Carl Friedrich Gauss: ascension to the throne. The great mathematician Gauss: biography, photos, discoveries

Mathematician and mathematical historian Jeremy Gray talks about Gauss and his enormous contributions to science, the theory of quadratic forms, the discovery of Ceres, and non-Euclidean geometry*

Portrait of Gauss by Eduard Riethmüller on the terrace of the Göttingen Observatory // Carl Friedrich Gauss: Titan of Science G. Waldo Dunnington, Jeremy Gray, Fritz-Egbert Dohe

Carl Friedrich Gauss was a German mathematician and astronomer. He was born to poor parents in Brunswick in 1777 and died in Göttingen in Germany in 1855, and by then everyone who knew him considered him one of the greatest mathematicians of all time.

Study of Gauss

How do we study Carl Friedrich Gauss? Well, when it comes to his early life, we have to rely on the family stories shared by his mother when he became famous. Of course, these stories are prone to exaggeration, but his remarkable talent was already noticeable when Gauss was in his early teens. Since then we have more and more records about his life.
As Gauss grew up and became noticed, we began to have letters about him from people who knew him, as well as official reports of various kinds. We also have a long biography of his friend, written from conversations they had towards the end of Gauss's life. We have his publications, we have a lot of his letters to other people, and he wrote a lot of material, but never published it. And finally, we have obituaries.

Early life and path to mathematics

Gauss's father was engaged in various activities, he was a worker, a construction site foreman and a merchant's assistant. His mother was intelligent but barely literate, and devoted herself entirely to Gauss until her death at the age of 97. It seems that Gauss was noticed as a gifted student while still at school, at age eleven, his father was persuaded to send him to the local academic school rather than force him to work. At that time, the Duke of Brunswick sought to modernize his duchy, and attracted talented people to help him with this. When Gauss turned fifteen, the Duke brought him to the Collegium Carolinum for his higher education, although by that time Gauss had already independently studied Latin and mathematics at the high school level. At the age of eighteen he entered the University of Göttingen, and at twenty-one he had already written his doctoral dissertation.

Gauss initially intended to study philology, a priority subject in Germany at the time, but he also carried out extensive research on the algebraic construction of regular polygons. Due to the fact that the vertices of a regular polygon of N sides are given by solving the equation (which is numerically equal to a completely new result, the Greek geometers were unaware of it, and the discovery caused a minor sensation - news of it was even published in the city newspaper.This success, which came when he was barely nineteen, made him decide to study mathematics.

But what made him famous were two completely different events in 1801. The first was the publication of his book entitled Arithmetical Reasoning, which completely rewrote the theory of numbers and led to the fact that it became, and still is, one of the central subjects of mathematics. It includes the theory of equations of the form x^n - 1, which is both very original and at the same time easy to understand, as well as a much more complex theory called the theory of quadratic form. This had already attracted the attention of two leading French mathematicians, Joseph Louis Lagrange and Adrien Marie Legendre, who recognized that Gauss had gone very far beyond anything they had done.

Second important event was Gauss's rediscovery of the first known asteroid. It was discovered in 1800 by Italian astronomer Giuseppe Piazzi, who named it Ceres after the Roman goddess of agriculture. He observed her for 41 nights before she disappeared behind the sun. This was a very exciting discovery, and astronomers were eager to know where it would appear again. Only Gauss calculated this correctly, which none of the professionals did, and this made his name as an astronomer, which he remained for many years to come.

Later life and family

Gauss's first job was as a mathematician in Göttingen, but after the discovery of Ceres and then other asteroids, he gradually switched his interests to astronomy, and in 1815 became director of the Göttingen Observatory, a position he held almost until his death. He also remained professor of mathematics at the University of Göttingen, but this does not seem to have required him to teach much, and the record of his contacts with younger generations was rather sparse. In fact, he appears to have been an aloof figure, more comfortable and sociable with the astronomers, and the few good mathematicians in his life.

In the 1820s, he led a massive exploration of northern Germany and southern Denmark and in the process rewrote the theory of surface geometry, or differential geometry as it is called today.

Gauss married twice, the first time quite happily, but when his wife Joanna died in childbirth in 1809, he married again to Minna Waldeck, but this marriage was less successful; She died in 1831. He had three sons, two of whom emigrated to the United States, most likely because their relationship with their father was troubled. As a result, there is an active group of people in the States who trace their origins to Gauss. He also had two daughters, one from each marriage.

Greatest contribution to mathematics

In considering Gauss's contributions to the field, we can start with the method of least squares in statistics, which he invented to understand Piazzi's data and find the asteroid Ceres. It was a breakthrough in averaging a large number of observations, all of which were slightly imprecise, to get the most reliable information out of them. As for number theory, we can talk about this for a very long time, but he made remarkable discoveries about what numbers can be expressed in quadratic forms, which are expressions of the form . You may think this is important, but Gauss turned what was a collection of disparate results into a systematic theory, and showed that many simple and natural hypotheses have proofs that lie in what is similar to other branches of mathematics in general. Some of the techniques he invented turned out to be important in other areas of mathematics, but Gauss discovered them before these branches were properly studied: group theory is an example.

His work on equations of the form and, more surprisingly, on the in-depth features of the theory of quadratic forms, opened up the use of complex numbers, for example, to prove results about integers. This suggests that a lot was happening below the surface of the object.

Later, in the 1820s, he discovered that there was a concept of surface curvature that was an integral part of the surface. This explains why some surfaces cannot be exactly copied onto others without transformation, just as we cannot make an accurate map of the Earth on a piece of paper. This freed the study of surfaces from the study of solids: you could have an apple peel without having to imagine the apple underneath.

A surface with negative curvature, where the sum of the angles of the triangle is less than that of the triangle on the plane //source:Wikipedia

In the 1840s, independently of the English mathematician George Green, he invented the subject of potential theory, which is a huge extension of the calculus of functions of several variables. It is the correct mathematics for the study of gravity and electromagnetism and has since been used in many areas of applied mathematics.

And we must also remember that Gauss discovered but did not publish quite a lot. No one knows why he made so much of himself, but one theory is that the flow of new ideas he had in his head was even more exciting. He convinced himself that Euclid's geometry was not necessarily true and that at least one other geometry was logically possible. The glory of this discovery went to two other mathematicians, Boyai in Romania-Hungary and Lobachevsky in Russia, but only after their deaths - it was so controversial at that time. And he worked a lot on what are called elliptic functions - you can think of them as generalizations of the sine and cosine functions of trigonometry, but more precisely, they are complex functions of a complex variable, and Gauss invented a whole theory of them. Ten years later, Abel and Jacobi became famous for doing the same thing, not knowing that Gauss had already done it.

Work in other areas

After his rediscovery of the first asteroid, Gauss worked hard to find other asteroids and calculate their orbits. It was difficult work in the pre-computer era, but he turned to his talents, and he seemed to feel that this work allowed him to repay his debt to the prince and the society that had educated him.

Additionally, while surveying in northern Germany, he invented heliotrope for precision surveying, and in the 1840s, he helped create and build the first electric telegraph. If he had also thought about amplifiers, he could have noted this as well, since without them the signals could not travel very far.

Lasting Legacy

There are many reasons why Carl Friedrich Gauss is still so relevant today. First of all, number theory has grown into a huge subject with a reputation for being very difficult. Since then, some of the best mathematicians have gravitated towards him, and Gauss gave them a way to approach him. Naturally, some of the problems he couldn't solve attracted attention, so you can say he created an entire field of research. It turns out that this also has deep connections to the theory of elliptic functions.

Moreover, his discovery of the intrinsic concept of curvature enriched the entire study of surfaces and inspired many years of work by subsequent generations. Anyone who studies surfaces, from enterprising modern architects to mathematicians, is in his debt.

The internal geometry of surfaces extends to the idea of ​​the internal geometry of higher order objects such as three-dimensional space and four-dimensional spacetime.

General theory Einstein's relativity and all of modern cosmology, including the study of black holes, were made possible by Gauss's breakthrough. The idea of ​​non-Euclidean geometry, so shocking in its time, made people realize that there might be many kinds of rigorous mathematics, some of which might be more accurate or useful - or just interesting - than those we knew about.

Non-Euclidean geometry //

Carl Gauss (1777-1855), - German mathematician, astronomer and physicist. He created the theory of “primordial” roots from which the construction of the 17-gon flowed. One of the greatest mathematicians of all time.
Carl Friedrich Gauss was born on April 30, 1777 in Brunswick. He inherited good health from his father's family, and a bright intellect from his mother's family.
At the age of seven, Karl Friedrich entered the Catherine Folk School. Since they started counting there in the third grade, they did not pay attention to little Gauss for the first two years. Students usually entered third grade at the age of ten and studied there until confirmation (age fifteen). Teacher Büttner had to work with children of different ages and different levels of training at the same time. Therefore, he usually gave some of the students long calculation tasks in order to be able to talk with other students. Once a group of students, among whom was Gauss, was asked to sum the natural numbers from 1 to 100. As they completed the task, the students had to place their slates on the teacher's table. The order of the boards was taken into account when grading. Ten-year-old Karl put down his board as soon as Büttner finished dictating the task. To everyone's surprise, only he had the correct answer. The secret was simple: the task was dictated for now. Gauss managed to rediscover the formula for the sum of an arithmetic progression! The fame of the miracle child spread throughout little Brunswick.
In 1788, Gauss entered the gymnasium. However, it does not teach mathematics. Classical languages ​​are studied here. Gauss enjoys studying languages ​​and makes such progress that he does not even know what he wants to become - a mathematician or a philologist.
Gauss is known at court. In 1791 he was introduced to Karl Wilhelm Ferdinand, Duke of Brunswick. The boy visits the palace and entertains the courtiers with the art of counting. Thanks to the Duke's patronage, Gauss was able to enter the University of Göttingen in October 1795. At first, he listens to lectures on philology and almost never attends lectures on mathematics. But this does not mean that he does not do mathematics.
In 1795, Gauss developed a passionate interest in integers. Unfamiliar with any literature, he had to create everything for himself. And here he again shows himself as an extraordinary calculator, paving the way into the unknown. In the autumn of the same year, Gauss moved to Göttingen and literally devoured the literature that he first came across: Euler and Lagrange.
“March 30, 1796 comes for him the day of creative baptism. - writes F. Klein. - Gauss had already been studying for some time the grouping of roots of unity on the basis of his theory of “primitive” roots. And then one morning, waking up, he suddenly clearly and distinctly realized that the construction of a 17-gon follows from his theory... This event was the turning point in Gauss's life. He decides to devote himself not to philology, but exclusively to mathematics.”
Gauss's work became an unattainable example of mathematical discovery for a long time. One of the creators of non-Euclidean geometry, János Bolyai, called it “the most brilliant discovery of our time, or even of all time.” How difficult it was to comprehend this discovery. Thanks to letters to the homeland of the great Norwegian mathematician Abel, who proved the unsolvability of equations of the fifth degree in radicals, we know about the difficult path that he went through while studying Gauss's theory. In 1825, Abel writes from Germany: “Even if Gauss - greatest genius, he obviously did not strive for everyone to understand this at once...” Gauss’s work inspires Abel to build a theory in which “there are so many wonderful theorems that it is simply unbelievable.” There is no doubt that Gauss also influenced Galois.
Gauss himself retained a touching love for his first discovery throughout his life.
“They say that Archimedes bequeathed to build a monument in the form of a ball and a cylinder over his grave in memory of the fact that he found the ratio of the volumes of a cylinder and a ball inscribed in it to be 3:2. Like Archimedes, Gauss expressed the desire to have a decagon immortalized in the monument on his grave. This shows the importance Gauss himself attached to his discovery. This drawing is not on Gauss’s gravestone; the monument erected to Gauss in Brunswick stands on a seventeen-sided pedestal, although barely noticeable to the viewer,” wrote G. Weber.
On March 30, 1796, the day when the regular 17-gon was built, Gauss's diary begins - a chronicle of his remarkable discoveries. The next entry in the diary appeared on April 8. It reported a proof of the quadratic reciprocity theorem, which he called the “golden” theorem. Special cases of this statement were proved by Ferm, Euler, and Lagrange. Euler formulated a general hypothesis, an incomplete proof of which was given by Legendre. On April 8, Gauss found a complete proof of Euler's conjecture. However, Gauss did not yet know about the work of his great predecessors. He walked the entire difficult path to the “golden theorem” on his own!
Gauss made two great discoveries in just ten days, a month before he turned 19! One of the most amazing aspects of the “Gauss phenomenon” is that in his first works he practically did not rely on the achievements of his predecessors, rediscovering, as it were, in a short period of time what had been done in number theory over a century and a half through the works of major mathematicians.
In 1801, Gauss's famous "Arithmetic Studies" were published. This huge book (over 500 large format pages) contains Gauss's main results. The book was published at the expense of the Duke and dedicated to him. In its published form, the book consisted of seven parts. There wasn't enough money for an eighth of it. In this part, we were supposed to talk about the generalization of the reciprocity law to degrees higher than the second, in particular, about the biquadratic reciprocity law. Gauss found a complete proof of the biquadratic law only on October 23, 1813, and in his diaries he noted that this coincided with the birth of his son.
Outside of the Arithmetic Studies, Gauss essentially no longer studied number theory. He only thought through and completed what was planned in those years.
“Arithmetic studies” had a huge impact on the further development of number theory and algebra. The laws of reciprocity still occupy one of the central places in algebraic number theory. In Braunschweig, Gauss did not have the literature necessary to work on Arithmetical Research." Therefore, he often traveled to neighboring Helmstadt, where there was a good library. Here, in 1798, Gauss prepared a dissertation devoted to the proof of the Fundamental Theorem of Algebra - the statement that every algebraic equation has a root, which can be a real or imaginary number, in one word - complex. Gauss critically examines all previous experiments and evidence and with great care carries out the idea to Lambert. An impeccable proof still did not work out, since there was a lack of a strict theory of continuity. Subsequently, Gauss came up with three more proofs of the Fundamental Theorem (the last time in 1848).
Gauss's "mathematical age" is less than ten years old. Wherein most Works that remained unknown to contemporaries (elliptic functions) took up time.
Gauss believed that he could not rush to publish his results, and this was the case for thirty years. But in 1827, two young mathematicians at once - Abel and Jacobi - published much of what they had obtained.
Gauss's work on non-Euclidean geometry became known only with the publication of a posthumous archive. Thus, Gauss provided himself with the opportunity to work calmly by refusing to make his great discovery public, causing ongoing debate to this day about the admissibility of the position he took.
With the advent of the new century, Gauss's scientific interests decisively shifted away from pure mathematics. He will occasionally turn to it many times, and each time he will get results worthy of a genius. In 1812 he published a paper on the hypergeometric function. Gauss's contribution to the geometric interpretation of complex numbers is widely known.
Gauss's new hobby was astronomy. One of the reasons he took up the new science was prosaic. Gauss occupied the modest position of privatdozent in Braunschweig, receiving 6 thalers a month.
A pension of 400 thalers from the patron duke did not improve his situation enough to support his family, and he was thinking about marriage. It was not easy to get a chair in mathematics somewhere, and Gauss was not very keen on active teaching. The expanding network of observatories made a career as an astronomer more accessible, and Gauss began to become interested in astronomy while still in Göttingen. He carried out some observations in Brunswick, and he spent part of the ducal pension on the purchase of a sextant. He is looking for a worthy computing problem.
A scientist calculates the trajectory of a proposed new large planet. The German astronomer Olbers, relying on Gauss's calculations, found a planet (it was called Ceres). It was a real sensation!
On March 25, 1802, Olbers discovers another planet - Pallas. Gauss quickly calculates its orbit, showing that it too is located between Mars and Jupiter. The effectiveness of Gauss's computational methods became undeniable for astronomers.
Recognition comes to Gauss. One of the signs of this was his election as a corresponding member of the St. Petersburg Academy of Sciences. Soon he was invited to take the place of director of the St. Petersburg Observatory. At the same time, Olbers makes efforts to save Gauss for Germany. Back in 1802, he proposed to the curator of the University of Göttingen to invite Gauss to the post of director of the newly organized observatory. Olbers writes at the same time that Gauss “has a positive aversion to the department of mathematics.” Consent was given, but the move took place only at the end of 1807. During this time, Gauss married. “Life seems to me like spring with always new bright colors", he exclaims. In 1806, the Duke, to whom Gauss was apparently sincerely attached, dies of his wounds. Now nothing is keeping him in Brunswick.
Gauss's life in Göttingen was not easy. In 1809, after the birth of his son, his wife died, and then the child himself. In addition, Napoleon imposed a heavy indemnity on Göttingen. Gauss himself had to pay an exorbitant tax of 2,000 francs. Olbers and, right in Paris, Laplace tried to pay for him. Both times Gauss proudly refused.
However, another benefactor was found, this time anonymous, and there was no one to return the money to. Only much later did they learn that it was the Elector of Mainz, a friend of Goethe. “Death is dearer to me than such a life,” writes Gauss between notes on the theory of elliptic functions. Those around him did not appreciate his work; they considered him, to say the least, an eccentric. Olbers reassures Gauss, saying that one should not count on people’s understanding: “they must be pitied and served.”
In 1809, the famous “Theory of Movement” was published. celestial bodies, revolving around the Sun along conical sections." Gauss outlines his methods for calculating orbits. To ensure the power of his method, he repeats the calculation of the orbit of the comet of 1769, which Euler had calculated in three days of intense calculation. It took Gauss an hour to do this. The book outlined the least squares method, which remains to this day one of the most common methods for processing observational results.
1810 saw a large number of honors: Gauss received the prize of the Paris Academy of Sciences and the gold medal of the Royal Society of London, and was elected to several academies.
Regular studies in astronomy continued almost until his death. The famous comet of 1812 (which “foreshadowed” the fire of Moscow!) was observed everywhere using Gauss’s calculations. On August 28, 1851, Gauss observed a solar eclipse. Gauss had many astronomer students: Schumacher, Gerling, Nikolai, Struve. The greatest German geometers Möbius and Staudt studied from him not geometry, but astronomy. He was in active correspondence with many astronomers on a regular basis.
By 1820, the center of Gauss's practical interests had shifted to geodesy. We owe it to geodesy that for a relatively short time Mathematics again became one of Gauss’s main concerns. In 1816, he thought about generalizing the basic problem of cartography - the problem of mapping one surface onto another "so that the mapping is similar to the one depicted in the smallest detail."
In 1828, Gauss's main geometric memoir, General Studies on Curved Surfaces, was published. The memoir is devoted to the internal geometry of a surface, that is, to what is associated with the structure of this surface itself, and not with its position in space.
It turns out that “without leaving the surface” you can find out whether it is curved or not. A “real” curved surface cannot be turned onto a plane by any bending. Gauss proposed a numerical characteristic of the measure of surface curvature.
By the end of the twenties, Gauss, who had passed the fifty-year mark, began to search for new areas of scientific activity. This is evidenced by two publications from 1829 and 1830. The first of them bears the stamp of thoughts about general principles mechanics (Gauss’s “principle of least constraint” is based here); the other is devoted to the study of capillary phenomena. Gauss decides to study physics, but his narrow interests have not yet been determined.
In 1831 he tried to study crystallography. This is a very difficult year in the life of Gauss,” his second wife dies, he begins to suffer from severe insomnia. In the same year, 27-year-old physicist Wilhelm Weber, invited on Gauss’ initiative, comes to Göttingen. Gauss met him in 1828 in Humboldt’s house. Gauss was 54 years old. , his reticence was legendary, and yet in Weber he found a scientific companion such as he had never had before.
The interests of Gauss and Weber lay in the field of electrodynamics and terrestrial magnetism. Their activities had not only theoretical, but also practical results. In 1833 they invent the electromagnetic telegraph. The first telegraph connected the magnetic observatory with the city of Neuburg.
The study of terrestrial magnetism was based both on observations at the magnetic observatory established in Göttingen and on materials collected in different countries ah "Union for the Observation of Terrestrial Magnetism", created by Humboldt after returning from South America. At the same time, Gauss created one of the most important chapters of mathematical physics - potential theory.
The joint studies of Gauss and Weber were interrupted in 1843, when Weber, along with six other professors, was expelled from Göttingen for signing a letter to the king, which indicated the latter’s violations of the constitution (Gauss did not sign the letter). Weber returned to Göttingen only in 1849, when Gauss was already 72 years old.

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Very important Gauss proved the fundamental theorem of algebra on the existence of a root of an algebraic equation in 1799. Based on this theorem, the following property of the equations is proven: “ Algebraic equation has as many real or complex roots as there are units in its exponent.” For the work in which these theorems were proven, Gauss received the title of privatdozent.
In the first part of the work "Arithmetic Studies" Gauss deeply analyzed the question of the so-called "quadratic surpluses" and for the first time proved an important theorem from number theory, which he called the "golden theorem" about the "quadratic reciprocity law". It can be said without exaggeration that number theory, as a science, began its true existence precisely from the research of Gauss. Gauss's "arithmetic studies" created an entire era in mathematical science, and Gauss was recognized as the greatest mathematician in the world.
In algebra, Gauss was primarily interested in the fundamental theorem. He returned to it more than once and gave more than six different proofs of it. All of them were published in the scientist’s works in 1808-1817. In these works, guidance was given regarding cubic and biquadratic excesses. Theorems on biquadratic surpluses are considered in works 1825-1831. These works significantly expanded number theory through the introduction of the so-called Gaussian integers, i.e. numbers of the form a+bi, Where A And b- whole numbers. In connection with astronomical calculations based on the expansion of integrals of the corresponding differential equations in endless rows. Gauss investigated the question of the convergence of infinite series, which he associated with the study of the so-called. hypergeometric series (“O hypergeometric series”, 1812). The main significance of this series is that it contains, as special cases, many of the well-known transcendental functions that have wide application. These studies of Gauss, together with the works of Cauchy and Abel, based on Gauss's studies, contributed to the significant development of the general theory of series.
Although Gauss worked fruitfully in various areas science, but he himself often said: “I am all devoted to mathematics.” He considered mathematics the queen of sciences, and arithmetic the queen of mathematics. He had no equal in mental calculations. He knew by heart the first decimal digits of many logarithms and used them for approximate calculations in his head. When solving complex problems, he rarely made mistakes and wrote numbers clearly. I checked the last decimal places without relying on tables. The discovery of Gauss did not make such a revolution as, for example, the discovery of Archimedes and Newton, but for their depth, diversity, and the discovery of new, previously unknown laws of nature in the field of physics, geodesy, and mathematics, contemporaries considered Gauss the best mathematician in the world. The medal, made in 1855 in his honor, is engraved with the inscription: “King of Mathematicians.”
Contributions to the field of astronomy
In 1807 he was awarded the title of extraordinary and later ordinary professor at the University of Göttingen. At the same time he was appointed director of the Göttingen Observatory. Gauss worked in the field of astronomy for about 20 years. In 1801, the Italian astronomer Piazzi discovered a small planet between the orbits of Mars and Jupiter, which he named Ceres. He observed this planet for 40 days, but Ceres quickly approached the Sun and disappeared in its bright rays. Piazzi's attempts to find her again were in vain. Gauss became interested in this phenomenon and, having studied Piazzi's observational materials, determined that three observations of Ceres were sufficient to determine the orbit of Ceres. After which it was necessary to solve an equation of the 8th degree, which Gauss did brilliantly: the orbit of the planet was calculated and Ceres itself was found. In the same way, Gauss calculated the orbit of another small planet, Pallas. In 1810, the French Astronomical Institute awarded him a gold medal for solving the problem of the motion of Pallas. During this period, the scientist wrote his fundamental work “The Theory of the Motion of Celestial Bodies Orbiting the Sun along Conical Sections” (1809).
Mathematical
Gauss was also interested in geometry. Certain issues, such as the most important problem of geometry - the problem of Euclid's V postulate, attracted his special attention. In his reasoning, he followed paths similar to those taken by Lobachevsky, but did not publish a single page. In a letter to the mathematician Bessel, Gauss wrote: “Apparently, I will not soon be able to process my research on this matter so that it can be published. It is even possible that I will not dare to do this throughout my life, because I am afraid of the outcry of the Boeotians that arises when I express my views.”
Gauss got acquainted with the results of Lobachevsky's research in a small brochure "Geometric Research on the Theory of Parallel Lines", written in German and published in 1840. He became interested in this work and, at the age of 62, decided to learn Russian in order to be able to read Lobachevsky’s works in the original. In letters to his friends, Gauss spoke with great praise about Lobachevsky's achievements. He wrote that Lobachevsky's work contains the foundations of a geometry that could and would be completely consistent if Euclid's geometry were not correct. He also wrote that for 54 years (since 1792) he had the same convictions. Gauss personally wrote a letter to Lobachevsky himself, in which he informed the Russian scientist that he had been elected a corresponding member of the Göttingen Mathematical Scientific Society.
Contributions to the field of physics
Gauss devoted the years 1830-1840 to theoretical physics. His research in this area was largely the result of close communication and collaboration. scientific work with W. Weber. Together with Weber, Gauss created an absolute system of electromagnetic units and designed the first electromagnetic telegraph in Germany in 1833. He was responsible for the creation of the general theory of magnetism, the foundations of potential theory, and many others. Therefore, it is difficult to indicate a branch of theoretical or applied mathematics to which Gauss did not make a significant contribution.
Due to his extremely high demands on himself, many of the outstanding mathematician’s studies remained unpublished during his life (essays, unfinished works, correspondence with friends). This scientific legacy of Gauss was very carefully studied at the Göttingen Scientific Society. As a result, 11 volumes of Gauss's works were published. A very interesting legacy of the scientist is his diary and research on non-Euclidean geometry and the theory of elliptic functions. In particular, from the published materials it is clear that Gauss came to the idea of ​​​​the possibility of existence, along with Euclidean geometry, of non-Euclidean geometry in 1818. However, fears that the ideas of non-Euclidean geometry would not be understood in the mathematical world, and perhaps insufficient awareness of their scientific importance were the reasons why that Gauss did not develop them further and did not publish anything on these issues during his lifetime. When he published non-Euclidean geometry M.I. Lobachevsky, Gauss reacted to this with great attention and proposed to elect Lobachevsky as a corresponding member of the Göttingen Scientific Society, but did not essentially give his own assessment of Lobachevsky’s great discovery.
In the archives of Gauss, materials were found with a unique theory of elliptic functions. However, the credit for its development and publication belongs to K. Jacobi and N. Abel. It should be noted that Gauss’s contemporaries already understood his greatness, as evidenced by the inscription on the medals minted in honor of Gauss - “King of Mathematicians.” In 1880, a bronze statue of Gauss was erected in Brunswick. In 1827, Gauss published a large work, “General Studies on Curved Surfaces,” the content of which concerns differential geometry.
Significant discoveries belonged to Gauss in the field of physics. He investigated and established a number of new laws in the theory of liquids, theory, magnetism, etc. Consequence important developments there were such works: “On one important law of mechanics” (1820), “ General beginnings theory of equilibrium of liquids" (1832), "General theory of terrestrial magnetism" (1838). In 1832, Gauss published an important article "On absolute measurement magnetic quantities." He designed a device for measuring magnetic quantities (magnetometer), and performed the first calculation of the position of the Earth's south magnetic pole, which gave a very small deviation from the current position. Gauss invented the electromagnetic method of communication (1834).
Other achievements
He worked no less successfully in the field of geodesy. In 1836, Gauss was offered to carry out geodetic measurements of the territory of the Kingdom of Hanover. After preparatory work the scientist personally began measurements. He worked on this for 14 years. He made a new one measuring device- heliotrope, which acted with the help of the sun's rays. At the same time, the practice of measurements prompted Gauss to conduct theoretical research. Their consequence was important theoretical work, which became the basis for the further development of geodesy.
Gauss's office
Gauss himself worked in a small office, there was a table, a desk painted in White color, a narrow sofa and a single chair. He was always dressed in a warm robe and cap, and luckily he was calm and cheerful. After hard work, Gauss loved to relax: he took walks to the literary museum, read fiction in German, English and Russian. Gauss highly appreciated Russian culture and respected the talented Russian people. In Russia, educated circles, in turn, highly valued Gauss as a scientist. The St. Petersburg Academy of Sciences was the first in the world to choose Gauss as its corresponding member.

Carl Friedrich Gauss(German: Carl Friedrich Gauß) - an outstanding German mathematician, astronomer and physicist, considered one of the greatest mathematicians of all time.

Carl Friedrich Gauss was born on April 30, 1777. in the Duchy of Brunswick. Gauss's grandfather was a poor peasant, his father was a gardener, mason, and canal caretaker. Gauss's early age showed extraordinary talent for mathematics. One day, while doing his father's calculations, his three-year-old son noticed an error in the calculations. The calculation was checked, and the number indicated by the boy was correct. Little Karl was lucky with his teacher: M. Bartels appreciated the exceptional talent of young Gauss and managed to get him a scholarship from the Duke of Brunswick.

This helped Gauss graduate from college, where he studied Newton, Euler, and Lagrange. Already there, Gaus made several discoveries in higher mathematics, including proving the law of reciprocity of quadratic residues. Legendre, however, discovered this most important law earlier, but failed to strictly prove it, and Euler also failed to do so.

From 1795 to 1798, Gauss studied at the University of Göttingen. This is the most fruitful period in Gauss's life. In 1796, Carl Friedrich Gauss proved the possibility of constructing a regular 17-gon using a compass and ruler. Moreover, he solved the problem of constructing regular polygons to the end and found a criterion for the possibility of constructing a regular n-gon using a compass and ruler: if n is a prime number, then it must be of the form n=2^(2^k)+1 (the number Farm). Gauss treasured this discovery very much and bequeathed that a regular 17-gon inscribed in a circle should be depicted on his grave.

On March 30, 1796, the day when the regular 17-gon was built, Gauss's diary begins - a chronicle of his remarkable discoveries. The next entry in the diary appeared on April 8. It reported on the proof of the quadratic reciprocity theorem, which he called the “golden” theorem. Gauss made two discoveries in just ten days, a month before he turned 19 years old.

Since 1799, Gauss has been a privatdozent at the University of Braunschweig. The Duke continued to patronize the young genius. He paid for the publication of his doctoral dissertation (1799) and awarded him a good scholarship. After 1801, Gauss, without breaking with number theory, expanded his range of interests to include the natural sciences.

Carl Gauss gained worldwide fame after developing a method for calculating the elliptical orbit of a planet. according to three observations. The application of this method to the minor planet Ceres made it possible to find it again in the sky after it had been lost.

On the night of December 31 to January 1, the famous German astronomer Olbers, using data from Gauss, discovered a planet called Ceres. In March 1802, another similar planet, Pallas, was discovered, and Gauss immediately calculated its orbit.

Karl Gauss outlined his methods for calculating orbits in his famous Theories of the motion of celestial bodies(lat. Theoria motus corporum coelestium, 1809). The book describes the least squares method he used, which to this day remains one of the most common methods for processing experimental data.

In 1806, his generous patron, the Duke of Brunswick, died from a wound received in the war with Napoleon. Several countries vied with each other to invite Gauss to serve. On the recommendation of Alexander von Humboldt, Gauss was appointed professor in Göttingen and director of the Göttingen Observatory. He held this position until his death.

The name of Gauss is associated with fundamental research in almost all the main areas of mathematics: algebra, mathematical analysis, theory of functions of a complex variable, differential and non-Euclidean geometry, probability theory, as well as in astronomy, geodesy and mechanics.

Published in 1809 Gauss's new masterpiece - "The Theory of the Motion of Celestial Bodies", where the canonical theory of taking into account orbital perturbations is outlined.

In 1810, Gauss received the Prize of the Paris Academy of Sciences and the Gold Medal of the Royal Society of London, was elected to several academies. The famous comet of 1812 was observed everywhere using Gauss's calculations. In 1828, Gauss's main geometric memoir, General Studies on Curved Surfaces, was published. The memoir is devoted to the internal geometry of a surface, that is, to what is associated with the structure of this surface itself, and not with its position in space.

Research in the field of physics, which Gauss was engaged in since the early 1830s, belongs to different branches of this science. In 1832 he created an absolute system of measures, introducing three basic units: 1 sec, 1 mm and 1 kg. In 1833, together with W. Weber, he built the first electromagnetic telegraph in Germany, connecting the observatory and the physical institute in Göttingen, carried out extensive experimental work on terrestrial magnetism, invented a unipolar magnetometer, and then a bifilar one (also together with W. Weber), created the foundations of potential theory , in particular, formulated the fundamental theorem of electrostatics (the Gauss–Ostrogradsky theorem). In 1840 he developed the theory of constructing images in complex optical systems. In 1835 he created a magnetic observatory at the Göttingen Astronomical Observatory.

In each scientific field his depth of penetration into the material, courage of thought and significance of the result were amazing. Gauss was called the “king of mathematicians.” He discovered the ring of complex Gaussian integers, created a theory of divisibility for them, and with their help solved many algebraic problems.

Gauss died on February 23, 1855 in Göttingen. Contemporaries remember Gauss as a cheerful, friendly person with an excellent sense of humor. The following names were named in honor of Gauss: a crater on the Moon, minor planet No. 1001 (Gaussia), a unit of measurement of magnetic induction in the GHS system, and the Gaussberg volcano in Antarctica.