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# Leonhard Euler

## Introduction

Leonhard Euler (April 15, 1707, Basel – September 18, 1783, St. Petersburg) was undoubtedly the most prolific of famous mathematicians. Born in Switzerland but most closely associated with the Berlin of Frederick the Great and the St. Petersburg of Catherine the Great. His career covered a highly productive period in mathematics when the newly discovered calculus was being extended in all directions at once, and he made significant contributions to all areas of mathematics. More than any other individual, Euler was responsible for the notation of modern mathematics, including $\displaystyle{ \large \pi }$,$\displaystyle{ \large e }$ and $\displaystyle{ \large i }$, the summation notation $\displaystyle{ \sum }$ and the standard function notation $\displaystyle{ f(x) }$, and was in fact the first to use the term 'function' in this context. He is the only mathematician to have two numbers named after him.

In the 17th century, the age of Fermat and Mersenne, there was a distinct lack of interest in providing logical proofs of the discoveries that were made. In an increasingly mechanised and industrial world, results were judged by their practical application and methods did not necessarily meet the ancient Greek ideal of proof exemplified by Euclid. Then, along came Euler, who provided explanations for many of the discoveries made by earlier mathematicians but which they had failed to account for. Euler's methods would play a significant role in opening up new windows onto our understanding, not just of the primes, but in many other areas of mathematics as well.

## Youth and Education

Leonhard Euler was the son of a Lutheran minister, Paul Euler, who had studied theology at Basel University and attended lectures given by Jacob Bernoulli, the founder of a whole dynasty of mathematician sons. In fact, Paul Euler had lived in the Bernoulli house while an undergraduate in Basel, and this association with a great mathematical family was to influence not just his life, but also that of his as yet unborn son. Leaving university, Paul Euler married Margaret Brucker, the daughter of another Lutheran minister and they made their home in Basel where, on 15th April 1707 Leonhard Euler was born.

When Leonhard was one year old the family moved to Riehen, not far from Basel, and this is where he was brought up. He was sent to school in Basel and lived with his maternal grandmother. His interest in mathematics already sparked by his father's teaching, Leonhard studied the subject in his own time, and it is quite possible that his father was unaware of this interest when in 1720 he sent his son to Basel University. The idea was that he would first receive a general education before moving on to the study of theology in preparation for joining his father in the church.

Leonhard quickly made contact with Johann Bernoulli, who had succeeded his father as professor of mathematics, and was given permission to visit him each Sunday afternoon for mathematical studies. In 1723 Euler completed the general education, obtaining a Master's degree in philosophy with a treatise in which he contrasted the philosophical ideas of Newton and Descartes. In the autumn of that year he continued to follow his father's wishes and commenced his study of theology. But he could not find the same enthusiasm for it that he found in mathematics, and with some help and encouragement from Johann Bernoulli, he obtained his father's permission to change from theology to mathematics.

Euler completed his studies in 1726 and the following year submitted an entry for the Grand Prix of the Paris Academy, which that year was for the best arrangement of masts on a ship. The Grand Prix went to Bouguer, an expert on ship design, but Euler's entry won him second prize and confirmed his entry into the world of mathematics. The mid-eighteenth century was a time of court patronage. Rulers throughout Europe were keenly aware of the potential for scientific and mathematical skills to boost their countries military and industrial worth, and surrounded themselves with intellectuals as a mark of their standing and enlightenment. It was therefore, somewhat fortuitous that in July 1726 Nicholas Bernoulli died in St. Petersburg creating a vacancy at the St. Petersburg Academy of Sciences. No doubt some family strings were pulled somewhere and Euler was offered the post. This was not, however, Euler's first choice. He applied for the chair in Physics at Basel University, and as part of his submission wrote a paper on acoustics that went on to become a classic in its field. The final decision was made by the drawing of lots, and with lady luck against him, on 5th April 1727 - just 10 days before his 20th birthday - Leonhard Euler left Basel for St. Petersburg.

## St. Petersburg, the first time

It took six weeks to complete the journey, taking a boat down the Rhine, crossing Germany in coaches and then another boat from Lubeck saw him arrive on 17th May in the sophisticated milieu of an intellectual academy just two years after it had been founded by Catherine I, the wife of Peter the Great. Already at the academy were the geometer Jakob Hermann, a distant relative who also hailed from Basel; Daniel Bernoulli, another of the vast Bernoulli clan whose interest lay in applied mathematics; Christian Goldbach, he of the Goldbach Conjecture; F. C. Maier working in trigonometry and the astronomer and geographer J. N. De-Lisle.

Euler served as a medical lieutenant in the Russian navy for the first three years, becoming Professor of Physics at the Academy in 1730. This allowed him to give up his naval post and he moved to St. Petersburg where he lived with Daniel Bernoulli, who held the senior chair in mathematics. Bernoulli left St. Petersburg to return to Basel in 1733 and Euler was appointed to replace him, boosting his finances sufficiently for him to marry, on 17th January 1734, Katharina Gsell, the daughter of a painter from St. Petersburg. They had thirteen children together although only five survived their infancy. In later life, Euler often claimed that he made many of his most important discoveries with a baby on his knee and several children playing round his feet.

The year after marrying, Euler's health problems began. In 1735 he had a severe fever, which he kept secret from friends and relatives in Basel. In his autobiographical writing Euler said that his eye problems began in 1738 with eyestrain caused by his cartographic work. At least one author, however, has argued that the fever was a symptom of the eyestrain, and another claimed that the eye problem was caused by the severity of the climate. What is certain is that by 1756 he was almost blind in one eye - the famous portrait of him by Emanuel Handmann clearly shows his right eye to be almost closed. Whether this was caused by or merely exacerbated by either cartographic work or the climate is pure conjecture but neither of these would have helped an already weakened eye. Soon after 1766 he suffered an unknown illness and became totally blind. There were a few days respite in 1771 when an operation to remove a cataract appeared to be successful, but it was not to be and Euler was totally blind for the rest of his life.

After taking up his post as professor of physics at the St. Petersburg Academy he carried out state projects in cartography, magnetism, fire engines and shipbuilding. At the same time the outline of his approach to mathematical research becomes apparent. He saw studies of number theory as being vital to the foundations of calculus, and that special functions and differential equations were essential to the solution of the more concrete problems of rational mechanics. In 1736 he published his first book, Mechanica, which for the first time translated Newtonian dynamics into the language of Mathematical Analysis.

## Berlin Interlude

By 1740, Euler's reputation had spread throughout Europe, not least by having twice won the Grand Prix of the Paris Academy, in 1738 and 1740, though on both occasions he shared the prize with others. This brought an invitation from Frederick the Great for him to go to Berlin to help found the planned Berlin Academy of Science. This proved an irresistible opportunity and Euler left St. Petersburg in June 1741 arriving in Berlin on 25th July. Soon after his arrival he wrote to a friend, "... the king calls me his professor, and I think I am the happiest man in the world.". His happiness was to last for 25 years.

Maupertuis was the president of the Berlin Academy and Euler was appointed Director of Mathematics. His work, however, covered much more than mathematics including supervising the observatory, the botanical gardens and the publication of calendars and maps. He sat on the committee for the academy library, did work for the government on lotteries, pensions and artillery, and supervised works on the plumbing system at Sans Souci, the royal summer palace. In 1749 he was charged by the king with a project to correct the flow of the Finow Canal.

Through all this he kept up a steady stream of mathematical work, writing books on the calculus of variations, on ballistics and artillery and on the calculation of planetary orbits. He also wrote over 350 other mathematical papers and articles during this period including an early example of popular mathematics, Letters to a Princess of Germany, published in three volumes between 1768 and 1772, and which remained a standard treatise on the subject for half a century. The princess of the title is popularly assumed to have been the Princess of Anhault-Dessau to whom Euler gave lessons on physics and mathematics.

In 1748 he published in two parts his Introduction to Infinite Analysis, which was intended to serve as an introduction to pure analytical mathematics. The first part contains the bulk of the subject matter found in modern textbooks under algebra, the theory of equations and trigonometry. In the algebra he paid particular attention to the expansion of various functions in series and to the summation of a given series. He explicitly stated that a series cannot be safely employed unless it is convergent. In the trigonometry he expanded on the idea of F. C. Maier (1727) and Johann Bernoulli (1734) that the subject was a branch of analysis and not simply an appendage of astronomy or geometry. He also introduced the trigonometric functions and showed that the trigonometric functions and exponential functions were connected.

The second part of the Infinite Analysis is on analytical geometry. Euler commenced this part by dividing curves into algebraic and transcendental curves, and established a variety of propositions that are true for all algebraic curves. He then applied these to the general equation of the second degree in two dimensions, showed that it represents the various conic sections, and deduced most of their properties from the general equation. He also considered the classification of cubic, quartic and other algebraic curves. He next discussed the question of surfaces represented by the general equation of the second degree in three dimensions, and how they may be discriminated one from the other: some of these surfaces had not been previously investigated. In the course of this analysis he laid down the rules for the transformation of co-ordinates in space. This was also the earliest attempt to bring the curvature of surfaces within mathematics, and the first complete discussion of tortuous curves.

The use of a single symbol to denote the number 3.14159... appears to have been introduced about the beginning of the eighteenth century. W. Jones in 1706 represented it by $\displaystyle{ \pi }$ a symbol that had been used by Oughtred in 1647, and by Barrow a few years later, to denote the circumference of a circle. Johann Bernoulli represented the number by c. Euler in 1734 denoted it by p, and in a letter of 1736 (in which he first stated the closed form of the sum of the reciprocals of the squares of the integers) he used the letter c. Christian Goldbach in 1742 used $\displaystyle{ \pi }$ and following the publication of Euler's Analysis in 1748 the symbol has been universally accepted.

In 1759 Maupertuis died and Euler assumed the role but not the title of President of the Academy. By 1763 Euler was no longer the king's favourite and the title of President was offered instead to d'Alembert, with whom Euler had argued over scientific matters. However, d'Alembert refused to move to Berlin and the king retained overall control of the academy. Disturbed by the king's interference, in 1766 Euler decided to return to St. Petersburg where, unbelievably, they had kept his post open for him for 25 years.

## Back to St. Petersburg

From this point in his life, when he was 59 years of age, he produced almost half his prolific output even though as previously noted he was totally blind for the rest of his life. On returning to St. Petersburg in 1766, Euler's son, Johann Albrecht Euler, was appointed to the chair of physics at the St. Petersburg Academy - becoming secretary of the academy in 1769 - and he also doubled as an unofficial assistant with practical aspects of his father's work. Euler also had two assistants appointed by the academy, W. L. Krafft and A. J. Lexell. In 1772 these were joined by Euler's grandson-in-law, Nicholas Fuss who was a young mathematician recently graduated from Basel University. These assistants were more than just secretaries taking dictation from their blind master. He discussed with them the outline of his work, explained his ideas and they were expected to calculate any tables and compile examples. Notwithstanding this help his output was more than prolific. To again quote Fuss from his eulogy, -He would give up his food before work, and now work became a perpetual habit-.

The academy of St. Petersburg was still bringing out un-published works by Euler 50 years after his death, and his complete works runs to over 80 volumes. It has been estimated that simply to write out Euler's life work by hand would take 50 years of eight-hour days, which leaves very little time for thinking, for conjecturing, or for playing with grandchildren.

On the morning of 18 September 1783 Euler gave a mathematics lesson to one of his grandchildren. After lunch he spent a couple of hours calculating the motion of balloons, then spent some time discussing with his assistants the orbit of the newly discovered planet Uranus. Around 5 O'clock, while seated in the garden playing with a grandson he suffered a brain haemorrhage and around 11 O'clock that night he died. He was survived by his second wife, Salome Abigail Gsell, the sister-in-law of his first wife who died in a fire in 1771, by three of his thirteen children and by twenty-six grandchildren.

## Mathematics

The basic language of mathematics, its symbols and notation owes so much to Euler; $\displaystyle{ \large e }$ for the base of natural logarithms in 1727, f(x) for functions in 1734, $\displaystyle{ \sum }$ for summation and as previously mentioned $\displaystyle{ \large\pi }$ for the ratio of the circumference of a circle to its diameter in 1755, $\displaystyle{ \large i }$ for the $\displaystyle{ \large\sqrt{-1} }$ in 1777, the notation for finite differences $\displaystyle{ \large\delta y }$ and $\displaystyle{ \large\delta^2 y }$ and many more.

He was the first to consider $\displaystyle{ \large \cos }$ and $\displaystyle{ \large \sin }$ as functions rather than as chords as all those since Ptolemy had done. He made fundamental contributions to number theory, calculus and geometry, and combined Newton's fluxions with Liebniz's differential calculus to almost single-handed create modern mathematical analysis.

Note: I have not finished this, I just wanted you to know I had not forgotten about it.