Lecture 30. Euler, Our Master in Everything
Figure 30.1 Leonhard Euler and Riehen
Leonhard Euler Leonhard Euler(1707-1783) was a Swiss
mathematician who madeenormous contributions to a wide range of
mathematics and physics including analytic ge-ometry, trigonometry,
geometry, calculus and number theory. Euler was the most
prolificmathematical writer of all times finding time to publish
over 800 papers in his lifetime.
Leonhard Eulers father Paul Euler was a friend of Bernoulli
family (see Lecture 29).In fact Paul Euler had studied theology at
the University of Basel and had attended JacobBernoullis lectures
there.
Leonhard Euler was born in Basel and was brought up in Riehen,
not far from Basel.
Living with his grandmother, Leonhard went to a local school in
Basel. This school wasso poor that Euler learned no mathematics at
all from the school. Paul Euler had somemathematical training and
he was able to teach his son elementary mathematics along withother
subjects. As a result, Leonhards interest in mathematics was
sparked by his fathersteaching, and he read mathematics texts on
his own and took some private lessons.
Eulers father wanted his son to follow him into the church and
sent him to the Universityof Basel to prepare for the ministry. As
he age of 14, Euler entered the University in 1720.Johann Bernoulli
soon discovered Eulers great potential for mathematics. Euler wrote
asfollows:
198
... I soon found an opportunity to be introduced to a famous
professorJohann Bernoulli. ... True, he was very busy and so
refused flatly to give meprivate lessons; but he gave me much more
valuable advice to start reading moredifficult mathematical books
on my own and to study them as diligently as Icould; if I came
across some obstacle or difficulty, I was given permission to
visithim freely every Sunday afternoon and he kindly explained to
me everything Icould not understand ...
In 1723 Euler completed his Masters degree in philosophy. As his
fathers wish, he beganhis study of theology in the same year, but
he could not find the enthusiasm for the studyof theology. After
persuaded by Johann Bernoulli, his father agreed to let him change
tomathematics.
Euler completed his studies at the University of Basel in 1726
where he had studiedmathematics. By 1726 Euler had already a paper
in print. In 1727 he published anotherarticle and submitted an
entry for the 1727 Grand Prize of the Paris Academy. Eulers
essaywon him second place which was a very good achievement for the
young graduate. SinceNicolaus(II) Bernoulli died in St Petersburg
in July 1726, it created a vacancy at ImperialRussian Academy of
Sciences in St Petersburg. Euler was offered the post, and he
acceptedthe post.
Figure 30.2 St Petersburg and Imperial Russian Academy of
Sciences there
He had a phenomenal memory, and once did a calculation in his
head to settle anargument between students whose computations
differed in the fiftieth decimal place. Eulerlost sight in his
right eye in 1735, and in his left eye in 1766. Nevertheless, aided
by hisphenomenal memory (and having practiced writing on a large
slate when his sight was failing
199
him), he continued to publish his results by dictating them. In
his life time, Euler publishedover 800 papers. He won the Paris
Academy Prize 12 times. When asked for an explanationwhy his
memoirs flowed so easily in such huge quantities, Euler is reported
to have repliedthat his pencil seemed to surpass him in
intelligence. Francois Arago 1 said of him hecalculated just as men
breathe, as eagles sustain themselves in the air.
Euler was the director of the Berlin Academys mathematics
section. With numerousmathematica papers, he became recognized as
the primier mathematician of Europe.
In 1766 Euler returned to the St. Petersburg Academy and spent
the rest of his life inRussia. Euler passed away in St. Petersburg
on Stetember 18, 1983.
Figure 30.3 Eulers grave at the Alexander Nevsky Lavra
Eulars mathematical contribution Euler worked in every field of
mathematics whichexisted in his day. Many of his results are of
fundamental interest. Eulers name is associatedwith a large number
of topics. Here are some of his works.
Prestigious textbooks Euler not only published his results in
articles of variedlength, but also in impressive number of large
textbooks.
In several fields Eulers presentation has been almost final. An
example is our presenttrigonometry with it conception of
trigonometric values as ratios and its useful nota-
1Arago (1786 - 1853) was an important French mathematician.
200
tion, which dates from Eulers Introductio in analysin
infinitorum (1748). The tremen-dous prestige of his textbooks
settled forever many notations. Lagrange, Laplace andGauss followed
Euler in all their works.
Our notation is almost Eulers Euler introduced and popularized
several nota-tional conventions through his numerous and widely
circulated textbooks, notations.He sharpened the concept of a
function and was the first to denote by () the function of a
variable . He used to denote the base of the natural logarithm (now
alsoknown as Eulers number). He used
to denote summations. He used to denote the
imaginary unit1. He used to denote the ratio of a circles
circumference to its di-
ameter. He also introduced the modern notation for the
trigonometric functions. MITProfessor Struk said2: Since Eulers
Latin is very simple and his notation is almostmodern or perhaps we
should better say that our notation is almost Eulers.
Basel problem Thanks to the influence of Bernoullis family,
studying calculusbecame the major focus of Eulers work. His ideas
led to many great advances. Euleris well-known for his frequent use
and development of power series, the expression offunctions as sums
of infinitely many terms, such as the power series expansions for
and for the inverse tangent function.
Figure 30.4 Euler
The Basel problem is a famous problem in number theory, first
posed by Pietro Mengoliin 1644, which asks for the precise
summation of the series
1
2
2D. Struk, A Concise History of Mathematics, fourth edt., Dover
Publications, Inc., 1987, p.124.
201
The series is approximately equal to 1.644934. Euler found the
exact sum to be 2
6
and announced this discovery in 1735. Since the problem had
attracted the leadingmathematicians of the day, Eulers solution
brought him immediate fame when he wastwenty-eight.
The most remarkable formula In 1697, Jacob Bernoulli studied
lim(1+ 1),which had been implicit in earlier work on natural
logarithms. In 1748, Euler definedthe two functions:3
= lim
(1 +
), = lim
(
1). (1)
and also proved his famous formula:
= + (2)
Euler proved that the infinite series (2) of both sides being
equal. 50 years later,the view of complex numbers as points in the
complex plane arose. Until Euler thetrigonometrical quantities
sine, cosine, tangent, etc., were regarded as lines connectedwith
the circle rather than functions. Even the derivation of the series
expansion forthe sine in dependence of the arc by Newton and
Leibniz did not change this view. Itwas Euler who introduced the
functional point of view. 4 A special case of the aboveformula is
known as Eulers identity:
+ 1 = 0.
called the most remarkable formula in mathematics by Richard
Feynman because ituses of the notions of
addition, multiplication, exponentiation, 0, 1, e, i, , sine,
and cosine.
In 1988, readers of the Mathematical Intelligence voted it the
Most Beautiful Math-ematical Formula Ever. In total, Euler was
responsible for three of the top fiveformulas in that poll.5
3Morris Kline, Mathematical Thought from Ancient to Modern
Times, volume 2, New York Oxford,Oxford University Press, 1972,
p.404.
4Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003,
p.115-116.5Here is the list of the top five: 1. Eulers formula; 2.
Eulers formula for a polyhedron: + = + 2.
3. The number of primes is infinite. 4. There are 5 regular
polyhedrons. 5. 1 + 122 +132 +
142 + ... =
2
6(Euler). cf., David Wells, Are these the most beautiful?
Mathematical Intelligencer 12(3)(1990), 37-41.
202
Figure 30.5 Eulers formulas
How to define log a? It was a difficulty problem how to define
when is anegative number. Between 1712 and 1713, Bernoulli held the
view that = ()because () =
, while Leibniz believed that () must be imaginary. Between
1727 and 1731, the question was taken up by Bernoulli and Euler
without at a solution.It was only in the period between 1749 and
1751, Euler developed ideas far enough tostudy logarithm so that it
leads to a satisfying solution. Eulers approach is to dealwith
logarithm of general complex numbers. Eulers argument is as
follows: by thedefinition (1), = lim (
1). For each positive integer , ( 1) has
different roots (Euler realized it!) so that the limit indeed
has infinitely many differentvalues. When = 1, he got 1 = 2, = 1,
2, 3, ....6In 1735, Euler define a constant:
:= lim
( =1
1
)which is the well-known Euler constant. Its numerical value to
50 decimal places is
0.57721566490153286060651209008240243104215933593992
6Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003,
p.117-118.
203
Gamma functions Euler introduced the gamma function:
() :=
0
1
which has the property: () = ( 1)! so that it is a natural
generalization of thefactorial.
Complex analysis He also found a way to calculate integrals with
complex limits,foreshadowing the development of modern complex
analysis.
Calculus of variations He invented the calculus of variations
including its best-known result, the Euler - Lagrange equation.
Number theory Euler proved that that the divergence of the
harmonic seriesimplied an infinite number of Primes. He factored
the fifth Fermat number (thusdisproving Fermats conjecture),
proving Fermats lesser theorem, and showing that was
irrational.
Three-body problem In 1772, he introduced a synoptic coordinates
(rotating) co-ordinate system to the study of the three-body
problem (especially the Moon ).
Topology The Euler characteristic was classically defined for
the surfaces of poly-hedral, according to the formula
= +
where , , and are respectively the numbers of vertices
(corners), edges and facesin the given polyhedron.
Any convex polyhedrons surface has Euler characteristic 2. This
result is known asEulers formula.
Graph theory Konigsberg was a city in Prussia situated on the
Pregel River,which served as the residence of the dukes of Prussia
in the 16th century (Today,the city is named Kaliningrad, and is a
major industrial and commercial center ofwestern Russia). The river
Pregel flowed through the town, creating an island, as inthe
following picture.
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Figure 30.6 Seven Bridges of Konigsberg
A famous problem concerning Konigsberg was whether it was
possible to take a walkthrough the town in such a way as to cross
over every bridge once, and only once.All who tried ended up in
failure, including Euler. However, Euler did succeed inexplaining
why such a journey was impossible, not only for the Konigsberg
bridges,but whether such a journey was possible or not for any
network of bridges anywhere.
This is the earliest work on graph theory by Leonhard Euler in
1736.
Ordinary differential equations Euler invented the Euler method
in ODE. Itis a first order numerical procedure for solving ordinary
differential equations (ODEs)with a given initial value. It is the
most basic kind of explicit method for numericalintegration for
ordinary differential equations. In Eulers textbook Institutiones
calculiintegralis, the section of linear, exact and homogeneous
equations is still themodel of our elementary texts on the this
subject.
Others He also made major contributions in optics, mechanics,
electricity, and mag-netism, astronomy, hydraulics, ship
construction, artillery ......
In a testament to Eulers proficiency in all branches of
mathematics, the great Frenchmathematician and celestial mechanic
Laplace told his students, Read Euler, read Euler,he is our master
in everything.
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