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Leonhard Euler (/ˈɔɪlər/ OY-lər; German pronunciation: [ˈɔʏlɐ] ( listen),
local pronunciation: [ˈɔɪlr̩] ( listen); 15 April 1707 – 18 September 1783) was
a pioneering Swiss mathematician and physicist. He made important discoveries in
fields as diverse as infinitesimal calculus and graph theory. He also introduced
much of the modern mathematical terminology and notation, particularly for
mathematical analysis, such as the notion of a mathematical function. He is
also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.
Euler is considered to be the pre-eminent mathematician of the 18th century and
one of the greatest mathematicians to have ever lived. He is also one of the
most prolific mathematicians; his collected works fill 60–80 quarto volumes.
He spent most of his adult life in St. Petersburg, Russia, and in Berlin,
Prussia.
A statement attributed to Pierre-Simon Laplace expresses Euler's influence on
mathematics: "Read Euler, read Euler, he is the master of us all."
Early years
Old Swiss 10 Franc banknote honoring Euler
Euler was born on 15 April 1707, in Basel to Paul Euler, a pastor of the
Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two younger
sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard,
the Eulers moved from Basel to the town of Riehen, where Euler spent most of his
childhood. Paul Euler was a friend of the Bernoulli family—Johann Bernoulli, who
was then regarded as Europe's foremost mathematician, would eventually be the
most important influence on young Leonhard. Euler's early formal education
started in Basel, where he was sent to live with his maternal grandmother. At
the age of thirteen he enrolled at the University of Basel, and in 1723,
received his Master of Philosophy with a dissertation that compared the
philosophies of Descartes and Newton. At this time, he was receiving Saturday
afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's
incredible talent for mathematics. Euler was at this point studying theology,
Greek, and Hebrew at his father's urging, in order to become a pastor, but
Bernoulli convinced Paul Euler that Leonhard was destined to become a great
mathematician. In 1726, Euler completed a dissertation on the propagation of
sound with the title De Sono. At that time, he was pursuing an (ultimately
unsuccessful) attempt to obtain a position at the University of Basel. In 1727,
he first entered the Paris Academy Prize Problem competition; the problem that
year was to find the best way to place the masts on a ship. Pierre Bouguer, a
man who became known as "the father of naval architecture" won, and Euler took
second place. Euler later won this annual prize twelve times.
St. Petersburg
Around this time Johann Bernoulli's two sons, Daniel and Nicolas, were working
at the Imperial Russian Academy of Sciences in St Petersburg. On 10 July 1726,
Nicolas died of appendicitis after spending a year in Russia, and when Daniel
assumed his brother's position in the mathematics/physics division, he
recommended that the post in physiology that he had vacated be filled by his
friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed
making the trip to St Petersburg while he unsuccessfully applied for a physics
professorship at the University of Basel.
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says:
250 years from the birth of the great mathematician, academician Leonhard Euler.
Euler arrived in the Russian capital on 17 May 1727. He was promoted from his
junior post in the medical department of the academy to a position in the
mathematics department. He lodged with Daniel Bernoulli with whom he often
worked in close collaboration. Euler mastered Russian and settled into life in
St Petersburg. He also took on an additional job as a medic in the Russian Navy.
The Academy at St. Petersburg, established by Peter the Great, was intended to
improve education in Russia and to close the scientific gap with Western Europe.
As a result, it was made especially attractive to foreign scholars like Euler.
The academy possessed ample financial resources and a comprehensive library
drawn from the private libraries of Peter himself and of the nobility. Very few
students were enrolled in the academy in order to lessen the faculty's teaching
burden, and the academy emphasized research and offered to its faculty both the
time and the freedom to pursue scientific questions.
The Academy's benefactress, Catherine I, who had continued the progressive
policies of her late husband, died on the day of Euler's arrival. The Russian
nobility then gained power upon the ascension of the twelve-year-old Peter II.
The nobility were suspicious of the academy's foreign scientists, and thus cut
funding and caused other difficulties for Euler and his colleagues.
Conditions improved slightly upon the death of Peter II, and Euler swiftly rose
through the ranks in the academy and was made professor of physics in 1731. Two
years later, Daniel Bernoulli, who was fed up with the censorship and hostility
he faced at St. Petersburg, left for Basel. Euler succeeded him as the head of
the mathematics department.
On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg
Gsell, a painter from the Academy Gymnasium. The young couple bought a house
by the Neva River. Of their thirteen children, only five survived childhood.
Berlin
Stamp of the former German Democratic Republic honoring Euler on the 200th
anniversary of his death. Across the centre it shows his polyhedral formula,
nowadays written as v − e + f = 2.
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on
19 June 1741 to take up a post at the Berlin Academy, which he had been offered
by Frederick the Great of Prussia. He lived for twenty-five years in Berlin,
where he wrote over 380 articles. In Berlin, he published the two works for
which he would become most renowned: The Introductio in analysin infinitorum, a
text on functions published in 1748, and the Institutiones calculi
differentialis, published in 1755 on differential calculus. In 1755, he
was elected a foreign member of the Royal Swedish Academy of Sciences.
In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt,
the Princess of Anhalt-Dessau and Frederick's niece. Euler wrote over 200
letters to her in the early 1760s, which were later compiled into a best-selling
volume entitled Letters of Euler on different Subjects in Natural Philosophy
Addressed to a German Princess. This work contained Euler's exposition on
various subjects pertaining to physics and mathematics, as well as offering
valuable insights into Euler's personality and religious beliefs. This book
became more widely read than any of his mathematical works, and was published
across Europe and in the United States. The popularity of the 'Letters'
testifies to Euler's ability to communicate scientific matters effectively to a
lay audience, a rare ability for a dedicated research scientist.
Despite Euler's immense contribution to the Academy's prestige, he was
eventually forced to leave Berlin. This was partly because of a conflict of
personality with Frederick, who came to regard Euler as unsophisticated,
especially in comparison to the circle of philosophers the German king brought
to the Academy. Voltaire was among those in Frederick's employ, and the
Frenchman enjoyed a prominent position within the king's social circle. Euler, a
simple religious man and a hard worker, was very conventional in his beliefs and
tastes. He was in many ways the antithesis of Voltaire. Euler had limited
training in rhetoric, and tended to debate matters that he knew little about,
making him a frequent target of Voltaire's wit. Frederick also expressed
disappointment with Euler's practical engineering abilities:
I wanted to have a water jet in my garden: Euler calculated the force of the
wheels necessary to raise the water to a reservoir, from where it should fall
back through channels, finally spurting out in Sanssouci. My mill was carried
out geometrically and could not raise a mouthful of water closer than fifty
paces to the reservoir. Vanity of vanities! Vanity of geometry!
A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the
right eyelid, and possible strabismus. The left eye, which here appears healthy,
was later affected by a cataract.
Eyesight deterioration
Euler's eyesight worsened throughout his mathematical career. Three years after
suffering a near-fatal fever in 1735, he became almost blind in his right eye,
but Euler rather blamed the painstaking work on cartography he performed for the
St. Petersburg Academy for his condition. Euler's vision in that eye worsened
throughout his stay in Germany, to the extent that Frederick referred to him as
"Cyclops". Euler later developed a cataract in his left eye, rendering him
almost totally blind a few weeks after its discovery in 1766. However, his
condition appeared to have little effect on his productivity, as he compensated
for it with his mental calculation skills and exquisite memory. For example,
Euler could repeat the Aeneid of Virgil from beginning to end without hesitation,
and for every page in the edition he could indicate which line was the first and
which the last. With the aid of his scribes, Euler's productivity on many areas
of study actually increased. He produced on average, one mathematical paper
every week in the year 1775.
Return to Russia
The situation in Russia had improved greatly since the accession to the throne
of Catherine the Great, and in 1766 Euler accepted an invitation to return to
the St. Petersburg Academy and spent the rest of his life in Russia. However,
his second stay in the country was marred by tragedy. A fire in St. Petersburg
in 1771 cost him his home, and almost his life. In 1773, he lost his wife
Katharina after 40 years of marriage. Three years after his wife's death, Euler
married her half-sister, Salome Abigail Gsell (1723–1794). This marriage
lasted until his death.
In St. Petersburg on 18 September 1783, after a lunch with his family, during a
conversation with a fellow academician Anders Johan Lexell, about the newly
discovered planet Uranus and its orbit, Euler suffered a brain hemorrhage and
died a few hours later. A short obituary for the Russian Academy of Sciences
was written by Jacob von Staehlin-Storcksburg and a more detailed eulogy was
written and delivered at a memorial meeting by Russian mathematician Nicolas
Fuss, one of Euler's disciples. In the eulogy written for the French Academy by
the French mathematician and philosopher Marquis de Condorcet, he commented,
il cessa de calculer et de vivre—... he ceased to calculate and to live.
He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky
Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard
Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone
on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the
headstone was moved in 1956, together with his remains, to the 18th-century
necropolis at the Alexander Nevsky Monastery.
Euler's grave at the Alexander Nevsky Monastery
Contributions to mathematics and physics Part of a series of articles on
The mathematical constant e Natural logarithm · Exponential function
Applications in: compound interest · Euler's identity & Euler's formula · half-lives
& exponential growth/decay Defining e: proof that e is irrational · representations
of e · Lindemann–Weierstrass theorem
People John Napier · Leonhard Euler Schanuel's conjecture
Euler worked in almost all areas of mathematics: geometry, infinitesimal
calculus, trigonometry, algebra, and number theory, as well as continuum physics,
lunar theory and other areas of physics. He is a seminal figure in the history
of mathematics; if printed, his works, many of which are of fundamental interest,
would occupy between 60 and 80 quarto volumes. Euler's name is associated
with a large number of topics.
Euler is the only mathematician to have two numbers named after him: the
immensely important Euler's Number in calculus, e, approximately equal to 2.71828,
and the Euler-Mascheroni Constant γ (gamma) sometimes referred to as just "Euler's
constant", approximately equal to 0.57721. It is not known whether γ is rational
or irrational.
Mathematical notation
Euler introduced and popularized several notational conventions through his
numerous and widely circulated textbooks. Most notably, he introduced the
concept of a function and was the first to write f(x) to denote the function
f applied to the argument x. He also introduced the modern notation for the
trigonometric functions, the letter e for the base of the natural logarithm (now
also known as Euler's number), the Greek letter Σ for summations and the letter i
to denote the imaginary unit. The use of the Greek letter π to denote the
ratio of a circle's circumference to its diameter was also popularized by Euler,
although it did not originate with him.
Analysis
The development of infinitesimal calculus was at the forefront of 18th Century
mathematical research, and the Bernoullis—family friends of Euler—were
responsible for much of the early progress in the field. Thanks to their
influence, studying calculus became the major focus of Euler's work. While some
of Euler's proofs are not acceptable by modern standards of mathematical rigour
(in particular his reliance on the principle of the generality of algebra), his
ideas led to many great advances. Euler is well known in analysis for his
frequent use and development of power series, the expression of functions as
sums of infinitely many terms, such as
Notably, Euler directly proved the power series expansions for e and the inverse
tangent function. (Indirect proof via the inverse power series technique was
given by Newton and Leibniz between 1670 and 1680.) His daring use of power
series enabled him to solve the famous Basel problem in 1735 (he provided a more
elaborate argument in 1741):
A geometric interpretation of Euler's formula
Euler introduced the use of the exponential function and logarithms in analytic
proofs. He discovered ways to express various logarithmic functions using power
series, and he successfully defined logarithms for negative and complex numbers,
thus greatly expanding the scope of mathematical applications of logarithms.
He also defined the exponential function for complex numbers, and discovered its
relation to the trigonometric functions. For any real number φ (taken to be
radians), Euler's formula states that the complex exponential function satisfies
A special case of the above formula is known as Euler's identity,
called "the most remarkable formula in mathematics" by Richard P. Feynman, for
its single uses of the notions of addition, multiplication, exponentiation, and
equality, and the single uses of the important constants 0, 1, e, i and π.
In 1988, readers of the Mathematical Intelligencer voted it "the Most Beautiful
Mathematical Formula Ever". In total, Euler was responsible for three of the
top five formulae in that poll.
De Moivre's formula is a direct consequence of Euler's formula.
In addition, Euler elaborated the theory of higher transcendental functions by
introducing the gamma function and introduced a new method for solving quartic
equations. He also found a way to calculate integrals with complex limits,
foreshadowing the development of modern complex analysis. He also invented the
calculus of variations including its best-known result, the Euler–Lagrange
equation.
Euler also pioneered the use of analytic methods to solve number theory problems.
In doing so, he united two disparate branches of mathematics and introduced a
new field of study, analytic number theory. In breaking ground for this new
field, Euler created the theory of hypergeometric series, q-series, hyperbolic
trigonometric functions and the analytic theory of continued fractions. For
example, he proved the infinitude of primes using the divergence of the harmonic
series, and he used analytic methods to gain some understanding of the way prime
numbers are distributed. Euler's work in this area led to the development of the
prime number theorem.
Number theory
Euler's interest in number theory can be traced to the influence of Christian
Goldbach, his friend in the St. Petersburg Academy. A lot of Euler's early work
on number theory was based on the works of Pierre de Fermat. Euler developed
some of Fermat's ideas, and disproved some of his conjectures.
Euler linked the nature of prime distribution with ideas in analysis. He proved
that the sum of the reciprocals of the primes diverges. In doing so, he
discovered the connection between the Riemann zeta function and the prime
numbers; this is known as the Euler product formula for the Riemann zeta
function.
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on
sums of two squares, and he made distinct contributions to Lagrange's four-square
theorem. He also invented the totient function φ(n), the number of positive
integers less than or equal to the integer n that are coprime to n. Using
properties of this function, he generalized Fermat's little theorem to what is
now known as Euler's theorem. He contributed significantly to the theory of
perfect numbers, which had fascinated mathematicians since Euclid. Euler also
conjectured the law of quadratic reciprocity. The concept is regarded as a
fundamental theorem of number theory, and his ideas paved the way for the work
of Carl Friedrich Gauss.
By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It
may have remained the largest known prime until 1867.
Graph theory
Map of Königsberg in Euler's time showing the actual layout of the seven bridges,
highlighting the river Pregel and the bridges.
In 1736, Euler solved the problem known as the Seven Bridges of Königsberg.
The city of Königsberg, Prussia was set on the Pregel River, and included two
large islands that were connected to each other and the mainland by seven
bridges. The problem is to decide whether it is possible to follow a path that
crosses each bridge exactly once and returns to the starting point. It is not
possible: there is no Eulerian circuit. This solution is considered to be the
first theorem of graph theory, specifically of planar graph theory.
Euler also discovered the formula V − E + F = 2 relating the number of vertices,
edges and faces of a convex polyhedron, and hence of a planar graph. The
constant in this formula is now known as the Euler characteristic for the graph
(or other mathematical object), and is related to the genus of the object.
The study and generalization of this formula, specifically by Cauchy and L'Huillier,
is at the origin of topology.
Applied mathematics
Some of Euler's greatest successes were in solving real-world problems
analytically, and in describing numerous applications of the Bernoulli numbers,
Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued
fractions and integrals. He integrated Leibniz's differential calculus with
Newton's Method of Fluxions, and developed tools that made it easier to apply
calculus to physical problems. He made great strides in improving the numerical
approximation of integrals, inventing what are now known as the Euler
approximations. The most notable of these approximations are Euler's method and
the Euler–Maclaurin formula. He also facilitated the use of differential
equations, in particular introducing the Euler–Mascheroni constant:
One of Euler's more unusual interests was the application of mathematical ideas
in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to
eventually incorporate musical theory as part of mathematics. This part of his
work, however, did not receive wide attention and was once described as too
mathematical for musicians and too musical for mathematicians.
Physics and astronomy Classical mechanics
History · Timeline · Branches
Statics
Dynamics · Kinetics ·
Kinematics
Applied mechanics
Celestial mechanics
Continuum mechanics
Statistical mechanics Formulations
Newtonian mechanics (Vectorial mechanics)
Analytical mechanics (Lagrangian mechanics Hamiltonian mechanics)
Fundamental concepts
Space · Time · Mass · Inertia ·
Velocity · Speed · Acceleration ·
Force · Momentum · Impulse ·
Torque / Moment / Couple
Angular momentum
Moment of inertia
Frame of reference
(kinetic · potential) ·
Mechanical work
Mechanical power
Virtual work
D'Alembert's principle Core topics
Rigid body
(dynamics · Euler's equations) ·
Motion (linear)
Euler's laws of motion
Newton's laws of motion
Newton's law of universal gravitation
Equations of motion
Inertial / Non-inertial reference frame
Fictitious force
Mechanics of planar particle motion
Displacement (vector)
Relative velocity
Friction
Simple harmonic motion
Harmonic oscillator
Vibration
Damping (ratio) Rotational motion
Circular motion
(uniform · non-uniform) ·
Rotating reference frame
Centripetal force
Centrifugal force
(rotating reference frame · reactive) ·
Coriolis force
Pendulum
Tangential / Rotational speed
Angular acceleration
Angular velocity
Angular frequency
Angular displacement Scientists
Galileo · Newton · Kepler ·
Horrocks · Halley · Euler ·
d'Alembert · Clairaut · Lagrange ·
Laplace · Hamilton · Poisson ·
Daniel / Johann Bernoulli · Cauchy · v · t · e ·
Euler helped develop the Euler–Bernoulli beam equation, which became a
cornerstone of engineering. Aside from successfully applying his analytic tools
to problems in classical mechanics, Euler also applied these techniques to
celestial problems. His work in astronomy was recognized by a number of Paris
Academy Prizes over the course of his career. His accomplishments include
determining with great accuracy the orbits of comets and other celestial bodies,
understanding the nature of comets, and calculating the parallax of the sun. His
calculations also contributed to the development of accurate longitude tables.
In addition, Euler made important contributions in optics. He disagreed with
Newton's corpuscular theory of light in the Opticks, which was then the
prevailing theory. His 1740s papers on optics helped ensure that the wave theory
of light proposed by Christiaan Huygens would become the dominant mode of
thought, at least until the development of the quantum theory of light.
In 1757 he published an important set of equations for inviscid flow, that are
now known as the Euler equations. In differential form, the equations are:
where
ρ is the fluid mass density,
u is the fluid velocity vector, with components u, v, and w,
E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energy per unit volume, with e being
the internal energy per unit mass for the fluid,
p is the pressure,
denotes the tensor product, and
0 being the zero vector.
Euler is also well known in structural engineering for his formula giving the
critical buckling load of an ideal strut, which depends only on its length and
flexural stiffness:
where
= maximum or critical force (vertical load on column),
= modulus of elasticity,
= area moment of inertia,
= unsupported length of column,
= column effective length factor, whose value depends on the conditions of end
support of the column, as follows.
For both ends pinned (hinged, free to rotate), = 1.0.
For both ends fixed, = 0.50.
For one end fixed and the other end pinned, = 0.699....
For one end fixed and the other end free to move laterally, = 2.0.
is the effective length of the column.
Logic
Euler is also credited with using closed curves to illustrate syllogistic
reasoning (1768). These diagrams have become known as Euler diagrams.
EULER'S DIAGRAM
An Euler diagram is a diagrammatic means of representing sets and their
relationships. Euler diagrams consist of simple closed curves (usually circles)
in the plane that depict sets. Each Euler curve divides the plane into two
regions or "zones": the interior, which symbolically represents the elements of
the set, and the exterior, which represents all elements that are not members of
the set. The sizes or shapes of the curves are not important: the significance
of the diagram is in how they overlap. The spatial relationships between the
regions bounded by each curve (overlap, containment or neither) corresponds to
set-theoretic relationships (intersection, subset and disjointness). Curves
whose interior zones do not intersect represent disjoint sets. Two curves whose
interior zones intersect represent sets that have common elements; the zone
inside both curves represents the set of elements common to both sets (the
intersection of the sets). A curve that is contained completely within the
interior zone of another represents a subset of it. Euler diagrams were
incorporated as part of instruction in set theory as part of the new math
movement in the 1960s. Since then, they have also been adopted by other
curriculum fields such as reading.
Personal philosophy and religious beliefs
Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and
the philosophy of Christian Wolff. Euler insisted that knowledge is founded in
part on the basis of precise quantitative laws, something that monadism and
Wolffian science were unable to provide. Euler's religious leanings might also
have had a bearing on his dislike of the doctrine; he went so far as to label
Wolff's ideas as "heathen and atheistic".
Much of what is known of Euler's religious beliefs can be deduced from his
Letters to a German Princess and an earlier work, Rettung der Göttlichen
Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine
Revelation against the Objections of the Freethinkers). These works show that
Euler was a devout Christian who believed the Bible to be inspired; the Rettung
was primarily an argument for the divine inspiration of scripture.
There is a famous legend inspired by Euler's arguments with secular
philosophers over religion, which is set during Euler's second stint at the St.
Petersburg academy. The French philosopher Denis Diderot was visiting Russia on
Catherine the Great's invitation. However, the Empress was alarmed that the
philosopher's arguments for atheism were influencing members of her court, and
so Euler was asked to confront the Frenchman. Diderot was informed that a
learned mathematician had produced a proof of the existence of God: he agreed to
view the proof as it was presented in court. Euler appeared, advanced toward
Diderot, and in a tone of perfect conviction announced this non-sequitur: "Sir,
, hence God exists—reply!" Diderot, to whom (says the story) all mathematics was
gibberish, stood dumbstruck as peals of laughter erupted from the court.
Embarrassed, he asked to leave Russia, a request that was graciously granted by
the Empress. However amusing the anecdote may be, it is apocryphal, given that
Diderot himself did research in mathematics. The legend was apparently first
told by Dieudonné Thiébault with significant embellishment by Augustus De
Morgan.
Commemorations
Euler was featured on the sixth series of the Swiss 10-franc banknote and on
numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was
named in his honor. He is also commemorated by the Lutheran Church on their
Calendar of Saints on 24 May—he was a devout Christian (and believer in biblical
inerrancy) who wrote apologetics and argued forcefully against the prominent
atheists of his time.
On 15 April 2013, Euler's 306th birthday was celebrated with a Google Doodle.
Selected bibliography
The title page of Euler's Methodus inveniendi lineas curvas.
Euler has an extensive bibliography. His best known books include:
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).
Two influential textbooks on calculus: Institutiones calculi differentialis (1755)
and Institutionum calculi integralis (1768–1770).
Letters to a German Princess (1768–1772).
Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive
solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin
title translates as a method for finding curved lines enjoying properties of
maximum or minimum, or solution of isoperimetric problems in the broadest
accepted sense.
A definitive collection of Euler's works, entitled Opera Omnia, has been
published since 1911 by the Euler Commission of the Swiss Academy of Sciences. A
complete chronological list of Euler's works is available at the following page:
The Eneström Index (PDF).