Leonhard Euler (17071783)
Leonhard Euler was the most published mathematician of all time. There is
probably not a single branch of mathematics known during his lifetime which he did not
influence. If a difficult problem arose, Euler was generally consulted, and could often
solve it.
Euler was born near Basel, Switzerland and raised in the village of Riehen. His
father was a Protestant Minister, and his mother was also from a clerical family.
He was expected to follow his father into the clergy. He was an able
student, mastering languages and mathematics and a memory for matters of detail.
He entered the University of Basel at the age of 14. A Professor
of mathematics there was Johann Bernoulli (16671748), arguably the world's
greatest active mathemtician. Euler became a good friend of Bernoulli, who
became his mentor. Both men appeared to have inspired each other greatly
during their regular meetings.
He obtained a Bachelor of Arts and Master of Philosophy Degree from Basel
University.
He did afterwards enter divinity school but found the call of mathematics to be greater.
Bernoulli's son Daniel (17001782) moved to Russia in 1725 to take up a position
at the newly formed St Petersburg Academy. In the following year Euler was invited to
join him and he arrived in the year 1727. Living at the same home as Daniel Bernoulli
Euler was able to discuss and collaborate with him extensively.
At about this time his work on exponential functions led him to introduce the constant e, the symbol for the important transcendental number 2.71828... .
He also discovered the result
linking e, π, and i, the
symbol he developed for the square root of 1.
In 1733 Bernoulli moved to a Chair in Switzerland. This enabled Euler
to move from a post in Physics to take up Bernoulli's Chair in Mathematics.
He married Katherine Gsell (d. 1773) and they had 13 children, only five of
whom reached adolescence and three of whom survived
him.
This was a period during which Euler did much consulting work for the Russian
Government and publishing many results, including the solution to the much
debated Basel Problem in 1735 (see below).
In 1736 Euler solved the Königsberg Bridges Problem, which is described
below. This solution established the branch of mathematics now known as Graph
Theory, and which is the basis of the understanding of networks, including
computer networks.
Whereas Euler's research continued at an astonishing pace, there were some
problems encountered during the next period, including the death of Catherine I,
a subsequent backlash against the foreigners who dominated the Academy, and in 1738
the first signs of failing eyesight, with the loss of sight from his right eye.
During this time he still produced groundbreaking works, including work on
shpbuilding, acoustics, music, Classical Number Theory in
collaboration with Christian Goldbach (16901764), Analytic Number Theory,
and a text Mechanica presenting Newtonian mechanics in
a framework of Calculus.
In 1741, while still in the employ of the St Petersburg Academy, Euler
and his family moved to Berlin at the invitation of Prussia's Frederick
the Great (17121786) to join the revitalised Berlin Academy. He was
to stay in Berlin until 1766.
In Berlin he published his most widely read book, Letters to a German Princess,
which contains over 200 "letters" inspired by the instruction he was required to
give to the Princess of Anhalt Dessau. The letters cover a wide range of topics
in mathematics and physics, including the explanation of commonly observed phenomena.
It is a classical example of excellent writing to explain science to the masses.
During his time in Berlin, Euler kept in excellent contact with the St Petersburg
Academy, which was still paying him, and fell out gradually with Frederick the Great.
While in Berlin he also fell out with the other leading identity Voltaire
(16941778) who was more in favour with the King and was rather
disdainful of Euler, who had not learned philosophy.
While absent the St Petersburg Academy had also been revitalised under
the influence of Catherine the Great (17291796) and in
1766 he returned to St Petersburg for the remainder of his life.
Euler's work in St Petersburg continued at a breathtaking pace despite the death
of his wife (he later married her halfsister) and the substantial loss
of sight in his good eye, forcing him to dictate all of his writings to scribes.
He died of a massive hemhorrhage on the afternoon of 18 September 1783, a day
on which he had still been working at his normal pace. The St Petersburg
Academy Journal had a massive backlog of his work to publish, a task which
took a further 48 years to complete.
The complete works of Euler, Omnia Opera, was only
published in the latter part of the twentieth century after a commitment
by the Swiss Academy of Science in 1909. It is very expensive and can only be found in major
research libraries. It comprises 29 volumes on mathematics, 31 on mechanics and astronomy,
12 on physics and other topics, 8 on correspondence. Further volumes on manuscripts
is still to appear.
Euler's work took him into virtually every branch of mathemtics and physics known
during his life. Here we briefly discuss some problems for which he became famous.
The individual problems discussed below indicate the flavour of Euler's
work and do not indicate his massive contribution to what we now call applied
mathematics.
The Königsberg Bridges Problem
Königsberg (now the Russian city of Kaliningrad,
on the Baltic Sea) was a city in East Prussia laid out on the River Pregel, which had split
into two courses forming two islands. The various regions of the
city were connected by bridges.
The left hand diagram shows the layout of the river and the seven bridges. The
citizens of Königsberg had tried unsuccessfully to find a route along which they
could tour the entire town, traversing each bridge exactly once.
Euler solved the problem by showing why such a route could not be found.
Essentially he showed that the region could topologically be considered as
having four regions A, B, C and D as shown in the left
hand diagram. He then demonstrated that a solution of the problem could
then be considered equivalent to finding paths through the network in the
right hand diagram. If such a solution was to be possible, each path of the network
would be travelled exactly once. The points A, B, C and D could be called
nodes of the network. Whereas a node could be visited more than once in a
successful tour Euler showed that successful tours depended on nodes being
arrived at and departed from different routes each time, requiring even numbers of routes
connecting each notes (number of arrivals matching number of departures).
In the case of the Königsberg Bridges problem, as can be seen from the right hand
diagram, all of the nodes have an odd number of connecting routes, making the
solution impossible.
As an extension of the above, Euler developed a theory
for networks, in which lines join nodes and enclose regions.
Euler developed the formula
which is satisfied by a network.
For example, this is obviously satisfied by the right hand diagram
describing the bridge network above, in which R=4, V=4 and L=7.
The Basel Problem
For several decades there was much speculation about the value of the sum of the
infinite series
This problem was known as the Basel
problem. It seemed clear that the real sum was a number
in the vicinity of 8/5. The problem had received much attention
from Pietro Mengoli (16251686) and Jakob Bernoulli (16541706), brother of
Johann and uncle of Daniel. Euler was able to solve this problem in 1735, when
he caused a major sensation by showing that the sum had the unexpected value
The card problem
In his work on infinite series, Euler also investigated the constant
which bears his name and showed how it can be used for estimating
the sum of the finite series
since
This applies to computing the expected number of packets
of chewing gum, cereal, etc one needs to buy when the manufactures place a hidden card
inside the packet, numbered as to form a collector's set.
If there are n cards in the set it can be shown
that the expected number of packets N one needs to buy before
completing the set is
For n=25, it turns out after much calculation that the exact value
is N=95.4. However Euler's formula gives with much less effort
the very accurate approximation
94.9.
The Gamma function (Extension of the factorial function)
Euler extended the concept of factorial, so useful in combinatorics,
infinite series and elsewhere, defined as
where n is integer. Euler was able to show
that the function
satisfied
for n integer and had wide power as a generalised factorial function,
that is for arguments which are not necessarily integer.
The function also has the unexpected value
Fermat's Last Theorem
Pierre de Fermat (16011665) posed one of the most famous Theorems
in Mathematics, stating that the equation
has no integer solutions for x, y and z when n is a positive integer for n greater than 2.
Fermat himself was able to construct an argument to show that there
was no solution for n=4. The next advance was not until 1765, when
Euler was able to announce a proof for the case n=3 to his friend
Christian Goldbach.
In later years Dirichlet, others and computers were able to extend the cases,
but it was not until virtually the end of the twentieth century that this
theorem was to be finally proved, by the English mathematician Andrew Wiles.
Written by Peter Taylor, June 1997,
revised March 2001.
References
 Boyer, Carl B, revised by Merzbach, Uta C, A History
of Mathematics, 2nd ed, Wiley, New York, 1991.
 Dunham, William, Euler: The Master of us all, Dolciani Mathematical
Exposition No 22, Mathematical Association of America, Washington DC, 1999.
 Green DR, Euler, Mathematical Spectrum, 15, 3, Sheffield, 1982/3.
 Gullberg, Jan, Mathematics: From the Birth of Numbers, Norton,
New York, 1997.
 Singh, Simon, Fermat's Last Theorem, Fourth Estate, London, 1997.
This T Shirt, which celebrates Euler's solution to the Königsberg
Bridges problem, is available from the AMT Publishing.
