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Liesbeth De Cock(left)           4ECWI

Ilse de Saegher(middle)          4ECWI       ilse_de_saegher@hotmail.com

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Natalie Van Eynde                6MTWE     vaneyndenatalie@hotmail.com

Liesbeth Van Raemdonck       6SPWE       liesje22@hotmail.com

llncabnelekl.jpg (10775 bytes)

Leonhard Euler.

Eulerfig_3.jpg (27634 bytes)

 

Born: 15 april 1707 in Basel, Switzerland

Died: 18 sept 1783 in St. Petersburg, Russia

Contents:

1. His life.
2. Euler's number e.
3. Euler's straight line.
4. The proposition of Euler or: Euler's formula.
5. The ninepointcircle
6. Other work from Euler
7. Sources

        1.His life.

Leonhard Euler was born in Basel (Switzerland), on 15 April 1707 and he died on 18 September 1783, in St Petersburg (Russia).

The family of Leonhard Euler moved to Riehen when Leonhard was one year old and it was in Riehen, not far from Basel, that Leonhard was brought up.

Leonhard was sent to school in Basel and during this time he lived with his grandmother. This school was a rather poor one, by all accounts and Euler learnt no mathematics at all from the school. However, his interest in mathematics had certainly been sparked by his father's teaching, Euler read mathematics texts on his own and he took some private lessons. Euler’s father wanted his son to follow him into the church and he sent him to the University of Basel to prepare his son for the ministry.

basel.jpg (54655 bytes)

He graduated from the University of Basel in 1724 where he studied theologie and Hebrew. During this time at the school, he was privately tutored in mathematics by Johann Bernoulli. Johann was so impressed by his pupil's ability that he convinced Euler's father to allow Leonhard to become a mathematician.

The fact that Euler’s father had been a friend of Johann Bernoulli’s in their undergraduate days undoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in 1726. He had studied many mathematical works during his time in Basel. These works were from Varignon, Descartes, Newton, Galileo, von Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis.

DEs8.jpg (14514 bytes)

René Descartes

Galileo_5.jpg (11134 bytes)

 Galileo Galilei

Euler took up a position at the Academy of Sciences in St Petersburg, Russia. Euler was offered the post when Nicolaus Bernoulli died.

He accepted the post in November 1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, 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 he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727 and he arrived 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. 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 Euler to bring him tea, coffee and other delicacies from Switzerland.

Euler became professor of physics at the academy in 1730 and, since this allowed him to became a full member of the Academy, he was able to give up his Russian navy post.

Now 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.

The financial improvement which came from this appointment allowed Euler to marry, which he did on 7 january 1734, marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children alltoghether although only five survived their infancy.

Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing around his feet. The publication of many articles and his book Mechanica, 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. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered.

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 (® the right)and the other (® the left) currently may be in the same danger. However, Euler’s eyesight problems certainly started earlier and that the severe fever of 1735 was a symptom of the eyestrain.

By 1740 Euler had a very high reputation, after he won the Grand Prize of the Paris Academy in 1738 and 1740. On both occasions he shared the first prize with others. He worked day and night for three days to solve a problem. This problem was a public contest. He discovered that the Czar's government was far from democratic as he was followed by secret police. He looked for a way out. He found it in 1741, when he moved his family to Berlin to take over as director of mathematics at the Academy of Sciences under Frederick the Great. While in Prussia, his home was destroyed by invading Russian armies, but he was held in such high esteem by both countries that he was compensated for more than he lost.

For this remuneration he bought books and instruments for the St Petersburg Academy, he continued to write scientific reports to them, and he educated young Russians. Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics. He deputises for Maupertuis in his absence and the two became great friends.

During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on the calculus of variations, on the calculation of planetary orbits, on artillery and ballistics, on analysis, on shipbuilding and navigation, on the motion of the moon, lectures on the differential calculus and a popular scientific publication Letters to a Princess of Germany.

In 1759 Maupertuis died. Euler, who 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.

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, 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. Because of his remarkable memory, Euler was able to continue with his work on optics, algebra and lunar motion.

Amazingly after his return to St Petersburg he produced almost half his total works despite the total blindness. 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 and Christoph Euler who had a military career. Euler was also helped by two other members of the Academy, WL Krafft and AJ 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.

stpetersburg.jpg (30958 bytes)

Euler died on 18 September 1783, in St Petersburg (Russia). After his death, the St Petersburg Academy continued to publish Euler’s unpublished work for nearly 50 more years.

 

2.Euler's number e.

When we look for a function f: Rà R that is the same as her derivative f', than she has to be the same as all the derivatives f(n) , when n is an element of the natural numbers. When we choose f'(0) = 1 than we call this function the  

'Natural Exponential Function'

We write these function as etotxt.gif (905 bytes) with gim31t.gif (1484 bytes)

Both limits, which are finite and equal, we call them the number e, being the first character of the word exponential or the first character of the name "Euler".

To determine the value of e, we can think of the following situation:

Suppose we find an great investment which gives us, starting with one unit, at the end of one year 100% intrest. Then at the end of the year our invested capital has grown to {1 + 1/1}1 ( units). When we could add the acquired intrest after half a year to our initial capital and let this new capital grow again at the same rate for the next half year, then we would end up with {1 + 1/2}². Even better is to add the acquired intrest after 1/3 of a year and let it grow again compoundedly.

  the number of periods
    of interest a year
  the capital grows to    at 4 decimals precision
              
               1  
          
               2                   

               3                   

               4         

               5                   

               6                   

               7                   

               8                   

               9                   

               10        

           

        
          {1 + 1/1}1           

          {1 + 1/2}²         

          {1 + 1/3}³           

          {1 + 1/4}4        

          {1 + 1/5}5        

          {1 + 1/6}6

          {1 + 1/7}7    

          {1 + 1/8}8  

          {1 + 1/9}9   

          {1 + 1/10}10         

   

                 
                2

                2,25

                2,3704

                2,4414

                2,4883

                2,5216

                2,5465

                  2,5668

                  2,5812

                2,5937

 

ey=.gif (1130 bytes)etabset.gif (1434 bytes)etab1.gif (1303 bytes)etab2.gif (1397 bytes)

When the number of periods of interest in a year increases, than the interest per period decreases inversely proportional and the total capital goes to a limit which can be proved to be finite which gives us :

The real number e = 2,71828182845904523536028747135... = The number of Euler

Euler's number e is irrational and in fact transcendental. 2,71828182845904523536028747135 is an approximation of e to 30 decimals.      

To get a better insight of the result of lim(hà0)(1+h)1/h we can make a list where the base 1+h goes to 1  for h positive and negative. 

h

(1+h)1/h

h

(1+h)1/h

0,1

2,593742

-0,1

2,867972

0,01

2,704814

-0,01

2,731999

0,001

2,716924

-0,001

2,719642

0,0001

2,718146

-0,0001

2,718418

eplush1.gif (1335 bytes)  eplush2.gif (1426 bytes)  eminh1.gif (1354 bytes)  eminh2.gif (1457 bytes)

   Proof:

We will show that the sequence is monotone increasing and bounded above. If that was true, then it must converge. Its limit, by definition, will be called e for Euler's number.

First, we can use the binomial theorem to expand the expression

gImage16t.gif (4776 bytes)

Similarly, we can replace n by n+1 in this expression to obtain

gImage20t.gif (2352 bytes)

The first expression has (n+1) terms, the second expression has (n+2) term. Each of the first (n+1) terms of the second expression is greater than or equal to each of the (n+1) terms of the first expression, because

gImage25t.gif (1730 bytes)

But then the sequence is monotone increasing, because we have shown that

gImage26t.gif (1457 bytes)

 Next, we need to show that the sequence is bounded.
Again, consider the expansion having the following properties :

gImage19kt.gif (2476 bytes) gim40t.gif (1244 bytes) and
gim41t.gif (1222 bytes) g2ekImage18t.gif (1279 bytes)

Now we only need to estimate the expression

gim30t.gif (1255 bytes)

If we define

gImage27t.gif (1321 bytes)

then

gim32t.gif (1161 bytes)

so that, finally,

gim33t.gif (1574 bytes) for all n

But then, putting everything together, we have shown that

gim34t.gif (1648 bytes) for all n

Hence, Euler's sequence is bounded by 3 for all n and since the sequence is monotone increasing it must converge.

If n becomes infinite, than also n-1, n-2, n-3....etc and  as a result the expression

gim42t.gif (1872 bytes) becomes getalekortt.gif (1256 bytes) = getalesomt.gif (1068 bytes) = 2,7182818284... 

                                                      

3. Euler's straight line.

Definition.

In each triangle is the intersection of the altitudes (the orthocenter, H), the intersection of the medians (the centroid, Z) and the intersection of the perpendicular bisectors (the centre of the circumcircle, M) situated on one line. This line is called: Euler's straight line.

Moreover is HZ/ZM = 2

recht.gif (4409 bytes) stomp2.gif (5257 bytes) scherp.gif (5226 bytes)

_____Medians       _____Altitudes        _____Perpendicular bisectors           _____Euler's straight line            _____Circumcircle

Euler's straight line can be drawn in any triangle : a right triangle, an obtuse triangle or an acute triangle.

 

4. The proposition of Euler.

Euler relates the number of faces (F), vertices (V) and edges (E) of a polyhedron in a three- dimension space by the following equation:

F + V = E + 2

A polyhedron is a connected, spatial figure limited by planes.

There are only five regular polyhdera for which one we can use and illustrate this equation.

This rule can be used to show that there are no other possibilities by this five regular polyhedra.

For a higher dimension applies an analogous of this equation that relate the number of faces, the edges, the vertices and the sides of higher dimension.

The five polyhedra
  Four plane

 

Eight plane Cube 12-plane 20-plane
V 4
6 8 20 12
E 6
12 12 30 30
F 4 8 6

 

12 20

For example the cube kubus.gif (4329 bytes)

 5. The ninepointcircle

In a triangle, the following points are located on one circle:

    • The midpoints of the sides.
    • The endpoints of the altitudes.
    • The midpoints of the segments connecting the orthocenter with the vertices.

->This circle is called: The ninepointcircle or the circle of Feuerbach.

The centre of this circle (F) is the midpoint of the line (HM) between the intersection of the altitudes (H) and the intersection of the perpendicular bisectors (M) (= Euler's straight line).

Feuerb.gif (8007 bytes)

 

6. Other work from Euler.

Leonhard Euler was the most prolific writer of mathematics of all time. 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.

He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz’s differential calculus and Newton’s method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics and music.

He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies. We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), p for pi, å for summation ( 1755), the notation for finite differences D y and D 2y and many others.

Bernoulli_Daniel_2.jpg (26066 bytes)

Daniël Bernoulli

Leonhard Euler’s work in number theory seems to have been stimulated by Goldbach. Euler studied unproved results of Fermat and he also proved them. Euler also proved a problem that had been studied by Jacob Bernoulli, Johann Bernoulli, Daniel Bernoulli, Leibniz, Stirling, de Moivre, ... The problem was to find a closed form for the sum of the infinite series V (2) = S (1/n2). In 1737, Euler proved the connection of the zeta function with the series of prime numbers giving the famous relation.

In 1739 Euler had found the rational coefficients C in V (2n)=Cp 2n in terms of the Bernoulli numbers. Euler also studied Fourier series and in 1744 was he 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 + ... in a letter to Goldbach. He discovered the Cauchy-Riemann equations in 1777, although d’Alembert had discovered them in 1752 while investigating hydrodynamics.

In 1755, Euler published Institutiones calculi differentialis. In this work, Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions.

He also studied beta and gamma functions, which he had introduced in 1729. 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. The work: Methodus inveniendi lineas curvas was published in 1740 and it is noted that Carathéodory considered the work as one of the most beautiful mathematical work ever written.

Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. 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.

In 1736, Euler published Mechanica. Mechanica was followed by another important work in rational mechanics, this time Euler’s two volume work on naval science. Euler gave a definitive version of the hydrostatics, that had been studied since Archimedes.

In 1756 Euler published another major work on mechanics Theoria motus corporum solidorum in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. Euler’s lunar theory was used Tobias Mayer in constructing his tables the moon.

Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739.

Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy’s geography section in 1735.

 

7. Sources

http://www.shu.edu/academic/arts_sci/Undergraduate/math_cs/sites/math/reals/history/euler.html

http://www.physics.ucla.edu/class/85HC_Gruner/bios/euler.html

http://www.maths.tcd.ie/pub/HistMath/People/Euler/RouseBall/RB_Euler.html

http://www.mathsoft.com/asolve/constant/euler/euler.html

http://www.mathstat.usouthal.edu/~lynn/mathematics/euler.html

http://euler.ciens.ucv.ve/English/mathematics/euler.html

http://mech.postech.ac.kr/fluidmech/history/Euler.html

http://www.encyclopedia.com/articles/04269.html

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