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[L Euler]

Leonhard Euler (1707-1783)

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 (1667-1748), 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 (1700-1782) 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

[relation]

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 ground-breaking works, including work on shp-building, acoustics, music, Classical Number Theory in collaboration with Christian Goldbach (1690-1764), 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 (1712-1786) 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 (1694-1778) 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 (1729-1796) 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 half-sister) 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.

[Königsberg Bridges precinct]

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

[Königsberg Bridges precinct]

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

[The Basel 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 (1625-1686) and Jakob Bernoulli (1654-1706), 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

[relation]

The card problem

In his work on infinite series, Euler also investigated the constant

[Gamma]

which bears his name and showed how it can be used for estimating the sum of the finite series

[relation]

since

[sum]

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

[relation]

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

[relation]

where n is integer. Euler was able to show that the function

[relation]

satisfied

[relation]

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

[relation]

Fermat's Last Theorem

Pierre de Fermat (1601-1665) posed one of the most famous Theorems in Mathematics, stating that the equation

[relation]

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.

[Euler T Shirt]

This T Shirt, which celebrates Euler's solution to the Königsberg Bridges problem, is available from the AMT Publishing.

 

 
 
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