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LEONHARD EULER

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QUICK INFO

Born 15 April 1707
Basel, Switzerland Died 18 September 1783
St Petersburg, Russia

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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.

View seventeen larger pictures

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BIOGRAPHY

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the
University of Basel and had attended Jacob Bernoulli's lectures there. In fact
Paul Euler and Johann Bernoulli had both lived in Jacob 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 in
1720, at the age of 14, 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account given in his unpublished autobiographical writings, see [1], is as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 professor Johann
> 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 ...

In 1723 Euler completed his Master's degree in philosophy having compared and
contrasted the philosophical ideas of Descartes and Newton. He began his study
of theology in the autumn of 1723, 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 after Johann Bernoulli
had used his persuasion. The fact that Euler's father had been a friend of
Johann Bernoulli's in their undergraduate days undoubtedly made the task of
persuasion much easier.

Euler completed his studies at the University of Basel in 1726. 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 of Johann
Bernoulli. They include works by Varignon, Descartes, Newton, Galileo, van
Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already
a paper in print, a short article on isochronous curves in a resisting medium.
In 1727 he published another article on reciprocal trajectories and submitted an
entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of
masts on a ship.

The Prize of 1727 went to Bouguer, 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 when Nicolaus(II) Bernoulli died in St Petersburg in July 1726 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
November 1726 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 was 19 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 on 5 April 1727. He travelled down the Rhine by boat, crossed the German
states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17
May 1727. He had joined the St Petersburg Academy of Sciences two years after it
had been founded by Catherine I the wife of Peter the Great. Through the
requests of Daniel Bernoulli and Jakob 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, geometer Jakob 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
> scholar Christian 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 from 1727 to 1730. In
St Petersburg he lived with Daniel 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 in 1730 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 in 1733 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 on 7 January 1734, marrying
Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium.
Katharina, like Euler, was from a Swiss family. They had 13 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:-

> ... after 1730 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 the
> calculus of variations; and rational mechanics. He viewed these three fields
> as intimately interconnected. Studies of number theory were vital to the
> foundations of calculus, and special functions and differential equations were
> essential to rational mechanics, which supplied concrete problems.

The publication of many articles and his book Mechanica (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 in 1735 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 in 1738 with overstrain due
to his cartographic work and that by 1740 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 of 1735 was a symptom of the
eyestrain. He also argues that a portrait of Euler from 1753 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.

By 1740 Euler had a very high reputation, having won the Grand Prize of the
Paris Academy in 1738 and 1740. 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 an Academy of Science was planned to
replace the Society of Sciences. He left St Petersburg on 19 June 1741, arriving
in Berlin on 25 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 the St Petersburg
Academy, he continued to write scientific reports for them, and he educated
young Russians.

Maupertuis was the president of the Berlin Academy when it was founded in 1744
with Euler as director of mathematics. He deputised for Maupertuis 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 in 1749 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 around 380 articles.
He wrote books on the calculus of variations; on the calculation of planetary
orbits; on artillery and ballistics (extending the book by Robins); on analysis;
on shipbuilding and navigation; on the motion of the moon; lectures on the
differential calculus; and a popular scientific publication Letters to a
Princess of Germany (3 vols., 1768-72).

In 1759 Maupertuis died and Euler assumed the leadership of the Berlin 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 with d'Alembert on scientific matters, was disturbed when
Frederick offered d'Alembert the presidency of the Academy in 1763. However
d'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.

In 1766 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. In 1771 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 in 1771, 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 was 59) 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 the Academy in St Petersburg in 1766 (becoming its secretary in 1769)
and Christoph Euler who had a military career. Euler was also helped by two
other members of the Academy, W L Krafft and A J Lexell, and the young
mathematician N Fuss who was invited to the Academy from Switzerland in 1772.
Fuss, who was Euler's grandson-in-law, became his assistant in 1776. 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 and Lexell for their help with his
775 page work on the motion of the moon, published in 1772. Fuss helped Euler
prepare over 250 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 in 1776.

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

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

> On 18 September 1783 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 with Lexell and
> Fuss 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 in 1783 the St Petersburg Academy continued to publish Euler's
unpublished work for nearly 50 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 as Ptolemy had done.

He made decisive and formative contributions to geometry, calculus and number
theory. He integrated Leibniz's differential calculus and Newton's method of
fluxions into mathematical analysis. He introduced beta and gamma functions, and
integrating factors for differential equations. He studied continuum mechanics,
lunar theory with Clairaut, the three body problem, elasticity, acoustics, the
wave theory of light, hydraulics, and music. He laid the foundation of
analytical mechanics, especially in his Theory of the Motions of Rigid Bodies
(1765).

We owe to Euler the notation f(x)f (x)f(x) for a function (1734), eee for the
base of natural logs (1727), iii for the square root of -1 (1777), π\piπ for pi,
∑\sum∑ for summation (1755), the notation for finite differences Δy\Delta yΔy
and Δ2y\Delta ^{2} yΔ2y 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 by Goldbach but probably originally
came from the interest that the Bernoullis had in that topic. Goldbach asked
Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n+12^{n} +
12n+1 were always prime if nnn is a power of 2. Euler verified this for nnn = 1,
2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case
232+1=42949672972^{32} + 1 = 4294967297232+1=4294967297 is divisible by 641 and
so is not prime. Euler also studied other unproved results of Fermat and in so
doing introduced the Euler phi function ϕ(n)\phi(n)ϕ(n), the number of integers
kkk with 1≤k≤n1 ≤ k ≤ n1≤k≤n and kkk coprime to nnn. He proved another of
Fermat's assertions, namely that if aaa and bbb are coprime then a2+b2a^{2} +
b^{2}a2+b2 has no divisor of the form 4n−14n - 14n−1, in 1749.

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)=∑1n2\zeta(2) = \sum
\Large\frac 1 {n^{2}}ζ(2)=∑n21 , a problem which had defeated many of the top
mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli.
The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre
and others. Euler showed in 1735 that ζ(2)=16π2\zeta(2) =
\large\frac{1}{6}\normalsize \pi^{2}ζ(2)=61 π2 but he went on to prove much
more, namely that ζ(4)=190π4,ζ(6)=1945π6,ζ(8)=19450π8,ζ(10)=193555π10\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}ζ(4)=901 π4,ζ(6)=9451 π6,ζ(8)=94501
π8,ζ(10)=935551 π10 and ζ(12)=691638512875π12\zeta(12) =
\large\frac{691}{638512875}\normalsize \pi^{12}ζ(12)=638512875691 π12. In 1737
he proved the connection of the zeta function with the series of prime numbers
giving the famous relation
ζ(s)=∑1ns=∏11−p−s\zeta(s) = \sum \Large\frac 1 {n^{s}}\normalsize = \prod
\Large\frac 1 {1- p^{-s}}ζ(s)=∑ns1 =∏1−p−s1
Here the sum is over all natural numbers nnn while the product is over all prime
numbers.

By 1739 Euler had found the rational coefficients CCC in ζ(2n)=Cπ2n\zeta(2n) =
C\pi^{2n}ζ(2n)=Cπ2n in terms of the Bernoulli numbers.

Other work done by Euler on infinite series included the introduction of his
famous Euler's constant γ, in 1735, which he showed to be the limit of
11+12+13+...+1n−log⁡en\large\frac{1}{1}\normalsize +
\large\frac{1}{2}\normalsize + \large\frac{1}{3}\normalsize + ... +
\large\frac{1}{n}\normalsize - \log_{e}n11 +21 +31 +...+n1 −loge n
as nnn tends to infinity. He calculated the constant γ to 16 decimal places.
Euler also studied Fourier series and in 1744 he was the first to express an
algebraic function by such a series when he gave the result
12π−12x=sin⁡x+12(sin⁡2x)+13(sin⁡3x)+...\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) + ...21 π−21 x=sinx+21 (sin2x)+31
(sin3x)+...
in a letter to Goldbach. Like most of Euler's work there was a fair time delay
before the results were published; this result was not published until 1755.

Euler wrote to James Stirling on 8 June 1736 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 later Stirling replied telling Euler that Maclaurin:-

> ... 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 Mr Maclaurin 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 of Fermat's Last
Theorem for the case of n=3n = 3n=3. Perhaps more significant than the result
here was the fact that he introduced a proof involving numbers of the form
a+b√−3a + b√-3a+b√−3 for integers aaa and bbb. Although there were problems with
his approach this eventually led to Kummer's major work on Fermats Last Theorem
and to the introduction of the concept of a ring.

One could claim that mathematical analysis began with Euler. In 1748 in
Introductio in analysin infinitorum Euler made ideas of Johann 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
eix=cos⁡x+isin⁡xe^{ix} = \cos x + i \sin xeix=cosx+isinx.
In Introductio in analysin infinitorum Euler dealt with logarithms of a variable
taking only positive values although he had discovered the formula
ln⁡(−1)=πi\ln(-1) = \pi iln(−1)=πi
in 1727. He published his full theory of logarithms of complex numbers in 1751.

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 the Cauchy-Riemann equations in 1777, although
d'Alembert had discovered them in 1752 while investigating hydrodynamics.

In 1755 Euler published Institutiones 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.

In Institutiones 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 in 1729. Legendre called these 'Eulerian integrals of the first and second
kind' respectively while they were given the names beta function and gamma
function by Binet and Gauss respectively. As well as investigating double
integrals, Euler considered ordinary and partial differential equations in this
work.

The calculus of variations is another area in which Euler made fundamental
discoveries. His work Methodus inveniendi lineas curvas Ⓣ(A method for
curves)... published in 1740 began the proper study of the calculus of
variations. In [12] it is noted that Carathéodory 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 the Bessel equation which he solved by
introducing Bessel functions.

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

In 1736 Euler published Mechanica which provided a major advance in mechanics.
As Yushkevich 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.

In Mechanica 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 since Archimedes, but Euler gave a
definitive version.

In 1765 Euler published another major work on mechanics Theoria motus corporum
solidorum Ⓣ(Theory of the motion of solid bodies) 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 the precession of the equinoxes.

Euler's work on fluid mechanics is also quite remarkable. He published a number
of major pieces of work through the 1750s setting up the main formulae for the
topic, the continuity equation, the Laplace velocity potential equation, and the
Euler equations for the motion of an inviscid incompressible fluid. In 1752 he
wrote:-

> However sublime are the researches on fluids which we owe to Messrs Bernoulli,
> Clairaut and d'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 the parallax 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 Paris Académie des Sciences, are
> concerned with celestial mechanics, which especially attracted scientists at
> that time.

In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables
of the moon. In 1765 Mayer'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 published Tentamen
novae theoriae musicae Ⓣ(A new musical theory) in 1739 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 the St Petersburg Academy's geography section in 1735. He had the
specific task of helping Delisle prepare a map of the whole of the Russian
Empire. The Russian Atlas was the result of this collaboration and it appeared
in 1745, consisting of 20 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.


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

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REFERENCES (SHOW)

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     York 1970-1990). See THIS LINK.
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     Mathematiker (Berlin, 1983).
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 7.  Leonhard Euler 1707-1783 : Beiträge zu Leben und Werk (Basel-Boston, 1983).
 8.  G du Pasquier, Leonhard Euler et ses amis (Paris, 1927).
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 51. N P Khomenko and T M Vyvrot, Euler and Kirchhoff - initiators of the main
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 54. E Knobloch, Eulers früheste Studie zum Dreikörperproblem, Amphora (Basel,
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ADDITIONAL RESOURCES (SHOW)

Other pages about Leonhard Euler:

 1.  Eulogy by Condorcet
 2.  Eulogy by Nicolas Fuss
 3.  Königsberg bridges
 4.  Carl B Boyer's Foremost Modern Textbook
 5.  Talk by Ferdinand Rudio
 6.  Title page of Introductio in analysin infinitorum (1784)
 7.  Another page from this work
 8.  Leonhard Euler's letters to a German Princess
 9.  Miller's postage stamps
 10. Heinz Klaus Strick biography

Other websites about Leonhard Euler:

 1.  Dictionary of Scientific Biography
 2.  Ian Bruce A translation of some early papers
 3.  Encyclopaedia Britannica
 4.  NNDB
 5.  Clark Kimberling
 6.  Rouse Ball
 7.  Plus Magazine (Calculus of variations)
 8.  Plus Magazine (The Basel problem)
 9.  Plus Magazine (Elements of Algebra)
 10. Google doodle
 11. Sci Hi blog
 12. Mathematical Genealogy Project
 13. MathSciNet Author profile
 14. zbMATH entry

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HONOURS (SHOW)

Honours awarded to Leonhard Euler

 1. Fellow of the Royal Society 1747
 2. Lunar features Crater Euler and Rima Euler
 3. Paris street names Rue Euler (8th Arrondissement)
 4. Popular biographies list Number 10
 5. Google doodle 2013

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CROSS-REFERENCES (SHOW)

 1.  History Topics: A history of Pi
 2.  History Topics: A history of Topology
 3.  History Topics: A history of the calculus
 4.  History Topics: African men with a doctorate in mathematics 1
 5.  History Topics: An overview of the history of mathematics
 6.  History Topics: Arabic mathematics : forgotten brilliance?
 7.  History Topics: Fermat's last theorem
 8.  History Topics: General relativity
 9.  History Topics: Light through the ages: Ancient Greece to Maxwell
 10. History Topics: Mathematical games and recreations
 11. History Topics: Orbits and gravitation
 12. History Topics: Pell's equation
 13. History Topics: Perfect numbers
 14. History Topics: Prime numbers
 15. History Topics: Quadratic, cubic and quartic equations
 16. History Topics: The Berlin Academy and forgery
 17. History Topics: The brachistochrone problem
 18. History Topics: The development of Ring Theory
 19. History Topics: The development of group theory
 20. History Topics: The four colour theorem
 21. History Topics: The function concept
 22. History Topics: The fundamental theorem of algebra
 23. History Topics: The number e
 24. History Topics: The real numbers: Stevin to Hilbert
 25. History Topics: The trigonometric functions
 26. Famous Curves: Catenary
 27. Famous Curves: Epicycloid
 28. Famous Curves: Hypocycloid
 29. Famous Curves: Lemniscate of Bernoulli
 30. Famous Curves: Tricuspoid
 31. Famous Curves: Trident of Newton
 32. Societies: Berlin Academy of Science
 33. Societies: Paris Academy of Sciences
 34. Societies: Swiss Mathematical Society
 35. Student Projects: Indian Mathematics - Redressing the balance: Chapter 14
 36. Student Projects: Indian Mathematics - Redressing the balance: Chapter 18
 37. Student Projects: Mathematics and Chess: Chapter 3
 38. Student Projects: Sofia Kovalevskaya: Chapter 12
 39. Student Projects: Sofia Kovalevskaya: Chapter 13
 40. Student Projects: Sofia Kovalevskaya: Chapter 15
 41. Other: 12th April
 42. Other: 15th June
 43. Other: 15th September
 44. Other: 1897 ICM - Zurich
 45. Other: 18th November
 46. Other: 1904 ICM - Heidelberg
 47. Other: 1908 ICM - Rome
 48. Other: 1912 ICM - Cambridge
 49. Other: 1932 ICM - Zurich
 50. Other: 19th April
 51. Other: 1st December
 52. Other: 2009 Most popular biographies
 53. Other: 27th August
 54. Other: 30th June
 55. Other: 3rd April
 56. Other: 5th December
 57. Other: 5th May
 58. Other: 7th March
 59. Other: 9th March
 60. Other: Earliest Known Uses of Some of the Words of Mathematics (A)
 61. Other: Earliest Known Uses of Some of the Words of Mathematics (B)
 62. Other: Earliest Known Uses of Some of the Words of Mathematics (C)
 63. Other: Earliest Known Uses of Some of the Words of Mathematics (D)
 64. Other: Earliest Known Uses of Some of the Words of Mathematics (E)
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 66. Other: Earliest Known Uses of Some of the Words of Mathematics (H)
 67. Other: Earliest Known Uses of Some of the Words of Mathematics (I)
 68. Other: Earliest Known Uses of Some of the Words of Mathematics (K)
 69. Other: Earliest Known Uses of Some of the Words of Mathematics (L)
 70. Other: Earliest Known Uses of Some of the Words of Mathematics (M)
 71. Other: Earliest Known Uses of Some of the Words of Mathematics (N)
 72. Other: Earliest Known Uses of Some of the Words of Mathematics (O)
 73. Other: Earliest Known Uses of Some of the Words of Mathematics (P)
 74. Other: Earliest Known Uses of Some of the Words of Mathematics (Q)
 75. Other: Earliest Known Uses of Some of the Words of Mathematics (R)
 76. Other: Earliest Known Uses of Some of the Words of Mathematics (T)
 77. Other: Earliest Uses of Function Symbols
 78. Other: Earliest Uses of Symbols for Constants
 79. Other: Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions
 80. Other: Earliest Uses of Symbols for Variables
 81. Other: Earliest Uses of Symbols from Geometry
 82. Other: Earliest Uses of Symbols in Probability and Statistics
 83. Other: Earliest Uses of Symbols of Calculus
 84. Other: Earliest Uses of Symbols of Number Theory
 85. Other: Earliest Uses of Symbols of Operation
 86. Other: Earliest Uses of Symbols of Relation
 87. Other: Jeff Miller's postage stamps
 88. Other: London individuals N-R
 89. Other: Most popular biographies – 2024
 90. Other: On Growth and Form
 91. Other: Popular biographies 2018

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Written by J J O'Connor and E F Robertson
Last Update September 1998
DJF/JOC/EFR
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