Реферат: Leonhard Euler
Born: 15 April 1707 in Basel, Switzerland
Died: 18 Sept 1783 in St Petersburg, Russia
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
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 Euler's own account given in his unpublished autobiographical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, 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 has reconstructed many of the works that Euler read with the advice of
Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo,
von 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
nor 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 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 Science 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
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 became a full member of
the Academy, he was able to give up his Russian navy post.
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  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
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
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 :-
... lost an eye and [the other] currently may be in
the same danger.
However, Calinger in  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.
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
... 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 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:-
.. 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.
describes the day of Euler's death in:-
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
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) for a function
(1734), e for the base of natural logs (1727), i for the square root of -1
(1777), p for
pi, for summation (1755), the notation for finite differences y and 2y
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 + 1 were always prime if n is a power of 2.
Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 at the latest,
showed that the next case 232 + 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 (n), the number of integers k with 1
k n and k coprime to n. He proved another of Fermat's assertions, namely that
if a and b are coprime then a2 + b2 has no divisor of the
form 4n - 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 (2) = (1/n2),
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 (2) = p2/6 but
he went on to prove much more, namely that (4) = p4/90, (6) = p6/945,
(8) = p8/9450,
(10) = p10/93555
and (12) = 691p12/638512875.
In 1737 he proved the connection of the zeta function with the series of prime
numbers giving the famous relation
(s) = (1/ns)
= (1 - p-s)-1
Here the sum is over all natural numbers n while the
product is over all prime numbers.
By 1739 Euler had found the rational coefficients C in
(2n) = Cp2n in
terms of the Bernoulli numbers.
Other work done by Euler on infinite series included
the introduction of his famous Euler's constant, in 1735, which he showed to be
the limit of
1/1 + 1/2
+ 1/3 + ... + 1/n - logen
as n tends to infinity. He calculated the constant 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
x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...
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 :-
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.
... 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 = 3. Perhaps more
significant than the result here was the fact that he introduced a proof
involving numbers of the form a + b-3 for integers a and b. 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 elementary functions rather than on geometric curves, as had been
done previously. Also in this work Euler gave the formula
eix= cos x + i sin x.
In Introductio in analysin infinitorum Euler dealt
with logarithms of a variable taking only positive values although he had
discovered the formula
ln(-1) = pi
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
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
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
... published in 1740 began the proper study of the calculus of variations.
It is noted that Carathéodory considered this
... one of the most beautiful mathematical works ever
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:-
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
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
establish 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 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 formulas 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:-
... 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 Tobias 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 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 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.
J J O'Connor and E F Robertson
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