Transcript Leonhard Euler (1707 - 1783) - Department of Mathmatics

By: Vivek Joseph
Department of Mathematics
BBDNIIT, Lucknow
1
Leonhard Euler, 1753
2
References
 “Euler: The Master of Us All” by William Dunham,
Mathematical Association of America
 “Leonhard Euler: His Life & His Faith” by Dr. George W
Benthein, 2008.
 “Leonhard Euler: His Life, The Man and his work” by
Walter Gautschi, SIAM Review, Vol 50 No. 1, pp 3 – 33,
2008.
3
“Read
Euler, read Euler, he is a master in Every thing” By:
Pierre Simon Laplace
Early Life
 On April 15th, 1707, Leonhard Euler was born in Basel,
Switzerland.
 His father named Paul Euler, who was pastor of the reformed
church and mother named Marguerite Brucker.
 Leonhard Euler had two younger sister Anna Maria and Maria
Magdalena.
4
Home of Leonhard Euler
5
Early Life
 Euler’s early formal education started in Basel where he lived
with his maternal grandmother.
 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 University in 1720 at the age of 13. Euler was
studying theology, Greek and Hebrew languages in order to
become a pastor.
 In meantime Leonhard Euler encountered its most famous
Professor, Johann Bernoulli (1667 -1748) the greatest active
mathematician at that time.
6
Johann Bernoulli
7
University of Basel
8
Academic Life
 At University, Euler completed a master’s degree in
Philosophy. Then fulfilling his apparent destiny . Euler
entered divinity school to study for the ministry. But the
call of mathematics was too strong. He left to others and
become a mathematician.
 His progress was rapid. At the age of 20, he earned
recognition in an international scientific competition for
the analysis of the placement of masts on a sailing ship.
9
Life in St. Petersburg’s Academy
 In 1725, Johann’s son Daniel Bernoulli (1700 – 1782)
arrived to assume a position in mathematics at the New
St. Petersburg Academy.
 In 1727 Euler wrote his dissertation in Physics, presenting
a theory of sound on the basis of which he applied for an
open physics professorship in Basel. Since this application
was unsuccessful, the young Euler decided an invitation
to the St. Petersburg Academy in the same year.
10
St. Petersburg Academy
11
Daniel Bernoulli
12
Life in St. Petersburg Academy
 In 1733 Daniel Bernoulli left for an academic post in Switzerland.
The position was occupied by Euler.
 In 1733 he married his Swiss compatriot Katharina Gsell, the
daughter of a well known painter. Euler fathered 13 children, only 5
of which reached adolescence. His children provided him with 38
grand children.
 Intellectual life at the St. Petersburg Academy suited Euler perfectly.
He became a scientific consultant to the Government in which
capacity he prepared maps, advised the Russian navy and even
design for the fire engines.
13
Only Picture of Leonhard Euler before he lost sight
14
Life in St. Petersburg Academy
 Meanwhile his frame was growing. One of his triumphs was a
solution of the so called “Basel Problem”.
 The issue was to determine the exact value of the infinite series.
1
1
1
1
 2  2  ....... 2
2
1
2
3
k
 The answer was given by Euler in 1735.
 With the Basel problem behind him and a promise of good things
ahead Euler pursued his research at a breath taking pace in St.
Petersburg Academy.
15
Euler lost his vision in One Eye
 In 1735, he fell seriously ill and almost lost his life. To the
great relief of all, he recovered, but suffered a repeat
attack three years later of (probably) the same infectious
disease. This time it cost him his right eye.
 The political turmoil in Russia that followed the death of
the czarina Anna Ivanovna induced Euler to seriously
consider, and eventually decide, to leave St. Petersberg
16
Life At Royal Academy at Berlin
 Political unrest in Russia led to a tense working environment at St.
Petersburg Academy, prompting Euler to accept an invitation from
Frederick the Great of Prussia to join his Royal Academy at Berlin in 1741.
 Frederick asked Euler to tutor one of his niece in the physical sciences and
Euler agreed.
 This resulted in a series of over 200 letters that have been collected under
the title “Letter of Euler to a German Princess on a Different subject in a
Natural Philosophy”.
 These letters explained in Laymen’s language the basic concepts of physics
as well as Euler’s view on philosophy and theology. These letters were later
published by the St. Petersburg Academy in two large illustrated volumes.
17
Berlin Academy
18
Frederick II
19
Letters written by Euler to Princesses
20
Contribution of Euler in Field of Mathematics During his
stay in Berlin
 In the last 25 years that Euler spent in Berlin he made
important discoveries in the Calculus of Variation.
 Established Euler Identity for complex numbers.
 Produced two treatises in analysis (Introduction to the
Analysis of the Infinite in 1748) and Foundations of
Differential Calculus with Applications to Finite Analysis and
series in 1755.
21
List of Publication During his stay in Berlin
22
Return of St. Petersburg Academy
23
Return of St. Petersburg Academy
 In 1762 , the politics in Russia changed. Empress Catherine II
later named changed Catherine the Great, came to the
thorne. The atmosphere in Russian society improved
dramatically.
 Catherine II aimed to create in Russia a regime of Educated
Absolutism. She invited many progressive people to Russia
and increased the budget of the St. Petersburgh Academy to
60000 rubles per year, which was much motre than the
budget of the Berlin Academy.
24
Return of St. Petersburg Academy
 Catherine II offered Euler an important post in the
mathematics department, conference-secretary of the
Academy, with a big salary. She instructed her
representative in Berlin to agree to Euler's terms if he
does not like her first offer.
 In 1766 he accepted the invitation of Catherine the Great to
return to the St. Petersburg Academy.
25
Lost of Eye Sight
 In 1771 Euler's home was destroyed by fire and he was able to save only
himself and his mathematical manuscripts.
 In September 1771, Euler had surgery to remove his cataract. The surgery
was very successful - the mathematicians vision was restored.
 Unfortunately, Euler didn’t take care of his eyes; he continued to work and
after a few days lost his vision again, this time without any hope of
recovery.
 He was virtually blind for the last seventeen years of his life. After losing
the ability to see he commented “Now I will have less distraction”.
26
Death of Euler
 On September 18, 1783, Euler passed away in St.
Petersburg after suffering a stroke, and was buried
with his wife in the Smolensk Lutheran
 Cemetery on Vasilievsky Island.
27
Tomb of Leonhard Euler
Tomb of Leonhard Euler in St. Petersburg Academy
28
List of Field in which Leonhard Euler worked























Differential and integral calculus
Logarithmic, exponential, and trigonometric functions
Differential equations, ordinary and partial
Elliptic functions and integrals
Hypergeometric integrals
Classical geometry
Number theory
Algebra
Continued fractions
Zeta and other (Euler) products
Infinite series and products
Divergent series
Mechanics of particles
Mechanics of solid bodies
Calculus of variations
Optics (theory and practice)
Hydrostatics
Hydrodynamics
Astronomy
Lunar and planetary motion
Topology
Graph theory
29
Bibliography
 Euler has an extensive bibliography but his best known
books are
 Elements of Algebra: This elementary algebra text starts
with a discussion of the nature of numbers and gives a
comprehensive introduction to algebra, including
formulae for solutions of polynomial equations.
 Introductio in analysin infinitorum (1748). English
translation Introduction to Analysis of the Infinite by John
Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag
1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
30
Bibliography
 Two influential textbooks on calculus: Institutiones calculi
differentialis (1755) and Institutiones calculi integralis
(1768-1770).
 Lettres a une Princesse d'Allemagne (Letters to a German
Princess)(1768-1772). English translation, with notes, and
a life of Euler, available online from Google Books:
Volume 1, Volume 2
 Methodus inveniendi lineas curvas maximi minimive
proprietate gaudentes, sive solutio problematis
isoperimetrici latissimo sensu accepti (1744). (Method for
finding curved lines enjoying properties of maximum or
minimum, or solution of isoperimetric problems in the
broadest accepted sense.)
31
Book of Euler on Differential Calculus
32
Books written by Euler on Mechanics
33
Book written by Euler on Navigation
34
Book written by Euler on curves
35
Book on Leonhard Euler
36
Character of Leonhard Euler
 Clifford Truesdell wrote “He was exceptionally generous, never
once making a claim of priority and in some cases actually
giving away discoveries that were his own. He was the first to
cite the work of others in what is now regarded as the just
way, that is, so as to acknowledge their worth”
 Euler was a committed Christian and frequently expressed awe
at the work of the creator. Euler was particularly impressed by
the design of eye.
37
Overview of some work of Leonhard Euler
Euler established the use of
 e for the base of Natural lograthim

for the ratio of circumference to the diameter of the
circle.
 f(x) for function value
 sinx and cosx for values of sine and cosine function
 i for Imaginary unit
 Σ for summation
 Δ for finite difference

38
Few works of Euler
 In December 1729, Goldbach wrote a letter to Euler in
regarding with the Fermat’s statement that all numbers of
the form 2 1 were primes. Euler became very much
2n
interested in number theory and afterwards disproved
the assertion about Fermat’s prime.
 Euler discovered that on this point Fermat was wrong for
2 1 = 4,294,697,297 is evenly divisible by 641
25
39
Few works of Euler
 In 1730 Euler established Gamma Function and Beta
Function.
 In 1735 Euler solved the famous problem known as Basel
Problem. Euler found that the sum of infinite series
1 1 1
2
1  2  2  2  .....
2 3 4
6
40
Periodical work of Euler
 Konigsberg Bridge
 The river Pregel, which flows through the Prussian city of
Konigsberg, divides the city into an island and three
distinct land masses, one in the north, one in the east,
and one in the south. There are altogether seven bridges
41
Periodical work of Euler
 Problem: Is it possible to follow a path that crosses each
bridge exactly once and returns to the starting point?
 Euler solved the problem in 1735, published as E53 in 1741, by
showing that such paths cannot exist.
 This solution is considered to be the first theorem of graph
theory and planar graph theory. Euler also introduced the
notion now known as the Euler characteristic of a space and a
formula relating the number of edges, vertices, and faces of a
convex polyhedron with this constant.
42
Few works of Euler
 Prime Numbers and the Zeta Function.
Let P = {2, 3, 5, 7,11,13,17......} be a set of prime numbers,
i.e., the integers > 1 that is only divisible by 1 or by
themselves. Euler’s fascination with prime numbers started
quite early and continued throughout his life. An example of
his profound insight into the theory of numbers is the
discovery in 1737 (E72) of the fabulous product formula
1
  s, s > 1, connecting prime numbers with zeta

s
pP 1  p
function.
43
Few works of Euler
 Euler in his Element of Algebra introduced
1
as imaginary unit.
 Euler stated the general result what we known as De- Moivre Theorem
for n ≥ 1
cos  i sin  
n
 cosn  i sin n 
 Euler with the help of De Moivre theorem developed an well known Euler
identity
eix  cos x  i sin x
 In 1728 Euler wrote a paper entitled On finding the equation of geodesic
curves. Later in 1744, he published a more general work entitled “A
method of discovering curved lines that enjoy the maximum and minimum
property
44
Few works of Euler
One thing he considered in
his paper the minimization of
b
integral of the form I   f x, y, y / dx
a
He showed that the necessary condition for the minimum
was Euler equation f  d  f   0
y dx  y 
45
Few works of Euler
 Euler’s Polyhedral Formula: In a 1752 study of polyhedra,
Euler observed that V + F = E + 2, where V is the number
of vertices, F is the number of faces, and E is the number
of edges of a solid figure. Because of the utter simplicity
of this relationship, Euler confessed that, “I find it
surprising that these general results in solid geometry
have not previously been noticed by anyone, so far as I
am aware. ” Of course, no previous mathematician had
had Euler’s penetrating insight.
46
Physics and Astronomy
 Euler helped develop the Euler-Bernoulli beam equation,
which became a cornerstone of engineering.
 He applied his analytic tools to problems in classical
mechanics and to celestial problems. His work in
astronomy was recognized by a number of Paris Academy
Prizes.
 His accomplishments in astronomy include determining
the orbits of comets and other celestial bodies,
understanding the nature of comets, and calculating the
parallax of the sun.
 In addition, Euler made important contributions in optics.
47
THANK YOU
48