Tag Archives: Euler

The Museum of Mathematics in the Collegius Maius

Photography by Carlos Dorce

Photography by Carlos Dorce

Another interesting museum located in the Collegius Maius of the Jagiellonian University is an exhibition about mathematics where children can play and learn a lot! There are old calculators from the 20th century…

Photography by Carlos Dorce

Photography by Carlos Dorce

…abacus and slide rulers:

Photography by Carlos Dorce

Photography by Carlos Dorce

Children can play with Geography and learn that straight lines in a map are not the shortest ways for the planes:

Photography by Carlos Dorce

Photography by Carlos Dorce

They can also learn the theorem of Pythagoras scrolling this interesting figure:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are polyhedra and a lot of geometrical and topological games:

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum is very small but all the tourist are inside Collegius Maius so you can be very quiet watching all the exhibited objects and toys, like the Rodin’s Thinker:

Photography by Carlos Dorce

Photography by Cristina Martínez

Finally… here you have my two children playing with Eulerian graphs! They are lovely! Aren’t they?

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Collegius Maius (map)

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Maria Gaetana Agnesi’s Doodle

Maria Gaetana Agnesi (Milan, 16 May, 1718 – 9 January, 1799) was born in a rich Italian family. She was a very clever girl who spoke Italian and French at five years of age and six years later she was able to speak Latin, Greek, Hebrew, German and Spanish. His father was mathematics teachers in the University of Bologna and he never stopped his daughter’s brightness so she wrote a very long speech about women’s rights to be educated when was only 9 y.o. She was always studing and she was very ill one year later because her great interest in all the sciences and humanistic subjects.

She studied Geometry and therefore her father invited her to his meetings with the most learned men in Bologna. Familiar problems let her to retire to study Mathematics and she could learn differential calculus. She became teacher at the university of Bologna and wrote the Instituzioni analitiche ad uso della gioventù italiana (Milan, 1748) where she show a great and didactic introduction to the differential and integral calculus and to the Euler’s works.

The doodle commemorates 296 years since her birthday and we can see the famous Witch of Agnesi. Quoting Wikipedia:

Starting with a fixed circle, a point O on the circle is chosen. For any other point A on the circle, the secant line OA is drawn. The point M is diametrically opposite to O. The line OA intersects the tangent of M at the point N. The line parallel to OM through N, and the line perpendicular to OM through A intersect at P. As the point A is varied, the path of P is the witch.

The curve is asymptotic to the line tangent to the fixed circle through the point O.

The Witch was studied by Pierre de Fermat in 1630 and by Guido Grandi in 1703 and Agnesi called it as “versiera” which is Grandi’s name.

306th anniversary of Euler’s birth

Doodle dedicated to Euler

Leonhard Euler was born on February 15, 1707. So, today marks the 306th anniversary of his birth.

Thanks Google for this homage!

Fermat’s theorem

Another of the important equations which stands out in the main entrance of Cosmocaixa in Barcelona is Fermat’s last theorem:

Photography by Carlos Dorce

Photography by Carlos Dorce

Fermat was reading Diophantus’ Arithmetic about the pythagorean triples x2 + y2 = z2 in 1637 when he noticed immediately that n = 2 was the only case which satisfies the equation xn + yn = zn. Then, he wrote in the margin of his edition of Diophantus’ work:

it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain

Source: Wikimedia Commons

Source: Wikimedia Commons

Probably, Fermat was only able to prove the new theorem for n = 4 using his proof by infinite descent: Fermat proved that there are not three numbers x, y, z such that  x4 + y4 = z2 . First of all, we can suppose that x, y, z are co-prime from (x2)2 + (y2)2 = z2. If <x,y,z> = d, then we can divide all the equation by d2 and we’ll have another equation (x2)2 + (y2)2 = z2 with <x,y,z> = 1.

Now, from the solution of the Pythagorean triples, there are pq such that x2 = 2pq, y2 = p2 – q2 and z = p2 + q2.

We can observe that  y2 = p2 – q2 implies that p2 = q2 – q2 is another Pythagorean triple so there are uv co-prime such that q = 2uvy = u2 – v2 and p = u2 + v2.

So, x2 = 2pq = 2(u2 + v2)(2uv) = 4(uv)(u2 + v2). Since <p,q> = 1, we know that <uv,u2 + v2> = 1 and since their product by 4 is a perfect square, they must be perfect squares too. So there exists a < p such that a2u2 + vp.

Finally, if we had (x2)2 + (y2)2 = z2 , we have obtain p and q co-prime with  z = p2 + q2. Then it is possible to obtain another pair u and v co-prime with p u2 + vand p < z, u < p and v < q. So we can iterate this algorithmic procedure so we’ll obtain a set of pairs of natural numbers each of them lower than the previous and this fact is false, since we’d have an infinite decreasing succession of natural numbers without end!

Nobody accepts that Fermat could have known any other case apart of n = 4! In 1753, Leonhard Euler wrote a letter to Christian Goldbach telling him that he had proved the case n = 3 but the demonstration that he published in his Algebra (1770) was wrong. He tried to find integers p, q, z such that (p2 + 3q2) = zand he found that:

p = a3 – 9ab2q = 3(a2b – b3) ⇒ p2 + 3q2 = (a2 + 3b2)3

He worked with numbers of the form a + b√-3 to find two new numbers a and b less than p and q such that p2 + 3q2 = cube and then he applied the method of infinite descent. Unluckily, he made some mistakes working with the new complex numbers  a + b√-3.

This story is so exciting but, almost quoting Fermat, this post is too short to contain everything. If you are interested in, you must read Simon Singh’s Fermat last theorem which is as interesting as the theorem itself.

Vassili Adadurov’s tomb

Adadurov's tombPhotography by Carlos Dorce

Adadurov’s tomb
Photography by Carlos Dorce

Vassili Evdokimovitch Adadurov (1709-1780) attended religious studies in Novgorod.  When he was 14 y.o., he joined the Slavic Greek Latin Academy (founded in 1685-1687) and graduated in 1726 to become a mathematics student and principal disciple of Jacques Bernoulli at the University. Because of his aptitudes and the recommendation of Bernoulli, Adadurov was appointed to assistant professor of mathematics and translator of German language and two years later he participated in the standardization of the Russian language. Regarding the area of mathematics, he was the translator of Euler’s works from German to Russian and Latin so he deserves to be buried next to Swiss grand-master in Laura Alexander Nevski in Saint Petersburg.

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Ekaterina Dashkova

Ekaterina DashkovaPhotography by Carlos Dorce

Ekaterina Dashkova
Photography by Carlos Dorce

There is an obligatory stop if you walk along Nevsky Prospect in St. Petersburg: the monument to the great Caternia II the Great of Russia. This Polish Empress reigned in Russia for 34 years to the late eighteenth century and one of the reasons why it deserves some attention is her patronage of culture (see: the Russian Academy of Sciences). She had contact with people like Voltaire or Euler and before news of a recess in the publication of the Encyclopedia, Diderot offered to finish the job in Russia.

The main reason to mention her here is the appearance in this monument of Ekaterina Dashmova (1743-1810) among her most influential men of his reign: in 1783, Catherine II named her director of the Russian Academy of Science. She was philologist but this woman was one of the most important women in science in front of a great country.

Monument to Catherine II in Nevski Prospekt (Saint Petersburg)
Photography by Carlos Dorce

Location: map

Leonhard Euler’s tomb

Euler and me

Euler and me

Today is 31 December 2012, the last day of this year. I’ve thought that the last post of 2012 must be related with my best mathematical moment of the year: I visited Euler’s tomb in Saint Petersburg in holidays! Euler was one of the best mathematician in the history of mathematics! According to D. M. Burton (The History of Mathematics, The McGraw-Hill Companies, Inc., 1991):

The key figure in eighteenth century mathematics was Leonhard Euler (1707-1783), and the scene of his activity was chiefly Germany and Russia.

A short Euler’s biography can be read in V. J. Katz (A History of Mathematics, Addison Wesley Longman, 1998):

Born in Basel, Switzerland, Euler showed his brilliance early, graduating with honors from the University of Basel when he was 15. Although his father preferred that he prepare for the ministry, Euler managed to convince Johann Bernoulli to tutor him privately in mathematics. The later soon recognized his student’s genius and persuaded Euler’s father to allow him to concentrate on mathematics. In 1726 Euler was turned down for a position at the university, partly because of his youth. A few years earlier, however, Peter the Great of Russia, on the urging of Leibniz, had decided to create the St. Petersburg Academy of Sciences as part of his efforts to modernize the Russian state. Among the earliest members of the Academy, appointed in 1725, were Nicolaus II (1695-726) and Daniel Bernoulli (1700-1782), two of Johann’s sons with whom Euler had developed a friendship. Although there was no position in mathematics available in St. Petersburg in 1726, they nevertheless recommended him for the vacancy in medicine and physiology, a position Euler immediately accepted. (He had studied these subjects during his time at Basel).

In 1733, after Nicolaus’ death and Daniel’s return to Switzerland, Euler was appointed the Academy’s chief mathematician. Late in the same year, he married Catherine Gsell with whom he subsequently had 13 children. The life of a foreign scientist was not always carefree in Russia at the time. Nevertheless, Euler was able generally to steer clear of controversies until the problems surrounding the succession to the Russian throne in 1741 convinced him to accept the invitation of Frederik II of Prussia to join the Berlin Academy of Sciences, founded by Frederik I, also on the advice of Leibniz. He soon became director of the Academy’s mathematics section, and with the publication of his texts in analysis as well as numerous mathematical articles, became recognized as the premier mathematician of Europe. In 1755 the Paris Academy of Sciences named him foreign member, partly in recognition of his winning their biennial prize competition 12 times. Ultimately, however, Frederik tired of Euler’s lack of philosophical sophistication. When the two could not agree on financial arrangements or on academic freedom, Euler returned to Russia in 1766 at the invitation of Empress Catherine the Great, whose succession to the throne marked Russia’s return to the westernizing policies of Peter the Great. With the financial security of his family now assured, Euler continued his mathematical activities even though he became almost totally blind in 1771. His prodigious memory enabled him to perform detailed calculations in his head. Thus he was able to dictate his articles and letters to his sons and others virtually until the day of his sudden death in 1783 while playing with one of his grandchildren.

Following with D. M. Burton again:

Without doubt, Euler was the most versatile and prolific writer in the entire history of mathematics. Fifty pages were finally required in his eulogy merely to list the titles of his published works. He wrote or dictated over 700 books and papers in his lifetime, and left so much unpublished material that the Saint Petersburg Academy did not finish printing all his manuscripts until 47 years after his death. The publication of Euler’s collected works was begun by the Swiss Society of Natural Sciences in 1911, and it is estimated that more than 75 large volumes will ultimately be required for the completion of this monumental project.

The tomb is located in Laura Alexander Nevsky which is an orthodox monastery built by Peter the Great in 1710. The building of the monastery itself has no special mathematical interest as the focus of the visit is focused on the search for the concrete tomb. When I was buying the tickets to the cemeteries, the lady in the entrance pointed to the door of Our Lady of Tikhvin cemetery and told me: “Tchaikovsky and Dostoevsky . But I wasn’t there to admire these two great Russian men! I went to St. Lazarus cemetery to locate Euler’s tomb:

Photography by Carlos Dorce

It was one of the best moments in my life! I took some photos and I hope that my mind could keep that magical moment. Remembering that famous Whitney Houston’s song: “One moment in time”. That moment was my “one moment in time”!

See you next year!

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The Russian Academy of Sciences

Russian Academy of Sciences (St. Petersburg)Photography by Carlos Dorce

Russian Academy of Sciences (St. Petersburg)
Photography by Carlos Dorce

In the period 1697-1698, Emperor Peter I of Russia traveled to England and the Netherlands and he could observe the development of culture and science in those countries. Peter I saw the necessity of promoting a cultural program in Russia to develop Russian science and began corresponding with Gottfried Wilhelm von Leibniz (1646-1716) about the state of Russian science. The result of this relationship was the establishment of a scientific academy in St. Petersburg which Leibniz would give precise instructions to found it as buying scientific books and founding museums and libraries. Leibniz also suggested that the new academy had to be conceived in the new Western European style so Russian science could have the same level as English, German or French cultures. Peter I decided to travel to Paris to visit the Paris Academy and he had the opportunity of attending one of its meetings in 1717 and this institution was the final model to establish the Russian Academy. So the new Academy would have a small group of scientists with direct royal support in spite of the decentralized model of the Royal Society of London and Peter I ordered to choose the best European scientits and to persuade them to joim his ambicious project in 1725. Peter I died on 28 January 1725 and his wife Enpress Catherine I took the responsability of the final founding of the Academy: the first meeting were held in November of that year with great scientisit as Nicolaus and Daniel Bernoulli, Johann Duvernoy, Christian Goldbach and Gerar Müller; Leonhard Eurler joined the Academy two years later in the physiology chair.

Catherine I died in 1727 only a few months after Euler’s arrival and she was succeeded by Peter II, grandson of Peter I, who was 12 y.o. His reign (1727-1730) was influenced by the royal conservative courtiers and after moving the Russian capital from St. Petersburg to Moscow, the Academy began a period of decline. His successor was Anna Ivanova (or Anna of Russia) whose reign ended at her death in 1740. She came from the German duchy of Courland in the Western Latvia) and she established herself as an autocratic and very unpopular ruler. She brought some advisers from Courland and her expensive and strange policies caused people to  think that the problem were her German courtiers so xenophobia started to increase among the citizens. The Empress was hated by her people but so were the foreign scientist who worked in the Academy. Furthermore, the leadership of the Academy was formed by German members meanwhile Russian mathematicians, physicists, astronomers,… occupied lower positions. Although the situation of the Academy was so bad, it became worse when the Empress died in 28 October 1740 and her grand-nephew Ivan VI Antonovich (born in 23 August 1740) was crowned as the new Emperor. Ivan’s mother Anna Leopoldovna was appointed to regent and in chaotic situation of the period 1740-1741, Euler and his family decided to move to Berlin to take up a position in the new German Academy: Russia had lost its great scientist!

In 1741 Elizaveta Petrovna was declared new Empress of Russia. She exiled the most unpopular German advisers of her court and started a period of anti-German policy. She was very extravagant but she loved the theatre, the music and the architecture so her reign was the perfect place to raise the cultural level. She transformed her court in a leading musical and theatre site and a lot of French, German and Italian actors, actresses and musicians traveled to Russia to play for the Empress. There also were a new revival for the Russian science and the  St. Petersburg Academy which was associated to an University. Despite the new situation of the institution, the scholars of the Academy didn’t have good conditions to work and they often were supervised by inquisitorial and restrictive rules. Elisaveta died in 1761 and she was succeeded by her nephew, Peter III who reigned in the period from  5 January to 9 July 1762. His wife Catherine led a coup against him and ordered his murder so she became Empress Catherine II in June 1762.

Portrait of Catherine II of Russia (1770) Fyodor Rokotov Hermitage Museum (St. Petersburg) Source: Wikimedia Commons

Portrait of Catherine II of Russia (1770)
Fyodor Rokotov
Hermitage Museum (St. Petersburg)
Source: Wikimedia Commons

Her ascension to the throne made a more  favorable environment for arts and science. She was a very cultivated and clever woman because she had been educated reading the French encyclopedists d’Alembert and Diderot and Voltaire too. Catherine issued the Statute of national Education in 1786 and during her reign, Leonhard Euler came back to the Academy from his position in Berlin. According to E. T. Bell (Men of Mathematics. New York: Simon and Schuster, 1961):

Catherine received the mathematician as if he were royalty, setting aside a fully furnished house for Euler and his eighteen dependents, and donating one of her own cooks to run the kitchen.

Euler lived in St. Petersburg until his death in September 1783. During his second Russian period, Euler wrote nearly half of his 856 listed works and the Academy published most of them posthumously.

Catherine II was succeeded by her son Paul I and in his only five years of reign he changed the reformist policy of his mother. For example, he decreed that no Russian could attend Western-style schools, attempted to prevent foreign books from reaching Russia, and cut off all funding for academic institutions. The Academy was saved by Paul I’s son alexander I who in 1803 presided its major reorganization. It was renamed Académie Impériale des Sciences and its annual publication also changed its name for its French version Mémoires de l’Académie Impériale des Sciences de St. Pétersbourg. Those years were a new height for Russian science until the Russian Revolution of 1917. In 1925, the Soviet government moved the Academy to Moscow and now the building placed in St. Petersburg is the headquarters of the Russian Academy of Sciences.

Location: Russian Academy of Sciences (map)