Leonhard Euler

Quick Info

Born

Died

18 September 1783
St Petersburg, Russia

Summary

Leonhard Euler was a Swiss mathematician who made enormous contibutions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory.

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Biography

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attendedJacob Bernoulli's lectures there. In fact Paul Euler andJohann Bernoulli had both lived inJacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister. Their son Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have mentioned, some mathematical training and he was able to teach his son elementary mathematics along with other subjects.

Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in1720, at the age of14, first to obtain a general education before going on to more advanced studies.Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings, see[1], is as follows:-

... I soon found an opportunity to be introduced to a famous professorJohann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

In1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas ofDescartes andNewton. He began his study of theology in the autumn of1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler obtained his father's consent to change to mathematics afterJohann Bernoulli had used his persuasion. The fact that Euler's father had been a friend ofJohann Bernoulli's in their undergraduate days undoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in1726. He had studied many mathematical works during his time in Basel, and Calinger[24] has reconstructed many of the works that Euler read with the advice ofJohann Bernoulli. They include works byVarignon,Descartes,Newton,Galileo,van Schooten,Jacob Bernoulli,Hermann,Taylor andWallis. By1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium. In1727 he published another article on reciprocal trajectories and submitted an entry for the1727 Grand Prize of theParis Academy on the best arrangement of masts on a ship.

The Prize of1727 went toBouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate. However, Euler now had to find himself an academic appointment and whenNicolaus(II) Bernoulli died in St Petersburg in July1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology. He accepted the post in November1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair. Almost certainly his youth(he was19 at the time) was against him. However Calinger[24] suggests:-

This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work.

As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on5 April1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on17 May1727. He had joined theSt Petersburg Academy of Sciences two years after it had been founded by Catherine I the wife of Peter the Great. Through the requests ofDaniel Bernoulli andJakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. At St Petersburg Euler had many colleagues who would provide an exceptional environment for him[1]:-

Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometerJakob Hermann, a relative;Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholarChristian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.

Euler served as a medical lieutenant in the Russian navy from1727 to1730. In St Petersburg he lived withDaniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland. Euler became professor of physics at the Academy in1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.

Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in1733 it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on7 January1734, marrying Katharina Gsell, the daughter of a painter from the St PetersburgGymnasium. Katharina, like Euler, was from a Swiss family. They had13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.

We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career. This is done in[24] as follows:-

... after1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place:number theory; infinitary analysis including its emerging branches,differential equations and thecalculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, andspecial functions and differential equations were essential to rational mechanics, which supplied concrete problems.

The publication of many articles and his bookMechanica(1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work.

Euler's health problems began in1735 when he had a severe fever and almost lost his life. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in1738 with overstrain due to his cartographic work and that by1740 he had[24]:-

... lost an eye and[the other] currently may be in the same danger.

However, Calinger in[24] argues that Euler's eyesight problems almost certainly started earlier and that the severe fever of1735 was a symptom of the eyestrain. He also argues that a portrait of Euler from1753 suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain.

By1740 Euler had a very high reputation, having won the Grand Prize of theParis Academy in1738 and1740. On both occasions he shared the first prize with others. Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil in Russia made the position of foreigners particularly difficult and contributed to Euler changing his mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where anAcademy of Science was planned to replace the Society of Sciences. He left St Petersburg on19 June1741, arriving in Berlin on25 July. In a letter to a friend Euler wrote:-

I can do just what I wish[in my research] ... The king calls me his professor, and I think I am the happiest man in the world.

Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for theSt Petersburg Academy, he continued to write scientific reports for them, and he educated young Russians.

Maupertuis was the president of theBerlin Academy when it was founded in1744 with Euler as director of mathematics. He deputised forMaupertuis in his absence and the two became great friends. Euler undertook an unbelievable amount of work for the Academy[1]:-

... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal.

During the twenty-five years spent in Berlin, Euler wrote around380 articles. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics(extending the book byRobins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publicationLetters to a Princess of Germany(3 vols.,1768-72).

In1759Maupertuis died and Euler assumed the leadership of theBerlin Academy, although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued withd'Alembert on scientific matters, was disturbed when Frederick offeredd'Alembert the presidency of the Academy in1763. Howeverd'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.

In1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind. Because of his remarkable memory he was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg(when Euler was59) he produced almost half his total works despite the total blindness.

Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at theAcademy in St Petersburg in1766(becoming its secretary in1769) and Christoph Euler who had a military career. Euler was also helped by two other members of the Academy, W L Krafft and A JLexell, and the young mathematicianN Fuss who was invited to the Academy from Switzerland in1772.Fuss, who was Euler's grandson-in-law, became his assistant in1776.Yushkevich writes in[1]:-

.. the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples.

For example Euler credits Albrecht, Krafft andLexell for their help with his775 page work on the motion of the moon, published in1772.Fuss helped Euler prepare over250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in1776.

He also wrote a eulogy of Euler, which you can see atTHIS LINK.

Yushkevich describes the day of Euler's death in[1]:-

On18 September1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed withLexell andFuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening.

After his death in1783 theSt Petersburg Academy continued to publish Euler's unpublished work for nearly50 more years.

Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords asPtolemy had done.

He made decisive and formative contributions to geometry, calculus and number theory. He integratedLeibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introducedbeta andgamma functions, andintegrating factors for differential equations. He studied continuum mechanics, lunar theory withClairaut, thethree body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in hisTheory of the Motions of Rigid Bodies(1765).

We owe to Euler the notation

f (x) f (x)

for a function(1734),

e e

for the base of natural logs(1727),

i i

for the square root of -1(1777),

π \pi

for pi,

\sum \sum

for summation(1755), the notation for finite differences

Δ y \Delta  y

and

Δ^{2} y \Delta ^{2} y

and many others.

Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated byGoldbach but probably originally came from the interest that the Bernoullis had in that topic.Goldbach asked Euler, in1729, if he knew ofFermat's conjecture that the numbers

2^{n} + 1 2^{n} + 1

were alwaysprime if

n n

is a power of2. Euler verified this for

n n

=1,2,4,8 and16 and, by1732 at the latest, showed that the next case

2^{32} + 1 = 4294967297 2^{32} + 1 = 4294967297

is divisible by641 and so is not prime. Euler also studied other unproved results ofFermat and in so doing introduced the Euler phi function

ϕ (n) \phi(n)

, the number of integers

k k

with

1 \leq k \leq n 1 ≤ k ≤ n

and

k k

coprime to

n n

. He proved another ofFermat's assertions, namely that if

a a

and

b b

are coprime then

a^{2} + b^{2} a^{2} + b^{2}

has no divisor of the form

4 n - 1 4n - 1

, in1749.

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

ζ (2) = \sum \frac{1}{n^{2}} \zeta(2) = \sum  \Large\frac 1 {n^{2}}

, a problem which had defeated many of the top mathematicians includingJacob Bernoulli,Johann Bernoulli andDaniel Bernoulli. The problem had also been studied unsuccessfully byLeibniz,Stirling,de Moivre and others. Euler showed in1735 that

ζ (2) = \frac{1}{6} π^{2} \zeta(2) = \large\frac{1}{6}\normalsize \pi^{2}

but he went on to prove much more, namely that

ζ (4) = \frac{1}{90} π^{4}, ζ (6) = \frac{1}{945} π^{6}, ζ (8) = \frac{1}{9450} π^{8}, ζ (10) = \frac{1}{93555} π^{10} \zeta(4) = \large\frac{1}{90}\normalsize \pi^{4}, \zeta(6) = \large\frac{1}{945}\normalsize \pi^{6}, \zeta(8) = \large\frac{1}{9450}\normalsize \pi^{8}, \zeta(10) = \large\frac{1}{93555}\normalsize \pi^{10}

and

ζ (12) = \frac{691}{638512875} π^{12} \zeta(12) = \large\frac{691}{638512875}\normalsize \pi^{12}

. In1737 he proved the connection of thezeta function with the series of prime numbers giving the famous relation

ζ (s) = \sum \frac{1}{n^{s}} = \prod \frac{1}{1 - p^{- s}} \zeta(s) = \sum  \Large\frac 1 {n^{s}}\normalsize = \prod \Large\frac 1 {1- p^{-s}}

Here the sum is over all natural numbers

n n

while the product is over all prime numbers.

By1739 Euler had found therational coefficients

C C

ζ (2 n) = C π^{2 n} \zeta(2n) = C\pi^{2n}

in terms of theBernoulli numbers.

Other work done by Euler on infinite series included the introduction of his famous Euler's constant γ, in1735, which he showed to be the limit of

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + . . . + \frac{1}{n} - \log_{e} n \large\frac{1}{1}\normalsize  + \large\frac{1}{2}\normalsize  + \large\frac{1}{3}\normalsize  + ... + \large\frac{1}{n}\normalsize  - \log_{e}n

n n

tends to infinity. He calculated the constant γ to16 decimal places. Euler also studiedFourier series and in1744 he was the first to express an algebraic function by such a series when he gave the result

\frac{1}{2} π - \frac{1}{2} x = \sin x + \frac{1}{2} (\sin 2 x) + \frac{1}{3} (\sin 3 x) + . . . \large\frac{1}{2}\normalsize \pi - \large\frac{1}{2}\normalsize x = \sin x + \large\frac{1}{2}\normalsize (\sin 2x) + \large\frac{1}{3}\normalsize (\sin 3x) + ...

in a letter toGoldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until1755.

Euler wrote toJames Stirling on8 June1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote[60]:-

Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.

He then goes on to describe what is now called the Euler-Maclaurin summation formula. Two years laterStirling replied telling Euler thatMaclaurin:-

... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.

Euler replied:-

... I have very little desire for anything to be detracted from the fame of the celebrated MrMaclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.

Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof ofFermat's Last Theorem for the case of

n = 3 n = 3

. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form

a + b √ - 3 a + b√-3

for integers

a a

and

b b

. Although there were problems with his approach this eventually led toKummer's major work onFermats Last Theorem and to the introduction of the concept of aring.

One could claim that mathematical analysis began with Euler. In1748 inIntroductio in analysin infinitorum Euler made ideas ofJohann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula

e^{i x} = \cos x + i \sin x e^{ix} = \cos x + i \sin x

InIntroductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula

\ln (- 1) = π i \ln(-1) = \pi i

in1727. He published his full theory of logarithms of complex numbers in1751.

Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography. He discovered theCauchy-Riemann equations in1777, althoughd'Alembert had discovered them in1752 while investigating hydrodynamics.

In1755 Euler publishedInstitutiones calculi differentialis which begins with a study of the calculus of finite differences. The work makes a thorough investigation of how differentiation behaves under substitutions.

InInstitutiones calculi integralis(1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in1729.Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function byBinet andGauss respectively. As well as investigating double integrals, Euler consideredordinary andpartial differential equations in this work.

The calculus of variations is another area in which Euler made fundamental discoveries. His workMethodus inveniendi lineas curvasⓉ... published in1740 began the proper study of the calculus of variations. In[12] it is noted thatCarathéodory considered this as:-

... one of the most beautiful mathematical works ever written.

Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to theBessel equation which he solved by introducingBessel functions.

Euler made substantial contributions todifferential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by Euler in this area were rediscovered byGauss. Other geometric investigations led him to fundamental ideas intopology such as the Euler characteristic of a polyhedron.

In1736 Euler publishedMechanica which provided a major advance in mechanics. AsYushkevich writes in[1]:-

The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.

InMechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium. He analysed the motion of a point mass under a central force and also considered the motion of a point mass on a surface. In this latter topic he had to solve various problems of differential geometry and geodesics.

Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science. It is described in[24] as:-

Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first established the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.

Of course hydrostatics had been studied sinceArchimedes, but Euler gave a definitive version.

In1765 Euler published another major work on mechanicsTheoria motus corporum solidorumⓉ in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He considered the Euler angles and studied rotational problems which were motivated by the problem of theprecession of the equinoxes.

Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of work through the1750s setting up the main formulae for the topic, the continuity equation, theLaplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid. In1752 he wrote:-

However sublime are the researches on fluids which we owe to Messrs Bernoulli,Clairaut andd'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ...

Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill. He did important work in astronomy including[1]:-

... determination of the orbits of comets and planets by a few observations, methods of calculation of theparallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the ParisAcadémie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time.

In fact Euler's lunar theory was used byTobias Mayer in constructing his tables of the moon. In1765Mayer's widow received £3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received £300 from the British government for his theoretical contribution to the work.

Euler also published on the theory of music, in particular he publishedTentamen novae theoriae musicaeⓉ in1739 in which he tried to make music:-

... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.

However, according to[8] the work was:-

... for musicians too advanced in its mathematics and for mathematicians too musical.

Cartography was another area that Euler became involved in when he was appointed director of theSt Petersburg Academy's geography section in1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. TheRussian Atlas was the result of this collaboration and it appeared in1745, consisting of20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography.

Quotations by Leonhard Euler
Other Mathematicians born in Switzerland
A Poster of Leonhard Euler

References(show)

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Written byJ J O'Connor and E F Robertson
Last Update September 1998

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