Our daily landscape is full of images like this one. Every day we see chains that bar our way and we assume them naturally. However, which shape do these chains take? Can we identify them with known curves? These questions have thrilled some of the greatest mathematicians of all times and have led to the development of elaborated techniques valid today.
The main characters of this post are the Bernoulli brothers, Jakob (1655-1705) and Johann (1667-1748). They studied together Gottfried W. Leibniz’s works until they mastered it and widened the basis of what is nowadays known as calculus. One of the typical problems was the study of new curves which were created in the 17th century to prove the new differential methods. René Descartes and Pierre de Fermat invented new algebraic and geometrical methods that allowed the study of algebraic curves (those which the coordinates x and y have a polynomial relation). Descartes didn’t consider in his Géométrie curves that were not of this kind, and he called “mechanical curves” to all those ones that are not “algebraic”. In order to study them, he developed non algebraic techniques that allowed the analysis of any kind of curves, either algebraic or mechanical. These mechanical curves had already been introduced a lot of time ago, and were used to solve the three classical problems: squaring the circle, angle trisection and doubling the cube.
When developing calculus, Leibniz’s objective was to develop this general method that Descartes asked for. When Bernoulli brothers started to study curves and its mechanical problems associated, calculus would become its principal solving tool. For instance, in 1690 Jakob solved in Acta Eruditorum, a new problem proposed by Leibniz. In this document, he proved that the problem was equivalent to solve a differential equation and the power of the new technique was also shown.
During the 17th century, mathematicians often proposed problems to the scientific community. The first challenge that Jakob exposed was to find the shape that takes a perfectly flexible and homogeneous chain under the exclusive action of its weight and it is fixed by its ends. This was an old problem that hadn’t been solved yet. As we can see in the picture, the shape taken by the chain is very similar to a parabola and, owing to the fact that it is a well-known curve for centuries, it comes easy to think that it is indeed a parabola (for example, Galileo Galilei thought that he had solved the problem with the parabola). However, in 1646 Christiaan Huygens (1629-1695) was capable to refute it using physical arguments, despite not being able to determine the correct solution.
When Jakob aunched the challenge, Huygens was already 60 years old and successed in finding the curve geometrically, while Johann Bernoulli and Leibniz used the new differential calculus. All of them reached the same result, and Huygens named the curve “catenary”, derived from the latin word “catena” for chain.
Hence the first picture shows a catenary which we observe without paying much attention although all its history. Nowadays, we know it can be described using the hyperbolic cosine, although any of all these great mathematicians couldn’t notice it, as the exponential function had not been introduced yet. Then… How did they do it? Using the natural geometric propreties of the curve so Huygens, Johann Bernoulli and Leibniz could construct it with high precision only usiny geometry. Jakob didn’t know the answer when he proposed the problem and neither found the solution by himself later. So Johann felt proud of himself for surpassing his brother, who had been his tutor (Jakob, who was autodidactic, introduced his younger brother to the world of mathematics while he was studying medicine). What is more, the catenary problem was one of the focusses of their rivalry.
Jakob and Johann were also interested in the resolution of another fmous problem: the braquistochrone. Now, Johann proposed the challenge of finding the trajectory of a particle which travels from one given point to another in the less possible time under the exclusive effect of gravity. In the deadline, only Leibniz had come up with a solution which was sent by letter to Johann with a request of giving more time in order to receive more answers. Johann himself had a solution and in this additional period Jakob, l’Hôpital and an anonymous english author. The answer was a cycloid, a well-known curve since the 1st half of the 17th century. Jakob’s solution was general, developing a tool that was the start of the variational calculus. Johann gave a more imaginative solution based on Fermat’s principle of maximums ans minimums and Snell’s law of refraction of light: he considered a light beam across a medium that changed its refraction index continuously. Given this diference between their minds, Johann enforced his believe that he was better due to his originality and brightness, in contrast with his brother, who was less creative and worked more generally.
A third controversial curve was the tautochrone which is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. This property was studied by Huygens and he applied it to the construccion of pendulum clocks.
While we can see catenaries in a park to prevent children from danger or in the entrance to this Catalan beach, we luckily do not see cycloid slides. Thus, children will not descend in the minimum time but will reach the floor safe and sound!
But… who was the misterious english author? For Johann Bernoulli the answer to this question did not involve any mistery: the solution carried inside Isaac Newton’s genius signature -to whom we apologise for having mentioned Leibniz as the calculus inventor-, and he expressed that in the famous sentence:
I recognize the lion by his paw.
Here we see one more example of a catenary in our daily life:
You have more information in this older post.
This post has been written by Bernat Plandolit and Víctor de la Torre in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
Location: Ocata beach (map)
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.
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:
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!
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.
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.