Reflections on the Story of leonhard euler

Leonhard euler is a Swiss mathematician and physicist. He is called one of the two greatest mathematicians in history (the other is C.F.Gauss). Euler was the first person to use the word "function" to describe expressions with various parameters, such as y = F(x) (the definition of function was given by Leibniz in 1694). He was one of the pioneers who applied calculus to physics. Euler was born and educated in Switzerland. Euler is a mathematical genius. As a math professor, he taught in St. Petersburg and Berlin and then returned to St. Petersburg. Euler is the most prolific mathematician in history. His complete works have 75 volumes. Euler actually ruled the mathematics in the18th century, and he derived many results for the newly invented calculus at that time. In the last seven years of her life, Euler was completely blind. Nevertheless, he finished half his work at an amazing speed. Euler lived a pious life. However, this widely circulated legend is not true. Legend has it that Euler challenged Denis Derot in the court of Catherine II: "Sir, (a+b) n/n = x; So God exists, that's the answer! " Euler's death was also very special: he left for work in the middle of a friend's party and finally lay quietly on his desk. The asteroid Euler 2002 was named in memory of Euler.

[Edit this paragraph] leonhard euler-Submission

"Euler's calculation seems effortless, just like a person breathing, like an eagle hovering in the wind (in arago). This letter from Leonard Euler (1770- 1783) is no exaggeration for his unparalleled mathematical talent. He is the most prolific mathematician in history. His contemporaries called him "the embodiment of analysis". Euler wrote a long academic paper as easily as a quick-witted writer wrote a letter to a close friend. Even if he was completely blind in the last 17 years of his life, it didn't stop his great fertility. If blindness has any effect, it is to improve his inner thinking imagination. It was not until 1936 that people knew exactly how many works Euler wrote. However, it is estimated that it will take 60 to 80 volumes to publish Euler's anthology. From 65438 to 0909, the Swiss Federation of Natural Sciences began to collect and publish the academic papers of Euler's anthology. This work has been funded by many individuals and mathematical groups all over the world. This just shows that Euler belongs to the whole civilized world, not just Switzerland. The budget carefully prepared for this work (1909 coins about 80,000 dollars) was completely broken by the unexpected discovery of a large number of Euler manuscripts in St. Petersburg (Leningrad).

[Edit this paragraph] leonhard euler Dietz

Euler's mathematical career began in the year when Newton died. For a genius like Euler, it is impossible to choose a more favorable era. Analytic geometry (1637) has been used for 90 years, calculus for about 50 years, and Newton's law of gravity, the key of physical astronomy, has been used in mathematics for 40 years. In each of these fields, a large number of isolated problems have been solved, and obvious attempts have been made to unify them everywhere. However, the whole mathematics, pure mathematics and applied mathematics have not been systematically studied as later. In particular, the powerful analytical methods of De Kratos, Newton and Leibniz have not been fully utilized as later, especially in mechanics and geometry. Algebra and trigonometry at that time had been systematized and developed at a lower level. Especially the latter, has been basically improved. In Fermat's Diophantine analysis and general integer properties, there can be no such "temporary perfection" (even now). But in this respect, Euler also proved that he is indeed a master. In fact, one of the most striking features of Euler's versatility is that he has the same ability in two branches of mathematics-continuous mathematics and discrete mathematics. As an mathematician, Euler has never been surpassed by anyone. Perhaps no one can approach his level except jacoby. Algorithmists are mathematicians who design algorithms to solve various special problems. For a simple example, we can assume (or prove) that any positive real number has a real square root. But how can we work out this root? There are many known methods, and algorithm scientists should design practical concrete steps. For example, in Diophantine analysis and integral calculus, it is often impossible to solve the problem until one or more variables are skillfully (often simply) transformed by the functions of other variables. Algorithmists are mathematicians, and they will naturally find this trick. They don't have any identical procedures to follow. Algorithmists are like people who can write limericks at will-they are born, not made. At present, fashion despises "little arithmeticians". However, when a truly great mathematician, such as Lo Manu Kuo of India, suddenly appears from somewhere, even experienced analysts will hail him as a gift from heaven: his magical insight into seemingly unrelated formulas will reveal hidden clues from one field to another. So that analysts can find new topics for them to find these clues. Algorithmists are "formulists", and they like beautiful forms for the formula itself.

[Edit this paragraph] leonhard euler-two factors affecting him.

Before talking about Euler's quiet and interesting life, we must introduce two environmental factors of his time, which promoted his amazing activity and guided his activities. In Europe in the18th century, universities were not the main centers of academic research. Without the classical tradition and its imaginable hostility to scientific research, universities could have become the main center. Mathematics was rigorous enough for ancient people to be valued; Physics is relatively new and is suspected. In addition, in universities at that time, mathematicians were expected to devote most of their energy to basic teaching. As for academic research, if it is carried out, it will be a useless luxury, just like ordinary universities in the United States today. At that time, researchers in British universities were able to do their own chosen topics quite well. However, they are seldom willing to choose any topic. Anyway, what they have achieved or what they have not achieved will not affect their jobs. Under such relaxation, or open hostility, there is no good reason to explain why those universities should be ahead in scientific development, but in fact they are not. The responsibility of taking the lead is borne by the royal college funded by generous or far-sighted rulers. Frederick the great of Prussia and Queen Catherine of Russia generously gave unrepentant support to mathematics. They made it possible for the development of mathematics to be in the most active period in the history of science for a whole century. For Euler, it was Berlin and St. Petersburg that provided the power of mathematical creation. These two centers of creativity should attribute the inspiration to Euler to Leibniz's enterprising ambition. It was Leibniz who drafted the plan. These two colleges provided Euler with the opportunity to become the most prolific mathematician in history. So, in a sense, Euler is a descendant of Leibniz. The Berlin Academy of Sciences has declined for 40 years due to lack of brains. Encouraged by frederick the great, Euler gave it a powerful impact and revived it. The St. Petersburg Academy of Sciences, which Peter the Great did not have time to establish according to Leibniz's plan before his death, was established by his successor. Unlike some colleges today, these two colleges do not take the evaluation of well-written excellent works and the granting of academician qualifications as their main responsibilities. They are research institutions that employ academicians to conduct scientific research. The salary and allowance are generous enough to ensure a comfortable life for the whole family. Euler's family was once no less than 18, and he was enough to make them all live a rich life. /kloc-the last attraction of academician's life in the 0/8th century is that as long as his children have any talents, they will certainly get a good chance to display them. Next, we will see the second factor that has a decisive influence on Euler's fruitful mathematical achievements. Rulers who provide economic support naturally hope that their money can be exchanged for something other than abstract culture. However, it must be emphasized that once the rulers get a proper return on their investment, they no longer insist that the employees spend the rest of their time on "productive" work. Academician Euler and Lagrange are free to do what they like. There is no obvious pressure to force anyone to produce something that the government can use directly. /kloc-the rulers of the 0/8th century were wiser than the directors of many research institutes today, letting science develop according to their own laws, but occasionally mentioning what they need now. They seem to instinctively realize that the so-called "pure" research, as long as appropriate hints are made from time to time, will make the urgent practical problems they expect become by-products. There is an important exception to this general statement, which neither proves nor denies this law. Just in Euler's time, the unsolved problems in mathematical research happened to be related to maritime hegemony, which was perhaps the first practical problem at that time. A country that surpasses all its rivals in navigation technology will inevitably control the ocean. The first problem of navigation is to accurately determine the position of the ship in the sea hundreds of nautical miles offshore, so that it can reach the place of naval battle faster than the enemy (unfortunately, that's why). As we all know, Britain controls the ocean. It can do this, to a great extent, because its navigators can apply the pure mathematical research results in celestial mechanics to practice in the18th century. This practical application is directly related to Euler. Newton was the founder of modern navigation, although he never bothered about this problem himself, and never (as far as people know) set foot on the deck of a ship. Determining the position of a ship at sea depends on the observation of celestial bodies (sometimes this includes Jupiter's satellites on special voyages). Newton's law of universal gravitation shows that, if necessary, with enough patience, the position of the traveling star and the moon phase profit and loss can be calculated in advance within a hundred years. Those who wish to control the ocean will arrange the calculator of the nautical almanac to work hard to compile a table of the future positions of the planets. In this very practical career, the moon has raised a particularly thorny problem, that is, Newton's law attracts three stars. When we enter the 20th century, this problem will recur many times. Euler was the first person to put forward a computable solution (moon theory) to this moon problem. Three related stars are the moon, the earth and the sun. Although there is nothing to talk about this problem here, it will be pushed to the following chapters, but we can say that this problem is one of the most difficult problems in the whole mathematics category. Euler did not specifically answer this question, but his approximate calculation method (replaced by a better method today) has enough practical value to make the British calculator calculate the monthly table for the British Admiralty. To this end, the calculator got 5000 pounds (which was a considerable sum at that time), and Euler got 300 pounds for his method.

[Edit this paragraph] leonhard euler-Young Euler

The son of LeonardEuler, PaulEuler and MargueriteBrucker is probably the greatest scientist in Switzerland. He was born in Basel. But the next year, my parents and I moved to nearby Richen Village. There, his father became a Calvinist priest. Paul Euler himself is an accomplished mathematician. He is a student in Jacob Bernoulli. Father wants Leonard to go his own way and inherit his position in the country church. But, thankfully, he made a "mistake" in teaching the boy math. Young Euler knew very early what he should do. But he was very filial to his father, so he went to university of basel to study theology and Hebrew. At this time, there was a considerable level of Euler in mathematics that attracted the attention of Johannes Bernoulli. He enthusiastically gives the young man a separate lesson once a week. Euler used the rest of the week to preview the next lesson, so that when listening to the teacher, the fewer problems the better. Soon, his diligence and outstanding ability attracted the attention of daniel bernoulli and Nicolaus Bernoulli, who became close friends of Euler. Leonard didn't get happy until 1724 got his master's degree, because before that, his father insisted that he give up mathematics and spend all his time on theology. Only when the father heard Bernoulli and his son say that his son was destined to be a great mathematician instead of a priest in Li Xing did he finally give in. Bernoulli's father and son's prophecy came true, but Euler's early religious training influenced his life. He never gave up a bit of Calvinism. In his later years, he even turned to his father's business to a considerable extent. He leads the whole family to say family prayers, which usually ends with a sermon. Euler's first independent job was completed at the age of 19. It is said that this first achievement also revealed the advantages and disadvantages of many of his later jobs. 1727, the Paris Academy of Sciences presented an award for the problem of ship masts. Euler's paper didn't win the prize, but it was praised. He later made up for this loss by winning the 12 bonus. His specialty lies in analysis-technical mathematics; Its weakness is that it is too divorced from reality, if any. If we remember the legendary jokes of the Swiss navy, we won't be surprised by the latter. Euler may have seen one or two ships in a Swiss lake, but he has never seen a warship. He is sometimes criticized for making mathematics divorced from reality. There's nothing wrong with that. For Euler, the material world is exactly what mathematics needs, but it is not a very interesting thing in itself. If the world is inconsistent with his analysis, it is that there is something wrong with the world. Euler knew that he was born a mathematician, so he applied for a professorship in Basel. After he couldn't find a job, he continued his studies, hoping to get in touch with daniel bernoulli in St. Petersburg and Gula Bernoulli, the tomb of Niola. The Bernoulli brothers enthusiastically offered to find him a position in St. Petersburg Academy of Sciences to keep him informed of the situation there. At this stage, Euler seems indifferent to what he does, as long as it is science. When the Bernoulli brothers wrote to tell him that there would be a vacancy in the medical department of St. Petersburg Academy of Sciences, Euler devoted himself to the study of physiology in Basel and attended the medical report meeting. However, even in this field, he can't be divorced from mathematics: auditory physiology raises some mathematical research problems, such as transmitting sound by waves in turn. This early work branched and expanded like a tree growing wildly in a nightmare, which ran through Euler's life. The Bernoulli brothers are agile people. 1727, Euler received an invitation to go to St. Petersburg to be a member of the medical department of the Academy of Sciences. According to a clever rule, each foreign member must bring two students-actually interns. Poor Euler, the joy soon disappeared without a trace. On the day he set foot on Russian soil, the enlightened Queen Catherine I died. Yekaterina was Peter the Great's mistress before he became his wife. In more than one way, she is already an open-minded person. It was she who, after only two years in office, realized Peter's wish to establish an academy of sciences. After yekaterina's death, when the little tsar was underage, power fell into the hands of a very authoritarian group (perhaps fortunately, the little tsar died before he came to power). Russia's new rulers regard the Academy of Sciences as an unnecessary luxury, and for months they even plan to cut it down and send all foreign academicians back to China. This is what happened when Euler arrived in St. Petersburg. In the confusion, there was no news about the position he was invited to take in the medical department. In despair, he almost accepted the title of captain of the navy, and then he slipped into the math department. After that, when the conditions were good, Euler concentrated on his work. He has been immersed in books for six years. This is not entirely because he is attracted by mathematics, but partly because there are spies everywhere, which makes him afraid to carry out normal communication activities.

[Edit this paragraph] leonhard euler-Anecdote

1733, daniel bernoulli was fed up with the sufferings of sacred Russia and returned to free Switzerland. At the age of 26, Euler won the first mathematics seat in the Academy of Sciences. He felt that his future life would be fixed in St. Petersburg, so he decided to get married, settle down and let nature take its course. His wife catherina is the daughter of gesell, a painter who was brought back to Russia by Peter the Great. Later, the political situation became worse. Euler desperately wanted to escape, but as children were born one after another soon, he felt more and more bound, and he went to endless work for comfort. Some biographers trace Euler's great productivity back to the time when he first lived in Russia; Ordinary caution forced him to become an unbreakable habit of hard work. Euler is one of the great mathematicians who can work anywhere and under any conditions. He likes children very much (he once had 13 children himself, but all but five died). When he writes a paper, he often holds a baby in his lap while the older children play around him. He wrote the most difficult math works with incredible ease. Many legends about his talent have been passed down to this day. Some of them are undoubtedly exaggerated, but it is said that Euler often writes a math paper in about half an hour when he is invited to dinner twice. As soon as the article is finished, it is put into the ever-increasing pile of manuscripts for the use of printers. When the Journal of Chinese Academy of Sciences needs materials, the printer takes them from the top pile. Therefore, the publication date of these articles is often contrary to the writing order. Because Euler is used to doing a theme repeatedly to understand or expand what he has done, the evil result is even more serious, so that sometimes a series of articles on a theme are published in the opposite order. 1730, the little tsar died, and Anna Alvanovna (Peter's niece) became the queen. As far as the Academy of Sciences is concerned, its work is much more active. Under the indirect rule of Anna's favorite Ernest John de Bilon, Russia suffered the bloodiest reign of terror in history. 10 years, Euler quietly buried himself in his work. During this period, he suffered his first great misfortune. In order to win the Paris Prize, he devoted himself to an astronomical problem, which took several influential mathematicians several months to work out (because Gauss had a similar problem, so I won't introduce it here), and Euler solved it three days later. However, overwork caused him to get a disease in which he lost his right eye. It should be noted that modern textual research that suspects all anecdotes in the history of mathematics has pointed out that Euler's blindness in his right eye cannot be attributed to that astronomical problem at all. As for how a learned textual research scholar (or anyone else) knows so much about the so-called law of causality, it is a secret to be solved for david hume (a contemporary of Euler). Let's talk about the story of the famous Euler and the atheist (perhaps the pantheist), the French professor Dennis Dilot (17 13- 1784). This is a bit out of chronological order, because it happened during Euler's second stay in Russia. Diderot was invited by Queen Catherine II to visit the court and made a living by propagating atheism to courtiers. Yekaterina was bored, so he asked Euler to shut up the boastful philosopher. This is easy, because the whole mathematics is a mystery to Diderot. De Morgan told this story (in his masterpiece (Paradox Set), 1872): Diderot was told that a learned mathematician had algebraic proof of the existence of God. If he wants to listen, the mathematician will announce it in front of the whole court. Diderot agreed happily. ..... Euler came to Diderot and said solemnly in a confident tone, "Sir, because, therefore, God exists. Please answer! " This makes Diderot sound reasonable. The poor man was humiliated by the awkward silence and had to ask yekaterina to return to France immediately. The queen generously promised him. Euler was not satisfied with this masterpiece, and he used the solemn proof of the intangible substance of the soul to gild the lily with great seriousness. It is said that these two proofs were written into theological papers at that time. As far as appreciating reality is concerned, these are probably the most outstanding representative works of Euler's talent. When Euler lived in Russia, mathematics itself did not exhaust all his abilities. Wherever he is asked to show his mathematical talents in a way that is not too far away from pure mathematics, he makes the government's money well spent. Euler compiled some elementary mathematics textbooks for Russian schools, managed the geographical departments of the government, helped reform weights and measures, and designed practical methods to test the balance. These are just some of his activities, but no matter how much other work Euler did, he always achieved results in mathematics. One of the most important works in this period is a paper on mechanics in 1736. There is no date of publication in the annotation, but there is a mark, which marks the centenary of the publication of analytic geometry by Decades. The contribution of Euler's paper to mechanics is just like that of Dries's paper to geometry, which breaks away from the bondage of comprehensive proof and makes it analytical. Newton's (principle) can be written by Archimedes; Euler's mechanics cannot be written by any Greek. The powerful calculus was first introduced into mechanics, and it entered a modern period of establishing basic science. In this respect, Euler was later surpassed by his friend Lagrange, but the honor of taking a decisive step belongs to Euler. 1740 After Anna's death, the Russian government became more enlightened, but Euler suffered enough and happily accepted frederick the great's invitation to the Berlin Academy of Sciences. The empress dowager likes Euler very much and tries to make him talk more. All she got was a monosyllabic answer. "Why don't you talk to me?" She asked. "Your Majesty," replied Euler, "I come from a country, and anyone who speaks will be hanged." Later, Euler spent 24 years in Berlin. Life is not always pleasant, because Frederick prefers smooth courtiers to pure Euler. Although Frederick felt responsible for sponsoring the development of mathematics, he looked down on the subject and was not familiar with it himself. But he still appreciated Euler's ability to solve practical problems such as coins, water pipes, canal navigation and annuity system. Russia never let Euler get rid of it completely, and even paid him part of his salary when he was in Berlin. Although Euler has a big family, he is still rich. Besides the house in Berlin, he also has a farm near Fort Charlotte. 1760 When Russia invaded Brandenburg, Euler's farm was looted. The commander-in-chief of the Russian army declared that "there was no war with science" and gave Euler compensation far greater than the actual loss. After hearing the news that Euler was robbed, Queen Elizabeth also gave him a considerable sum of money that exceeded the compensation needs. One of the reasons why Euler was unpopular in Frederick's court was that he could not stay out of the debate on philosophical issues, and he knew nothing about it. I just want to flatter Voltaire of Frederick all day, and I like to tease the unfortunate Euler with metaphysical questions with other people who are good at talking about words around Frederick. Euler responded with all his good temper and laughed at his funny mistakes with the noise of others. But Frederick became angry gradually, and he began to try to find a more eloquent philosopher to lead his academy of sciences and add joy to his court. D'Alembert was invited to Berlin to inspect the situation. He and Euler have differences in mathematics. But D'Alembert is not the kind of person who let personal discord affect his judgment. He told Frederick bluntly that it was an insult to put any other mathematician above Euler. This suggestion only made Frederick more angry and stubborn than before, and Euler's situation became unbearable. He felt that his children would have no future in Prussia. Finally, at the age of 59 (1766), he packed his bags and moved to St. Petersburg again at the warm invitation of Catherine II. Yekaterina welcomed the mathematician like royalty, allocated a fully furnished house to Euler and his 18 family members, and gave one of his chefs to Euler to manage his meals. At this time, Euler's remaining eye began to go blind (due to cataract), and he soon became completely blind. During his gradual loss of vision, Lagrange, D'Alembert and other great mathematicians at that time expressed shock and sympathy in their correspondence. Euler himself is also very calm in the face of blindness. There is no doubt that his deep religious beliefs help him face the future. But he didn't give in to silence and darkness, and soon began to remedy his irreparable eyesight. Before the last light disappeared, he used to write the formula on the slate with chalk, and then his children (especially [AlbertEuler]) became scribes, and then he dictated the explanation of the formula. His new mathematics books have not decreased, but increased. Luckily for Euler, he has an extraordinary memory all his life. He recited Virgil's Aeneas. Although he seldom read this book since he was a child, he can always tell the true story of his version. His memory is visual and auditory. He also has amazing mental arithmetic ability. He can not only calculate arithmetic problems, but also calculate difficult problems that require advanced algebra and calculus. At that time, the main formulas in the whole mathematical field were accurately installed in his mind. As an example of his mental arithmetic ability, condorcet said that two students of Euler summed up the first 65,438+07 terms of a complex convergence series (the specific value of a variable), and the result was only a single digit difference in the 50th place. In order to decide which pair is right, Euler did mental arithmetic, and people were convinced that his answer was correct. This ability has now helped Euler, making him less suffering from blindness. But even so, it is inconceivable that he was blind 17 years. This is the theory of the movement of the moon-the only problem that bothered Newton-which was thoroughly studied by Euler for the first time. The whole complicated analysis process is completely carried out in his mind. Five years after Euler returned to St. Petersburg, another disaster befell him. In the fire of 177 1, his house and all the furniture were burned down. Thanks to the bravery of Swiss servant peter green, Euler survived. Green risked his life to save the sick blind master from the fire. The collection of books was set on fire. Thanks to Count orlov, all Euler's manuscripts were preserved. Queen Catherine immediately compensated Euler for all his losses, and he soon went back to work. 1776 (that is, when he was 69 years old), Euler suffered even greater losses and his wife died. The next year, he got married again. Salom Abigail and gesell, the second wives, are half-sisters of the first wife. His greatest misfortune was the failure of the operation to restore the vision of his left eye (probably due to the negligence of the surgeon), which was originally the only eye with a little hope. The operation was successful, and Euler was happy for a while. But soon the infection began, and after what he described as "terrible" pain, he fell into darkness again. Go back and browse Euler's voluminous works. At first glance, we may be inclined to think that anyone with talent can make full use of it as easily as Euler. However, compared with today's mathematics, we will soon correct our mistakes and think that it is a single book in seven languages. This also shows that the public's interest in science has not only increased recently, but sometimes we tend to imagine it this way. Euler remained full of energy and clear-headed until his death. That was at 1783.