History of Gauss. Biography of Carl Friedrich Gauss. Normal distribution law

German mathematician, astronomer and physicist, participated in the creation of Germany's first electromagnetic telegraph. Until his old age, he got used to doing most of the calculations in his head...

According to family legend, he is already in 3 for years he knew how to read, write, and even corrected his father’s calculation errors in the payroll for workers (my father worked either at a construction site or as a gardener...).

"At eighteen he did amazing discovery, concerning the properties of the decagon; this has not happened in mathematics for 2000 years since the ancient Greeks (this success was decided by the choice of Karl Gauss: what to study next: languages ​​or mathematics in favor of mathematics - Note by I.L. Vikentyev). His doctoral dissertation on the topic “New evidence that every whole rational function one variable can be represented by the product real numbers first and second degrees" is devoted to solving the fundamental theorem of algebra. The theorem itself was known before, but he proposed a completely new proof. Glory Gauss was so great that when French troops approached Göttingen in 1807, Napoleon ordered to take care of the city in which “the greatest mathematician of all time” lives. This was very kind of Napoleon, but fame also has a downside. When the victors imposed an indemnity on Germany, they demanded from Gauss 2000 francs This corresponded to approximately 5,000 today's dollars - quite a large sum for a university professor. Friends offered help Gauss refused; while the bickering was going on, it turned out that the money had already been paid by the famous French mathematician Maurice Pierre de Laplace(1749-1827). Laplace explained his action by saying that he considered Gauss, who was 29 years younger than him, “the greatest mathematician in the world,” that is, he rated him slightly lower than Napoleon. Later, an anonymous admirer sent Gauss 1000 francs to help him pay off Laplace.”

Peter Bernstein, Against the Gods: Taming Risk, M., Olympus Business, 2006, p. 154.

10 year old Karl Gauss very lucky to have an assistant math teacher - Martin Bartels(he was 17 years old at the time). He not only appreciated the talent of young Gauss, but managed to get him a scholarship from the Duke of Brunswick to enter the prestigious school Collegium Carolinum. Later Martin Bartels was a teacher and N.I. Lobachevsky…

“By 1807, Gauss had developed a theory of errors (errors), and astronomers began to use it. Although in all modern physical measurements error indication required, outside of astronomy physics Not error estimates were reported up until the 1890s (or even later).”

Ian Hacking, Representation and Intervention. Introduction to the philosophy of natural sciences, M., “Logos”, 1998, p. 242.

“In recent decades, among the problems of the foundations of physics, the problem of physical space has acquired particular importance. Research Gauss(1816), Bolyai (1823), Lobachevsky(1835) and others led to non-Euclidean geometry, to the realization that the classical geometric system of Euclid, which has so far reigned supreme, is only one of an infinite number of logically equal systems. Thus, the question arose which of these geometries is the geometry of real space.
Gauss also wanted to solve this issue by measuring the sum of angles big triangle. Thus, physical geometry turned into an empirical science, a branch of physics. These problems were further considered in particular Riemann (1868), Helmholtz(1868) and Poincare (1904). Poincare emphasized, in particular, the relationship between physical geometry and all other branches of physics: the question of the nature of real space can only be resolved within the framework of some general system of physics.
Then Einstein found one common system, within the framework of which this question was answered, an answer in the spirit of a specific non-Euclidean system."

Rudolf Carnap, Hans Hahn, Otto Neurath, Scientific worldview - Vienna circle, in Collection: Journal “Erkenntnis” (“Knowledge”). Favorites / Ed. O.A. Nazarova, M., “Territory of the Future”, 2006, p. 70.

In 1832 Carl Gauss“... built a system of units in which three arbitrary, independent from each other basic units were taken as a basis: length (millimeter), mass (milligram) and time (second). All other (derived) units could be defined using these three. Later, with the development of science and technology, other systems of units appeared physical quantities, built according to the principle proposed by Gauss. They were based on metric system measures, but differed from each other in basic units. The question of ensuring uniformity in the measurement of quantities reflecting certain phenomena material world, has always been very important. The lack of such uniformity created significant difficulties for scientific knowledge. For example, until the 80s of the 19th century there was no unity in the measurement electrical quantities: 15 different units of electrical resistance were used, 8 units electromotive force, 5 units electric current etc. The current situation made it very difficult to compare the results of measurements and calculations performed by various researchers.”

Golubintsev V.O., Dantsev A.A., Lyubchenko B.S., Philosophy of Science, Rostov-on-Don, “Phoenix”, 2007, p. 390-391.

« Carl Gauss, like Isaac Newton, often Not published scientific results. But all the published works of Carl Gauss contain significant results - there are no crude or pass-through works among them.

“Here it is necessary to distinguish the research method itself from the presentation and publication of its results. Let's take as an example three great, one might say brilliant, mathematicians: Gauss, Euler And Cauchy. Gauss, before publishing any work, subjected his presentation to the most careful processing, exerting extreme care for the brevity of the presentation, the elegance of methods and language, without leaving at the same time, traces of the rough work that he achieved before these methods. He used to say that when a building is built, they do not leave the scaffolding that served for the construction; therefore, he not only was in no hurry to publish his works, but left them to mature not just for years, but for decades, often returning to this work from time to time in order to bring it to perfection. […] He did not bother to publish his studies on elliptic functions, the main properties of which he discovered 34 years before Abel and Jacobi, for 61 years, and they were published in his “Heritage” approximately another 60 years after his death. Euler did exactly the opposite of Gauss. Not only did he not dismantle the scaffolding around his building, but sometimes he even seemed to clutter it up with them. But he shows all the details of the very method of his work, which is so carefully hidden in Gauss. Euler did not bother with finishing; he worked straight away and published it as the work turned out; but he was far ahead of the Academy's printed media, so that he himself said that academic publications would have enough of his works for 40 years after his death; but here he was wrong - they lasted more than 80 years. Cauchy He wrote so many works, both excellent and hasty, that neither the Paris Academy nor the mathematical journals of that time could contain them, and he founded his own mathematical journal, in which he published only his works. Gauss put it this way about the most hasty of them: “Cauchy suffers from mathematical diarrhea.” It is not known whether Cauchy said in retaliation that Gauss suffered from mathematical constipation?

Krylov A.N., My memories, L., “Shipbuilding”, 1979, p. 331.

«… Gauss was a very reserved person and led a reclusive lifestyle. He Not published a lot of his discoveries, and many of them were re-done by other mathematicians. In his publications, he paid more attention to the results, without attaching much importance to the methods for obtaining them and often forcing other mathematicians to spend a lot of effort on proving his conclusions. Eric Temple Bell, one of the biographers Gauss, believes that his unsociability delayed the development of mathematics for at least fifty years; half a dozen mathematicians could have become famous if they had obtained the results that had been kept in his archive for years, or even decades.”

Peter Bernstein, Against the Gods: Taming Risk, M., Olympus Business, 2006, p.156.

Mathematician and mathematical historian Jeremy Gray talks about Gauss and his enormous contributions to science, theory 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 times.

Study of Gauss

How do we study Carl Friedrich Gauss? Well, when it comes to his early life, we have to rely on family stories, which his mother shared 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 adolescence. 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 various matters, 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 who would help him with this. When Gauss turned fifteen, the Duke brought him to the College Carolinum to receive his higher education, although by that time Gauss had already independently studied Latin and mathematics at the level high school. 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 peaks regular polygon of N sides are given by solving the equation (which is numerically equal to . Gauss discovered that with n = 17 the equation factorizes in such a way that a regular 17-sided polygon can only be constructed using a ruler and compass. This was completely new result, the Greek geometers were unaware of this, and the discovery caused a minor sensation - news about 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.

The 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 seems to have been an aloof figure, more comfortable and sociable with astronomers, and 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

Considering Gauss's contribution to this field, we can start with the method 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 up the study of surfaces from studying solids: You can 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 stream 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 so-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 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 objects more high order, such as three dimensional space and four-dimensional space-time.

Einstein's general theory of 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 //

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Karl Gauss (1777 - 1855) - great German mathematician, mechanic, physicist, surveyor.

He is considered one of the greatest mathematicians of all time and is nicknamed the "King of Mathematics".

Gauss discovered many laws in algebra and geometry, gave the first rigorous proofs of the Fundamental Theorem of Algebra, discovered the ring of complex integers called Gaussian, and formulated and proved a huge number of theorems.

At the same time, Gauss was incredibly strict in relation to his publications: he never published his works, even impeccable ones, if he considered them unfinished.

This led to the fact that priority in a number of discoveries made by him went to other scientists who made them at the same time as him or even decades later:

Despite this, Gauss's mathematical merits are by no means diminished. Many of his students subsequently also became outstanding scientists.

Child prodigy

Kar Gauss was born on February 30, 1777. Kar Gauss showed brilliant mental abilities from the age of two. At the age of three he knew how to write and read, and he counted on an equal basis with his father and even corrected his mistakes.

There is a legend that once at school the teacher had to leave for a long time. To keep the students busy, he gave them a task - to calculate the sum of all numbers from 1 to 100. While the rest of the schoolchildren were painstakingly adding, Gauss noticed that numbers from opposite ends add up to the same sums, that is, 100 + 1 = 101, 99 + 2 = 101 and so on.

He instantly found the required amount, multiplying 101 by 50, which turned out to be 5050. It is not known how true this story is, but until old age Gauss made most of the calculations in his head.

Language expert

In addition to mathematics, Gauss was interested in philology. He hesitated between these two disciplines, but eventually entered the Faculty of Mathematics. Gauss knew many languages, including Russian, which he learned out of love for Russian literature and in order to read Lobachevsky’s works in the original. He liked Latin, so he wrote a significant part of his works in this language.

Normal distribution law

Normal Law distributions are a frequently occurring phenomenon in nature associated with probability distributions. The graph of this phenomenon is often called a Gaussian, despite the fact that Gauss was not the discoverer of this law. He only studied it, but he studied it very carefully.

Gauss and astronomy

Separate works of Gauss are devoted to astronomy. In them, he studied celestial mechanics, studied the orbits of small planets and discovered a way to determine orbital elements from three known quantities.

Gauss gun

An electromagnetic gun is also named after Gauss - a device that fires a metal projectile due to electromagnetic energy. Gauss is the discoverer of electromagnetism, hence the name of the gun.

Carl Friedrich Gauss, the son of a poor man and an uneducated mother, independently solved the riddle of the date of his own birth and determined it as April 30, 1777. From childhood, Gauss showed all the signs of genius. The young man completed the main work of his life, “Arithmetic Research,” back in 1798, when he was only 21 years old, although it would not be published until 1801. This work was of paramount importance for improving the theory of numbers as scientific discipline, and presented this field of knowledge as we know it today. The amazing abilities of Gauss so impressed the Duke of Brunswick that he sent Charles to study at the Charles College (now the Brunswick College). Technical University), which Gauss visited from 1792 to 1795. In 1795-1798. Gauss moves to the University of Götting. During his university years, the mathematician proved many significant theorems.

Start of work

1796 turns out to be the most successful year both for Gauss himself and for his theory of numbers. One after another, he does important discoveries. On March 30, for example, he reveals the rules for constructing a regular heptagon. It improves modular arithmetic and greatly simplifies manipulations in number theory. April 8 Gauss proves the law of reciprocity of quadratic residues, which allows mathematicians to find a solution to any quadratic equation of modular arithmetic. On May 31, he proposes the prime number theorem, thereby providing an accessible explanation of how prime numbers distributed among integers. On July 10, a scientist makes the discovery that any whole positive number can be expressed as the sum of no more than three triangular numbers.

In 1799, Gauss defended his dissertation in absentia, in which he presented new proofs of the theorem stating that every entire rational algebraic function with one variable can be represented by the product of real numbers of the first and second degrees. It confirms the fundamental theorem of algebra, which states that every non-constant polynomial in one variable with complex coefficients has at least one complex root. His efforts greatly simplify the concept of complex numbers.

Meanwhile, Italian astronomer Giuseppe Piazzi discovers the dwarf planet Ceres, which instantly disappears in the solar glow, but a few months later, when Piazzi expects to see it again in the sky, Ceres does not appear. Gauss, who had just turned 23 years old, having learned about the astronomer's problem, took up the task of solving it. In December 1801, after three months of hard work, he determined the position of Ceres on starry sky with an error of only half a degree.

In 1807, the brilliant scientist Gauss received the post of professor of astronomy and head astronomical observatory Gottingen, which he will occupy for the rest of his life.

Later years

In 1831, Gauss met professor of physics Wilhelm Weber, and this acquaintance turned out to be fruitful. Their joint work leads to new discoveries in the field of magnetism and the establishment of Kirchhoff's rules in the field of electricity. Gauss formulated the law own name. In 1833, Weber and Gauss invented the first electromechanical telegraph, which connected the observatory with the Göttingen Institute of Physics. Following this, in the courtyard of the astronomical observatory, a magnetic observatory was built, in which Gauss, together with Weber, founded the “Magnetic Club”, which was engaged in measurements magnetic field Lands in different points planets. Gauss also successfully developed a technique for determining the horizontal component of the Earth's magnetic field.

Personal life

Gauss's personal life was a succession of tragedies, beginning with the premature death of his first wife, Joanna Ostoff, in 1809, and the subsequent death of one of their children, Louis. Gauss marries again, to the best friend of his first wife, Frederica Wilhelmina Waldeck, but she, too, dies after a long illness. Gauss had six children from two marriages.

Death and legacy

Gauss died in 1855 in Göttingen, Hanover (now Lower Saxony in Germany). His body was cremated and buried in Albanifridhof. According to the study of his brain by Rudolf Wagner, Gauss's brain had a mass of 1.492 g and a brain cross-sectional area of ​​219.588 mm² (34.362 square inches), which scientifically proves that Gauss was a genius.

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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 up integers 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 for himself the formula for the sum 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.
They learn about Gauss 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 is the 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 impossible to I believe it." 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, having, as it were, rediscovered short term what has 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 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. At the same time, most of the time was occupied by works that remained unknown to contemporaries (elliptic functions).
Gauss believed that he could take his time in publishing 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 nova big 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 1769 comet, 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.
In 1810 it was big number honors: Gauss received the Prize of the Paris Academy of Sciences and gold medal Royal Society of London, 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 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 about curved surfaces." 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 for himself 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 by Gauss, 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"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.