## Brief Chronology

** 1707 **
Leonhard Euler born in Basel (15th April).

** 1721 **
Entered his studies at Basel University

**1726 **
Won a prize from the French Academie des Sciences for a work on masting of ships.
(12 times won the biennial prize offered by the Paris Academy. 1742? shared with McLaurin and Daniel Bernoulli for essay on tides)

** 1726 **
Applied for Professorship of Physics at Basel.

**1727**
Moved to Petersburg

** 1741 **
Moved to Berlin

** 1766 **
Moved back to Petersburg

### Early Years

His father was a preacher who wanted him to follow in his footsteps and study theology, which he originally studied at University. He soon came under the wing of Johann Bernoulli.

### Basel University

In 1723 he received a Masters degree in philosophy and then began to study to be a clergyman, but he seemingly studied mathematics further under Johann Bernouilli, informally and inofficially. Johann apears to have been a major factor in persuading Euler snr to agree to a switch to Maths.

### After University

By as early as 1726 Euler had a paper in print, a short article on isochronous curves in a resisting medium.

In 1726, he won second place in the grand Prix of the French Academie des Sciences for a work on the placement of masts on ships.

Although he had accepted a place at the Petersburg Academy, he wrote an article on acoustics (now considered a classic) as part of an attempt to fill the vacant post of Professor of Physics at basel University but he was not chosen to go forward to the final stage (where the post was filled by lot)

From 1727-31, he was engaged in correspondence with Johann Bernoulli over the issue of logarithms of complex numbers.

he called complex numbers impossible numbers

### Petersburg

The Academy of Sciences was founded 1724, and for the first 20 years of its existence it employed no Russians. It was run originally by Johann Schumacher.

In 1726, Euler obtained a post in Petersburg with the help of Daniel and Nicholas Bernoulli, the sons of Johann.

He left Basel on 5 April 1727. He travelled down the Rhein, crossed the German states by post coach, then by sea from Lübeck, arriving in St Petersburg on 17 May 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.

He served as a medical lieutenant in the Russian navy from 1727 to 1730. In Petersburg he lived with Daniel Bernoulli.

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 to Basel in 1733, Euler was appointed to this senior chair of mathematics, at 26. The financial improvement which came from this appointment allowed Euler to marry on 7 January 1734 to Katharina Gsell, the daughter of an artist at the St Petersburg Gymnasium.

Euler's health problems began in 1735 when he had a severe fever and almost lost his life.

In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had :-

*
... lost an eye and [the other] currently may be in the same danger.*

i.e. he had lost the right eye .

He had received an offer to go to Berlin, but initially he preferred to remain - however a change in the political situation in Russia after Catherine 1 made the position of foreigners particularly difficult and (and along with problems with Schumacher) resulted in Euler changing his mind. At the invitation of Friedrich II ("the Great"), he went to Berlin where an Academy of Science was planned to replace the Society of Sciences. He left St Petersburg on 19 June 1741, arriving in Berlin on 25 July.

1731 Euler found a simplified method of arriving at an approx 1.644934

In 1734, he showed that

1735 Basel Problem, posed by Pietro Mengoli in 1694. Brought to attention by Jakob in 1689.

1735 exact, Wallis formula as corollary

### Berlin

1741 Political problems in Russia led to him moving to Berlin.

Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics. He deputised for Maupertuis in his absence and the two became great friends. Euler undertook an unbelievable amount of work for the Academy [1]:-

*
... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.
*

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.

By 1744, he had recognized the difference between transcendental and algebraic numbers.

1744 - director of mathematical section of Berlin Academy.

In a prize paper of 1748 on the subject of inequalities of Jupiter and saturn he gave the full systematic treatment of the trigonometric functions.

In 1748, the Introductio appeared

Started off by integrating

Introductio - introduced the parametric representation of curves, where x and y are represented in terms of a third variable.

Introduced radian measure of angles (I)

state problems of insurance

design of canals and waterworks

He was still sending papers to be published in Petersburg.

He gave lessons to the princess of Anhalt-Dessau, niece of Friedrich. These lessons were
published as *Letters to a German Princess*.

In 1765, he stated the polar coordinates for radial and normal components of acceleration moving along a plane curve.

1766 first paper developed purely to p.d.e.s

Separation of variables

Euler-McLaurin generalization of Bernoulli's formula

Latin Square Sudoku

Mapertuis

Received abuse from Voltaire

### Return to Petersburg

In 1766, he returned to Petersburg at the request of Catherine the Great, who proclaimed "I am sure the Academy will revive due to this important acquisition and I can congratulate myself on the return of this great man".

Soon afterwards, in 1771, he became completely blind, but his mathematical output did not seem to suffer (his total output doubled after he became completely blind.

In 1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. 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

In 1773 Katharina died - in 1776 he married Katharina's half sister. .

Mambers of the Academy included 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. Yushkevich writes in [1]:-

After 1777, he used **i** to indicate the square root of minus 1 - previously he had used
the same letter to denote something else completely.

### Number Theory

Christian Goldbach. He became Professor of Maths at St Petersburg in 1725, moving to Moscow three years later to tutor the Tsar' son - the future Peter II. The first mention of the Goldbach Conjecture is made in letter he wrote to Euler in 1742. Every even number greater than 2 can be written as the sum of two prime numbers.

Amicable numbers. These are numbers (m and n, say) such that the sum of the proper divisors of m is n, and vice-versa. Only 3 had been discovered previously - 220 and 284 are the smallest. Euler supplied 59 more pairs.

### Complex Number Theory

In 1743, he published the formulae (in *Miscellanioa Berolinesia*)

cos θ = ½ (e^{iθ} + e^{-iθ})

sin θ = (1/2*i*) (e^{iθ} - e^{-iθ})

In 1748 he re-discovered a result of Roger Cotes from 1714 (which follows from above also)

*i*x = log_{e} (cos x + *i* sin x)

leading to

e^{ix} = cos x + *i* sin x
which is known as the **Euler Identity **

when x = *π*

e^{iπ} = -1

He stated De Moivre's Theorem in its 'modern' form, and also generalized it for all real n - De Moivre assumed it was only valid for n an integer > 0.

### Logarithms

In 1749, he produced an article on the logarithms of complex numbers, entitled
*'De la contoversie entre Messrs Leibnitz et Bernouilli sur les logarithmes negatifs
et imaginaires'*. He started off from

d(ln x ) = dx/x

but disagreed with Leibniz's belief that this was only applicable for positive x. He showed it was applicable for both positive and negative x, but pointed out that this just showed that ln(-x) and ln(x) differ by a constant. Furthermore

ln (-x) = ln (-1.x) = ln (-1) + ln x

and stated that although previously Bernoulli had assumed that log(-1)=0, stemming from the conclusion that ln(-x) = ln x, this had not been proved specifically.

Leibniz had argued from the series for log(1+x) that log(-1) is not equal to 0, but the series he had produced was diverging and Euler showed that Leibniz's procedure was invalid.

When Euler came up with the correct form for the logarithm, it was not generally accepted.

Use

^{iπ}log e

^{iπ}log (-1) = -π

e^{x} = &\Sigma;^{ÝE;}_{r=0}

arrived at series of Numbers and M

ln(1+x)

-1 < x <= 1

ln((1+x)/(1-x))

### Alpha and Beta Functions

In 1781, Euler let t=log x, giving

The integral already considered by Wallis

was referred to as the first Eulerian integral by Legendre, but became standardized as

the Beta Function.

The expression

was originally called the second Eulerian integral. Legendre christened it the Gamma function. Euler prersented it in its full modern form

Euler also discovered the relationship between the two integrals

### Algebra

Fermat's Last Theorem, Euler proved it correct for n=3 and n=4

Proof of fundamental theorem of algeabra

### Geometry

Euler Line. Centtroid, orthocenter and circumcenter fall on the same line

### Trigonometry

He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done.

### Euler Coefficients

### Differential Equations

#### Euler-Cauchy Equation

### Partial Differential Equations

In 1727, Johann Bernoulli by using a 'string of beads'. D'Alembert modified this to produce wave ewquation as we now know it

-Equation HERE

1746

shortly after actually seeing this paper 'On the Vibration of Strings' of May 1748, man diff at end between Euler and D'Alembert. Euler > all kinds of initial curves i.e. non-analytic solutions. D'Alembert only analytic curves and solutions.

### Mechanics

Analytical Mechanics, Rigid Body Mechanics

### Acoustics

1759 three papers on sound 'On the Propagation of Sound'

### Optics

influenced design of telescopes and microscopes

He favored the wave theory of light rather than particle theory, at a time when very few people took the wave theory seriously.

Wave equation (cf. d'Alembert)

### Fluids

Found the d.e. for the motion of an ideal flow. The differential equation of continuity

differential equation of motion

In 1752 'Principle of the Motion of Fluids', incompressible

1755 General Principle of Motion of Fluids

Studied compressibility, stated that resistance increased at high speed because of compressibility of air and advocated pointed objects (in line with Robins, experiments by Benjamin Robins)

### Moon

Tobias Mayer was able to use Euler's Equations to produce tables of the Moon, which he sent to London in 1754. Trials were interrupted by the Seven Year's War but Mayer produced improved tables before his death in 1762. His widow was eventually awarded £3000 by the British Government, and Euler also received £300, which was apparently a great surprise for him. The tables allowed Nevil Maskelyne in 1766 to publish the first edition of *The Nautical Almanac*, which appeared annually with the intention of allowing sailors to determine their longitude by the method of lunar distances.

### Euler's Constant

γ = 0.577218

Still do not know whether it is rational or irrational.

### Symbols

e appeared in print for the first time in *Mechanica*

If you play agame of snap with two ordinary packs of cards, the chance of getting thru the pack without a pair of identical cards being turned up at the same time is almost exactly 1/e.

primarily responsible for the symbol π

Used in print in 1777, although the book was not published until 1794. The symbol was adopted by Gauss.

f(x)

Sigma

### Family

His father, Paul, was a clergyman who had been taught some maths by Jakob Bernooulli while a student at Basel University. He married Margaret Brucker and moved to Riehen when Leonhard was one (Riehen is now a part of Basel). He had 13 children although only five survived infancy (3 sons and 2 daughters). Johann Albrecht Euler became the professor of physics at the Academy in St Petersburg in 1766 (becoming its secretary in 1769) and Christoph Euler had a military career

### Books

** Mechanica (in full - Mechanica sive motus scientia analytice exposita - Mechanics or motion explained with analytical science i.e. calculus)).
**

*(1736)*Newtonian mechanics presented in analytic form for the first time

** Essay on Fire **
*(1738)* heat

**Neue Grundsatze der Artillerie **
*(1738)* translation of book published in London in 1742 by Benjamin Robins. 5 times larger at 720 pages.

** Introductio in Analyses Infinitorum **
*(1748)*

** Scientia Navalis **
*(1749)* theory of tides and design and sailing of ships

** Institutiones Calculi Differentialis **
*(1755)*

** Vollständige Anleitung zur Algebra **
*(1768, Russian; 1770 in Deutsch)*

** Institutiones Calculi Integralis **
*(1768-70)*

** Elements of Algebra **
*(1770)*

** Letters to a German Princess **
based on the lessons he gave to the Princess of Anhalt-Dessau, niece of
Friedrich II.

** Theorie complete de la construction et de la manoeuvre des vaisseaux **
*(1773)* theory of tides and design and sailing of ships

** Meditato in experimenta explosione tormentorum nuper instituta **
*(1862)* vertical motion of spherical bodies

** Opera Omnia **
*(1911)* Teubner and Füsli

### Grand prix of Paris

- 1738, shared the first prize
- 1740, shared the first prize
### Glossary

### References

- He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis.
- He introduced beta and gamma functions
- integrating factors for differential equations.
- lunar theory with Clairaut
- the three body problem
- He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).
- the notation f(x) for a function (1734)
- e for the base of natural logs (1727)
- i for the square root of -1 (1777),
- π for pi
- σ for summation
- $delta; etc , notation for finite differences

His work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime. Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function k

He proved another of Fermat's assertions, namely that if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1, in 1749.

In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation

(1/ns) = to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result p/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...

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. In particular he wrote [60]:- Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.

He then goes on to describe what is now called the Euler-Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:- ... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.

Euler replied:- ... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.

Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b300 from the British government for his theoretical contribution to the work.

Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:- ... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.

However, according to [8] the work was:- ... for musicians too advanced in its mathematics and for mathematicians too musical.

Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. 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.