Карл Фридрих Гаусс

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Карл Фридрих Гаусс
Портрет Кристиана Альбрехта Йенсена , 1840 г. (копия с Готлиба Бирмана, 1887 г.)
Рожденный	Иоганн Карл Фридрих Гаусс ( 1777-04-30 ) 30 апреля 1777 г. Брауншвейг , Княжество Брауншвейг-Вольфенбюттель, Священная Римская империя
Умер	23 февраля 1855 г. ) ( 1855-02-23 ) ( 77 лет Геттинген , Королевство Ганновер, Германская Конфедерация
Альма-матер	Кэролайн Колледж Геттингенский университет Университет Хельмштедта (доктор философии)
Известный	Полный список
Супруги	Йоханна Остхофф ( м. 1805; умер в 1809 г.) Минна Вальдек ( м. 1810; умер в 1831 г.)
Дети	6
Награды	Премия Лаланда (1809 г.) Медаль Копли (1838 г.)
Научная карьера
Поля	Математика, Астрономия, Геодезия, Магнетизм
Учреждения	Геттингенский университет
Диссертация	Новая демонстрация...   (1799)
Докторантура	Иоганн Фридрих Пфафф
Докторанты	show   Richard Dedekind Christian Ludwig Gerling Wilhelm Klinkerfues Johann Benedict Listing Bernhard Riemann August Ritter Karl von Staudt
Другие известные студенты	show   Gotthold Eisenstein Johann Franz Encke Carl Wolfgang Benjamin Goldschmidt Adolph Theodor Kupffer August Ferdinand Möbius Moritz Stern Georg Frederik Ursin Moritz Ludwig Wichmann
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Иоганн Карл Фридрих Гаусс (нем. Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ; Латынь : Карол Фридерикус Гаусс ; 30 апреля 1777 — 23 февраля 1855) — немецкий математик , астроном , геодезист и физик, внесший вклад во многие области математики и естественных наук. Он был директором Геттингенской обсерватории и профессором астрономии с 1807 года до своей смерти в 1855 году.

Во время учебы в Геттингенском университете он выдвинул несколько математических теорем . Гаусс завершил свои шедевры Disquisitiones Arithmeticae и Theoria motus corporum coelestium как частный ученый. Он дал второе и третье полные доказательства основной теоремы алгебры , внес вклад в теорию чисел и разработал теории бинарных и троичных квадратичных форм.

Гаусс сыграл важную роль в идентификации Цереры как карликовой планеты. Его работа по движению планетоидов, возмущенных большими планетами, привела к введению гравитационной постоянной Гаусса и метода наименьших квадратов , который он открыл до того, как Адриан-Мари Лежандр опубликовал его. Гаусс отвечал за обширные геодезические исследования Ганноверского королевства вместе с проектом измерения дуги с 1820 по 1844 год; он был одним из основоположников геофизики и сформулировал основные принципы магнетизма . Плодами его практической деятельности стали изобретения гелиотропа в 1821 году, магнитометра в 1833 году и – вместе с Вильгельмом Эдуардом Вебером – первого электромагнитного телеграфа в 1833 году.

Гаусс отказался публиковать неполные работы и оставил несколько работ на редактирование посмертно . Он считал, что наибольшее удовольствие доставляет сам процесс обучения, а не обладание знаниями. Гаусс признался, что не любит преподавать, но некоторые из его учеников стали влиятельными математиками.

Биография

Молодежь и образование

Дом рождения в Брансуике (разрушен во время Второй мировой войны)

Дом студента Гаусса в Геттингене.

Иоганн Карл Фридрих Гаусс родился 30 апреля 1777 года в Брауншвейге (Брауншвейг) в герцогстве Брауншвейг-Вольфенбюттель (ныне часть федеральной земли Германии Нижняя Саксония ) в семье с более низким социальным статусом. Его отец Гебхард Дитрих Гаусс работал на нескольких должностях: мясником, каменщиком, садовником и казначеем фонда пособий по случаю смерти. Гаусс охарактеризовал своего отца как благородного и уважаемого человека, но грубого и властного дома. Он имел опыт письма и вычислений, тогда как его вторая жена Доротея, мать Карла Фридриха, была почти неграмотной. У него был старший брат от первого брака отца.

Гаусс был вундеркиндом в математике. Когда учителя начальной школы заметили его интеллектуальные способности, они привлекли к нему внимание герцога Брауншвейгского , который отправил его в местный Коллегиум Каролинум . который он посещал с 1792 по 1795 год под руководством Эберхарда Августа Вильгельма фон Циммермана в качестве одного из своих учителей. После этого герцог предоставил ему ресурсы для изучения математики, естественных наук и классических языков в Геттингенском университете до 1798 года. Его профессором по математике был Авраам Готхельф Кестнер , которого Гаусс называл «ведущим математиком среди поэтов и ведущим поэтом среди математиков» из-за его эпиграмм . Астрономию преподавал Карл Феликс Зейффер , с которым Гаусс продолжал переписываться после окончания учебы; Ольберс и Гаусс высмеивали его в своей переписке. С другой стороны, он высоко ценил Георга Кристофа Лихтенберга , своего учителя физики, и Кристиана Готлоба Хейне , лекции которого по классике Гаусс посещал с удовольствием. Сокурсниками этого времени были Иоганн Фридрих Бензенберг , Фаркас Боляи и Генрих Вильгельм Брандес .

Вероятно, он был студентом-самоучкой по математике, поскольку самостоятельно заново открыл несколько теорем. Он решил геометрическую задачу, которая занимала математиков со времен древних греков , когда в 1796 году он определил, какие правильные многоугольники можно построить с помощью циркуля и линейки . математики вместо филологии . Это открытие в конечном итоге привело Гаусса к выбору карьеры Математический дневник Гаусса, сборник коротких заметок о его результатах за период с 1796 по 1814 год, показывает, что многие идеи его математического великого труда Disquisitiones Arithmeticae (1801) относятся к этому времени.

Частный ученый

Гаусс получил степень доктора философии в 1799 году, а не в Геттингене, как иногда утверждают. но по специальному запросу герцога Брауншвейгского из Университета Хельмштедта, единственного государственного университета герцогства. Иоганн Фридрих Пфафф защитил докторскую диссертацию, а Гаусс получил степень заочно, без дальнейшего устного экзамена. Затем герцог предоставил ему возможность жить частным ученым в Брауншвейге. Впоследствии Гаусс отказался от звонков из Российской академии наук в Санкт-Петербурге и Ландсхутского университета . Позже герцог пообещал ему основать обсерваторию в Брауншвейге в 1804 году. Архитектор Петер Йозеф Краэ разработал предварительные проекты, но одна из войн Наполеона отменила эти планы: Герцог был убит в битве при Йене в 1806 году. В следующем году герцогство было упразднено, и финансовая поддержка Гаусса прекратилась.

Когда Гаусс рассчитывал орбиты астероидов в первые годы века, он установил контакт с астрономическим сообществом Бремена и Лилиенталя , особенно с Вильгельмом Ольберсом , Карлом Людвигом Гардингом и Фридрихом Вильгельмом Бесселем , как часть неформальной группы астрономов, известной как Небесная полиция . Одной из их целей было открытие новых планет. Они собрали данные об астероидах и кометах и ​​легли в основу исследований Гаусса их орбит, которые он позже опубликовал в своем астрономическом выдающемся труде Theoria motus corporum coelestium (1809 г.).

Professor in Göttingen

Old Göttingen observatory, c. 1800

In November 1807, Gauss followed a call to the University of Göttingen, then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory, and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt, later discovered to be Prince-primate Dalberg, paid the sum.

Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach, and a heliometer from Fraunhofer.

The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade.

Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about the burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics, and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor, Dedekind, Dirksen, Encke, Gould, Heine, Klinkerfues, Kupffer, Listing, Möbius, Nicolai, Riemann, Ritter, Schering, Scherk, Schumacher, von Staudt, Stern, Ursin; as geoscientists Sartorius von Waltershausen, and Wappäus.

Gauss did not write any textbook and disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German. He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians.

The new Göttingen observatory of 1816; Gauss's living rooms were in the western wing (right)

Wilhelm Weber and Heinrich Ewald (first row) as members of the Göttingen Seven

Gauss on his deathbed (1855) (daguerreotype from Philipp Petri)

In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by a strong calculus as the sole tasks of astronomy. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures, the physicist Mayer, known for his textbooks, his successor Weber since 1831, and in the observatory Harding, who took the main part of lectures in practical astronomy. When the observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations.

Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which the years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834.

Nevertheless, Gauss twice refused the opportunity to solve the problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, and in his later years he was one of the best-paid professors of the university.

When Gauss was asked for help by his colleague and friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title, Gauss provided a doctorate honoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. He also gave successful support to the mathematician Gotthold Eisenstein in Berlin.

Gauss was loyal to the House of Hanover. After King William IV died in 1837, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the "Göttingen Seven", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss was deeply affected by this quarrel but saw no possibility to help them.

Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. Being entrusted with the widow's pension fund of the university, he dealt with actuarial science and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years.

Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of a heart attack in Göttingen; and was interred in the Albani Cemetery there. Heinrich Ewald, Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer, gave eulogies at his funeral.

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms.

Gauss's brain

The day after Gauss's death his brain was removed, preserved, and studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams (52.6 oz). Wagner's son Hermann, a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of both persons. Thus, all investigations on Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs.

Family

Gauss's second wife Wilhelmine Waldeck

Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after the birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids.

On 4 August 1810, the widower married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than a decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald. Gauss's mother Dorothea lived in his house from 1817 until she died in 1839.

The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in the summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways. In 1836 he studied the railroad system in the US for some months.

Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the American Fur Company in the Midwest. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel; he then moved to Missouri, started as a farmer and became wealthy in the shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children.

• 
  Joseph Gauss
• 
  Sophie Gauss née Erythropel
  Joseph's wife
• 
  Wilhelmina Gauss
• 
  Heinrich Ewald
  Wilhelmina's husband
• 
  Eugen (Eugene) Gauss
• 
  Henrietta Gauss née Fawcett
  Eugene's wife
• 
  Wilhelm (Charles William) Gauss
• 
  Louisa Aletta Gauss née Fallenstein
  William's wife
• 
  Therese Gauss
• 
  Constantin Staufenau
  Therese's husband

Personality

Scholar

A student draws his professor of mathematics: Caricature of Abraham Gotthelf Kästner by Gauss (1795)

A student draws his professor of mathematics: Gauss sketched by his student Johann Benedict Listing, 1830

In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into the French language.

Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like Leonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, Gauss however introduced a new style of direct and complete explanation that did not attempt to show the reader the author's train of thought.

Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments.

— Klein 1894, p. 101

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows:

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

— Dunnington 2004, p. 416

The posthumous papers, his scientific diary, and short glosses in his own textbooks show that he worked to a great extent in an empirical way. He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked the results by masterly estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used a lot of mathematical tables, examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination. It has been taken as a curious feature of his working style that he carried out calculations with a high degree of precision much more than required, and prepared tables with more decimal places than ever requested for practical purposes. Very likely, this method gave him a lot of material which he used in finding theorems in number theory.

Gauss's seal with his motto Pauca sed Matura

Gauss refused to publish work that he did not consider complete and above criticism. This perfectionism was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.

On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not the first to publish" differed from that of his scientific contemporaries. In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of the history of science and more time than he wished to spend.

Private man

Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on the summit of science". His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host. Gauss abominated polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on a university chair in Göttingen, "because he was always involved in some polemic."

Gauss's residence from 1808 to 1816 in the first floor

Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody, the most personal surviving document of Gauss. The situation worsened when tuberculosis ultimately destroyed the health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings".

By reason of his wife's illness, both younger sons were educated for some years in Celle, far from Göttingen. The military career of his elder son Joseph ended after more than two decades with the rank of a poorly paid first lieutenant, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. The second son Eugen shared a good measure of his father's talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused a scandal in public, Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life.

Collecting numerical data on very different things, useful or useless, became a habit in his later years, for example, the number of paths from his home to certain places in Göttingen, or the number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days.

Similar to his excellent knowledge of Latin he was also acquainted with modern languages. At the age of 62, he began to teach himself Russian, very likely to understand scientific writings from Russia, among them those of Lobachevsky on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in the original languages. His favorite English author was Walter Scott, his favorite German Jean Paul. Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of the contemporary school of Naturphilosophie.

Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto "mundus vult decipi". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the Revolutions of 1848, though he agreed with some of their aims, such as the idea of a unified Germany. As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors.

Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded – not on account of my hard efforts, but by the grace of the Lord." Gauss was a member of the Lutheran church, like most of the population in northern Germany. It seems that he did not believe all dogmas or understand the Holy Bible quite literally. Sartorius mentioned Gauss's religious tolerance, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions.

Scientific work

Algebra and number theory

Fundamental theorem of algebra

German stamp commemorating Gauss's 200th anniversary: the complex plane or Gauss plane

In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Disquisitiones Arithmeticae

Main article: Disquisitiones Arithmeticae

In the preface to the Disquisitiones, Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself. The Disquisitiones Arithmeticae, written since 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces the triple bar symbol (≡) for congruence and uses it for a clean presentation of modular arithmetic. It deals with the unique factorization theorem and primitive roots modulo n. In the main chapters, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms.

The Disquisitiones include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares, he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards the end of the fifth chapter, it appears that Gauss already knew the class number formula in 1801.

In the last chapter, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or the product of a power of 2 and any number of distinct Fermat primes. In the same chapter, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields, which amounts to counting integral points on an elliptic curve. An unfinished eight chapter was found among left papers only after his death, consisting of work done during 1797–1799.

Further investigations

One of Gauss's first results was the empirically found conjecture of 1792 – the later called prime number theorem – giving an estimation of the number of prime numbers by using the integral logarithm.

When Olbers encouraged Gauss in 1816 to compete for a prize from the French Academy on proof for Fermat's Last Theorem (FLT), he refused because of his low esteem on this matter. However, among his left works a short undated paper was found with proofs of FLT for the cases n = 3 and n = 5. The particular case of n = 3 was proved much earlier by Leonhard Euler, but Gauss developed a more streamlined proof which made use of Eisenstein integers; though more general, the proof was simpler than in the real integers case.

Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centers of the spheres form a cubic face-centered arrangement, when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms. Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.

In two papers on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers $\mathbb {Z} [i]$, showed that it is a unique factorization domain. and generalized some key arithmetic concepts, such as Fermat's little theorem and Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.

In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.

Analysis

One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through the Landen's transformation, and a diary entry recorded the discovery of the connection of Gauss's constant to lemniscatic elliptic functions, a result that Gauss stated that "will surely open an entirely new field of analysis". He also made early inroads into the more formal issues of the foundations of complex analysis, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – Cauchy's integral theorem – and understood the notion of complex residues when integrating around poles.

Euler's pentagonal numbers theorem, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on Jacobi theta functions, culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case. His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.

Several mathematical fragments in his Nachlass indicate that he knew parts of the modern theory of modular forms. In his work on the multivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM to its two "simplest values". In his unpublished writings he recognized and made a sketch of the key concept of fundamental domain for the modular group. One of Gauss's sketches of this kind was a drawing of a tessellation of the unit disk by "equilateral" hyperbolic triangles with all angles equal to $\pi /4$.

An example of Gauss's insight in the fields of analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the lemniscate curve, which inspired Abel's theorem on lemniscate division. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of quadratic Gauss sum, in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem out of his work on elliptic functions theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic functions theory, and only the later work of mathematicians such as Jacobi and Hermite has exposed the crux of his argument.

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general hypergeometric function $F(\alpha ,\beta ,\gamma ,x)$, and shows that many of the functions known at the time are special cases of the hypergeometric function. This work is the first one with an exact inquiry of convergence of infinite series in the history of mathematics. Furthermore, it deals with infinite continued fractions arising as ratios of hypergeometric functions which are now called Gauss continued fractions.

In 1823, Gauss won the prize of the Danish Society with an essay on conformal mappings, which contains several developments that pertain to the field of complex analysis. Gauss stated that angle-preserving mappings in the complex plane must be complex analytic functions, and used the later called Beltrami equation to prove the existence of isothermal coordinates on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an ellipsoid of revolution.

Numeric analysis

Gauss often deduced theorems inductively from numerical data he had collected empirically. As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to numeric analysis, as the method of Gaussian quadrature published in 1816.

In a private letter to Gerling from 1823, he described a solution of a 4X4 system of linear equations by using Gauss-Seidel method – an "indirect" iterative method for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.

Gauss invented an algorithm for calculating what is now called discrete Fourier transforms, when calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey found their similar Cooley–Tukey FFT algorithm. He developed it as a trigonometric interpolation method, but the paper Theoria Interpolationis Methodo Nova Tractata was published only posthumously in 1876, preceded by the first presentation by Joseph Fourier on the subject in 1807.

Chronology

The first publication following the doctoral thesis dealt with the determination of the date of Easter (1800), an elementary matter of mathematics. Gauss aimed to present a most convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usually required terms of golden number, epact, solar cycle, domenical letter, and any religious connotations. Biographers speculated on the reason why Gauss dealt with this matter, but it is likely comprehensible by the historical background. The replacement of the Julian calendar by the Gregorian calendar had caused confusion in the Holy Roman Empire since the 16th century, and was not finished in Germany until 1700, when the difference of eleven days was deleted, but the difference in calculating the date of Easter remained between Protestant and Catholic territories. A further agreement of 1776 equalized the confessional way of counting; thus in the Protestant states like the Duchy of Brunswick the Easter of 1777, five weeks before Gauss's birth, was the first one calculated in the new manner. The public difficulties of replacement may be the historical background for the confusion on this matter in the Gauss family (see chapter: Anecdotes). For being connected with the Easter regulations, an essay on the date of Pesach followed soon in 1802.

Astronomy

Main article: Discovery of Ceres

Carl Friedrich Gauss 1803 by Johann Christian August Schwartz

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called Titius–Bode law, and named it Ceres. He could track it only for a short time until it disappeared behind the glare of the Sun. The mathematical tools of the time were not sufficient to extrapolate a position from the few data for its reappearance. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree when Franz Xaver von Zach on 7 and 31 December at Gotha, and independently Heinrich Olbers on 1 and 2 January in Bremen, identified the object near the predicted position.

Gauss's method leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.

The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum. It introduced the Gaussian gravitational constant.

Since the new asteroids had been discovered, Gauss occupied himself with the perturbations of their orbital elements. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was Pallas, because of its great eccentricity and orbital inclination, whereby Laplace's method did not work. Gauss used his own tools: the arithmetic–geometric mean, the hypergeometric function, and his method of interpolation. He found an orbital resonance with Jupiter in proportion 18:7 in 1812; Gauss gave this result as cipher, and gave the explicit meaning only in letters to Olbers and Bessel. After long years of work, he finished it in 1816 without a result that seemed sufficient to him. This marked the end of his activities in theoretical astronomy.

Göttingen observatory seen from the North-west (by Friedrich Besemann, c. 1835)

One fruit of Gauss's research on Pallas perturbations was the Determinatio Attractionis... (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". It introduced an averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time taking the planet to follow the corresponding orbital arcs. Gauss presents the method of evaluating the gravitational attraction of such an elliptic ring, which includes several steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral.

While Gauss's contributions to theoretical astronomy came to an end, more practical activities in observational astronomy continued and occupied him during his entire career. Even early in 1799, Gauss dealt with the determination of longitude by use of the lunar parallax, for which he developed more convenient formulas than those were in common use. After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables for nutation and aberration, the solar coordinates, and refraction. He made many contributions to spherical geometry, and in this context solved some practical problems about navigation by stars. He published a great number of observations, mainly on minor planets and comets; his last observation was the solar eclipse of 28 July 1851.

Theory of errors

Gauss likely used the method of least squares for calculating the orbit of Ceres to minimize the impact of measurement error. The method was published first by Adrien-Marie Legendre in 1805, but Gauss claimed in Theoria motus (1809) that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of normally distributed errors (Gauss–Markov theorem), in the two-part paper Theoria combinationis observationum erroribus minimis obnoxiae (1823).

In the first paper he proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of Gauss-Winckler inequality). He derived lower and upper bounds for the variance of sample variance. In the second paper, Gauss described recursive least squares methods. His work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert to the Gauss-Helmert model.

Gauss also contributed to problems in probability theory that are not directly concerned with the theory of errors. One example appears as a diary note where he tried to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in (0,1). He derived this distribution, now known as the Gauss-Kuzmin distribution, as a by-product of the discovery of the ergodicity of the Gauss map for continued fractions. Gauss's solution is the first-ever result in the metrical theory of continued fractions.

Arc measurement and geodetic survey

Order of King George IV from 9 May 1820 to the triangulation project (with the additional signature of Count Ernst zu Münster below)

The heliotrope

Gauss's vice heliotrope, a Troughton sextant with additional mirror

Gauss was busy with geodetic problems since 1799 when he helped Karl Ludwig von Lecoq with calculations during his survey in Westphalia. Beginning in 1804, he taught himself some geodetic practise with a sextant in Brunswick, and Göttingen.

Since 1816, Gauss's former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg as head of an observatory, carried out a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south. This project was the basis for map production but also aimed at determining the geodetic arc between the terminal sites. Data from geodetic arcs were used to determine the dimensions of the earth geoid, and long arc distances brought more precise results. Schumacher asked Gauss to continue this work further to the south in the Kingdom of Hanover; Gauss agreed after a short time of hesitation. Finally, in May 1820, King George IV gave the order to Gauss.

An arc measurement needs a precise astronomical determination of at least two points in the network. Gauss and Schumacher used the favourite occasion that both observatories in Göttingen and Altona, in the garden of Schumacher's house, laid nearly in the same longitude. The latitude was measured with both their instruments and a zenith sector of Ramsden that was transported to both observatories.

Gauss and Schumacher had already determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818. During the summers of 1821 until 1825 Gauss directed the triangulation work personally, from Thuringia in the south to the river Elbe in the north. The triangle between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum size of 107 km (66.5 miles). In the thinly populated Lüneburg Heath without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.

For pointing signals, Gauss invented a new instrument with movable mirrors and a small telescope that reflects the sunbeams to the triangulation points, and named it heliotrope. Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope. Gauss got assistance by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in the baseline measurement (Braak Base Line) of Schumacher in the village of Braak near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.

An additional result was a better value of flattening of the approximative Earth ellipsoid. Gauss developed the universal transverse Mercator projection of the ellipsoidal shaped Earth (what he named conform projection) for representing geodetical data in plane charts.

When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover with a Royal decree from 25 March 1828. The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Gauss, who applied his mathematical inventions such as the method of least squares and the elimination method to it. The project was finished in 1844, and Gauss sent a final report of the project to the government; his method of projection was not edited until 1866.

In 1828, when studying differences in latitude, Gauss first defined a physical approximation for the figure of the Earth as the surface everywhere perpendicular to the direction of gravity; later his doctoral student Johann Benedict Listing called this the geoid.

Differential geometry

Main article: Theorema Egregium

The geodetic survey of Hanover fueled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led him in 1828 to the publication of a memoir that marks the birth of modern differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as cartesian graphs of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, regardless of the embedding of the surface in three-dimensional or two-dimensional space.

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended manifolds; it clarifies the distinction between the intrinsic properties of the manifold (the metric) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing projections for geographical maps. A portion of this essay is dedicated to a profound study of geodesics. In particular, Gauss proves the local Gauss-Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.

Gauss's memoir from 1828 lacks the conception of geodesic curvature. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its invariance under isometric transformations, a result that was later obtained by Ferdinand Minding and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by Pierre Ossian Bonnet in 1848 and known as Gauss-Bonnet theorem.

Non-Euclidean geometry

Main article: Non-Euclidean geometry

Lithography by Siegfried Bendixen (1828)

In the lifetime of Gauss, a vivid discussion on the Parallel postulate in Euclidean geometry was going on. Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it. Gauss thought about the basics of geometry since the 1790s years, but in the 1810s he realized that a non-Euclidean geometry without the parallel postulate could solve the problem. In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry", but he strongly forbade Taurinus to make any use of it.

The first publications on non-Euclidean geometry in the history of mathematics were authored by Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832. In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion. Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai claiming that these were congruent to his own thoughts of some decades. However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his letter remarks are only vague and obscure.

Sartorius mentioned Gauss's work on non-Euclidean geometry firstly in 1856, but only the edition of left papers in Volume VIII of the Collected Works (1900) showed Gauss's ideas on that matter, at a time when non-Euclidean geometry had yet grown out of controversial discussion.

Early topology

Gauss was also an early pioneer of topology or Geometria Situs, as it was called in his lifetime. The first proof of the fundamental theorem of algebra in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.

Gauss bust by Heinrich Hesemann (1855)

Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the celestial sphere in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are linked, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid 7 Iris, he published a further qualitative discussion of the Zodiacus.

In Gauss's letters of 1820–1830, he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of knots. To do so he devised a symbolical scheme, the Gauss code, that in a sense captured the characteristic features of tract figures.

In a fragment from 1833, Gauss defined the linking number of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. On the same note, he lamented the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as braids and tangles.

Gauss's influence in later years to the emerging field of topology, which he held in high esteem, was through occasional remarks and oral communications to Mobius and Listing.

Minor mathematical accomplishments

Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography, he stated the fundamental theorem of axonometry, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers. He described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane, and gave a proof for the geometric theorem that the altitudes of a triangle always meet in a single orthocenter.

Gauss was concerned with John Napier's "Pentagramma mirificum" – a certain spherical pentagram – for several decades; he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given quadrilateral, and discovered a surprising result about the computation of area of pentagons.

Magnetism and telegraphy

Geomagnetism

Gauss-Weber monument in Göttingen by Ferdinand Hartzer (1899)

The Gauss-Weber magnetometer

Gauss had been interested in magnetism since 1803. After Alexander von Humboldt visited Göttingen in 1826, both scientists began intensive research on geomagnetism, partly independently, partly in productive cooperation. In 1828, Gauss was Humboldt's guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquainted with the physicist Wilhelm Weber.

When Weber got the chair for physics in Göttingen as successor of Johann Tobias Mayer by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of magnetism with a representation for the unit of magnetism in terms of mass, charge, and time. They founded the Magnetic Association (German: Magnetischer Verein), an international working group of several observatories, which supported measurements of Earth's magnetic field in many regions of the world with equal methods at arranged dates in the years 1836 to 1841.

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the British dominions with a letter to the Duke of Sussex, then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods. Together with other instigators, this led to a global program known as "Magnetical crusade" under the direction of Edward Sabine. The dates, times, and intervals of observations were determined in advance, the Göttingen mean time was used as standard. 61 stations on all five continents participated in this global program. Gauss and Weber founded a series for publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to Leipzig in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Association activity.

Following Humboldt's example, Gauss ordered a magnetic observatory to be built in the garden of the observatory, but the scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity, but discriminated Humboldt's concept of magnetic intensity to the terms of "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of intensity of the magnetic field, and constructed a suitable magnetometer to measure absolute values of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus. The precision of the magnetometer was about ten times higher than of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities.

Gauss carried out a General Theory of Terrestrial Magnetism (1839), in what he believed to describe the nature of magnetic force; according to Felix Klein, this work is a presentation of observations by use of spherical harmonics rather than a physical theory. The theory predicted the existence of exactly two magnetic poles on the Earth, thus Hansteen's idea of four magnetic poles became obsolete, and the data allowed to determine their location with rather good precision.

Gauss influenced the beginning of geophysics in Russia, when Adolph Theodor Kupffer, one of his former students, founded a magnetic observatory in St. Petersburg, following the example of the observatory in Göttingen, and similarly, Ivan Simonov in Kazan.

Electromagnetism

Town plan of Göttingen with course of the telegraphic connection

The discoveries of Hans Christian Ørsted on electromagnetism and Michael Faraday on electromagnetic induction drew Gauss's attention to these matters. Gauss and Weber found rules for branched electric circuits, which were later found independently and firstly published by Gustav Kirchhoff and benamed after him as Kirchhoff's circuit laws, and made inquiries on electromagnetism. They constructed the first electromechanical telegraph in 1833, and Weber himself connected the observatory with the institute for physics in the town centre of Göttingen, but they did not care for any further development of this invention for commercial purposes.

Gauss's main theoretical interests in electromagnetism were reflected in his attempts to formulate quantitive laws governing electromagnetic induction. In notebooks from these years, he recorded several innovative formulations; he discovered the idea of vector potential function (independently rediscovered by Franz Ernst Neumann in 1845), and in January 1835 he wrote down an "induction law" equivalent to Faraday's law, which stated that the electromotive force at a given point in space is equal to the instantaneous rate of change (with respect to time) of this function.

Gauss tried to find a unifying law for long-distance effects of electrostatics, electrodynamics, electromagnetism, and induction, comparable to Newton's law of gravitation, but his attempt ended in a "tragic failure".

Potential theory

Since Isaac Newton had shown theoretically that the Earth and rotating stars assume non-spherical shapes, the problem of attraction of ellipsoids gained importance in mathematical astronomy. In his first publication on potential theory, the "Theoria attractionis..." (1813), Gauss provided a closed-form expression to the gravitational attraction of a homogeneous triaxial ellipsoid at every point in space. In contrast to previous research of Maclaurin, Laplace and Lagrange, Gauss's new solution treated the attraction more directly in the form of an elliptic integral. In the process, he also proved and applied some special cases of the so-called Gauss's theorem in vector analysis.

In the General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances (1840) Gauss gave the baseline of a theory of the magnetic potential, based on Lagrange, Laplace, and Poisson; it seems rather unlikely that he knew the previous works of George Green on this subject. However, Gauss could never give any reasons for magnetism, nor a theory of magnetism similar to Newton's work on gravitation, that enabled scientists to predict geomagnetic effects in the future.

Optics

Gauss's calculations enabled instrument maker Johann Georg Repsold in Hamburg to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was the nonprecise knowledge of the refractive index and dispersion of the used glass types. In a short article from 1817 Gauss dealt with the problem of removal of chromatic aberration in double lenses, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician Carl August von Steinheil, who in 1860 introduced the achromatic Steinheil doublet, partly based on Gauss's calculations. Many results in geometrical optics are only scattered in Gauss's correspondences and hand notes.

In the Dioptrical Investigations (1840), Gauss gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). He characterized optical systems under a paraxial approximation only by its cardinal points, and he derived the Gaussian lens formula, applicable without restrictions in respect to the thickness of the lenses.

Mechanics

Gauss's first business in mechanics concerned the earth's rotation. When his university friend Benzenberg carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as an effect of the Coriolis force, he asked Gauss for a theory-based calculation of the values for comparison with the experimental ones. Gauss elaborated a system of fundamental equations for the motion, and the results corresponded sufficiently with Benzenberg's data, who added Gauss's considerations as an appendix to his book on falling experiments.

After Foucault had demonstrated the earth's rotation by his pendulum experiment in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum. It is described in the Gauss–Gerling correspondence, and Weber made some experiments with this apparatus in 1853, but no data were published.

Gauss's principle of least constraint of 1829 was established as a general concept to overcome the division of mechanics into statics and dynamics, combining D'Alembert's principle with Lagrange's principle of virtual work, and showing analogies to the method of least squares.  

Metrology

In 1828, Gauss was appointed to head of a Board for weights and measures of the Kingdom of Hanover. He provided the creation of standards of length and measures. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical preparation. In the correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision. He submitted the final reports on the Hanoverian foot and pound to the government in 1841. This work got more than regional importance by the order of a law of 1836 that connected the Hanoverian measures with the English ones.

Anecdotes

Parochial registration of Gauss's christening on 4 May 1777 with later added birth date

Several stories of his early genius have been reported. Carl Friedrich Gauss's mother had never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which occurs 39 days after Easter. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.

In his memorial on Gauss, Wolfgang Sartorius von Waltershausen tells a story about the three-year-old Gauss, who corrected a math error his father made. The most popular story, also told by Sartorius, tells of a school exercise: the teacher Büttner and his assistant Martin Bartels ordered students to add an arithmetic series. Out of about a hundred pupils, Gauss was the first to solve the problem correctly by a significant margin. Although (or because) Sartorius gave no details, over time many versions of this story have been created, with more and more details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100 – and the circumstances in the classroom.

Honours and awards

Copley Medal for Gauss (1838)

The first membership of a scientific society was given to Gauss in 1802 by the Russian Academy of Sciences. Further memberships (corresponding, foreign or full) were awarded from the Academy of Sciences in Göttingen (1802/ 1807), the French Academy of Sciences (1804/ 1820), the Royal Society of London (1804), the Royal Prussian Academy in Berlin (1810), the National Academy of Science in Verona (1810), the Royal Society of Edinburgh (1820), the Bavarian Academy of Sciences of Munich (1820), the Royal Danish Academy in Copenhagen (1821), the Royal Astronomical Society in London (1821), the Royal Swedish Academy of Sciences (1821), the American Academy of Arts and Sciences in Boston (1822), the Royal Bohemian Society of Sciences in Prague (1833), the Royal Academy of Science, Letters and Fine Arts of Belgium (1841/1845), the Royal Society of Sciences in Uppsala (1843), the Royal Irish Academy in Dublin (1843), the Royal Institute of the Netherlands (1845/ 1851), the Spanish Royal Academy of Sciences in Madrid (1850), the Russian Geographical Society (1851), the Imperial Academy of Sciences in Vienna (1848), the American Philosophical Society (1853), the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem.

Both the University of Kazan and the Philosophy Faculty of the University of Prague appointed him honorary member in 1848.

Gauss received the Lalande Prize from the French Academy of Science in 1809 for the theory of planets and the means of determining their orbits from only three observations, the Danish Academy of Science prize in 1823 for his memoir on conformal projection, and the Copley Medal from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".

Gauss was appointed Knight of the French Legion of Honour in 1837, and was taken as one of the first members of the Prussian Order Pour le Merite (Civil class) when it was established in 1842. He received the Order of the Crown of Westphalia (1810), the Danish Order of the Dannebrog (1817), the Hanoverian Royal Guelphic Order (1815), the Swedish Order of the Polar Star (1844), the Order of Henry the Lion (1849), and the Bavarian Maximilian Order for Science and Art (1853).

The Kings of Hanover appointed him the honorary titles "Hofrath" (1816) and "Geheimer Hofrath" (1845). In 1949, on the occasion of his golden doctor degree jubilee, he got the honorary citizenship of both towns of Brunswick and Göttingen. Soon after his death a medal was issued by order of King George V of Hanover with the back inscription dedicated "to the Prince of Mathematicians".

The "Gauss-Gesellschaft Göttingen" ("Göttingen Gauss Society") was founded in 1964 for research on life and work of Carl Friedrich Gauss and related persons and edits the Mitteilungen der Gauss-Gesellschaft (Communications of the Gauss Society).

Names and commemorations

• List of things named after Carl Friedrich Gauss

Selected writings

Mathematics and astronomy

Statue of Gauss in Brunswick (1880), made by Hermann Heinrich Howaldt, designed by Fritz Schaper

• 1799: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse [New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors of the first or second degree]. Helmstedt: C. G. Fleckeisen. (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) Original book
• 1816: "Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 107–134. Original
• 1816: "Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 135–142. Original
• 1850: "Beiträge zur Theorie der algebraischen Gleichungen". Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 4: 34–35. Original (Lecture from 1849)
  • Die vier Gauss'schen Beweise für die Zerlegung ganzer algebraischer Funktionen in reelle Faktoren ersten und zweiten Grades. (1799–1849) [The four Gaussian proofs of the fundamental theorem of algebra]. Translated by Netto. Leipzig: Wilhelm Engelmann. 1890. (German)
• 1800: "Berechnung des Osterfestes" [Calculation of Easter]. Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde (in German). 2: 121–130. Original
• 1801: Disquisitiones Arithmeticae. Leipzig: Gerh. Fleischer jun.
  • Gauss, Carl Friedrich (1986). Disquisitiones Arithmeticae & other papers on number theory. Translated by Clarke, Arthur A. (2nd, corrected ed.). New York: Springer. doi:10.1007/978-1-4939-7560-0. ISBN 978-0-387-96254-2. (translated from the second German edition, Göttingen 1860)
• 1802: "Berechnung des jüdischen Osterfestes" [Calculation of Jewish Easter]. Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde (in German). 5: 435–437. Original
• 1804: "Über die Grenzen der geocentrischen Oerter der Planeten" [On the limits of the geocentric places of the planets]. Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde (in German). 10: 171–193. Original (on the Zodiacus)
• 1808: "Theorematis arithmetici demonstratio nova". Commentationes Societatis Regiae Scientiarum Gottingensis. Comm. Math. 16: 69–74. Original (Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity)
• 1808: Methodus peculiaris elevationem poli determinandi (in Latin). Göttingen.
• 1809: Theoria motus corporum coelestium in sectionibus conicis solem ambientium (in Latin). Hamburg: Friedrich Perthes & Johann Heinrich Besser. Original book
  • Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections. Translated by Davis, Charles Henry. Little, Brown & Co. 1857.
• 1811: "Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1806, 1807, 1808, 1809". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Math. 1: 1–26. Original (from 1810) (Orbit of Pallas)
• 1811: "Summatio quarundam serierum singularium". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 1: 1–40. Original (from 1808) (Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity)
• 1813: "Disquisitiones generales circa seriem infinitam $1+{\frac {\alpha \beta }{\gamma .1}}+{\mbox{etc.}}$". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 2: 1–42. Original (from 1812, contains the Gauss's continued fraction)
• 1816: "Methodus nova integralium valores per approximationem inveniendi". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 39–76. Original (from 1814)
• 1818: "Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 4: 3–20. Original (from 1817) (Fifth and sixth proofs of quadratic reciprocity)
• 1818: "Determinatio attractionis, quam in punctum positionis datae exerceret planeta, si eius massa per totamorbitam, ratione temporis, quo singulae partes describuntur, uniformiter esset dispertita". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 4: 21–48. Original (Only reference to the – mostly unpublished – work on the algorithm of the arithmetic-geometric mean.)
• 1823: "Theoria combinationis observationum erroribus minimis obnoxiae. Pars Prior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 5: 33–62. Original (from 1821)
• 1823: "Theoria combinationis observationum erroribus minimis obnoxiae. Pars Posterior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 5: 63–90. Original
• 1825: "Allgemeine Auflösung der Aufgabe die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird". Astronomische Abhandlungen. 3. Altona. (Prize winning essay from 1822 on conformal mapping)
• 1828: Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector [Determination of the Difference in Latitude between the Observatories of Göttingen and Altona by Observations with Ramsden's Zenith sector] (in German). Göttingen: Vandenhoeck und Ruprecht. 1828. Original book
• 1828: Gauss, Carl Friedrich (1828). "Supplementum theoriae combinationis observationum erroribus minimis obnoxiae". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 57–98. Bibcode:1828stco.book.....G. (from 1826)
  • Gauss, Carl Friedrich; Stewart, G. W. (1995). Theory of the Combination of Observations Least Subject to Errors. Part One, Part Two, Supplement (Classics in Applied Mathematics). Translated by G. W. Stewart. Philadelphia: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971248. ISBN 978-0-89871-347-3. (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
• 1828: "Disquisitiones generales circa superficies curvas". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 99–146. Original (from 1827)
  • General Investigations of Curved Surfaces (PDF). Translated by J. C. Morehead and A. M. Hiltebeitel. The Princeton University Library. 1902.
• 1828: "Theoria residuorum biquadraticorum, Commentatio prima". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 27–56. Original (from 1825)
• 1832: "Theoria residuorum biquadraticorum, Commentatio secunda". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 7: 89–148. Original (from 1831) (Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i)
• 1845: "Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung". Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band, von den Jahren 1842–1844: 3–46. Original (from 1843)
• 1847: "Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung". Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band, von den Jahren 1845–1847: 3–44. Original (from 1846)
• 1848: Gauss (1848). "Schreiben des Herrn Geheimen Hofrathes Gauss an den Herausgeber" [Letter of Mr. Secret Councillor of the Court Gauss to the editor]. Astronomische Nachrichten (in German). 27: 1–3. Bibcode:1848AN.....27....1G. doi:10.1002/asna.18480270102. Original
• 1903: Wissenschaftliches Tagebuch (Klein, Felix, ed. (1903). "Gauß' wissenschaftliches Tagebuch 1796–1814". Mathematische Annalen (in Latin and German). 57: 1–34. doi:10.1007/BF01449013. S2CID 119641638.) Original book (from 1847, on the Zodiacus)
  • Jeremy Gray (1984). "A commentary on Gauss's mathematical diary, 1796–1814". Expositiones Mathematicae. 2: 97–130.

Physics

• 1804: Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde ( in original book: Benzenberg, Johann Friedrich. Versuche über das Gesetz des Falls, über den Widerstand der Luft und über die Umdrehung der Erde [Experiments on the Law of falling Bodies, on the Resistance of Air, and of the Rotation of the Earth]. Dortmund: Gebrüder Mallinckrodt. pp. 363–371. Original)
• 1813: "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 2: 1–24. Original (contains Gauss's theorem of vector analysis)
• 1817: "Ueber die achromatischen Doppelobjective besonders in Rücksicht der vollkommnern Aufhebung der Farbenzerstreuung" [On achromatic double lenses with special regard to a more complete dispersion of colours]. Zeitschrift für Astronomie und verwandte Wissenschaften (in German). IV: 345–351.
• 1829: "Über ein neues allgemeines Grundgesetz der Mechanik" [On a new General Fundamental Law of Mechanics]. Journal für die reine und angewandte Methematik (Crelle's Journal). 1829 (4): 232–235. 1829. doi:10.1515/crll.1829.4.232. S2CID 199545985.
• 1830: "Principia generalia theoriae figurae fluidorum in statu aequilibrii". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 7: 39–88. Original (from 1829)
• 1841: «Интенсивность магнитной силы Земли вернулась к абсолютному измерению» . Недавние комментарии Королевского общества геттингенских наук . 8 : 3–44. Оригинал (с 1832 г.)
  • Интенсивность магнитной силы Земли, приведенная к абсолютному измерению. Перевод Сьюзан П. Джонсон.
• 1836: Земной магнетизм и магнитометр (Оригинальная книга: ХК Шумахер (ред.). Ежегодник за 1836 год (на немецком языке). Том 1836. Тюбинген: книжный магазин JGCotta'sche. стр. 1–47. )
• 1840: Общие теоремы о силах притяжения и отталкивания, действующих в обратном соотношении квадрату расстояния (Оригинальная книга: о силах притяжения и отталкивания, действующих в пропорциях, обратных квадратам расстояний ( Общие теоремы на немецком языке). Лейпциг: Книжный магазин Weidmannsche. 1840.
• 1843: « Диоптрические исследования». Трактаты Королевского общества наук в Геттингене (на немецком языке). Первый том: 1–34. Оригинал (с 1840 г.)

Вместе с Вильгельмом Вебером

• 1837–1839: Результаты наблюдений магнитной ассоциации в 1836–1838 гг. (на немецком языке). Геттинген: книжный магазин Дитрих.
• 1840–1843: Результаты наблюдений магнитной ассоциации в 1839–1841 гг. (на немецком языке). Лейпциг: Weidmannsche Verlagsbuchhandlung.
• 1840: Атлас земного магнетизма составлен по элементам теории. Дополнение к результатам наблюдений магнитной ассоциации (на немецком языке). Лейпциг: Weidmannsche Verlagsbuchhandlung.

Собрание сочинений

• Королевская прусская академия наук, изд. Карл Фридрих Гаусс. Сочинения (на латыни и немецком языке). Том 1-12. Геттинген: (разные издательства). (включая неопубликованное литературное наследие)

Переписка

• Королевская Прусская академия наук, изд. Переписка Гаусса и Бесселя (на немецком языке). Лейпциг: Вильгельм Энгельманн. (письма с декабря 1804 г. по август 1844 г.)
• Шенберг, Эрих; Перлик, Альфонс (1955). Неизвестные письма от К. Ф. Гаусса и г-жи В. Бессель . Трактаты Баварской академии наук, Матем.-нат. Класс, Новая серия, №. 71 (на немецком языке). Мюнхен: Издательство Баварской академии наук. стр. 5–21. (письма Богуславскому с февраля 1835 г. по январь 1848 г.)
• Швемин, Фридхельм, изд. Переписка Карла Фридриха Гаусса и Иоганна Элерта Боде . Acta Historica Astronomica (на немецком языке). Том 53. Лейпциг: Akademische Verlaganstalt. ISBN  978-3-944913-43-8 . (письма с февраля 1802 г. по октябрь 1826 г.)
• Франц Шмидт, Пауль Штекель, изд. Переписка между Карлом Фридрихом Гауссом и Вольфгангом Бояи (на немецком языке). Лейпциг: Б. Г. Тойбнер. (письма с сентября 1797 г. по февраль 1853 г.; добавлены письма других корреспондентов)
• Аксель Виттманн, изд. Хотя и однако. Переписка Карла Фридриха Гаусса и Иоганна Франца Энке (на немецком языке). Ремаген: Издательство Кесселя. ISBN  978-3945941379 . (письма с июня 1810 г. по июнь 1854 г.)
• Клеменс Шефер, изд. Переписка между Карлом Фридрихом Гауссом и Кристианом Людвигом Герлингом (на немецком языке). Берлин: Отто Эльснер. (письма с июня 1810 г. по июнь 1854 г.)
• Карл Кристиан Брюнс , изд. Письма между А. в. Гумбольдт и Гаусс (на немецком языке). Лейпциг: Вильгельм Энгельманн. (письма с июля 1807 г. по декабрь 1854 г.; добавлены письма других корреспондентов)
• Рич, Карин ; Русанова, Елена (2018). Карл Крейль и земной магнетизм. Его переписка с Карлом Фридрихом Гауссом в историческом контексте . Издания Комиссии истории естествознания, математики и медицины. 68 (на немецком языке). Вена: Издательство Австрийской академии наук. (письма с 1835 по 1843 год)
• Джерарди, Тео, изд. Переписка между Карлом Фридрихом Гауссом и Карлом Людвигом фон Лекоком . Трактаты Академии наук в Геттингене, Математик-физический класс, № 1. 4 (на немецком языке). Геттинген: Ванденхук и Рупрехт. стр. 37–63. (письма с февраля 1799 г. по сентябрь 1800 г.)
• Форбс, Эрик Г. (1971). «Переписка между Карлом Фридрихом Гауссом и преподобным Невилом Маскелином (1802–05)» . Анналы науки . 27 (3): 213–237. дои : 10.1080/00033797100203767 .
• Каннингем, Клиффорд (2004). «Обнаружение пропавшей переписки между Карлом Фридрихом Гауссом и преподобным Невилом Маскелином (1802–05)» . Анналы науки . 61 (4): 469–481. дои : 10.1080/00033790310001660164 .
• Карл Шиллинг, изд. Переписка Ольберса и Гаусса: Первый раздел . Вильгельм Ольберс . Его жизнь и его творчество. Второй том (на немецком языке). Берлин: Юлиус Шпрингер. (письма с января 1802 г. по октябрь 1819 г.)
• Карл Шиллинг, изд. Переписка Ольберса и Гаусса: Второй раздел . Вильгельм Ольберс. Его жизнь и его творчество. Второй том (на немецком языке). Берлин: Юлиус Шпрингер. (письма с января 1820 г. по май 1839 г.; добавлены письма других корреспондентов)
• Кристиан Август Фридрих Петерс , изд. Переписка между К.Ф. Гауссом и Х.К. Шумахером (на немецком языке). Альтона: Густав Эш.
  • Тома 1+2 (письма с апреля 1808 г. по март 1836 г.)
  • Тома 3+4 (письма с марта 1836 г. по апрель 1845 г.)
  • Тома 5+6 (письма с апреля 1845 г. по ноябрь 1850 г.)
• Позер, Ганс, изд. Переписка Карла Фридриха Гаусса и Эберхарда Августа Циммермана . Трактаты Академии наук в Геттингене, Математик-физический класс, Эпизод 3, № 1. 39 (на немецком языке). Геттинген: Ванденхук и Рупрехт. ISBN  978-3525821169 . (письма с 1795 по 1815 гг.)

Геттингенская академия наук и гуманитарных наук предоставляет полную коллекцию известных писем Карла Фридриха Гаусса и к нему, доступную в Интернете. Литературное наследие хранится и предоставляется Геттингенской государственной и университетской библиотекой . Письменные материалы Карла Фридриха Гаусса и членов его семьи также можно найти в муниципальном архиве Брауншвейга.

Ссылки

Источники

• Бахманн, Пол (1922). «О теоретико-числовых работах Гаусса» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 1).
• Больца, Оскар (1921). «Гаусс и вариационное исчисление» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 5).
• Брендель, Мартин (1929). «Об астрономических трудах Гаусса» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том XI, 2 (Документ 3).
• Бюлер, Вальтер Кауфманн (1981). Гаусс: Биографическое исследование . Издательство Спрингер. ISBN  978-0-387-10662-5 .
• Даннингтон, Дж. Уолдо (2004). Карл Фридрих Гаусс: Титан науки . Математическая ассоциация Америки. ISBN  978-0-88385-547-8 . OCLC   53933110 . Первое издание: Карл Фридрих Гаусс: Титан науки. Исследование его жизни и творчества . Нью-Йорк: Экспозиционная пресса. 1955.
  • Грей, Джереми (1955). «Введение в «Гаусс» Даннингтона ». Карл Фридрих Гаусс: Титан науки. Исследование его жизни и творчества . Нью-Йорк: Экспозиционная пресса. стр. XIX – XXVI. С критическим взглядом на стиль и оценки Даннингтона.
• Галле, Андреас (1924). «О геодезических трудах Гаусса» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том XI, 2 (Документ 1).
• Гепперт, Харальд (1933). «О работах Гаусса по механике и теории потенциала» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 7).
• Кляйн, Феликс (1894). «Развитие математики в немецких университетах» . Лекции по математике . Нью-Йорк, Лондон: Macmillan and Co., стр. 99–101.
• Кляйн, Феликс (1979) [1926]. Лекции о развитии математики в XIX веке. Часть 1 [ Лекции о развитии математики в XIX веке ]. Основные учения математических наук 24 (на немецком языке). Берлин, Гейдельберг, Нью-Йорк: Springer Verlag. ISBN  3-540-09234-Х .
• Меннхен, Филипп (1930). «Гаусс как числовой калькулятор» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 6).
• Островский, Александр (1920). «О первом и четвёртом гауссовских доказательствах основной теоремы алгебры» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 3).
• Сарториус фон Вальтерсхаузен, Вольфганг (1856 г.). Гаусс по памяти (на немецком языке). С. Хирзель.
  • Карл Фридрих Гаусс. Мемориал (PDF) . Перевод Хелен Уортингтон Гаусс. Колорадо-Спрингс. 1966. {{cite book}}: CS1 maint: отсутствует местоположение издателя ( ссылка )
• Шааф, Уильям Л. (1964). Карл Фридрих Гаусс: принц математиков . Нью-Йорк: Франклин Уоттс.
• Шефер, Клеменс (1929). «О физических работах Гаусса» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том XI, 2 (Документ 2).
• Шлезингер, Людвиг (1933). «О работах Гаусса по теории функций» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 2).
• Штекель, Пауль (1917). «Гаусс как геометр» . В Королевской прусской академии наук (ред.). Карл Фридрих Гаусс. Работы (на немецком языке). Том X, 2 (Документ 4).
• Стулофф, Николай (1964). Историческая комиссия Баварской академии наук (ред.). Гаусс, Карл Фридрих . Новая немецкая биография (на немецком языке). Том 6. Берлин: Данкер и Хамблот. стр. 101–107. ISBN  3-428-00187-7 .
• Вуссинг, Ганс (1982). Карл Фридрих Гаус (на немецком языке) (4-е изд.). Лейпциг: BSB Б.Г. Тойбнер.

Дальнейшее чтение

• Мерцбах, Ута К. ; Бойер, Карл Б. (2011). История математики (3-е изд.). Нью-Джерси: Джон Уайли и сыновья. ISBN  978-0470630563 .
• Холл, Торд (1970). Карл Фридрих Гаусс: Биография . Кембридж, Массачусетс: MIT Press . ISBN  978-0-262-08040-8 . OCLC   185662235 .
• Нахин, Пол Дж . (2010). Воображаемая сказка: История √-1 . Издательство Принстонского университета. ISBN  9 78-1-4008-3389-4 .
• Симмонс, Дж. (1996). Гигантская книга учёных: 100 величайших умов всех времён . Сидней: Книжная компания.
• Палатка, Маргарет (2006). Принц математики: Карл Фридрих Гаусс . АК Петерс. ISBN  978-1-56881-455-1 .

Вымышленный

• Кельманн, Дэниел (2005). Измерение мира (на немецком языке). Ровольт. ISBN  978-3-498-03528-0 . OCLC   144590801 .
  • Кельманн, Дэниел (2006). Измерение мира . Перевод Джейнвей, Кэрол Браун .

Внешние ссылки

• Работы Карла Фридриха Гаусса или о нем в Internet Archive
• Публикации К.Ф. Гаусса в системе астрофизических данных
• «Некролог» . Ежемесячные уведомления Королевского астрономического общества . 16 (80). 1856 год . Проверено 8 апреля 2024 г.
• «Карл Фридрих Гаусс» . Би-би-си 4 . Краткая история математики . Проверено 8 апреля 2024 г.
• «Разум важнее математики: как Гаусс определил дату своего рождения» . american_almanac.tripod.com . Проверено 8 апреля 2024 г.
• «О замощении Гаусса единичного диска» . Проверено 8 апреля 2024 г.
• «Карл Фридрих Гаусс» . Геттингенский университет . Проверено 8 апреля 2024 г.
• « Потомки Гаусса » . www.coloradocollege.edu . Архивировано из оригинала 28 мая 2010 года . Проверено 4 июня 2024 г.
• «Гаусс» . www.gausschildren.org . Проверено 8 апреля 2024 г.
• Карл Фридрих Гаусс – Следы его жизни (Места, используемые как точки для триангуляции)