Physica D 237 (2008) 1887–1893www.elsevier.com/locate/physd

Euler, the historical perspective

E. Knobloch∗

Technische Universitat Berlin, GermanyBerlin-Brandenburgische Akademie der Wissenschaften, Germany

Available online 18 March 2008

Abstract

This paper is meant to give some interesting details of Euler’s unusual life and extraordinary creativity. A fast-rising scientist, he becameEurope’s teacher of mathematics by his numerous textbooks. Yet, he, too, had to manage the problems of daily academic and private life. He wasa pioneer in solving special cases of the famous three-body problem: the problem of two gravitational centers and the collinear configuration.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Euler

3 Sandifer, 2007: p. 5.4 Euler, 1728.5 Euler, 1727.

1. Introduction

The Irish satirist Jonathan Swift once said1:

Elephants are drawn always smaller than life, but a flea always larger.

Whoever would like to speak about Euler has to solveexactly this problem: How to do justice to this mathematician,“universal, richly detailed and inexhaustible”2? The followingessay is meant to emphasize some less well-known details ofEuler’s unusual life and work, especially his pioneering workin celestial mechanics regarding the three-body problem.

2. The fast-rising scientist

1720 13 years old, Leonhard Euler enrolls at the University ofBasel;

1721 14 years old, he obtains the Bachelor’s degree;1722 still 14 years old, he is for the first time opponent in an

appointment procedure for a professorship (of logic);1722 15 years old, he is for the second time opponent in an

appointment procedure for a professorship (of history oflaw);

∗ Corresponding address: Technische Universitat Berlin, Germany.E-mail address: Eberhard.Knobloch@TU-Berlin.DE.

1 Cf. Fellmann, 2007: p. XIII.2 Simmons, 2007: p. 168.

0167-2789/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2008.03.005

1723 16 years old, he obtains his Master’s degree (A.L.M. =

Artium Liberalium Magister);1726 18 years old, he publishes his first (faulty) paper on

isochronic curves3;1726 19 years old, he submits his paper on ship’s masts, thus

gaining an honourable mention by the French Academyof sciences4;

1727 19 years old, he submits his habilitation thesis withouthaving obtained the Ph.D. degree5;

1727 20 years old, he begins his work in St. Petersburg.

Euler submitted the thesis in order to receive the vacantprofessorship of physics at the University of Basel. Its completetitle reads6:

Q.F.F.Q.S.7 Physical dissertation on sound which Leonhard Euler, Master of

the liberal arts submits to the public examination of the learned in the juridical

lecture-room on February 18, 1727 at 9 o’clock looking at the free professorship

of physics by order of the magnificent and wisest class of philosophers whereby

6 Euler, 1727: p. 181 (“Q.F.F.Q.S. [=Quod felix faustumque sit] Dissertatiophysica De sono, quam annuente numine divino jussu magnifici et sapientissimiphilosophorum ordinis pro vacante professione physica ad d. 18. Febr. A.MDCCXXVII. In Auditorio Juridico hora 9. Publico Eruditorum Examinisubjicit Leonhardus Eulerus A.L.M. Respondente Praestantissimo AdolescenteErnesto Ludovico Burcardo Phil. Cand.”).

7 Quod felix faustumque sit (May it bring you happiness and good fortune).

1888 E. Knobloch / Physica D 237 (2008) 1887–1893

the divine will is nodding assent. The most eminent young man Ernst Ludwig

Burchard, candidate of philosophy, is responding.

But all imploring was in vain: Euler did not get the position.In the appendix he raised the following problem: What wouldhappen if a stone dropped into a straight tunnel drilled to thecenter of the earth and onward to the other side of the planet?According to Euler it reaches infinite velocity at the centerand immediately returns to the same point from which it hadfallen down. Only in his Mechanica did Euler justify this falsesolution saying8:

This seems to differ from truth . . . . However that may be, here we have to

confide more in the calculation than in our judgement and have to confess that

we do not understand at all the jump if it is done from the infinite into the finite.

Euler’s result was the consequence of his mathematicalmodelling of the situation (a non-permitted commutation oflimits). Benjamin Robins put it as follows9:

When y, the distance of the body from the center, is made negative, the terms

of the distance expressed by yn , when n may be any number affirmative,

or negative, whole number or problem are sometimes changed with it. The

centripetal force being as some power of the fraction; if, when y is supposed

negative, yn be still affirmative, the solution gives the velocity of the body in its

subsequent ascent from the center; but if yn by this supposition becomes also

negative, the solution exhibits the velocity, after the body has passed the center,

upon condition, that the centripetal force becomes centrifugal; and when on this

supposition yn becomes impossible, the determination of the velocity beyond

the center is impossible, the condition being so.

Such mistakes are not uncommon in the writings of greatmen. Curiously, Euler never recanted.

3. Euler’s publications and posthumous works

Euler published more than 800 books or papers, mainly inLatin or French, some in German or Russian. His posthumousworks are kept in the archives of the Russian Academyof Sciences in St. Petersburg. The Euler Archives in Baseldispose of microfilms of all preserved Eulerian manuscripts.They are described in a volume published in Moscow andLeningrad.10 The twelve mathematical notebooks covering theperiod 1725 to 1783 are of special interest. They consist of 2300sheets of paper written nearly exclusively in Latin. Russian,German, and English surveys appeared in 1988, 1989, and2007, respectively.11 The notebooks will not be published inthe Opera omnia. Their digitization is planned.

Euler dealt with all aspects of pure and applied mathematicsand likewise with philosophy and theology.12 Differential andintegral calculus; logarithmic, exponential, and trigonometric

8 Euler, 1736: p. 88. (“Hoc quidem veritati minus videtur consentaneum;. . . Quicquid autem sit, hic calculo potius quam nostro iudicio est fidendumatque statuendum, nos saltum, si fit ex infinito in finitum, penitus noncomprehendere.”)

9 Robins, 1739: p. 12.10 Kopelevic, Krutikova, Mikhailov, Raskin, 1962.11 Knobloch, 1988, 1989, 2007.12 Cf. Varadarajan, 2006: p. 2.

functions; ordinary and partial differential equations; ellipticfunctions and integrals; hypergeometric integrals; classicalgeometry (theorem on polyhedra); number theory; algebra;continued fractions; Zeta and other (Euler) products; infiniteseries and products (Basel problem); divergent series;mechanics of particles and solid bodies; calculus of variations;theory and practice of optics; hydrostatics; hydrodynamics;astronomy; lunar and planetary motions; topology; graphtheory (Konigsberg bridge problem); philosophy; theology;shipbuilding; engineering; music theory.

The following enumeration gives a survey of Euler’s mostimportant monographs or textbooks. They are chronologicallyordered according to the date of publication and assignedto Euler’s three stays in St. Petersburg (1727–1741), Berlin(1741–1766), and again in St. Petersburg (1766–1783).

3.1. St. Petersburg (1727–1741)13

• Mechanics or the science of motion set forth analytically,1736 (so-called First Mechanics)

• Introduction to the art of arithmetic for the use of thehigh school at the Imperial Academy of Sciences in St.Petersburg, 1738

• Essay of a new theory of music set forth clearly according tothe most certain principles of harmony, 1739

• Naval science or treatise on the construction and navigationof ships, 1749 (so-called First naval theory, already writtenin 1738 while still in St. Petersburg).

3.2. Berlin (1741–1766)14

• Method of finding curvilinear lines having a property to ahighest or smallest degree or solution of the isoperimetricproblem understood in the largest sense, 1744

• Theory of the motion of planets and comets, 1744• New principles of gunnery, 1745• Introduction into the analysis of the infinite, 1748• Theory of the motion of the moon, setting forth all of

its inequalities, 1753 (so-called First lunar theory; itspublication was paid for by the Russian Academy ofSciences)

• Elements of instruction of the differential calculus togetherwith its application in the analysis of the finite and theoryof series, 1755 (its publication was paid for by the RussianAcademy of Sciences)

• Theory of the motion of solid or rigid bodies stabilizedaccording to the first principles of our cognition andaccommodated to all motions that can fall in such bodies,1765 (so-called Second Mechanics)

• Letters to a German princess on diverse subjects of physicsand philosophy, 1768–1772 (already written in the years1760–1762 while still in Berlin)

• Elements of instruction of the integral calculus, 1768–1770,1794 (already written in 1763 while still in Berlin)

13 Euler, 1736, 1738, 1739, 1749.14 Euler, 1744a,b, 1745, 1748, 1753, 1755a, 1765c, 1768–1772, 1794.

E. Knobloch / Physica D 237 (2008) 1887–1893 1889

3.3. St. Petersburg (1766–1783)15

• Dioptrics, 1769–1771• Complete introduction to algebra, 1770• Theory of the motions of the moon dealt with by a new

method together with astronomical tables, 1772 (so-calledSecond lunar theory)

• Complete theory of the construction and navigation of ships,1773 (so-called Second naval theory).

Through these textbooks Euler became Europe’s teacher notonly in his own time, but also for mathematicians of the 19thcentury.16

4. The troubles of daily life

In spite of all intellectual flights of fancy Euler had tomanage the problems of daily academic and private life. Threeexamples may illustrate this aspect of his activities.

4.1. The quadrature of the circle

In his capacity as director of the mathematical class of theBerlin Academy he had to evaluate mathematical writings andprojects, for example the writing of a certain Thibault fromAvignon about the quadrature of the circle.17 The report datesfrom the 15th of March, 1750 (cf. Fig. 1). It begins by saying18:

After reading the writing of Mr. Thibault where he pretends to have found the

quadrature of the circle, I doubt very much that one has ever seen a paper on

this subject being just as absurd as this one.

The report ends by saying19:

This suffices to demonstrate that the author not only does not have the slightest

notion of the question he is dealing with but that he does not know either

anything about the first elements of geometry.

4.2. The supply of dead bodies

Since the departure of Maupertuis from Berlin Euler washis proxy. He had to inform the administrator David Kohlerof the Academy’s financial affairs to pay the due honorariumto the widow of the grave-digger Schunemann for supplyingthe Anatomy with dead bodies.20 The Anatomical Theatre hadbeen constructed in 1713.

15 Euler, 1769–1771, 1770, 1772, 1773.16 Spieß, 1929: p. 206.17 W. Knobloch, 1984: p. 27, no. 64. Publication of the following citations by

courtesy of the Archives of the Berlin-Brandenburg Academy of Sciences andHumanities.18 “Ayant lu l’ecrit de Mr. Thibault, ou il pretend d’avoir trouve la quadrature

du cercle, je doute fort qu’on ait jamais vu une piece aussi absurde sur ce sujetque celle-cy.”19 “Cela suffit pour faire voir, que l’Auteur n’a non seulement aucune idee de

la question, dont il s’agit, mais qu’il est meme entierement ignorant dans lespremiers elemens de Geometrie.”20 W. Knobloch, 1984: p. 252, no. 1430; p. 270, no. 1553; p. 315, no.

1857–1860.

Fig. 1. Euler’s report on Thibault’s quadrature of the circle dating from March15, 1750; Archives of the Berlin-Brandenburg Academy of Sciences andHumanities I–M 101, sheet 1.

4.3. The plundering of Euler’s estate

During the Seven Years War between England and Prussiaon the one side, Russia, Austria, and France on the other side,Euler’s estate in the village Lietzow (outside of but near toBerlin in those days) was plundered by Saxon troops, alliesof the Russians. The still existing list of damages elaboratedby the mayor of Lietzow enumerates 1 Wispel, 5 Scheffel rye(1 Wispel = 24 Scheffel, 1 Scheffel = 54,73 l), 1 Wispel,6 Scheffel barley/oat, 30 metric hundred-weights, two horses,thirteen cows, seven pigs, twelve sheep (cf. Fig. 2).21

In his letter to the Russian secretary Gerhard FriedrichMuller in St. Petersburg Euler spoke of four horses thusdoubling the damage.22

5. Euler’s work in celestial mechanics

When Euler published his Mechanica in 1736, it waspreceded by the copperplate engraving presented in Fig. 3.

The head of the celestial deity carries the sun revolved bythe six planets that can be seen with the naked eye. The planetUranus discovered only in 1781 is still absent. In the righthand the deity holds an opened book. The figures representthe whole or partial elliptic orbit of a planet. The message isclear: Celestial mechanics is a part of mechanics wherein theNewtonian law of gravitation plays the crucial role.

21 Brandenburgisches Landeshauptarchiv, Kurmarkische Kriegs- undDomanenkammer, Stadte-Registratur: De anno 1760, Nr. S 3498; reproductionby courtesy of these archives.22 Fellmann, 2007: p. 101f.

1890 E. Knobloch / Physica D 237 (2008) 1887–1893

23 Kirsten, 1977: p. 9 (“Die wahre Theorie der Astronomie bestehetaber hauptsachlich in einer grundlichen Erkenntnuß der sogenanntenNewtonianischen Philosophie, als welche nicht nur alle schon erkannten MotusCoelestes sehr herrlich erklaret, sondern auch Anlaß gibt in der Astronomie jelanger je mehr Entdeckungen zu machen, und die wahren Bewegungen allerHimmlischen Corper genauer zu erkennen. Durch diese Wissenschaft wirdein Astronom in Stand gesetzt, nicht nur alle seine Observationen auf einengewissen Endzweck zu dirigiren, sondern daraus auch allen moglichen Nutzenzu ziehen.”)24 Wilson, 2007.25 Kopelevic/Krutikova/Mikhailov/Raskin 1962–1965, vol. 1: no. 401, fol.

76v; no. 402, fol. 49r–50r, 78v, 93v.26 Euler, 1765b: p. 281.27 Euler, 1764b, 1765b.28 Wilson, 1994: p. 1054; cf. Subbotin, 1958; Volk, 1983.29 Euler, 1760, 1764a, 1765a.30 This is explained in Euler, 1760.

Fig. 2. List of damages regarding the village Lietzow. It was elaborated bythe mayor of Lietzow dating from October 24, 1760. Lietzow was plunderedby Saxon troops commanded by a Russian general. The fourth line of the listenumerates “Prof. Euler’s” damages. Brandenburgisches LandeshauptarchivPotsdam, Rep. 2 Kurmarkische Kriegs- und Domanenkammer Nr. S 3498.

Fig. 3. Copperplate in Euler’s Mechanics (published in 1736) at the beginningof the dedication to Johann Albert Korff (Euler 1736: p. 5).

The message on the right part of the copperplate is not soevident. Euler himself does not give any explanation. Yet, a boyobviously throws rings into a water basin by means of a slingthat he holds in his left hand. He observes the curved line therings are describing sinking to the ground of the water basin.This might be an illustration of a motion in a resisting mediumas dealt with in the second volume of Euler’s Mechanica.

When Euler came to Berlin in 1741, he at once elaborateda corresponding research program for astronomy at the Berlin

Academy of Sciences. He defined the true theory of astronomyas follows23:

The true theory of astronomy mainly consists of a thorough understanding of

the so-called Newtonian philosophy which does not only explain all already

known celestial motions but it also shows the reason why one makes more and

more new discoveries in the long run and recognizes more precisely the true

motions of all celestial bodies. By this science an astronomer can relate all of

his observations to a final aim and derive all kinds of profit from them.

Euler’s own contributions to celestial mechanics can bedivided into three groups: 1. Planetary perturbations, 2. Lunartheories, 3. Three-body problem. The first two subjects havebeen dealt with by Curtis Wilson very recently.24 Hence thissection will confine itself to the third subject.

First trials to solve the three-body problem where the bodiesare moving in the same plane are to be found in Euler’smathematical notebooks dating from 1750 to 1755.25 In hispublications he emphasized the importance of the problemsaying that we have to solve the famous three-body problem inorder to arrive at the culmination of astronomy.26 The solutionturned out to be extremely difficult. Yet, Euler did not questionthe solvability. He only stated that first we have to study specialcases, to introduce certain restrictions before we can hope tosolve the general problem.27 Euler was indeed “the first toinvestigate restricted forms of the three-body problem with aview to obtaining exact integrals”.28 He considered two specialcases: the problem of two gravitational centers and the collinearconfiguration.

5.1. The problem of two gravitational centers

It can be explained in the following way: Two fixed bodiesA, B of masses a, b act on a third body Z according to theNewtonian law of gravitation. What will be the curve describedby Z?

Euler dealt with it in three papers.29 In the first two of themhe presupposed that the curve described by Z lies in the sameplane as the two centers of gravitation. In the third paper hedropped this restriction (cf. Fig. 4).

His method of solving the problem consisted of four steps30:

E. Knobloch / Physica D 237 (2008) 1887–1893 1891

34 Euler to Lagrange, November 9, 1762, in Euler, Opera omnia, ser. 4A,5, p. 450 (“Je suis extremement ravi, Monsieur, que mes recherches sur lemouvement d’un corps attire a deux centres de forces fixes aient merite votre

Fig. 4. The configuration of the problem of two gravitational centers as dealtwith by Euler in Euler, 1765a: p. 248. A, B are the gravitational centers, themoving body Z describes a curve that does not lie in the same plane as the twocenters.

1. Find the general differential equations of the second orderwhich determine the motion of the body.

2. Integrate these equations in order to obtain differentialequations of the first order.

3. Apply separation of variables to these equations in order toconstruct the solution.

4. Determine the cases where the described curve becomesalgebraic.

Eventually, he was able to deduce an equation with twoelliptic integrals with separated variables, and recognizedthe advantage of introducing the sum and difference of thedistances v, u of the centers A, B from Z as new variables. IfZ moves in a plane where A, B are to be found, the curve is anellipse or a hyperbola. Dropping this condition Euler discussedthe case where the curve lies on a hyperbolic conoid or on anelliptic spheroid.

He elaborated the first paper31 in 1759, in my opinionpresumably because he was stimulated by Clairaut’s paperpublished exactly in this year.32 Therein Clairaut abandoned theplan of finding the complete solution of the three-body problemin favour of approximative solutions:

Now integrate who will be able to do it! I have found the six equations which

I have just found since the first times that I have considered the three-body

problem. But I made only few efforts to solve them because they always seemed

to me to be hardly manageable.

In 1762, Euler, too, was inclined to set aside exact integralsand worked out an iteration method based on series expansionspraising its simplicity, practicality, and generality.33

On the 9th of November of the same year, Euler wrote toLagrange about his relative researches on the subject:

I am utmost delighted, Sir, that my investigations on the motion of a body

attracted by two fixed centers of force have deserved your attention. But you

have only seen what has been inserted into the Memoirs of Berlin and what

31 Euler, 1764a.32 Clairaut, 1759: p. 566. (“Integre maintenant qui pourra ! J’ai trouve les six

equations que je viens de trouver des les premiers temps que j’ai envisage leprobleme des trois corps, mais je n’ai jamais fait que peu d’efforts pour lesresoudre, parce qu’elles m’ont toujours paru peu traitables.”)33 Euler, 1763a.

Fig. 5. The configuration of the problem of two gravitational centers as dealtwith by Lagrange in Lagrange, 1766–1769a: p. 73.

mainly regards the algebraic curves included in my solution. Yet, I have written

still two other memoirs on that subject. One of them is to be found in the 10th

volume of our Commentaries and the other in the 11th volume.

Only in 1767 did Lagrange come back to this problem, whenall three Eulerian papers had already been printed.34 In his ownpaper35 Lagrange at once considered the generalized case dealtwith in Euler’s third paper and used v + u, v − u as variables(cf. Fig. 5). Apparently in order to avoid unpleasant suspicions,he claimed that he had written his paper before he knew Euler’sthird paper. The reader will be able to judge whose method wasmore direct or simpler.36

5.2. The collinear configuration

The three bodies A, B, C with masses a, b, c remain on astraight line that turns uniformly around itself.

Euler investigated this collinear case in four papers.37 Itpresents the first particular solution of the three-body problem.Either the two distances between A, B and B, C remain con-stant. Then they can be determined thanks to the quintic equa-tion

1 − 2x + x2− mx2

− x3+ 2x4

− x5= 0 (1)

with m =n2c3

d3 = constant, c mean distance of the ‘moon’ fromthe ‘earth’, n:1 ratio of the mean motion of the ‘moon’ to themean motion of the ‘sun’, d mean distance between ‘sun’ and‘earth’.

Or the ratio n of the distances p, q between the threebodies remains constant. On the understanding that an arbitraryangular velocity is given, the mutual distances are periodicalfunctions of time and can be determined thanks to the quintic

attention; mais vous n’en avez vu que ce qui a ete insere dans les Memoires deBerlin et qui regarde principalement les courbes algebriques que ma solutionrenferme. Or j’en ai compose encore deux autres memoires, dont l’un se trouvedans le Xe Volume de nos Commentaires et l’autre dans le XIe.”); Lagrange toEuler, October 29, 1767, in Euler, Opera omnia, ser. 4A, 5, p. 460.35 Lagrange, 1766–1769a, 1766–1769b.36 Lagrange, 1766–1769a: p. 94.37 Euler, 1764b, 1765b, 1763b, 1785.

1892 E. Knobloch / Physica D 237 (2008) 1887–1893

equation

(a + b)n5+ (3a + 2b)n4

+ (3a + b)n3

− (b + 3c)n2− (2b + 3c)n − b − c = 0. (2)

Euler derived Eq. (1) in his first paper, Eq. (2) in all threesubsequent papers.

When in 1771 Lagrange submitted his famous prize ‘Essayon the three-body problem’,38 he did not employ a new method(as he claimed), using only the distances between the threebodies in order to determine the orbits. He applied Euler’smethod to a more general case than Euler had considered.

He investigated the two cases that the distances remainconstant or that they maintain a constant ratio. Both conditionscan only be fulfilled again in two cases: the bodies movealong the same straight line (collinear case) or they form anequilateral triangle (triangular solution). No wonder that hederived again Euler’s quintic equation for a constant ratio inthe collinear case.

Nowadays we know that the triangular configuration isapproximately realized in the solar system by the sun, Jupiter,and the Trojan group of the asteroids Achilles, Patrocles,Hector, and Nestor.

One might say that Euler paved the way, Lagrange gatheredthe fruits. The three-body problem demonstrates how heinitiated new inquiries. Other fields of knowledge coulddemonstrate how he invented new methods (Zeta-function),defended new ideas (divergent series), developed new theories(theory of music). In his eyes mathematical problems weresolvable. If necessary they have to be formulated in such a waythat they become solvable. Or to put it as Eduard Fueter in 1941:“For where mathematical reason did not suffice, for Euler beganthe kingdom of God.”39

6. Conclusion

Fueter’s affirmation is especially true, too, of Euler’sepochal contributions to hydromechanics that were compre-hensively described by Truesdell in 1954.40 In 1983, GlebMikhailov41 analysed the complicated relationship betweenDaniel Bernoulli’s, John Bernoulli’s, and Euler’s achievementsin this respect. Euler praised John Bernoulli’s Hydraulicaprinted in 1742 (it appeared only in 1743) because thereinBernoulli had calculated the force acting on an infinitesimal el-ement. This essential idea helped Euler to create his generaltheory of fluids. Euler completed and perfected classical hy-dromechanics. His Scientia navalis begins with the fundamen-tal lemma that the pressure which the water exerts upon a sub-merged body in its several points is normal to the surface ofthe body. A long series of papers followed wherein Euler rein-troduced internal pressure as a means to derive the motion of

38 Lagrange, 1772.39 Fellmann, 2007: p. 172.40 Truesdell, 1954.41 Mikhailov, 1983.

fluid elements. This series culminated in the three French writ-ten treatises forming the core of Euler’s general theory of fluids.They appeared in 1757. The second treatise Principes generauxdu mouvement des fluides (General principles of the motion offluids)42 first introduced the famous Euler equations of fluidmotion. The 250th anniversary of its publication gave rise tothe conference that took place in Aussois. The history of thisdevelopment is thoroughly analysed and reconstructed by Dar-rigol and Frisch.43

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