Leonhard Euler, a German princess and topology O18

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Leonhard Euler, a German princess and topology
P. Pieranski
1
Laboratoire de Physique des Solids, Université Paris-Sud, 91405, Orsay, France
“Lisez Euler, lisez Euler, c’est notre maître à tous »
Pierre-Simon Laplace
The 36 Topical Meeting on Liquid Crystals in Magdeburg gives me a marvellous pretext to
share with you my admiration to Leonhard Euler, a scientific genius from the Age of Enlightenment.
Indeed, “Lettres à une princesse d’Allemagne”, the most popular of Euler’s writings,
have been addressed to the princess Sophie Friederika Charlotte Leopoldine von Anhalt-
Dessau who lived here. To be more precise, Friedrich II, who in 1741 invited Euler to join the
Berlin Academy, asked him to be a scientific tutor of his niece. Euler accomplished this task
by writing, from Berlin, between 1760 and 1762, 234 letters in French to the princess Sophie
who lived in Magdeburg.
Figure 1: Leonhard Euler and the title’s page of his “Lettres à une princesse d’Allemagne”.
We don’t know whether the princess red and liked Euler’s letters or not. Fortunately, they
have been published in Saint Petersburg in 1768 as a book (in three volumes) (see fig.1)
which immediately became a best-seller and ran through many editions [1] and translations.
Even today, it can be read with pleasure and a lot of benefit not only by princesses.
For scientists and scholars, it is a marvelous example of fertile scientific thought process and
clear expression. It is so interesting to see, from today’s perspective, how Euler tackled problems
such as :
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1. concepts of space, time or reference frame,
2. the nature of sound, its speed or its production by vibrating cords and membranes,
3. the nature of light and its speed, laws of reflection and refraction and their use in optical
instruments such as eye, telescopes or microscopes,
4. the Newton’s law of the universal gravitation and its application to the motions of planets
and comets as well as for explanation of tides
5. laws of the fluid’s flows and their application for sailing of ships,
6. the nature of magnetism and electricity
7. etc
The lecture of “Lettres ...“ encouraged me to look for other Euler’s writings and a striking
surprise was waiting for me. Following William Dunham [2], “Euler was a veritable Niagara,
one who wrote mathematics faster then most people can absorb it“. Indeed, Eulers was
the most prolific mathematician and natural scientist of all time. He wrote about 900 papers
and books (about 800 pages per year). Publication of his collected works entitled “Opera Omnia“
started in 1911 and is still going on. It counts about 80 volumes divided into four series :
Series I Opera mathematica (In 29 volumes; 30 volume-parts)
Series II Opera mechanica et astronomica (In 31 volumes; 32 volume-parts)
Series III Opera physica, Miscellanea (In 12 volumes)
Series IV A. Commercium epistolicum (10 volumes)
Beside “Opera Omnia“ that can be found in some academic libraries, original versions of Euler’s
papers are available as pdf files on Web pages of the Euler Archive
(http://www.math.dartmouth.edu/~euler/). The index of this archive counts 866 entries.
For the purpose of my talk here in Magdeburg, I looked for Euler’s papers dealing with topological
and geometrical properties of surfaces that we use today, for example, for description
of bicontinuous lyotropic phases. For instance, the definition of principal curvatures is due to
Euler as it shown in Figure 2.
Figure 2 : Euler’s expression for the radius of curvature of surfaces. Quotation from “Recherches
sur la courbure des surfaces“ [3].
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It is also interesting (and touching) to see (but harder to read) the paper, written in Latin, in
which Euler writes for the first time his famous formula relating the number of vortices (numerus
angulorum solidorum), the number of edges (numerus acierum) and the number of faces
(numerus hedradum) of polyhedra (see Figure 3).
Figure 3: Quotation from the original Euler’s paper on topological properties of polyhedra
[4].
In the Euler’s work we fin another source of topology - the celebrated problem of “Seven
Kônigsberg’s bridges“ (see Figure. 4) - as well as the first definition of this new branch of
mathematics :
“Besides that part of geometry which deals with quantities and has always been studiously
cultivated, there is another branch which is still virtually unknown; this was first mentioned
by Leibniz, who called it the “geometry of position” [geometric situs]. It is concerned, he
says, just with the determination of position and the investigation of its properties, and for
these goals no quantities are to be considered nor any calculations with quantities used.
However it has not yet been clearly defined what problems belong to this geometry of position
and what method must be used to solve them. Hence, when a certain problem was recently
mentioned to me which seemed to belong to geometry but did not ask for the determination of
any quantities and did not admit a solution by the calculation of quantities either, I did not
doubt that this belongs to the geometry of position – all the more since in its solution only
positions are considered and no use is made of calculations. I propose therefore to explain
here the method which I invented for the solution of this kind of problems, as a specimen of
the geometry of position.”
It is also striking to learn that Euler who was born in … Switzerland, in Basel, worked almost
all his life in Russia (Saint Petersburg) and Prussia (Berlin) and wrote almost all of his papers
in Latin and in French. Since that time, things changed a little…
How pleasant is to read in the Letter XLI these words: “Maintenant, je me vois en état
d’expliquer à Votre Altesse (La Princesse Sophie) de quelle manière se fait la vision dans les
yeux des hommes et de tous les animaux, ce qui est sans doute la chose la plus merveilleuse à
laquelle l’esprit humain a pu pénétrer.” !
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Figure 4 : The original drawing from the Euler’s paper on the problem of “Seven Königsberg’s
bridges”. [5].
Acknowledgments
I would to thank François Rothen for illuminating remarks about Euler, Bernoulli’s family
and the beauty of pure and applied mathematics.
References
[1] Leonhard Euler, “Lettres à une princesse d’Allemagne”, Presses Polytechniques et Universitaires
Romandes, Lausanne 2003 (the last edition in French)
[2] William Durham, “The Master of Us All”, The Mathematical Association of America
1999
[3] Leonhard Euler, “Recherches sur la courbure des surfaces“, originally published in
Memoires de l'academie des sciences de Berlin 16, 1767, pp. 119-143, Opera Omnia:
Series 1, Volume 28, pp. 1 – 22
[4] Leonhard Euler, “Elementa doctrinae solidorum“, originally published in Novi
Commentarii Academiae Scientiarum Petropolitanae 4, 1758, pp. 109-140, Opera
Omnia: Series 1, Volume 26, pp. 71 – 93
[5] Leonhard Euler, “Solutio problematis ad Geometriam Situs pertinentis“, originally
published in Commentarii Academiae Scientiarum Petropolitanae 8, 1741, p 1736,
Opera Omnia: Series 1, Volume 7, p. 1
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