**Leonhard Euler** was a Swiss mathematician who made enormous contibutions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory.

- Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there.
- Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel.
- This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school.
- Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered.
- In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton.
- Euler completed his studies at the University of Basel in 1726.
- By 1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium.
- The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate.
- However, Euler now had to find himself an academic appointment and when Nicolaus(II) Bernoulli died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology.
- Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair.
- As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727.
- Through the requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered.
- Euler served as a medical lieutenant in the Russian navy from 1727 to 1730.
- In St Petersburg he lived with Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland.
- Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.
- Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics.
- We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career.
- The publication of many articles and his book Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work.
- Euler's health problems began in 1735 when he had a severe fever and almost lost his life.
- He also argues that a portrait of Euler from 1753 suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind.
- Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain.
- By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740.
- Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg.
- Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where an Academy of Science was planned to replace the Society of Sciences.
- Even while in Berlin Euler continued to receive part of his salary from Russia.
- Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics.
- The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ...
- During the twenty-five years spent in Berlin, Euler wrote around 380 articles.
- In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not the title of President.
- The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour.
- Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in 1763.
- However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.
- In 1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure.
- Soon after his return to Russia, Euler became almost entirely blind after an illness.
- A cataract operation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind.
- Amazingly after his return to St Petersburg (when Euler was 59) he produced almost half his total works despite the total blindness.
- Euler of course did not achieve this remarkable level of output without help.
- He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in 1766 (becoming its secretary in 1769) and Christoph Euler who had a military career.
- Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772.
- Fuss, who was Euler's grandson-in-law, became his assistant in 1776.
- The scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples.
- For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the motion of the moon, published in 1772.
- Fuss helped Euler prepare over 250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in 1776.
- After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.
- Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it.
- Let us examine in a little more detail some of Euler's work.
- Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem.
- Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755.
- Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series.
- He then goes on to describe what is now called the Euler-Maclaurin summation formula.
- Some of Euler's number theory results have been mentioned above.
- One could claim that mathematical analysis began with Euler.
- In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions.
- Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography.
- In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences.
- In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions.
- Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively.
- As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.
- The calculus of variations is another area in which Euler made fundamental discoveries.
- Problems in mathematical physics had led Euler to a wide study of differential equations.
- When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.
- Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces.
- Many unpublished results by Euler in this area were rediscovered by Gauss.
- Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.
- In 1736 Euler published Mechanica which provided a major advance in mechanics.
- Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.
- In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium.
- Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science.
- Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.
- Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version.
- In 1765 Euler published another major work on mechanics "Theoria motus corporum solidorum" Ⓣ(Theory of the motion of solid bodies) in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion.
- He considered the Euler angles and studied rotational problems which were motivated by the problem of the precession of the equinoxes.
- Euler's work on fluid mechanics is also quite remarkable.
- He published a number of major pieces of work through the 1750s setting up the main formulae for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid.
- Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill.
- Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon.
- In 1765 Mayer's widow received £3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received £300 from the British government for his theoretical contribution to the work.
- Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735.
- Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography.

Born 15 April 1707, Basel, Switzerland. Died 18 September 1783, St Petersburg, Russia.

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Parts: 1

Proofs: 2

Theorems: 3 4 5 6

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive