official first names, in full, were "Gaspard Gustave".
Remarks on Coriolis’ influence
It is perhaps tempting to assume that Coriolis must have influenced later authors simply because they published their work after 1835 – a way of reasoning that belongs to the category of fallacies known as "post hoc ergo propter hoc".
A case in point is Poisson’s 1838 paper, of which some claim that it was "directly influenced" by Coriolis. However, the evidence points to the contrary: in Poisson’s paper, the name Coriolis is never mentioned nor is there any trace of Coriolis’ work. Poisson instead refers to Laplace and Gauss when he discusses the eastward deflection of falling bodies. His actual derivation was apparently made independently of earlier work.
The association of the deflecting force with "Coriolis" – so natural for us today – seems to have become common only in the late 19th century. For example, Kamerlingh Onnes, in his thesis on Foucault’s pendulum, published in 1879, refers to the deflecting force as "de nevenkrachten van Coriolis".
However, in earlier periods the connection was not made, most probably because Coriolis’ paper had passed unnoticed. For example, in the important papers by Foucault and Binet, both from 1851 (see above), the name "Coriolis", or his work, features nowhere.
In meteorology, the deflecting force was, in its full form, introduced by Ferrel in the 1850s. He, too, draws his inspiration from Laplace’s tidal theory, and there is no awareness of Coriolis’ work. It is only half a century later, in Brillouin’s collection of papers, that we encounter Coriolis’ name in the meteorological context:
In this book, Brillouin collected articles from Hadley to Helmholtz on atmospheric circulation, including those by Ferrel. He translated them into french and added useful corrections and remarks. When Ferrel discusses the two ’traditional’ deflecting forces, borrowed from Laplace’s tidal theory, Brillouin adds in a note that these "sont les deux composantes les plus importantes de l’accélération centrifuge composée de Coriolis" (p. 23).
In the end, it is natural that Laplace was much more influential than Coriolis. Firstly, because Laplace’s work concerned the Earth’s rotation, which, ever since Borelli, was the primary object of study in effects of deflection (the references mentioned above testify to that). By contrast, Coriolis’ work was more abstract, and the applications he envisaged were rooted rather in the industrial revolution, with rotating devices like waterwheels. It is a wholly different context. Secondly, Laplace’s derivation of the deflecting force preceded that by Coriolis by four decades. Historically, it would be more correct to speak of the "Laplace force". But, then, Laplace has already got so many things named after him, that it should be considered a matter of fairness to speak of the "Coriolis force" !
Text and bibliographic research : T Gerkema
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