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
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 Euler's own account given in his unpublished autobiographical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 mathematicalphysical 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 JN 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
(173637), 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 twentyfive 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., 176872).
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 grandsoninlaw, 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.
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) for a function
(1734), e for the base of natural logs (1727), i for the square
root of 1 (1777), ? for pi, 誇 for
summation (1755), the notation for finite differences ?퀉 and ??SUP>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 2^{n} + 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 2^{32} + 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 a^{2} + b^{2} 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瓘(2) = 誇
(1/n^{2}), 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 瓘(2) = ?^{2}/6 but he went on to prove much
more, namely that 瓘(4) =
?^{4}/90, 瓘(6) =
?^{6}/945, 瓘(8) =
?^{8}/9450, 瓘(10) =
?^{10}/93555 and 瓘(12) =
691?^{12}/638512875. In 1737 he proved the connection of the zeta
function with the series of prime numbers giving the famous relation
瓘(s) = 誇 (1/n^{s}) = ? (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
瓘(2n) =
C?^{2n} 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} 
log_{e}n
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
?/2  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 [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
EulerMaclaurin 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 selfsame 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 = 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
of elementary functions rather than on geometric curves, as had been done
previously. Also in this work Euler gave the formula
e^{ix} = 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) = ?i
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 CauchyRiemann 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 (176870) 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. 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 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 占?000 from Britain for the contribution
the tables made to the problem of the determination of the longitude, while
Euler received 占?00 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 [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.
Article by: J J O'Connor and E F Robertson
September 1998
