Carl Friedrich Gauss

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Carl Friedrich Gauss
Carl Friedrich Gauss 1840 by Jensen.jpg
Portrait by Christian Albrecht Jensen, 1840 (copy from Gottlieb Biermann, 1887) [1]
Born
Johann Carl Friedrich Gauss

(1777-04-30)30 April 1777
Died23 February 1855(1855-02-23) (aged 77)
Göttingen, Kingdom of Hanover, German Confederation
Alma mater
Known for Full list
Spouses
Johanna Osthoff
(m. 1805;died 1809)
Minna Waldeck
(m. 1810;died 1831)
Children6
Awards
Scientific career
FieldsMathematics, Astronomy, Geodesy, Magnetism
InstitutionsUniversity of Göttingen
Thesis Demonstratio nova...  (1799)
Doctoral advisor Johann Friedrich Pfaff
Doctoral students
Other notable students
Signature
Carl Friedrich Gauss signature.svg

Johann Carl Friedrich Gauss (German: Gauß [kaʁlˈfʁiːdʁɪçˈɡaʊs] ; [2] [3] Latin : Carolus Fridericus Gauss; 30 April 1777 23 February 1855) was a German mathematician, astronomer, geodesist, and physicist who contributed to many fields in mathematics and science. Gauss 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.

Contents

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, developed the theories of binary and ternary quadratic forms, and is credited with inventing the fast Fourier transform algorithm. 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 new 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 and refused to publish incomplete work. Although he 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 was known to dislike teaching, but some of his students became influential mathematicians.

Biography

Youth and education

House of birth in Brunswick (destroyed in World War II) Braunschweig Brunswick Geburtshaus CF Gauss (1914).jpg
House of birth in Brunswick (destroyed in World War II)
Caricature of Abraham Gotthelf Kastner by Gauss (1795) Carl Friedrich Gauss, Karikatur von Abraham Gotthelf Kastner, 1795.jpg
Caricature of Abraham Gotthelf Kästner by Gauss (1795)

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig) in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony Germany), to a family of lower social status. [4] His father Gebhard Dietrich Gauss (1744–1808) worked in several jobs, as butcher, bricklayer, gardener, and as treasurer of a death-benefit fund. Gauss characterized his father as an honourable and respected man but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea (1743–1839), Carl Friedrich's mother, was nearly illiterate. [5] He had one elder brother from his father's first marriage. [6]

Gauss was a child prodigy in the field of mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum, [lower-alpha 1] which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the Hanoverian University of Göttingen until 1798. [7] It is not known why Gauss went to Göttingen and not to the University of Helmstedt near his native town Brunswick; it is assumed that the large library of Göttingen, where students were allowed to borrow books and take them home, was the decisive reason. [8] His professor in mathematics was Abraham Gotthelf Kästner, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams. [9] Gauss depicted him in a drawing showing a lecture scene where he produced errors in a simple calculation. Astronomy was taught by Karl Felix Seyffer, with whom Gauss stayed in correspondence after graduation; [10] Olbers and Gauss mocked him in their correspondence. [11] On the other hand, he thought highly of Georg Christoph Lichtenberg, his teacher of physics, and of Christian Gottlob Heyne, whose lectures in classics Gauss attended with pleasure. [10] Fellow students of this time were Johann Friedrich Benzenberg, Farkas Bolyai, and Heinrich Wilhelm Brandes. [10]

Though being a registered student at university, it is evident that he was a self-taught student in mathematics since he independently rediscovered several theorems. [12] He succeeded with a breakthrough in a geometrical problem that had occupied mathematicians since the days of the Ancient Greeks, when he determined in 1796 which regular polygons can be constructed by compass and straightedge. This discovery was the subject of his first publication and ultimately led Gauss to choose mathematics instead of philology as a career. [13] Gauss' mathematical diary shows that many ideas for his mathematical magnum opus Disquisitiones arithmeticae (1801) date from this time. [14]

Private scholar

Gauss graduated as a Doctor of Philosophy in 1799. He did not graduate from Göttingen, as is sometimes stated, [lower-alpha 2] [15] but rather, at the Duke of Brunswick's special request, from the University of Helmstedt, the only state university of the duchy. There, Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without the further oral examination that was usually requested. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss showed his gratitude and loyalty for this benefit when he refused several calls from the Russian Academy of Sciences in St. Peterburg and from Landshut University. [16] Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans: [17] the Duke was mortally wounded in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped. He then followed a call to the University of Göttingen, an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory. [18]

Studying the calculation of asteroid orbits, Gauss established contact with the astronomical community of Bremen and Lilienthal, especially Wilhelm Olbers, Karl Ludwig Harding, and Friedrich Wilhelm Bessel, as part of the informal group of astronomers known as the Celestial police. [19] One of their aims was the discovery of further planets, and they assembled data on asteroids and comets as a basis for Gauss's research. Gauss was thereby able to develop new, powerful methods for the determination of orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809). [20]

Professor in Göttingen

Old Gottingen observatory, circa 1800 Goettingen Sternwarte 01.jpeg
Old Göttingen observatory, circa 1800
Gauss on his deathbed (1855) Carl Friedrich Gauss on his Deathbed, 1855.jpg
Gauss on his deathbed (1855)

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, [21] paid the sum. [18]

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, [22] with usable, but partly out-of-date instruments. [23] 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, [24] but Gauss could not move to his new place of work until September 1816. [16] He got new up-to-date instruments, for instance two meridian circles from Repsold [25] and Reichenbach, [26] and a heliometer from Fraunhofer. [27]

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. [28]

Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, Gauss died of a heart attack in Göttingen; [9] 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. [29]

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). [30] [31] The cerebral area was determined by Wagner's son Hermann in his doctoral thesis to be 219,588 square millimetres (340.362 sq in). [32] Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation for his genius. [33] 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. [34]

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. [35] 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. [36]

Family

Gauss' second wife Wilhelmine Waldeck Minna Gauss geb. Waldeck, 002.jpg
Gauss' second wife Wilhelmine Waldeck

Gauss married Johanna Osthoff (1780–1809) on 9 October 1805 in St. Catherine's church. [37] 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. [38]

On 4 August 1810, Gauss 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. [39] 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. [40] Her sister Wilhelmina married the orientalist Heinrich Ewald. [41] Gauss' mother Dorothea lived in his house from 1817 until her death in 1839. [7]

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. [42] [lower-alpha 3]

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. [42] Wilhelm married a niece of the astronomer Bessel and also moved to Missouri in 1837, [45] starting as a farmer and later becoming wealthy in the shoe business in St. Louis. [46] Eugene and William have numerous descendants in America, but the descendants left in Germany all derive from Joseph, as the Gauss daughters had no children. [42]

Personality

The scholar

Gauss' seal GaussSiegel1777-1855.png
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. [lower-alpha 4] The mathematical mainstream mainly dealt with solving practical problems in mechanics, astronomy, geodesy, etc. [47] 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. [48] [49]

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: [50]

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". [51] Nevertheless, he published many short communications of urgent content in various journals, but the "Collected Works" contain a considerable literary estate, too. [52] [53] Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", [54] 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. [55]

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. [56] 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. [51]

A sketch of Gauss by his student Johann Benedict Listing, 1830 Carl Friedrich Gauss, 003.jpg
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. Gauss was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precis controlling, but checked the results by masterly estimation. Nevertheless, his calculations, especially in geodesy and astronomy, were not always free from mistakes. [57] He coped with the enormous workload by using skillful tools. [58] Gauss used a lot of mathematical tables, examined their exactness, and constructed new tables on various matters for personal use. [59] He developed new tools for effective calculation, for example the Gaussian elimination. [60] 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. [61] Very likely, this method gave him a lot of material which he used in finding theorems in number theory. [58]

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. [62] 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. [63] In all these 47 years of teaching, most of his lectures dealt with astronomy, geodesy, and applied mathematics, [64] and only three lectures on subjects of pure mathematics. [63] [lower-alpha 5] Some of Gauss' students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor, Dedekind, Dirksen, Encke, Gould, [lower-alpha 6] Heine, Klinkerfues, Kupffer, Listing, Möbius, Nicolai, Riemann, Ritter, Schering, Scherk, Schumacher, von Staudt, Stern, Ursin; as geoscientists Sartorius von Waltershausen, and Wappäus. [63]

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 [66] and the essay Erdmagnetismus und Magnetometer of 1836. [51] Gauss published his papers and books exclusively in Latin or in German. [lower-alpha 7] [lower-alpha 8] He wrote Latin in a merely classical style, but used some customary modifications set by contemporary mathematicians. [69]

The new Gottingen Observatory of 1816; Gauss' living rooms were in the western wing (right) Universitats-Sternwarte Gottingen 02.jpg
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. [64] 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. [16] 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. [16] [lower-alpha 9] Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except some very seldom joint observations. [71] The years since 1820 were evaluated as a "period of lower astronomical activity". [72] 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. [70] [71] [lower-alpha 10]

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. [70] Gauss' salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, [16] and in his later years, Gauss was one of the best-paid professors of the university. [42]

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. [lower-alpha 11] Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. [75] He also gave successful support for the talented mathematician Gotthold Eisenstein in Berlin. [76]

Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. [77] 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. [77]

The private man

Soon after Gauss' death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw Gauss as a serene and forward-striving man with childlike modesty, [78] but also of "iron character" [79] with an unshakeable strength of mind. [80] He was noted for a sense of justice [81] and religious tolerance. [82] Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on the summit of science". [83] 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. [51]

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'. [84] [85] 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. [86] 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". [51]

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. [42] The second son Eugen shared a good measure of Gauss' 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, [87] 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. [42] 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. [40]

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. [88]

Wilhelm Weber and Heinrich Ewald (first row) as members of the Gottingen Seven Die Gottinger Sieben von Eduard Ritmuller.jpg
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. [89] Gauss read both classical and modern literature, the English and French in the original languages. [90] [lower-alpha 12] His favorite English author was Walter Scott, his favorite German Jean Paul. [92] Gauss liked singing and went to concerts. [93] He was a busy newspaper reader, and in his last years he used to visit an academic press salon of the university every noon. [94] 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 . [95]

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". [94] 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. [79] [lower-alpha 13] 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. [94]

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 Gauss' 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. [96]

Gauss' religious beliefs have been a subject of speculation by some of his biographers. He sometimes said "God arithmetizes" [97] and "I succeeded - not on account of my hard efforts, but by the grace of the Lord." [98] 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. [99] Sartorius mentioned Gauss' religious tolerance, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions. [82]

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, but he disapproved of the idea of paper money. [100] After his death a great sum of money was found hidden in his rooms. [101]

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. [102]

Disquisitiones Arithmeticae

German stamp commemorating Gauss' 200th anniversary: the complex plane DBP 1977 928 Carl Friedrich Gauss.jpg
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. [103] 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, [104] which allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, and develops the theories of binary [105] and ternary quadratic forms. [106]

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. [107] From several analytic results on class numbers that Gauss gives without proof towards the end of the fifth chapter, [108] it appears that Gauss already knew the class number formula in 1801. [109]

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. [110] He shows that a regular polygon is constructible if the number of its sides is a product of distinct Fermat primes and a power of 2. 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. [111] 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. [112] [lower-alpha 14]

Gauss intended to include an eighth chapter that would treat the topic of higher congruences modulo a prime number in its full generality, the unfinished chapter was found among left papers only after his death, consisting of work done during the years 1797–1799. [115] [116]

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; [117] [lower-alpha 15]

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. [119] 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. [120]

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. [121] He showed this in a review of a book written by Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms, [122] 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. [123]

In two papers on biquadratic residues (1828, 1832) Gauss introduces the ring of Gaussian integers , shows that it is a unique factorization domain. [124] 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 [124] – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws. [125]

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. [126]

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. [127]

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. [128] 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". [129] 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. [111] [130]

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, [111] culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case, [131] two decades before Jacobi's "Fundamenta nova" appeared in 1829. In his publication from 1811 on the determination of the sign of quadratic Gauss sum, Gauss solved the problem by introducing Gaussian binomial coefficients and by using a line of reasoning that somehow "hides" its origin in theta function theory, as later mathematicians have shown.

Several mathematical fragments in his Nachlass indicate that he knew quite well parts of the modern theory of modular forms. [111] 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". [129] 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. [lower-alpha 16] [132] 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 . [133]

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 , 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. [134] This work is the first one with an exact inquiry of convergence of infinite series in the history of mathematics. [135] 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. [136] [137] 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. [138]

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. [139]

In a private letter to Gerling from 1823, [140] 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". [141]

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. [142] He developed it as a trigonometric interpolation method, but the paper Theoria Interpolationis Methodo Nova Tractata was published only posthumously in 1876, [143] preceded by the first presentation by Joseph Fourier on the subject in 1807. [144]

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. [145] 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. [146] 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. [147]

Carl Friedrich Gauss 1803 by Johann Christian August Schwartz Carl Friedrich Gauss, Pastellgemalde von Johann Christian August Schwartz, 1803, ohne Rahmen.jpg
Carl Friedrich Gauss 1803 by Johann Christian August Schwartz

Astronomy

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. [148] 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. [149] [lower-alpha 17] 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. [15]

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. [150]

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. [151] It introduced the Gaussian gravitational constant. [64]

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. [152] 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. [153] [154] [lower-alpha 18] 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. [156]

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. [157] 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. [lower-alpha 19] [158] 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. [159]

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. [160] 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. [161] He made many contributions to spherical geometry, and in this context solved some practical problems about navigation by stars. [162] He published a great amount of observations, mainly on minor planets and comets; his last observation was the solar eclipse of July 28, 1851. [163]

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. [164] 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. [165] [166] [167] In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". [56] 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). [168]

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). [169] He derived lower and upper bounds for the variance of sample variance. [lower-alpha 20] 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. [170]

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. [171]

Order of King George IV to the triangulation project Georg IV Erlass Landvermessung.jpg
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. [172] Later since 1804, he taught himself some geodetic practise with a sextant in Brunswick, [173] and Göttingen. [174]

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. [lower-alpha 21] 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. [175]

Gauss and Schumacher had yet determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818. [176] 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. [177]

The heliotrope Gauss Heliotrop@20170430.JPG
The heliotrope
Gauss' vice heliotrope, a Troughton sextant with additional mirror Vize-Heliotrop Gauss-Ausstellung Bomann-Museum (1).jpg
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 . [178] Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope. [179] 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. [180]

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. [181] [lower-alpha 22]

An additional result was a better value of flattening of the approximative Earth ellipsoid. [182] [lower-alpha 23] Gauss developed the universal transverse Mercator projection of the ellipsoidal shaped Earth (what he named conform projection) [184] 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. [185] [186]

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; [187] later his doctoral student Johann Benedict Listing called this the geoid . [188]

Differential geometry

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. [189]

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. [189] 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. [190]

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. [lower-alpha 24] [191] [192]

Non-Euclidean geometries

Lithography by Siegfried Bendixen (1828) Bendixen - Carl Friedrich Gauss, 1828.jpg
Lithography by Siegfried Bendixen (1828)

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. [193] 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. [194] [195] He is the one who coined the term "non-Euclidean geometry". [196]

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 – , [193] Gauss himself put down his ideas, but avoided any influence to the contemporary scientific discussion, because he did not publish about it. [194] [197] Gauss commended the ideas of Janos Bolyai in a letter to his father, [198] claiming that these were congruent to his own thoughts since some decades. [199] [200] 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. [193]

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. [194]

In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen from three proposals. [201] [202] On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement. [203]

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). [204]

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. [205]

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. [206] To do so he devised a symbolical scheme, the Gauss code, that in a sense captured the characteristic features of tract figures. [207] [208]

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. [205]

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 material by Gauss from this period, his influence was made mainly through occasional remarks and oral communications. [lower-alpha 25] For example, an indirect report by Mobius referred to a surface constructed by Gauss, which Gauss called "double ring" and sayed something about its connectivity properties. [209] This report is consistent with a fragment of Gauss, written around 1840, which sketched a theory of the order of connectivity of surfaces. [210] Listing expressed his indebtness to Gauss's influence several times. [211]

Gauss bust by Heinrich Hesemann (1855) Carl Friedrich Gauss, Buste von Heinrich Hesemann, 1855.jpg
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. [212]

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. [213] 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 the 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. [214] [215]

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, [216] and an analytic solution for the Problem of Apollonius. [217] [218] Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given quadrilateral, [219] [220] and discovered a surprising result about the computation of area of pentagons. [221] [222]

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. [223] 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. [224] In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane. [225]

Magnetism and telegraphy

Geomagnetism

Gauss-Weber monument in Gottingen Gottingen-Gauss-Weber-Monument.01.JPG
Gauss-Weber monument in Göttingen
The Gauss-Weber magnetometer A magnetometer used by Carl Friedrich Gauss, from Gerlach und F. Traumuller, 1899.png
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. [226] 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. [227]

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. [228] 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. [229] 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. [230] 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. [231]

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. [231] [232] 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. [230]

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. [233] The theory predicted the existence of exactly two magnetic poles on the Earth, thus Hansteen's idea of four magnetic poles became obsolete, [234] and the data allowed to determine their location with rather good precision. [235] 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; [233] it seems rather unlikely that he had knowledge of the previous works of George Green on this subject. [236] 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. [230]

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. [234]

Electromagnetism

Town plan of Gottingen with course of the telegraphic connection Gauss-Weber-Telegraf Paulinerkirche 02.jpg
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. [236] Gauss and Weber found rules for branched electric circuits, later benamed as Kirchhoff's circuit laws, [237] 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, [lower-alpha 26] but they did not care for any further development of this invention with regard to commercial purposes. [238] [239] [240]

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. [lower-alpha 27] [241]

Gauss tried to find a unifying law for long-distance effects of electrostatics, electrodynamics, electromagnetism, and induction, comparable to Newton's law of gravitation, [242] but his attempt ended in a "tragic failure". [230]

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. [243] Many results in geometrical optics are scattered in Gauss's correspondences and handnotes. [244]

In the Dioptrical Investigations (1840), Gauss gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). [245] Gauss demonstrated, that under a paraxial approximation an optical system can be characterized by its cardinal points, [246] and he derived the Gaussian lens formula, applicable without restrictions in respect to the thickness of the lenses. [247] [248]

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. [249]

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. [250] [251]

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. [252]

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. [146] In the correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision. [253] 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. [146]

Anecdotes

Parochial registration of Gauss' christening on 4 May 1777 with later added birth date Gauss keresztlevele.jpg
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. [254] 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. [255]

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. [256] 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. [257] [lower-alpha 28]

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. [259]

Honours and awards

Copley Medal for Gauss (1838) Carl Friedrich Gauss, Copley-Medaille, 1838.jpg
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 from the Academy of Sciences in Göttingen (1802/ 1807), [260] the French Academy of Sciences (1804/ 1820), [261] the Royal Society of London (1804), [262] the Royal Prussian Academy in Berlin (1810), [263] the National Academy of Science in Verona (1810), [264] the Royal Society of Edinburgh (1820), [265] the Bavarian Academy of Sciences of Munich (1820), [266] the Royal Danish Academy in Copenhagen (1821), the Royal Astronomical Society in London (1821), [267] the Royal Swedish Academy of Sciences (1821), the American Academy of Arts and Sciences in Boston (1822), [268] the Royal Bohemian Society of Sciences in Prague (1833), the Royal Academy of Science, Letters and Fine Arts of Belgium (1841/ 1845), [269] the Royal Society of Sciences in Uppsala (1843), the Royal Irish Academy in Dublin (1843), the Royal Institute of the Netherlands (1845/ 1851), [270] the Spanish Royal Academy of Sciences in Madrid (1850), [271] the Russian Geographical Society (1851), the Imperial Academy of Sciences in Vienna (1848), the American Philosophical Society (1853), [272] the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem. [273]

Gauss was an honorary member of the University of Kazan and of the Philosophy Faculty of the University of Prague since 1849. [274]

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, [275] 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". [273]

Gauss was appointed Knight of the French Legion of Honour [276] in 1837 and was one of the first members of the Prussian Order Pour le Merite (Civil class) when it was established in 1842. [277] 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). [273]

The Kings of Hanover appointed him the honorary titles "Hofrath" (1816) [77] and "Geheimer Hofrath" [lower-alpha 29] (1845). On the occasion of his golden doctor degree jubilee he got the honorary citizenship of both towns of Brunswick and Göttingen in 1849. [273] 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. [278]

The ″Gauss-Gesellschaft Göttingen″ (Gauss Society) was founded in 1964 for researches on life and work of Carl Friedrich Gauss and related persons and edits the ″Mitteilungen der Gauss-Gesellschaft″ (Communications of the Gauss Society). [279]

Selected writings

Mathematics and astronomy

Statue of Gauss in Brunswick (1880), made by Hermann Heinrich Howaldt, designed by Fritz Schaper Gauss-Denkmal - Braunschweig, Germany - DSC04436.JPG
Statue of Gauss in Brunswick (1880), made by Hermann Heinrich Howaldt, designed by Fritz Schaper

Physics

together with Wilhelm Weber

Collected works

Correspondence

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. [65] The literary estate is kept and provided by the Göttingen State and University Library. [280] Written estate from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick. [281]

Names and Commemorations

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References

Notes

  1. The Collegium Carolinum was the preceding institution of the Technische Hochschule Braunschweig, now Braunschweig Institute of Technology, but at Gauss' time not equal to a university.
  2. This error occurs for example in Marsden (1977). [15]
  3. On this journey he met the geodesist Ferdinand Rudolph Hassler, who was a scientific correspondent of Carl Friedrich Gauss. [43] [44]
  4. Gauss himself, in a letter to Bolyai, complained about "the shallowness that is so dominating in our contemporary mathematics". [12]
  5. The totally 195 lectures announced by Gauss divide in 70 per cent astronomical, 15 per cent mathematical, 9 per cent geodetical, and 6 per cent physical subjects. [64]
  6. The index of correspondence shows that Benjamin Gould was presumably the last correspondent who, on 13 February 1855, sent a letter to Gauss in his lifetime. It was an actual letter of farewell, but it is uncertain whether it reached the addressee just in time. [65]
  7. After his death, a discourse on the perturbations of Pallas in French was found among his papers, probably as a contribution to a prize competition of the French Academy of Science. [67]
  8. The Theoria motus... was completed in German language in 1806, but on request of editor Friedrich Christoph Perthes Gauss translated it into Latin. [68]
  9. Both Gauss and Harding dropped only veiled hints on this personal problem in their correspondence. A letter to Schumacher indicates that Gauss tried to get rid of his colleague and searched for a new position for him outside of Göttingen, but without result. Apart from that, Charlotte Waldeck, Gauss' mother-in-law, pleaded with Olbers to try to provide Gauss with another position far from Göttingen. [70]
  10. Gauss' first assistant was Benjamin Goldschmidt, and his second Wilhelm Klinkerfues, who later became one of his successors. [64]
  11. Bessel never got a university education. [73] [74]
  12. The first book he loaned from the university library in 1795 was the novel Clarissa from Samuel Richardson. [91]
  13. The political background was the confusing situation of the German Confederation with 39 nearly independent states, the sovereigns of three of them being Kings of other countries (Netherlands, Danmark, United Kingdom), whereas the Kingdom of Prussia and the Austrian Empire extended widely over the frontiers of the Confederation.
  14. Especially entry 146 in his diary, the only one using complex numbers, [113] is now seen as a prelude to the modern developments associated with Weil conjectures and the tools of algebraic geometry. [114]
  15. Gauss told the story later in detail in a letter to Encke. [118]
  16. Gauss already dealt implicitly with fundamental domains in the Disquisitiones Arithmeticae (1801), in the context of reduction theory of binary quadratic forms; this geometric sketch appears for the first time in his analytic work on the AGM.
  17. The unambigeous identification of a cosmic object as planet among the fixed stars requires at least two observations with interval.
  18. Brendel (1929) thought this cipher to be insoluble, but actually decoding was very easy. [153] [155]
  19. This article is the only reference to the —mostly unpublished— work on the AGM algorithm.
  20. In 1947, Andrey Kolmogorov corrected a mistake he found in Gauss's formula for the lower bound.
  21. Lauenburg was the southernmost town of the Duchy of Holstein, that was held in personal union by the King of Denmark.
  22. This Ramsden sector was loaned by the Board of Ordnance, and had earlier been used by William Mudge in the Principal Triangulation of Great Britain. [181]
  23. The new value of about 1/298.39 what was a significant improvement against the former value of 1/302.78 from Walbeck (1820). The calculation was done by Eduard Schmidt, private lecturer at Göttingen University. [183]
  24. Oskar Bolza mentions a transformation for the expression of geodesic curvature, that Gauss gives at the last section of this manuscript, which is "the actual core not only of Gauss's theorem about total curvature, but also of Bonnet's later generalization".
  25. As a result, there are only indirect reports by his colleagues and students Mobius, Listing and Riemann as testimony to his influence.
  26. A thunderstorm damaged the cable in 1845. [238]
  27. Printed in the Collected Works, Volume 5, pp. 609-610.
  28. Some authors, such as Joseph J. Rotman, question whether it ever happened. [258]
  29. literally translation: Secrete Councillor of the court

Citations

  1. Axel D. Wittmann, Inna V. Oreshina (2009). "On Jensen's Paintings of C. F. Gauss". Mitteilungen der Gauss-Gesellschaft (46): 57–61.
  2. Duden – Das Aussprachewörterbuch [The Pronunciation Dictionary] (in German) (7th ed.). Berlin: Dudenverlag. 2015 [1962]. pp. 246, 381, 391. ISBN   978-3-411-04067-4.
  3. Krech, Eva-Maria; Stock, Eberhard; Hirschfeld, Ursula; Anders, Lutz-Christian (2009). Deutsches Aussprachewörterbuch [German Pronunciation Dictionary] (in German). Berlin: W alter de Gruyter. pp. 402, 520, 529. ISBN   978-3-11-018202-6.
  4. Rudolf Borch (1929) : Ahnentafel des Mathematikers Carl Friedrich Gauß [Genealogical table]. Ahnentafeln Berühmter Deutscher, Vol. 1, pp. 63–65. Zentralstelle für Deutsche Personen- und Familiengeschichte (ed.)
  5. Dunnington 2004, p. 8.
  6. Dunnington 2004, p. 8–9.
  7. 1 2 Dunnington, Waldo (1927). "The Sesquicentennial of the Birth of Gauss". The Scientific Monthly . 24 (5): 402–414. Bibcode:1927SciMo..24..402D. JSTOR   7912. Archived from the original on 26 February 2008. Also available at "T he Sesquicentennial of the Birth of Gauss". Retrieved 23 February 2014. Comprehensive biographical article.
  8. Dunnington 2004, pp. 398–404.
  9. 1 2 Dunnington 2004, p. 24.
  10. 1 2 3 Dunnington 2004, p. 26.
  11. Wattenberg, Diedrich (1994). Wilhelm Olbers im Briefwechsel mit Astronomen seiner Zeit (in German). Stuttgart: GNT - Verlag für Geschichte der Naturwissenschaften und der Technik. p. 41. ISBN   3-928186-19-1.
  12. 1 2 Ullrich, Peter (2005). "Herkunft, Schul- und Studienzeit von Carl Friedrich Gauß". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 17–29. ISBN   3-930457-72-5.
  13. Dunnington 2004, p. 28.
  14. Dunnington 2004, p. 37.
  15. 1 2 3 Marsden, Brian G. (1 August 1977). "Carl Friedrich Gauss, Astronomer". Journal of the Royal Astronomical Society of Canada . 71: 309–323. Bibcode:1977JRASC..71..309M. ISSN   0035-872X.
  16. 1 2 3 4 5 Beuermann, Klaus (2005). "Carl Friedrich Gauß und die Göttinger Sternwarte" (PDF). In Beuermann, Klaus (ed.). Grundsätze über die Anlage neuer Sternwarten unter Beziehung auf die Sternwarte der Universität Göttingen von Georg Heinrich Borheck. Göttingen: Universitätsverlag Göttingen. pp. 37–45. ISBN   3-938616-02-4.
  17. Michling, Horst (1966). "Zum Projekt einer Gauß-Sternwarte in Braunschweig". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 24.
  18. 1 2 Dunnington 2004, p. 85–87.
  19. Dunnington 2004, pp. 50, 54–55, 74–77.
  20. Dunnington 2004, pp. 91–92.
  21. Dunnington 2004, pp. 86–87.
  22. Brendel 1929, pp. 81–82.
  23. Brendel 1929, p. 49.
  24. Brendel 1929, p. 83.
  25. Brendel 1929, p. 84.
  26. Brendel 1929, p. 119.
  27. Brendel 1929, p. 56.
  28. Klein 1979, p. 7.
  29. Sartorius von Waltershausen 1856, p. 104.
  30. Wagner, Rudolf (1860). Über die typischen Verschiedenheiten der Windungen der Hemisphären und über die Lehre vom Hirngewicht, mit besondrer Rücksicht auf die Hirnbildung intelligenter Männer. Vorstudien zu einer wissenschaftlichen Morphologie und Physiologie des menschlichen Gehirns als Seelenorgan, Vol. 1. Göttingen: Dieterich.
  31. Wagner, Rudolf (1862). Über den Hirnbau der Mikrocephalen mit vergleichender Rücksicht auf den Bau des Gehirns der normalen Menschen und der Quadrumanen. Vorstudien zu einer wissenschaftlichen Morphologie und Physiologie des menschlichen Gehirns als Seelenorgan, Vol. 2. Göttingen: Dieterich.
  32. Wagner, Hermann (1864). Maassbestimmungen der Oberfläche des grossen Gehirns [Measurements of the surface of the large brain] (in German). Cassel & Göttingen: Georg H. Wigand.
  33. Bardi, Jason (2008). The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons, Inc. p. 189. ISBN   978-0-470-46736-7.
  34. Hänicke, Wolfgang; Frahm, Jens; Wittmann, Axel D. (1999). "Magnetresonanz-Tomografie des Gehirns von Carl Friedrich Gauß". MPI News (in German). 5 (12). Archived from the original on 19 July 2011. Retrieved 3 September 2023.
  35. Schweizer, Renate; Wittmann, Axel; Frahm, Jens (2014). "A rare anatomical variation newly identifies the brains of C.F. Gauss and C.H. Fuchs in a collection at the University of Göttingen". Brain. 137 (4): e269. doi: 10.1093/brain/awt296 . hdl: 11858/00-001M-0000-0014-C6F0-6 . PMID   24163274. (with further references)
  36. "Unravelling the true identity of the brain of Carl Friedrich Gauss". Max Planck Society.
  37. Dunnington 2004, p. 66.
  38. Wußing 1982, p. 44.
  39. Cajori, Florian (19 May 1899). "Carl Friedrich Gauss and his children". Science. New Series. 9 (229). American Association for the Advancement of Science: 697–704. Bibcode:1899Sci.....9..697C. doi:10.1126/science.9.229.697. JSTOR   1626244. PMID   17817224.
  40. 1 2 Dunnington 2004, p. 374.
  41. Dunnington 2004, p. 206.
  42. 1 2 3 4 5 6 Gerardy, Theo (1966). "C. F. Gauß und seine Söhne". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 25–35.
  43. Gerardy, Theo (1977). "Geodäten als Korrespondenten von Carl Friedrich Gaus". Allgemeine Vermessungs-Nachrichten (in German) (84): 150–160. p. 157
  44. Dunnington 2004, p. 286.
  45. Wolf, Armin (1964). "Der Pädagoge und Philosoph Johann Conrad Fallenstein (1731–1813) – Genealogische Beziehungen zwischen Max Weber, Gauß und Bessel". Genealogie (in German). 7: 266–269.
  46. Weinberger, Joseph (1977). "Carl Friedrich Gauß 1777–1855 und seine Nachkommen". Archiv für Sippenforschung und alle verwandten Gebiete (in German). 43/44 (66): 73–98.
  47. Schubring, Gert (1993). "The German mathematical community". In Fauvel, John; Flood, Raymond; Wilson, Robin (eds.). Möbius and his band: Mathematics and Astronomy in Nineteenth-century Germany. Oxford University Press. pp. 21–33.
  48. Klein 1979, p. 5-6.
  49. Dunnington 2004, p. 217.
  50. Letter from Gauss to Bolyai from 2 September 1808
  51. 1 2 3 4 5 Biermann, Kurt-R. (1966). "Über die Beziehungen zwischen C. F. Gauß und F. W. Bessel". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 7–20.
  52. Klein 1979, p. 29.
  53. Dunnington 2004, p. 420–430.
  54. Sartorius von Waltershausen 1856, p. 79.
  55. Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics . Washington, DC: Joseph Henry Press. p.  202. ISBN   978-0-309-08549-6.
  56. 1 2 Stigler, Stephen M. (1981). "Gauss and the Invention of Least Squares". Annals of Statistics . 9 (3): 465–474. doi: 10.1214/aos/1176345451 .
  57. Maennchen 1930, p. 64–65.
  58. 1 2 Maennchen 1930, p. 4–9.
  59. Reich, Karin (2005). "Logarithmentafeln – Gauß' "tägliches Arbeitsgeräth"". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 73–86. ISBN   3-930457-72-5.
  60. Althoen, Steven C.; McLaughlin, Renate (1987), "Gauss–Jordan reduction: a brief history", The American Mathematical Monthly , 94 (2), Mathematical Association of America: 130–142, doi:10.2307/2322413, ISSN   0002-9890, JSTOR   2322413
  61. Maennchen 1930, p. 3.
  62. Dunnington 2004, p. 405–410.
  63. 1 2 3 Reich, Karin (2000). "Gauß' Schüler". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (37): 33–62.
  64. 1 2 3 4 5 Wittmann, Axel (2005). "Carl Friedrich Gauß und sein Wirken als Astronom". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 131–149. ISBN   3-930457-72-5.
  65. 1 2 "The complete correspondence of Carl Friedrich Gauss". Akademie der Wissenschaften zu Göttingen. Retrieved 10 March 2023.
  66. Gauss, C. F. (1874). Königlich Preußische Akademie der Wissenschaften (ed.). Carl Friedrich Gauss. Werke (in German). Vol. VI. pp. 73–86.
  67. Brendel 1929, p. 211.
  68. Dunnington 2004, p. 90.
  69. Dunnington 2004, p. 37–38.
  70. 1 2 3 Küssner, Martha (1978). "Friedrich Wilhelm Bessels Beziehungen zu Göttingen und Erinnerungen an ihn". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (15): 3–19.
  71. 1 2 Brendel 1929, pp. 106–108.
  72. Brendel 1929, pp. 7, 128.
  73. Hamel, Jürgen (1984). Friedrich Wilhelm Bessel. Leipzig: BSB B.G.Teubner Verlagsgesellschaft. p. 29.
  74. Dunnington 2004, p. 76.
  75. Mackinnon, Nick (1990). "Sophie Germain, or, Was Gauss a feminist?". The Mathematical Gazette . 74 (470). The Mathematical Association: 346–351. doi:10.2307/3618130. JSTOR   3618130. S2CID   126102577.
  76. Biermann, Kurt-R. (1964). "Gotthold Eisenstein". Journal für die reine und angewandte Mathematik (in German). 214: 19–30. doi:10.1515/crll.1964.214-215.19.
  77. 1 2 3 Dunnington 2004, p. 288.
  78. Sartorius von Waltershausen 1856, p. 102.
  79. 1 2 Sartorius von Waltershausen 1856, p. 95.
  80. Sartorius von Waltershausen 1856, p. 8.
  81. Sartorius von Waltershausen 1856, p. 68.
  82. 1 2 Sartorius von Waltershausen 1856, p. 100.
  83. Wußing 1982, p. 41.
  84. "Letter from Carl Friedrich Gauss to Johanna Gauss, 23. October 1809". Der komplette Briefwechsel von Carl Friedrich Gauss. Akademie der Wissenschaften zu Göttingen. 23 October 1809. Retrieved 26 March 2023.
  85. Dunnington 2004, pp. 94–95.
  86. Dunnington 2004, p. 206, 374.
  87. "Letter: Charles Henry Gauss to Florian Cajori – 21 December 1898" . Retrieved 25 March 2023.
  88. Sartorius von Waltershausen 1856, p. 71.
  89. Lehfeldt, Werner (2005). "Carl Friedrich Gauß' Beschäftigung mit der russischen Sprache". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 302–310. ISBN   3-930457-72-5.
  90. Dunnington 2004, p. 241.
  91. Reich, Karin (2005). "Gauß' geistige Väter: nicht nur "summus Newton", sondern auch "summus Euler"". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 105–115. ISBN   3-930457-72-5.
  92. Wußing 1982, p. 80.
  93. Wußing 1982, p. 81.
  94. 1 2 3 Sartorius von Waltershausen 1856, p. 94.
  95. Wußing 1982, p. 79.
  96. Dunnington 2004, p. 195-200.
  97. Sartorius von Waltershausen 1856, p. 97.
  98. "Letter from Carl Friedrich Gauss to Wilhelm Olbers, 3 September 1805". Der komplette Briefwechsel von Carl Friedrich Gauss. Akademie der Wissenschaften zu Göttingen. 23 October 1809. Retrieved 26 March 2023.
  99. Dunnington 2004, p. 300.
  100. Sartorius von Waltershausen 1856, p. 90.
  101. Gronwald, W. (1955). "Carl Friedrich Gauss – Ein Lebensbild". In Niedrsächsische Vermessungs- und Katasterverwaltung (ed.). Carl Friedrich Gauss und die Landesvermessung in Niedersachsen (in German). Hannover. pp. 1–16.{{cite book}}: CS1 maint: location missing publisher (link)
  102. Basu, Soham; Velleman, Daniel J. (21 April 2017). "On Gauss's first proof of the fundamental theorem of algebra". arXiv: 1704.06585 [math.CV].
  103. Bachmann 1922, p. 8–9.
  104. Bachmann 1922, p. 16–25.
  105. Bachmann 1922, p. 14–16, 25.
  106. Bachmann 1922, p. 25–28.
  107. Bachmann 1922, p. 29.
  108. Bachmann 1922, p. 22-23.
  109. Bachmann 1922, p. 66-69.
  110. Denker, Manfred; Patterson, Samuel James (2005). "Gauß – der geniale Mathematiker". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 53–62. ISBN   3-930457-72-5.
  111. 1 2 3 4 Stuhler, Ulrich (2005). "Arithmetisch-geometrisches Mittel und elliptische Integrale: Gauß und die komplexe Analysis". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 62–72. ISBN   3-930457-72-5.
  112. Weil, André (1949). "Numbers of solutions of equations in finite fields" (PDF). Bulletin of the American Mathematical Society . 55 (5): 497–508. doi:10.1090/S0002-9904-1949-09219-4.
  113. Bachmann 1922, p. 56.
  114. Königlich Preußische Akademie der Wissenschaften, ed. (1917). Carl Friedrich Gauss. Werke (in Latin). Vol. X, 1. Göttingen. p. 571.{{cite book}}: CS1 maint: location missing publisher (link)
  115. Dunnington 2004, p. 44.
  116. Frei, Günter (2007). "The Unpublished Section Eight: On the Way for Function Fields over a Finite Field". In Goldstrein, Catherine; Schappacher, Norbert; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae. Berlin, Heidelberg, New York: Springer. pp. 159–198. doi:10.1007/978-3-540-34720-0. ISBN   978-3-540-20441-1.
  117. Koch, H.; Pieper, H. (1976). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. pp. 6, 124.
  118. Bachmann 1922, p. 4.
  119. Kleiner, I. (2000). "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem" (PDF). Elemente der Mathematik . 55: 19–37. doi:10.1007/PL00000079. S2CID   53319514. Archived from the original (PDF) on 8 June 2011.
  120. Bachmann 1922, p. 60-61.
  121. Hales, Thomas C. (2006). "Historical overview of the Kepler conjecture". Discrete & Computational Geometry . 36 (1): 5–20. doi: 10.1007/s00454-005-1210-2 . ISSN   0179-5376. MR   2229657.
  122. Seeber, Ludwig August (1831). Untersuchungen über die Eigenschaften der positiven ternaeren quadratischen Formen. Mannheim.
  123. "Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber". Göttingische gelehrte Anzeigen (108): 1065–1077. July 1831.
  124. 1 2 Kleiner, Israel (1998). "From Numbers to Rings: The Early History of Ring Theory". Elemente der Mathematik . 53 (1): 18–35. doi: 10.1007/s000170050029 . Zbl   0908.16001.
  125. Lemmermeyer, Franz (2000). Reciprocity Laws: from Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer. p. 15. doi:10.1007/978-3-662-12893-0. ISBN   3-540-66957-4.
  126. Bachmann 1922, p. 52, 57-59..
  127. Bell, E.T. (1944). "Gauss and the Early Development of Algebraic Numbers". National Mathematics Magazine . 18 (5): 188–204. doi:10.2307/3030061. JSTOR   3030061.
  128. Schlesinger 1933, p. 41–57.
  129. 1 2 Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement mathématique . 30 (2): 275–330.
  130. Letter Gauss to Bessel from 18 December 1811, partly printed in the Collected Works, Volume 8, pp. 90-92.
  131. Roy, Ranjan (2021). Series and Products in the Development of Mathematics (PDF) (2 ed.). Cambridge: Cambridge University Press. pp. 20–22.
  132. Schlesinger 1933, p. 101–106.
  133. Printed in the Collected Works, Volume 8, p. 104.
  134. Schlesinger 1933, p. 136–142.
  135. Schlesinger 1933, p. 142.
  136. Monna, A. F. (1975). Dirichlet's principle : a mathematical comedy of errors and its influence on the development of analysis (PDF). Utrecht. pp. 11–12.{{cite book}}: CS1 maint: location missing publisher (link)
  137. Printed in the Collected Works Volume 10.1, pp. 311–324
  138. Schlesinger 1933, p. 193.
  139. Gautschi, Walter (1981). "A Survey of Gauss-Christoffel Quadrature Formulae". In Butzer, Paul B.; Fehér, Franziska (eds.). E.B. Christoffel. The Influence of his Work on Mathematics and the Physical Science (1 ed.). Birkhäuser, Basel: Springer. pp. 72–147. doi:10.1007/978-3-0348-5452-8_6. ISBN   978-3-0348-5452-8.
  140. Letter Gauss to Gerling from 26 December 1823
  141. Yousef Saad (2 August 2019). "Iterative Methods for Linear Systems of Equations: A Brief Historical Journey". arXiv: 1908.01083v1 [math.HO].
  142. Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series". Mathematics of Computation . 19 (90): 297–301. doi: 10.2307/2003354 . JSTOR   2003354.
  143. Gauss, C.F. (1876). Theoria Interpolationis Methodo Nova Tractata (in Latin). Göttingen: K. Gesellschaft der Wissenschaften zu Göttingen. pp. 265–327.
  144. Heideman, Michael T.; Johnson, Don H.; Burrus, C. Sidney (1984). "Gauss and the history of the fast Fourier transform" (PDF). IEEE ASSP Magazine. 1 (4): 14–21. doi:10.1109/MASSP.1984.1162257. S2CID   10032502. Archived (PDF) from the original on 19 March 2013.
  145. Maennchen 1930, p. 49–63.
  146. 1 2 3 Olesko, Kathryn (2005). "Der praktische Gauß – Präzisionsmessung für den Alltag". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 236–253. ISBN   3-930457-72-5.
  147. Dunnington 2004, p. 69.
  148. Forbes, Eric G. (1971). "Gauss and the Discovery of Ceres". Journal for the History of Astronomy . 2 (3): 195–199. Bibcode:1971JHA.....2..195F. doi:10.1177/002182867100200305. S2CID   125888612. Archived from the original on 18 July 2021.
  149. Teets, Donald; Whitehead, Karen (1965). "The discovery of Ceres. How Gauss became famous". Mathematics Magazine . 19 (90): 83–91.
  150. Klein 1979, p. 8.
  151. Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.
  152. Brendel 1929, p. 194–195.
  153. 1 2 Brendel 1929, p. 206.
  154. Taylor, D. B. (1982). "The secular motion of Pallas". Monthly Notices of the Royal Astronomical Society . 199 (2): 255–265. Bibcode:1982MNRAS.199..255T. doi: 10.1093/mnras/199.2.255 .
  155. Schroeder, Manfred R. (2005). "Gauß, die Konzertsaalakustik und der Asteroid Palls". In Mittler, Elmar (ed.). "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen (PDF). Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 259–260. ISBN   3-930457-72-5.
  156. Brendel 1929, p. 254.
  157. Brendel 1929, p. 253–254.
  158. Schlesinger 1933, p. 169-170.
  159. Hill, George William (1882). "On Gauss's Method of Computing Secular Perturbations, with an Application to the Action of Venus on Mercury". Astronomical Papers of the American Ephemeris. 1: 315–361. Bibcode:1882USNAO...1..315H.
  160. Brendel 1929, pp. 8–9.
  161. Brendel 1929, p. 3.
  162. Brendel 1929, p. 54.
  163. Brendel 1929, p. 144.
  164. Stigler, Stephen M. (1981). "Gauss and the Invention of Least Squares". The Annals of Statistics. 9 (3): 465–474. doi: 10.1214/aos/1176345451 . ISSN   0090-5364. JSTOR   2240811.
  165. Schaaf 1964, p. 84.
  166. Plackett, R.L. (1972). "The discovery of the method of least squares" (PDF). Biometrika. 59 (2): 2 39–251. doi:10.2307/2334569. JSTOR   2334569.
  167. Lim, Milton (31 March 2021). "Gauss, Least Squares, and the Missing Planet". Actuaries Digital. Retrieved 14 October 2023.
  168. Plackett, R. L. (1949). "A Historical Note on the Method of Least Squares". Biometrika . 36 (3/4): 458–460. doi:10.2307/2332682. JSTOR   2332682. PMID   15409359.
  169. Avkhadiev, F. G. (2005). "A Simple Proof of the Gauss-Winckler Inequality". The American Mathematical Monthly . 112 (5): 459–462. doi:10.2307/30037497. JSTOR   30037497.
  170. Schaffrin, Burkhard; Snow, Kyle (2010). "Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth". Linear Algebra and Its Applications. 432 (8). Elsevier BV: 2061–2076. doi: 10.1016/j.laa.2009.09.014 . ISSN   0024-3795.
  171. Sheynin, O. B. (1979). "C. F. Gauss and the Theory of Errors". Archive for History of Exact Sciences . 20 (1): 21–72. doi:10.1007/BF00776066. JSTOR   41133536.
  172. Galle 1924, pp. 16–18.
  173. Galle 1924, p. 22.
  174. Galle 1924, p. 28.
  175. Galle 1924, p. 32.
  176. Galle 1924, p. 60.
  177. Galle 1924, p. 75.
  178. Schaaf 1964, p. 81.
  179. Galle 1924, p. 69.
  180. Dunnington 2004, p. 121.
  181. 1 2 Galle 1924, p. 61.
  182. Galle 1924, pp. 37–38, 49–50.
  183. Galle 1924, p. 49-50.
  184. Dunnington 2004, p. 164.
  185. Galle 1924, p. 129.
  186. Schreiber, Oscar (1866). Theorie der Projectionsmethode der Hannoverschen Landesvermessung (in German). Hannover: Hahnsche Buchhandlung.
  187. Gauß, C.F. (1828). Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector (in German). Vandenhoeck und Ruprecht. p. 73.
  188. Listing, J.B. (1872). Ueber unsere jetzige Kenntniss der Gestalt und Grösse der Erde (in German). Göttingen: Dieterich. p. 9.
  189. 1 2 Stäckel 1917, p. 110-119.
  190. Stäckel 1917, p. 105-106.
  191. Bolza 1921, p. 73-74.
  192. Minimal surfaces (2010), p. 50
  193. 1 2 3 Bühler 1981, p. 100–102.
  194. 1 2 3 Klein 1979, pp. 57–60.
  195. Schaaf 1964, p. 128–140.
  196. Winger, R. M. (1925). "Gauss and non-Euclidean geometry". Bulletin of the American Mathematical Society. 31 (7): 356–358. doi: 10.1090/S0002-9904-1925-04054-9 . ISSN   0002-9904.
  197. Jenkovszky, László; Lake, Matthew J.; Soloviev, Vladimir (12 March 2023). "János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword". Symmetry. 15 (3): 707. arXiv: 2303.17011 . Bibcode:2023Symm...15..707J. doi: 10.3390/sym15030707 . ISSN   2073-8994.
  198. Letter from Gauss to Bolyai from 6 March 1832
  199. Klein 1979, p. 59.
  200. Krantz, Steven G. (2010). An Episodic History of Mathematics: Mathematical Culture through Problem Solving. The Mathematical Association of America. pp. 171f. ISBN   978-0-88385-766-3 . Retrieved 9 February 2013.
  201. Monastyrsky, Michael (1987). Riemann, Topology, and Physics. Birkhäuser. pp. 21–22. ISBN   978-0-8176-3262-5.
  202. Riemann, Bernhard. On the Hypotheses which lie at the Bases of Geometry (PDF). Translated by William Kingdon Clifford.
  203. Bühler 1981, p. 154.
  204. Ostrowski 1920, p. 1-18.
  205. 1 2 Epple, Moritz (1998). "Orbits of asteroids, a braid, and the first link invariant". The Mathematical Intelligencer . 20 (1): 45–52. doi:10.1007/BF03024400. S2CID   124104367.
  206. Epple, Moritz (1999). "Geometric Aspects in the Development of Knot Theory" (PDF). In James, I.M. (ed.). History of Topology. Amsterdam: Elseviwer. pp. 301–357.
  207. Lisitsa, Alexei; Potapov, Igor; Saleh, Rafiq (2009). "Automata on Gauss Words" (PDF). In Dediu, Adrian Horia; Ionescu, Armand Mihai; Martín-Vide, Carlos (eds.). Language and Automata Theory and Applications. Lecture Notes in Computer Science. Vol. 5457. Berlin, Heidelberg: Springer. pp. 505–517. doi:10.1007/978-3-642-00982-2_43. ISBN   978-3-642-00982-2.
  208. Stäckel 1917, p. 50-51.
  209. Stäckel 1917, p. 51-53.
  210. Stäckel 1917, p. 55.
  211. Stäckel 1917, p. 54.
  212. Stäckel 1917, p. 70-74.
  213. Eastwood, Michael; Penrose, Roger (2000). "Drawing with Complex Numbers". The Mathematical Intelligencer. 22 (4): 8–13. arXiv: math/0001097 . doi:10.1007/BF03026760. S2CID   119136586.
  214. Pujol, Jose (2012). "Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: a Historical Reassessment". Commununications in Mathematical Analysis. 13 (2): 1–14.
  215. Mutationen des Raumes in Collected Works Volume 8, pp. 357–362.
  216. Carl Friedrich Gauss: Zusätze.II. In: Carnot, Lazare (1810). Geometrie der Stellung (in German). Translated by H.C. Schumacher. Altona: Hammerich. pp. 363–364. (Text by Schumacher, algorithm by Gauss), republished in Collected Works Volume 4, p. 393
  217. Carl Friedrich Gauss: Zusätze.II. In: Carnot, Lazare (1810). Geometrie der Stellung (in German). Translated by H.C. Schumacher. Altona: Hammerich. pp. 371–372. (Text by Schumacher, algorithm by Gauss), republished in Collected Works Volume 4, pp. 399–400
  218. Röttgen-Burtscheidt, Johannes (2007). "Das Apollonische BerührungsprobleJohannes" (PDF). University of Wuppertal. pp. 119–121. Retrieved 22 April 2024.
  219. Bestimmung der größten Ellipse, welche die vier Ebenen eines gegebenen Vierecks berührt, printed in Collected Works Volume 4, pp. 385-392; original in Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, Volume 22, 1810, pp. 112–121
  220. Stäckel 1917, p. 71-72.
  221. Printed in Collected Works Volume 4, pp. 406–407
  222. Stäckel 1917, p. 76.
  223. Coxeter, H. S. M. (1971). "Frieze patterns" (PDF). Acta Arithmetica . 18: 297–310. doi: 10.4064/aa-18-1-297-310 .
  224. Pentagramma mirificum, printed in Collected Works Volume III, pp. 481–490
  225. Schechtman, Vadim (2013). "Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)". Annales de la faculté des sciences de Toulouse Mathématiques. 22 (2): 353–375. doi: 10.5802/afst.1375 .
  226. Reich, Karin (2011). "Alexander von Humboldt und Carl Friedrich Gauss als Wegbereiter der neuen Disziplin Erdmagnetismus". Humboldt Im Netz (in German). 12 (22): 33–55.
  227. Dunnington 2004, p. 136.
  228. Dunnington 2004, p. 161.
  229. Humboldt, Alexander von (1836). "Letter of the Baron von Humboldt to His Royal Highness the Duke of Sussex,..., on the Advancement of the Knowledge of Terrestrial Magnetism, by the Establishment of Magnetic Stations and correspionding Observations". Philosophical Magazine (9): 42–53.
  230. 1 2 3 4 Rupke, Nicolaas (2005). "Carl Friedrich Gauß und der Erdmagnetismus". In Mittler, Elmar (ed.).