كارل فريدريش غاوس

"Gauss" redirects here. For other uses, see Gauss (توضيح).

كارل فريدريش گاوس Johann Carl Friedrich Gauss
كارل فريدريش گاوس، كما رسمه كريستيان ألبرخت ينسن، 1840 (copy from Gottlieb Biermann, 1887)
وُلِدَ	Johann Carl Friedrich Gauss 30 أبريل 1777 براون‌شڤايگ، إمارة براون‌شڤايگ-ڤولفن‌بوتل، الإمبراطورية الرومانية المقدسة
توفي	23 فبراير، 1855 گوتنگن، مملكة هانوڤر، الاتحاد الألماني
الجنسية	ألماني
المدرسة الأم	كلية كارولينوم جامعة گوتنگن جامعة هلم‌شتت (دكتوراه)
اللقب	القائمة الكاملة
الزوج	يوهانا أوستهوف ​ ​ (m. 1805; died 1809)​ مينا ڤالدك ​ ​ (m. 1810; died 1831)​
الأنجال	6
الجوائز	Lalande Prize (1809) وسام كوپلي (1838)
السيرة العلمية
المجالات	رياضيات وفلك وجيوديسيا ومغناطيسية
الهيئات	جامعة گوتنگن
أطروحة	Demonstratio nova... (1799)
المشرف على الدكتوراه	يوهان فريدرش پفاف
طلاب الدكتوراه	فريدرش بسل كريستوف گودرمان Christian Ludwig Gerling ي. ڤ. ريتشارد ددكيند يوهان إنكه يوهان لستنگ برنارد ريمان August Ritter Karl von Staudt
طلاب بارزون آخرون	Gotthold Eisenstein Johann Franz Encke Carl Wolfgang Benjamin Goldschmidt Adolph Theodor Kupffer August Ferdinand Möbius Moritz Stern Georg Frederik Ursin Moritz Ludwig Wichmann
التوقيع

كارل فريدرش گاوس ( Carl Friedrich Gauß ؛ ألمانية: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs]  ( listen); لاتينية: Carolus Fridericus Gauss؛ 30 أبريل 1777 - 23 فبراير 1855) عالم ألماني معروف بأبحاثه المتميزة في حقول الرياضيات والفيزياء وعلم الفلك وعلم قياسات الأرض وعلم فيزياء الأرض، تُنسب إليه النظرية الحديثة في الأعداد، وأبدى نبوغاً في الرياضيات واللغات. He ranks among history's most influential mathematicians and has been referred to as the "Prince of Mathematicians". He was director of the Göttingen Observatory and professor for astronomy for nearly half a century, from 1807 until his death in 1855. ويُروى أنه أدهش أستاذه عندما كان في الثامنة من عمره حين عرض عليه جمع الأعداد من الواحد حتى المئة باستخدام عملية الضرب من دون الجمع. ومازال يعود له الفضل الكبير في عدد من المجالات العلمية في وقتنا الحاضر.

While still a student at the University of Göttingen, he propounded several mathematical theorems. Gauss completed his masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium as a private scholar. He gave the second and third complete proofs of the fundamental theorem of algebra, made contributions to number theory, and developed the theories of binary and ternary quadratic forms. He is considered one of the discoverers of non-Euclidean geometry alongside Nikolai Lobachevsky and János Bolyai and coined that term.

Gauss was instrumental in the identification of the newly discovered Ceres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and the method of least squares, which he had discovered before Adrien-Marie Legendre published on the method.

Gauss was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844, did much of the fieldwork, and provided the complete scientific evaluation. Furthermore, he was one of the founders of geophysics while formulating the fundamental principles of magnetism, and did basic practical research in this field. Fruits of his practical works were the inventions of the heliotrope in 1821, a magnetometer in 1833 and, alongside Wilhelm Eduard Weber, the first electromagnetic telegraph in 1833.

Gauss was a careful author who refused to publish incomplete work, and although having published extensively during his life, he left behind several works to be edited posthumously. He believed that the act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to dislike teaching, but some of his students became influential mathematicians.

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 السيرة

 Youth and education

House of birth in Brunswick (destroyed in World War II)

Caricature of Abraham Gotthelf Kästner by Gauss (1795)

عاش العلامة الفذ الذي ولد في مدينة براونشڤايگ عام 1777 مدة خمسين عاماً في مدينة گوتنگن، توفى عام 1855 م, حيث كان طالباً في جامعة گوتنگن ومن ثم أصبح في عام 1807 أستاذاً ومديراً لمرصد گوتنگن. تكريماً لهذا العالم المشهور قامت جامعة گوتنگن بالتنسيق والتعاون مع مدينة گوتنگن وجمعية گاوس بالاحتفال بعام گاوس 2005. وشمل الاحتفال على معارض وسلسلة من المحاضرات والجولات ومهرجان النجوم.

لقد صاغ هذا العالم طريقة أصغر المربعات وأستنبط حلاً للمعادلات ذات الحدين وأثبت قانون التبادل التربيعي وأسس النظرية الرياضية للكهرباء.

أطلق اسمه على الوحدة الكهرومغناطيسية المستخدمة لقياس الحث المغناطيسي گاوس (وحدة قياس).

ترعرع گاوس في عائلة فقيرة تمارس الفلاحة، وأعانه على تلقّي تعليمه دوق المدينة التي ولد فيها حتى التحق بجامعة گوتنگن Göttingen، وتخرج فيها، وعندما بلغ كارل التاسعة عشرة من عمره بحث عن حل لمسألة حيَّرت العلماء منذ عهد إقليدس؛ ألا وهي رسم مضلع منتظم ذي سبعة أضلع ضمن دائرة باستخدام مسطرة وفرجار، فبين استحالة ذلك إلا أنه في الوقت ذاته وضَّح إمكانية رسم مضلع منتظم ذي سبعة عشر ضلعاً أو 257 ضلعاً.

حاز گاوس شهادة الدكتوراه من جامعة هلمشتت Helmstedt عام 1799، وقد برهن في أطروحته على أن لكل كثير حدود جذراً واحداً على الأقل، وعرفت هذه النظرية باسم «النظرية الأساسية في الجبر»، وقد نشرت إسهاماته الأساسية في نظرية الأعداد عام 1801.

وفي العام ذاته أعلن الفلكي جوزپى پياتسي Giuseppe Piazzi أنه عثر على جرم من القدر الثامن eighth magnitude ظن في بادئ الأمر أنه كوكب جديد، ولدى تتبع مساره على طول واحد من أربعين من مساره اختفى الجرم في ضوء الشمس ولم يتمكن فلكيون آخرون من متابعته، وكان أن تصدى غوص لتحديد مسار الجرم السماوي من ثلاث عمليات رصد له، ونشر طريقته عام 1809، واعتمدت على طريقة أصغر المربعات التي سبق له أن اكتشفها عام 1795. وكان الجرم السماوي الذي حدد مساره هو الكويكب سيريز Ceres، وهو أكبر ما اكتشف وأولها من كويكبات عدة، وقد تنبأ بظهور الكويكب ثانية في زمن محدد، وأثبتت الأرصاد الفلكية صحة نبوءته.

 أستاذ في گوتنگن

Old Göttingen observatory, 1800ح. 1800

Gauss on his deathbed (1855)

عُيِّن گاوس عام 1807 أستاذاً للرياضيات بجامعة گوتنگن ومديراً لمرصدها، وهو المنصب الذي استمر على تولِّيه حتى تاريخ وفاته. وقام في أثناء ذلك بإنتاج علمي وفير تناول الهندسة التفاضلية وعلم الاحتمالات، وإليه يُنسب توزيع گاوس Gauss distribution.

قام گاوس في الفترة بين عامي 1818 و1848 بأعمال جيوديسية مستخدماً طريقة التثليث triangulation فيما كان يدعى حينئذ بمملكة هانوفر. وشارك مع فيبر W.Weber بأبحاث في المغناطيسية والكهرباء؛ فوضعا نظرية المغناطيسية الأرضية التقليدية ونظرية الكمون وابتكر مقياس المغناطيسية ذا خيْطَي التعليق bifilar magnetometer. وفي عام 1833 ابتكر طريقة البرق الكهرمغناطيسي electromagnetic telegraph التي تستخدم فيها إشارات مورس.

Gauss arrived at Göttingen in November 1807, and in the following years he was confronted with the demand for two thousand francs from the Westphalian government as a war contribution. Without having yet received his salary, he could not raise this enormous amount. 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, for instance 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 in the fourth decade he occupied himself with physics, mainly magnetism.

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.

استمر گاوس في عطائه العلمي مدة تزيد على نصف قرن، ولدى الاحتفال باليوبيل الذهبي لحيازته شهادة الدكتوراه في گوتنگن عرض برهاناً رابعاً لنظريته الأساسية في علم الجبر. ويُعدّ من أعظم علماء الرياضيات في العصور الحديثة بل أميزهم. ولإسهاماته القيِّمة في الفيزياء، ولاسيما في مجال المغناطيسية، فقد استخدمت واحدة الغوص ورمزها G لقياس شدة الحقل المغناطيسي، كما أن له إسهامات في البصريات، ولاسيما فيما يرتبط بضم العدسات، ومن مكتشفاته في هذا المضمار عينية تُعرف باسمه هي عينية گاوس.

 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). The cerebral area was determined by Wagner's son Hermann in his doctoral thesis to be 219,588 square millimetres (340.362 sq in). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation for his genius. After various previous investigations, a magnetic resonance study of 1998, done at the Max Planck Institute for Biophysical Chemistry in Göttingen, gave no results which could be used to explain his mathematical abilities.

In 2013, a neurobiologist at the same institute discovered that Gauss's brain had been mixed up, 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 either person. 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' second wife Wilhelmine Waldeck

Gauss married Johanna Osthoff (1780–1809) 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.

On 4 August 1810, the widower married Wilhelmine (Minna) Waldeck (1788–1831), 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 the actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald. Gauss' mother Dorothea lived in his house from 1817 until her death in 1839.

The eldest son Joseph, whilst still a schoolboy, helped his father as an assistant during the survey campaign in summer 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, where he learned the Sioux language. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel; then moved to Missouri, too, 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
  (1806–1873)

• 

  Sophie Gauss née Erythropel
  (1818–1883)
  Joseph's wife

• 

  Wilhelmina Gauss
  (1808–1840)

• 

  Heinrich Ewald
  (1803–1875)
  Wilhelmina's husband

• 

  Eugene Gauss
  (1811–1896)

• 

  Henrietta Gauss née Fawcett
  (1817–1909)
  Eugene's wife

• 

  William Gauss
  (1813–1879)

• 

  Louisa Aletta Gauss née Fallenstein
  (1813–1883)
  William's wife

• 

  Therese Gauss
  (1816–1864)
  (painting by Ludwig Becker)

• 

  Constantin Staufenau
  (1809–1886)
  Therese's husband

 Personality

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 The scholar

Gauss' seal

At the end of the 18th century, German academic mathematics were in a poor condition: the prolific mathematicians of that time worked in France and other European countries. The mathematical mainstream mainly dealt with solving practical problems in mechanics, astronomy, geodesy, etc. In this scientific environment, Gauss can be seen — following Felix Klein— as typical of both 18th and 19th-century mathematicians. His interest in practical applicability, for example in geodesy and astronomy, qualified Gauss to be taken as a typical applied mathematician of the century of enlightenment. On the other hand, he began research in numerous parts of mathematics without defined links to practical purposes, and thus showed himself as a pioneer of what was later called "pure mathematics". In contrast to earlier mathematicians, such as Leonhard Euler—who let their readers take part in their reasoning as they developed new ideas, and included certain erroneous deviations from the correct path—Gauss developed 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 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

Gauss refused to publish work which 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"). His personal diary indicates that several mathematical discoveries may be found by him years or decades before contemporaries firstly published on them. 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 the "Collected Works" contain 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 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 the 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.

A sketch of Gauss by his student Johann Benedict Listing, 1830

Though Gauss is seen as a master of axiomatic presentation, it became obvious from his posthumous papers, his diary, and short glosses in his own textbooks, 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, especially in geodesy and astronomy, 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. Very likely, this method gave him a lot of material which he used in finding theorems in number theory.

It was well known to his close colleagues that Gauss disliked giving academic lectures. He first stated this to Olbers in 1802, so this aversion was not the result of bad experience. Thus he refused to accept any academic position with teaching duties during his years as a private scholar. But from the start of his academic career at Göttingen in 1807, 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 one or other student as talented. In all these 47 years of teaching, most of his lectures dealt with astronomy, geodesy, and applied mathematics, and only three lectures on subjects of pure mathematics. Some of Gauss' 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 —unlike his friends Bessel, Humboldt, and Olbers— he disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or in German. He wrote Latin in a merely classical style, but used some customary modifications set by contemporary mathematicians.

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

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: for instance the brilliant Thibaut in mathematics, in physics Weber and Mayer, well known for his successful textbooks, and Harding, who took the main part of lectures in 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. Once they had been on friendly terms with another, but in the course of 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 some very seldom joint observations. The years since 1820 were evaluated as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as others; Gauss' 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's death 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 by reason of the family's difficult situation. Gauss' 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 for the talented mathematician Gotthold Eisenstein in Berlin.

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, even in his last year of life.

 The private man

Soon after Gauss' 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 grumpy behaviour, but a short time later his mood could change, and he became a charming, open-minded host.

Gauss' life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the death of their third child, he plunged into a depression from which he never fully recovered. Soon after her death he wrote a last letter to her in the style of an ancient threnody, the most personal surviving document of Gauss'. The situation worsened when tuberculosis afflicted and ultimately destroyed, the health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Both younger sons were educated for some years in Celle far from Göttingen. Gauss himself gave only slight hints of his personal distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings".

Gauss grew to dominate his children and eventually had conflicts with his sons, because he did not want any of them to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements. 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 for starting, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties to get an appropriate education, and eventually emigrated as well. Only Gauss' 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 numbers 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.

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

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, the English and French 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, and 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, in accordance with the motto "mundus vult decipi". He disliked Napoleon and his system, and all kind 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.

Gauss was loyal to the House of Hanover. After King William IV's death in 1837, the personal union between the kingdoms of Great Britain and Ireland and Hanover ceased. In the same year, the new Hanoverian King Ernest Augustus annulled the constitution given to the state by his brother in 1833. Seven prominent professors, later known as the "Göttingen Seven", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss' son-in-law Heinrich Ewald. All of them were dismissed, three of them were expelled, but Ewald and Weber could stay in Göttingen. Ewald took a position at the University of Tübingen in 1838, where Gauss' daughter Wilhelmina died soon afterwards in 1840, and Weber went to the University of Leipzig in 1843; both of them returned to their Göttingen positions in 1849 as the only ones of the Göttingen Seven. Gauss was deeply affected by this quarrel but saw no possibility to help them.

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 to be true quite literally. Sartorius mentioned Gauss' religious tolerance, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions.

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, but he disapproved of the idea of paper money. After his death a great sum of money was found hidden in his rooms.

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 Scientific work

 Algebra and number theory

 Fundamental theorem of algebra

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' 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

المقالة الرئيسية: Disquisitiones Arithmeticae

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

The entries in Gauss' Mathematical diary indicate that he was busy with the subject of number theory at least since 1796. A detailed study of previous researches showed him that some of his findings had been already done by other scholars. In the years 1798 and 1799 Gauss wrote a voluminous compilation of all these results in the Disquisitiones Arithmeticae, published in 1801, that consolidated number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces, among other things, the triple bar symbol (≡) for congruence and uses it in 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, which allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, and develops the theories of binary and ternary quadratic forms.

Highlights of these theories include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as 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 a proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical to an algebraic problem. He shows that a regular polygon is constructible if the number of its sides is 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. Some 150 years later, Andre Weil remarked that this particular result, together with some other unpublished results of Gauss, led him to formulate what is now called Weil conjectures.

Gauss intended to include an eighth chapter that would treat the topic of higher congruences modulo a prime number in its full generality, but the unfinished chapter was found among left papers only after his death, consisting of work done during the years 1797–1799.

 Further investigations

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

When Olbers encouraged Gauss in 1816 to compete for a prize of the French Academy on a proof for Fermat's Last Theorem (FLT), he refused by reason of his low esteem on this matter. However, among his left works a short undated paper was found with proves 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 made a significant contribution on solving the Kepler conjecture in 1831, when he proved that —in the three-dimensional case— the greatest packing density of spheres is given when the centers of the spheres form a cubic face-centered arrangement. He showed this in a review of a book written by Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms, with accordance with the program outlined in Gauss's Disquisitiones. Having noticed that Seeber had not proved a central theorem of his theory, he simplified many of his lengthy arguments, proved the central conjecture, and remarked that this theorem is equivalent to Kepler conjecture for regular arrangements.

In two papers on biquadratic residues (1828, 1832) Gauss introduces the ring of Gaussian integers $\mathbb {Z} [i]$, shows that it is a unique factorization domain. and generalizes 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 states the general law of biquadratic reciprocity, and proves several special cases of it, but a proof of the general theorem is lacking, despite Gauss's statements that he found such a proof around 1814. He promised a third paper with a general proof, which has never appeared. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claims the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.

Gauss's publications on biquadratic residues opened the way for boundless enlargement of the theory of numbers, and are memorable for the wealth of investigations in "higher arithmetic" that they led to.

 Analysis

One of Gauss's first independent discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. Systematic investigations on the AGM led him to discover an unusually rich mathematical landscape, and to obtain plenty of new results associated with it. 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.

Another source of inspiration for Gauss's early work in analysis was the acquaintance with Euler's pentagonal numbers theorem. This 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, two decades before Jacobi's "Fundamenta nova" appeared in 1829. In addition, his works show that since 1808 he knew remarkable modular transformations of order 3,5,7 for elliptic functions.

Several mathematical fragments in his Nachlass indicate that he knew quite well parts of the modern theory of modular forms. In his work on the multivalued AGM of two complex numbers, he discovered a very deep connection between the infinitely many values of the AGM to its two "simplest values". His unpublished writings include several drawings that show he was quite aware of the geometric side of the theory; in the context of his work on the complex AGM 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 $\pi /4$.

Having chosen not to publish these results, Gauss' impact in this emerging field of analysis was limited, and the mathematical community became aware of them only through independent efforts of other mathematicians to recover the hidden ideas laying beneath some of his theorems and proofs. One example is his 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 his lifetime, Gauss published almost nothing about those more modern theories of elliptic functions, but he published most of the results on the related theme of the hypergeometric function. 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 to science at the time, such as the elementary functions and some special functions, are a special case 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 1822, 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. In this essay, Gauss made explicit the insight 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. In addition, in unpublished fragments from the years 1834-1839 he investigated and solved the more difficult task of explicitly constructing a conformal mapping from the interior of an ellipse to the unit disk. His solution, which combined an early work on elliptic functions and with later ideas on potential theory, reveals his mastery of the theory of logarithmic potential, and the final results corresponded to the formula found by Hermann Schwarz in 1870.

 Numeric analysis

Gauss often deduced theorems inductively from numerical data he had collected in an empirical way. As such, the use of efficient algorithms to facilitate calculations was vital to his researches, and he made many contributions to numeric analysis. In 1816, he published an article on numeric integration, in which he described the method of Gaussian quadrature, that was a significant improvement over existing methods, such as Newton–Cotes formulas. Walter Gautschi regarded the appearance of this paper of Gauss as a major event of 19th century numeric analysis that inspired much work made by later mathematicians.

In a private letter to Gerling from 1823, he described a solution of a certain 4X4 system of linear equations by using Gauss-Seidel method – an "indirect" iterative method for the solution of linear systems, that in some cases converges very rapidly to the exact solution. Gauss recommended it over the usual method (what is called "direct elimination") for systems of more than 2 equations, stating that it can be done "while half asleep, or while thinking about other things".

Gauss invented an algorithm for calculating what are 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), a very elementary matter of mathematics. Gauss aimed to present a most convenient algorithm for people without any knowledge in 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 to the hundreds of states of the Holy Roman Empire since the 16th century, and was finished in Germany not until the year 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' 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.

Carl Friedrich Gauss 1803 by Johann Christian August Schwartz

 Astronomy

المقالة الرئيسية: Discovery of Ceres

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could track Ceres for only somewhat more than a month, following it for three degrees across the night sky, less than 1% of the total orbit, until it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data. Gauss tackled the problem within three months of intense work, and predicted a position for Ceres in December 1801. This turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 7 and 31 December at Gotha, and independently by Heinrich Olbers on 1 and 2 January in Bremen. This confirmation eventually led to the classification of Ceres as minor-planet designation 1 Ceres; that was taken as the predicted planet between Mars and Jupiter by the most speculative Titius–Bode law.

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. In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. 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, too.

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 a useful 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 complicated steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral. In the late 19th century Gauss's method was adapted by American astronomer George William Hill, who applied it directly to the problem of secular perturbation induced by Venus on Mercury orbit.

While Gauss's contributions to theoretical astronomy came to a marked end, more practical activities in observational astronomy continued and occupied him during his entire career. Even early in 1799, Gauss dealt with 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 amount of observations, mainly on minor planets and comets; his last observation was the solar eclipse of July 28, 1851.

 Theory of errors

It is likely that Gauss 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 the method 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, which was relatively little known in the English speaking world in the first century after its publication, he stated and 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 that went unnoticed until 1950, when his work was rediscovered as a consequence of the growing demand of quick estimation for various new technologies. Gauss's work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert to the Gauss-Helmert model.

Gauss made several striking contributions to problems in probability theory that are not directly concerned with the theory of errors, but offer a glimpse into his broad minded view on the applicability of probabilistic thinking. One example appears as a diary note and is concerned with a very unusual problem that came to his mind: 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.

Order of King George IV to the triangulation project

 Arc measurement and geodetic survey

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

Since 1816, Gauss' former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg, made a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south. The aim was not only the foundation of map production, but also the determination of the geodetic arc of that distance. Schumacher asked Gauss to continue this work further to the south and said he could find support for this project directly from the government of Hanover. Finally in May 1820, King George IV gave the order to Gauss.

Gauss and Schumacher had yet 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 personally, that reached from Thuringia in the south to the river Elbe in the north. The triangel 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 side of 107 km (66.5 miles). In the thin populated Lüneburg Heath without significant natural summits or artificial buildings, he had difficulties to find suitable triangulation points, sometimes cutting lanes through the vegetation was necessary or even the erection of signal towers.

The heliotrope

Gauss' vice heliotrope, a Troughton sextant with additional mirror

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.

The arc measurement needed a precise astronomical determination of 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 own instruments and a zenith sector of Ramsden that was transported to both observatories.

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 intended the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover. The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Carl Friedrich Gauss, who applied own mathematical inventions as the method of least squares and the elimination method to it. The project was finished in 1844, and Gauss send 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

المقالة الرئيسية: Theorema Egregium

The geodetic survey of Hanover fueled Gauss' 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. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-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 (the embedding) 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, what causes a fundamental problem in designing projections for geographical maps. An additional significant 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.

One key differential geometric conception was lacking from Gauss's memoir, that of geodesic curvature. However, the posthumous papers show that this notion did not escape his mind, and in the years of composing his memoir he also wrote up a manuscript in which he introduced it and referred to it as "side curvature" (in German: "Seitenkrümmung"). More importantly, he proved its invariance under isometric transformations, a result later obtained by Ferdinand Minding. Based on this evidence and the announcement in the memoir of further investigations on the curvature integral, it is very likely that he knew the more general version of the Gauss-Bonnet theorem proved by Pierre Ossian Bonnet in 1848, which is closer in spirit to the global version of this theorem.

 Non-Euclidean geometries

Lithography by Siegfried Bendixen (1828)

المقالة الرئيسية: Non-Euclidean geometry

In the lifetime of Gauss a vivid discussion on Euclid's parallel axiom was going on. Numerous mathematicians made efforts to prove it, whereas some of them discussed the possibility of geometrical systems without it. Gauss himself was only interested in the geometrical aspects of the physical space, but did not care about the philosophical aspects of an enlarged geometry. In 1816, he gave a first short public comment on this matter in a book review, and in the following time he occasionally made some remarks in letters to his correspondents. He is the one who coined the term "non-Euclidean geometry".

Not until Lobachevsky (1829) and Janos Bolyai (1832) had published their ideas of a non-Euclidean geometry – for the first time in history of mathematics – , Gauss himself put down his ideas, but avoided any influence to the contemporary scientific discussion, because he did not publish about it. Gauss commended the ideas of Janos Bolyai in a letter to his father, claiming that these were congruent to his own thoughts since some decades. But it is not clear to what extent he preceded Lobachevsky and Bolyai, for he gave only vague and obscure remarks on it in his letters.

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

In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen from three of Riemann's proposals. On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.

 Early topology

One of the lesser known aspects of Gauss's work is that he was also an early pioneer of topology, or as it was called in his lifetime, Geometria Situs. The first proof of the fundamental theorem of the algebra contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem (in 1849).

His earliest "serious" encounter with topological notions occurred to him in the course of his astronomical work, and in a small article from 1804 he determined the limits of the region on the celestial sphere in which comets and asteroids might appear, region which he termed "Zodiacus". He determined this region, and observed 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 another short article in which he further elaborated the qualitative discussion of the Zodiacus.

From Gauss's letters during the period of 1820–1830, one can learn that 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. In the same note, he lamented on 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.

In later years, Gauss held the emerging field of topology in a very high esteem and expected great future developments for it, but since there is so few written, but unpublished material by Gauss on these matters, his influence was made mainly through occasional remarks and oral communications to his colleagues and students Mobius, Listing, and Riemann.

Gauss bust by Heinrich Hesemann (1855)

 Minor mathematical accomplishments

Gauss's work did not only initiate significant mathematical theories, as he was also the author of many little "gems" in mathematics, especially in elementary geometry and algebra. In his way, he helped spread the new mathematical ideas by demonstrating how they illuminate and shorten the solution of small mathematical problems.

For example, he was a vivid spirit in applying complex numbers to various problems, and used them in his work on perspective and projective geometry: in a short 1836 note on "Projections of the Cube", 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. In an unpublished 1819 note entitled "the Sphere", he conceived of the complex plane extended by a point at infinity as the stereographic projection of a sphere (the Riemann sphere), and described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane. Gauss seems to have in his foresight the algebraic system of quaternions, the later discovery of William Rowan Hamilton. In 1819, Gauss drafted an unpublished short treatise on "Rotations of Space", in which he elaborated on the use of quadruples of real numbers (of which he called "scales") to describe 3D rotations.

Occasionally Gauss worked even on problems of elementary geometry. For instance, he gave a proof that the altitudes of a triangle always meet in a single orthocenter, and an analytic solution for the Problem of Apollonius. 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.

One of his investigations was concerned with John Napier's "Pentagramma mirificum" — a certain spherical pentagram whose properties intrigued and occupied Gauss's mind for several decades. In studies of the Pentagramma 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.

 Magnetism and telegraphy

 Geomagnetism

Gauss-Weber monument in Göttingen

The Gauss-Weber magnetometer

Gauss' interest in magnetism is obvious since the first decade of the 19th century. Since 1826, when Alexander von Humboldt visited him in Göttingen, both scientists began intensive research on geomagnetism, partly independent, partly in productive cooperation. In 1828, Gauss was Humboldt's personal guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquaintance with the physicist Wilhelm Weber.

When Weber got the chair for physics in Göttingen as successor of Johann Tobias Mayer by Gauss' 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 was helpful to organize the worldwide spread of observatories including the British dominions with a letter to the Duke of Sussex, then president of the Royal Society, wherein he asked for support for a program of global research based on Gauss' methods. Together with other instigators, this led to a global programm known as "Magnetical crusade" under directory of Edward Sabine. The dates, times, and intervalls of observations were determined in advance, the Göttingen mean time was used as standard. Finally 61 stations 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 both 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 one who derived 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; following Felix Klein, this work is actually 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 got influence on 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 Götttingen, and similar 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' 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 with regard to 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 an homogeneous triaxial ellipsoid at every point in space. This problem was previously tackled and solved by Maclaurin, Laplace and Lagrange, but their procedures either relied on development into slowly converging infinite series or did not solve it in full generality. 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 had knowledge of 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

Instrument maker Johann Georg Repsold in Hamburg asked Gauss in 1807 for help to construct an achromatic lens system. Based on Gauss' calculations, Repsold succeeded with a new objective in 1810. A main problem, among other difficulties, was the non precise 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 made calculations about 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 indroduced the achromatic Steinheil doublet, partly based on Gauss's calculations. Many results in geometrical optics are scattered in Gauss's correspondences and handnotes.

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

 Mechanics

Gauss' first and last 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' considerations as appendix to his book on falling experiments.

After Foucault had demonstrated his pendulum 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; the time of oscillation was 3.1 seconds. It is described in the Gauss–Gerling correspondence, and Weber made some experiments with this obviously working 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' christening on 4 May 1777 with later added birth date

Several stories of his early genius have been reported. Carl Friedrich Gauss' 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. Gauss felt sorry for his new born daughter Wilhelmine, because she was born on the leap day in 1808 and thus would celebrate her birthday only every four years.

In his memorial on Gauss, Wolfgang Sartorius von Waltershausen tells a story about the three-years-aged 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, in the course of 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.

Gauss' favorite English author was Walter Scott; when he sometimes read the words "the moon rises broad in the nord west", he was very amused.

 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 study of angle-preserving maps", 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). On the occasion of his golden doctor degree jubilee he got the honorary citizenship of both towns of Brunswick and Göttingen in 1849. Soon after his death a medal was issued by order of King George V of Hanover with the back side inscription: GEORGIVS V REX HANNOVERAE MATHEMATICORVM PRINCIPI and the circumscription: ACADEMIAE SVAE GEORGIAE AVGVSTAE DECORI AETERNO.

The ″Gauss-Gesellschaft 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 الألمانية). 2: 121–130. Original
• 1801: Disquisitiones Arithmeticae. Leipzig: Gerh. Fleischer jun.
  • Disquisitiones Arithmeticae & other papers on number theory. Translated by Clarke, Arthur A. (2nd, corrected ed.). New York: Springer. 1986. 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 الألمانية). 5: 435–437. Original
• 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 اللاتينية). Göttingen.
• 1809: Theoria motus corporum coelestium in sectionibus conicis solem ambientium (in اللاتينية). 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. Altona. 3. (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 الألمانية). Göttingen: Vandenhoeck und Ruprecht. 1828. Original book
• 1828: "Supplementum theoriae combinationis observationum erroribus minimis obnoxiae". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 57–98. 1828. Bibcode:1828stco.book.....G. (from 1826)
  • 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. 1995. 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)
• 1903: Wissenschaftliches Tagebuch (Klein, Felix, ed. (1903). "Gauß' wissenschaftliches Tagebuch 1796–1814". Mathematische Annalen (in اللاتينية and الألمانية). 57: 1–34. doi:10.1007/BF01449013. S2CID 119641638.) Original book
  • 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 الألمانية). 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: "Intensitas vis magneticae terrestris ad mensuram absolutam revocata". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 8: 3–44. Original (from 1832)
  • The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement. Translated by Susan P. Johnson.
• 1836: Erdmagnetismus und Magnetometer (Original book: H.C. Schumacher (ed.). Jahrbuch für 1836 (in الألمانية). Vol. 1836. Tübingen: J.G.Cotta'sche Buchhandlung. pp. 1–47.)
• 1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte (Original book: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte [General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances] (in الألمانية). Leipzig: Weidmannsche Buchhandlung. 1840.
• 1843: "Dioptrische Untersuchungen" [Dioptrical Investigations]. Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen (in الألمانية). Erster Band: 1–34. Original (from 1840)

 together with Wilhelm Weber

• 1837–1839: Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1836–1838 (in الألمانية). Göttingen: Dieterichsche Buchhandlung.
• 1840–1843: Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839–1841 (in الألمانية). Leipzig: Weidmannsche Verlagsbuchhandlung.
• 1840: Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins (in الألمانية). Leipzig: Weidmannsche Verlagsbuchhandlung.

• 

  Title page Disquisitiones Arithmeticae (1801)

• 

  Title page of Theorema arithmetici demonstratio nova (1808)

• 

  Title page of Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (1809)

• 

  Title page to the English Translation of Theoria Motus by Charles Henry Davis (1857)

• 

  Title page of Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata (1833)

• 

  Volume II of "Carl Friedrich Gauss Werke," 1876

 Collected works

• Königlich Preußische Akademie der Wissenschaften, ed. (1863–1933). Carl Friedrich Gauss. Werke (in اللاتينية and الألمانية). Vol. 1–12. Göttingen: (diverse publishers).

 Correspondence

• Königlich Preußische Akademie der Wissenschaften, ed. (1880). Briefwechsel zwischen Gauss und Bessel (in الألمانية). Leipzig: Wilhelm Engelmann. (letters from December 1804 to August 1844)
• Schoenberg, Erich; Perlick, Alfons (1955). Unbekannte Briefe von C. F. Gauß und Fr. W. Bessel. Abhandlungen der Bayerischen Akademie der Wissenschaften, Math.-nat. Klasse, Neue Folge, No. 71 (in الألمانية). Munich: Verlag der Bayerischen Akademie der Wissenschaften. pp. 5–21. (letters to Boguslawski from February 1835 to January 1848)
• Schwemin, Friedhelm, ed. (2014). Der Briefwechsel zwischen Carl Friedrich Gauß und Johann Elert Bode. Acta Historica Astronomica (in الألمانية). Vol. 53. Leipzig: Akademische Verlaganstalt. ISBN 978-3-944913-43-8. (letters from February 1802 to October 1826)
• Franz Schmidt, Paul Stäckel, ed. (1899). Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai (in الألمانية). Leipzig: B.G. Teubner. (letters from September 1797 to February 1853; added letters of other correspondents)
• Axel Wittmann, ed. (2018). Obgleich und indeßen. Der Briefwechsel zwischen Carl Friedrich Gauss und Johann Franz Encke (in الألمانية). Remagen: Verlag Kessel. ISBN 978-3945941379. (letters from June 1810 to June 1854)
• Clemens Schaefer, ed. (1927). Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling (in الألمانية). Berlin: Otto Elsner. (letters from June 1810 to June 1854)
• Karl Christian Bruhns, ed. (1877). Briefe zwischen A. v. Humboldt und Gauss (in الألمانية). Leipzig: Wilhelm Engelmann. (letters from July 1807 to December 1854; added letters of other correspondents)
• Reich, Karin; Roussanova, Elena (2018). Karl Kreil und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext. Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68 (in الألمانية). Vienna: Verlag der Österreichischen Akademie der Wissenschaften. (letters from 1835 to 1843)
• Gerardy, Theo, ed. (1959). Briefwechsel zwischen Carl Friedrich Gauß und Carl Ludwig von Lecoq. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 (in الألمانية). Göttingen: Vandenhoeck & Ruprecht. pp. 37–63. (letters from February 1799 to September 1800)
• Forbes, Eric G. (1971). "The Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802-05)". Annals of Science. 27 (3): 213–237. doi:10.1080/00033797100203767.
• Cunningham, Clifford (2004). "Discovery of the Missing Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802-05)". Annals of Science. 61 (4): 469–481. doi:10.1080/00033790310001660164.
• Carl Schilling, ed. (1900). Briefwechsel zwischen Olbers und Gauss: Erste Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in الألمانية). Berlin: Julius Springer. (letters from January 1802 to October 1819)
• Carl Schilling, ed. (1909). Briefwechsel zwischen Olbers und Gauss: Zweite Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in الألمانية). Berlin: Julius Springer. (letters from January 1820 to May 1839; added letters of other correspondents)
• Christian August Friedrich Peters, ed. (1860–1865). Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (in الألمانية). Altona: Gustav Esch.
  • Volumes 1+2 (letters from April 1808 to March 1836)
  • Volumes 3+4 (letters from March 1836 to April 1845)
  • Volumes 5+6 (letters from April 1845 to November 1850)
• Poser, Hans, ed. (1987). Briefwechsel zwischen Carl Friedrich Gauß und Eberhard August Zimmermann. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 (in الألمانية). Göttingen: Vandenhoeck & Ruprecht. ISBN 978-3525821169. (letters from 1795 to 1815)

The Göttingen Academy of Sciences and Humanities provides a complete collection of the yet known letters from and to Carl Friedrich Gauss that is accessible online. The literary estate is kept and provided by the Göttingen State and University Library. Written estate from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.

 References

 Sources

• Bachmann, Paul (1922). "Über Gauss' zahlentheoretische Arbeiten". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 1).
• Bolza, Oskar (1921). "Gauss und die Variationsrechnung". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 5).
• Brendel, Martin (1929). "Über die astronomischen Arbeiten von Gauss". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. XI, 2 (Abhandlung 3).
• Bühler, Walter Kaufmann (1981). Gauss: A Biographical Study. Springer-Verlag. ISBN 978-0-387-10662-5.
• Dunnington, G. Waldo (2004). Carl Friedrich Gauss: Titan of Science. The Mathematical Association of America. ISBN 978-0-88385-547-8. OCLC 53933110. First edition: Carl Friedrich Gauss: Titan of Science. A Study of his Life and Work. New York: Exposition Press. 1955.
  • Gray, Jeremy (1955). "Introduction to Dunnington's "Gauss"". Carl Friedrich Gauss: Titan of Science. A Study of his Life and Work. New York: Exposition Press. pp. xix–xxvi. With a critical view on Dunnington's style and appraisals
• Galle, Andreas (1924). "Über die geodätischen Arbeiten von Gauss". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. XI, 2 (Abhandlung 1).
• Geppert, Harald (1933). "Über Gauss' Arbeiten zur Mechanik und Potentialtheorie". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 7).
• Klein, Felix (1894). "The Development of Mathematics at the German Universities". Lectures on Mathematics. New York, London: Macmillan and Co. pp. 99–101.
• Klein, Felix (1979) [1926]. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Teil 1 [Lectures on the Development of Mathematics in the 19th Century]. Grundlehren der mathematischen Wissenschaften 24 (in الألمانية). Berlin, Heidelberg, New York: Springer-Verlag. ISBN 3-540-09234-X.
• Maennchen, Philipp (1930). "Gauss als Zahlenrechner". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 6).
• Ostrowski, Alexander (1920). "Über den ersten und vierten Gaussschen Beweis des Fundamentalsatzes der Algebra". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 3).
• Sartorius von Waltershausen, Wolfgang (1856). Gauss zum Gedächtniss (in الألمانية). S. Hirzel.
  • Carl Friedrich Gauss. A Memorial (PDF). Translated by Helen Worthington Gauss. Colorado Springs. 1966.{{cite book}}: CS1 maint: location missing publisher (link)
• Schaaf, William L. (1964). Carl Friedrich Gauss: Prince of Mathematicians. New York: Franklin Watts.
• Schaefer, Clemens (1929). "Über Gauss' physikalische Arbeiten". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. XI, 2 (Abhandlung 2).
• Schlesinger, Ludwig (1933). "Über Gauss' Arbeiten zur Funktionentheorie". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 2).
• Stäckel, Paul (1917). "Gauss als Geometer". In Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in الألمانية). Vol. X, 2 (Abhandlung 4).
• Stuloff, Nikolai (1964). Historische Kommmission der Bayerischen Akademie der Wissenschaften (ed.). Gauß, Carl Friedrich. Neue Deutsche Biographie (in الألمانية). Vol. 6. Berlin: Duncker & Humblot. pp. 101–107. ISBN 3-428-00187-7.
• Wußing, Hans (1982). Carl Friedrich Gauß (in الألمانية) (4 ed.). Leipzig: BSB B. G. Teubner.

 Further reading

• Merzbach, Uta C.; Boyer, Carl B. (2011). A History of Mathematics (3rd ed.). New Jersey: John Wiley & Sons. ISBN 978-0470630563.
• Hall, Tord (1970). Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press. ISBN 978-0-262-08040-8. OCLC 185662235.
• Nahin, Paul J. (2010). An Imaginary Tale: The Story of √-1. Princeton University Press. ISBN 9 78-1-4008-3389-4.
• Simmons, J. (1996). The Giant Book of Scientists: The 100 Greatest Minds of All Time. Sydney: The Book Company.
• Tent, Margaret (2006). The Prince of Mathematics: Carl Friedrich Gauss. A. K. Peters. ISBN 978-1-56881-455-1.

 fictional

• Kehlmann, Daniel (2005). Die Vermessung der Welt (in الألمانية). Rowohlt. ISBN 978-3-498-03528-0. OCLC 144590801.
  • Kehlmann, Daniel (2006). Measuring the World. Translated by Janeway, Carol Brown.

 المصادر

• Asimov, I. (1972). Biographical Encyclopedia of Science and Technology; the Lives and Achievements of 1195 Great Scientists from Ancient Times to the Present, Chronologically Arranged. New York: Doubleday.
• Bell, E. T. (1986). "Ch. 14: The Prince of Mathematicians: Gauss". Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York: Simon and Schuster. pp. pp. 218–269. ISBN 0-671-46400-0. {{cite book}}: |pages= has extra text (help)
• "Carl Friedrich Gauss". {{cite web}}: Unknown parameter |accessmonthday= ignored (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)
• Carl Friedrich Gauss على بلانيت ماث
• Dunnington, G. Waldo. (May, 1927). "The Sesquicentennial of the Birth of Gauss". Scientific Monthly. XXIV: 402–414. Retrieved 29 June. {{cite journal}}: Check date values in: |accessdate= and |date= (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help) مقالة شاملة عن سيرته.
• Dunnington, G. Waldo. (June 2003). Carl Friedrich Gauss: Titan of Science. The Mathematical Association of America. ISBN 0-88385-547-X.
• Gauss, Carl Friedrich (1965). Disquisitiones Arithmeticae. tr. Arthur A. Clarke. Yale University Press. ISBN 0-300-09473-6.
• Hall, T. (1970). Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press. ISBN 0-262-08040-0.
• "Gauss and His Children". {{cite web}}: Unknown parameter |accessmonthday= ignored (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)
• Simmons, J. (1996). The Giant Book of Scientists: The 100 Greatest Minds of All Time. Sydney: The Book Company.

 الوصلات الخارجية

Wikisource has original works written by or about:

Carl Friedrich Gauss

• Gauss biography
• O'Connor, John J.; Robertson, Edmund F., "كارل فريدريش غاوس", MacTutor History of Mathematics archive 
• كارل فريدريش غاوس at the Mathematics Genealogy Project
• Carl Friedrich Gauss, Biography at Fermat's Last Theorem Blog.
• Gauss: mathematician of the millennium, by Juergen Schmidhuber
• Gauss, general information, submit your site about Gauss.
• Obituary: MNRAS 16 (1856) 80
• A discussion of childhood problem and the sources
• Complete works
• Carl Friedrich Gauss on the 10 Deutsch Mark banknote.

 قراءات اضافية

• Kehlmann, Daniel (2005). Die Vermessung der Welt. Rowohlt. ISBN 3-498-03528-2.

جـوائـز
سبقه Antoine César Becquerel و John Frederic Daniell	ميدالية كوپلي 1838 بالاشتراك مع مايكل فراداي	تبعه روبرت براون