Template:Short description Template:Redirect Template:Good article Template:Pp-pc Template:Use dmy dates Template:Infobox scientist Johann Carl Friedrich Gauss (Template:IPAc-en;<ref>"Gauss". Random House Webster's Unabridged Dictionary.</ref> Template:Langx {{#invoke:IPA|main}};<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Template:Langx; 30 April 1777Template:Spaced ndash23 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855.
While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science and mathematics, more than 100 mathematical and scientific concepts are named after him.
Gauss was instrumental in the identification of 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 it. Gauss led the geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders of geophysics and formulated the fundamental principles of magnetism. His practical work led to the invention of the heliotrope in 1821, a magnetometer in 1833 and – with Wilhelm Eduard Weber – the first electromagnetic telegraph in 1833.
Gauss was the first to discover and study non-Euclidean geometry, which he also named. He developed a fast Fourier transform some 160 years before John Tukey and James Cooley.
Gauss refused to publish incomplete work and left several works to be edited posthumously. He believed that the act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss was not a committed or enthusiastic teacher, generally preferring to focus on his own work. Nevertheless, some of his students, such as Dedekind and Riemann, became well-known and influential mathematicians in their own right.
BiographyEdit
Youth and educationEdit
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Gauss was born on 30 April 1777 in Brunswick in the Duchy of Brunswick-Wolfenbüttel (now in the German state of Lower Saxony). His family was of relatively low social status.<ref>Template:Cite book</ref> His father Gebhard Dietrich Gauss (1744–1808) worked variously as a butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate.Template:Sfn He had one elder brother from his father's first marriage.Template:Sfn
Gauss was a child prodigy in 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,Template:Efn which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers.Template:SfnTemplate:Sfn<ref name="Ullrich">Template:Cite book</ref> Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the University of Göttingen until 1798.<ref name="scientificmonthly">Template:Cite journal Also available at {{#invoke:citation/CS1|citation |CitationClass=web }} Retrieved 23 February 2014. Comprehensive biographical article.</ref> 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.Template:SfnTemplate:Efn Astronomy was taught by Karl Felix Seyffer, with whom Gauss stayed in correspondence after graduation;Template:Sfn Olbers and Gauss mocked him in their correspondence.<ref>Template:Cite book</ref> 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.Template:Sfn Fellow students of this time were Johann Friedrich Benzenberg, Farkas Bolyai, and Heinrich Wilhelm Brandes.Template:Sfn
He was likely a self-taught student in mathematics since he independently rediscovered several theorems.<ref name="Ullrich" /> He solved a geometrical problem that had occupied mathematicians since the Ancient Greeks when he determined in 1796 which regular polygons can be constructed by compass and straightedge. This discovery ultimately led Gauss to choose mathematics instead of philology as a career.Template:Sfn Gauss's mathematical diary, a collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time.Template:Sfn
As an elementary student, Gauss and his class were tasked by their teacher, J.G. Büttner, to sum the numbers from 1 to 100. Much to Büttner's surprise, Gauss replied with the correct answer of 5050 in a vastly faster time than expected.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Gauss had realised that the sum could be rearranged as 50 pairs of 101 (1+100=101, 2+99=101, etc.). Thus, he simply multiplied 50 by 101.<ref>Template:Cite book</ref> Other accounts state that he computed the sum as 100 sets of 101 and divided by 2.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Private scholarEdit
Gauss graduated as a Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated,Template:Efn<ref name="Marsden" /> but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of the duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination.<ref name="Ullrich" /> The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the Russian Academy of Sciences in St. Peterburg and Landshut University.<ref name="Reich2000" /><ref name="Beuermann">Template:Cite book</ref> 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:<ref>Template:Cite journal</ref> the Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped.
When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical communities of Bremen and Lilienthal, especially Wilhelm Olbers, Karl Ludwig Harding, and Friedrich Wilhelm Bessel, forming part of the informal group of astronomers known as the Celestial police.Template:Sfn One of their aims was the discovery of further planets. They assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809).Template:Sfn
Professor in GöttingenEdit
In November 1807, Gauss was hired by the University of Göttingen, then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory,Template:Sfn and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt, later discovered to be Prince-primate Dalberg,Template:Sfn paid the sum.Template:Sfn
Gauss took on the directorship of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower,Template:Sfn with usable, but partly out-of-date instruments.Template:Sfn 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,Template:Sfn but Gauss could not move to his new place of work until September 1816.<ref name="Beuermann" /> He got new up-to-date instruments, including two meridian circles from RepsoldTemplate:Sfn and Reichenbach,Template:Sfn and a heliometer from Fraunhofer.Template:Sfn
The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade.Template:Sfn
Gauss made no secret of his aversion to giving academic lectures.<ref name="Reich2000">Template:Cite journal</ref><ref name="Beuermann" /> But from the start of his academic career at Göttingen, he continuously gave lectures until 1854.Template:Sfn He often complained about the burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented.<ref name="Reich2000" /> Most of his lectures dealt with astronomy, geodesy, and applied mathematics,<ref name="Wittmann">Template:Cite book</ref> and only three lectures on subjects of pure mathematics.<ref name="Reich2000" />Template:Efn Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor, Dedekind, Dirksen, Encke, Gould,Template:Efn Heine, Klinkerfues, Kupffer, Listing, Möbius, Nicolai, Riemann, Ritter, Schering, Scherk, Schumacher, von Staudt, Stern, Ursin; as geoscientists Sartorius von Waltershausen, and Wappäus.<ref name="Reich2000" />
Gauss did not write any textbook and disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836.<ref name="Biermann" /> Gauss published his papers and books exclusively in Latin or German.Template:EfnTemplate:Efn He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians.Template:Sfn
Gauss gave his inaugural lecture at Göttingen University in 1808. He described his approach to astronomy as based on reliable observations and accurate calculations, rather than on belief or empty hypothesizing.<ref name="Wittmann" /> At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures,<ref>Template:Cite book</ref> the physicist Mayer, known for his textbooks,<ref>Template:Cite book</ref> his successor Weber since 1831, and in the observatory Harding, who took the main part of lectures in practical astronomy. When the observatory was completed, Gauss occupied the western wing of the new observatory, while Harding took the eastern.<ref name="Beuermann" /> They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer.<ref name="Beuermann" />Template:Efn Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations.Template:Sfn
Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which the years since 1820 are taken as a "period of lower astronomical activity".Template:Sfn The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834.<ref name="Küssner">Template:Cite journal</ref>Template:SfnTemplate:Efn
Nevertheless, Gauss twice refused the opportunity to solve the problem, turning down offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation.<ref name="Küssner" /> Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2500 Reichsthaler in 1824,<ref name="Beuermann" /> and in his later years he was one of the best-paid professors of the university.<ref name="Gerardy" />
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.Template:Efn Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it.<ref>Template:Cite journal</ref> He also gave successful support to the mathematician Gotthold Eisenstein in Berlin.<ref>Template:Cite journal</ref>
Gauss was loyal to the House of Hanover. After King William IV died in 1837, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the "Göttingen Seven", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss was deeply affected by this quarrel but saw no possibility to help them.Template:Sfn
Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy.Template:Sfn 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.Template:Sfn
Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of a heart attack in Göttingen;Template:Sfn 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.Template:Sfn
Gauss was a successful investor and accumulated considerable wealth with stocks and securities, amounting to a value of more than 150,000 Thaler; after his death, about 18,000 Thaler were found hidden in his rooms.Template:Sfn
Gauss's brainEdit
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 Template:Convert.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Wagner's son Hermann, a geographer, estimated the cerebral area to be Template:Convert in his doctoral thesis.<ref>Template:Cite book</ref> In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen a few months after Gauss.<ref>Template:Cite journal (with further references)</ref> A further investigation showed no remarkable anomalies in the brains of either person. Thus, all investigations of Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
FamilyEdit
Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick.Template:Sfn 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.Template:Sfn Gauss chose the first names of his children in honour of Giuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids.Template:Sfn
On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than a decade.<ref>Template:Cite journal</ref> Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau.Template:Sfn Her sister Wilhelmina married the orientalist Heinrich Ewald.Template:Sfn Gauss's mother Dorothea lived in his house from 1817 until she died in 1839.<ref name="scientificmonthly"/>
The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in the summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network into 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.<ref name="Gerardy">Template:Cite journal</ref>Template:Efn
Eugen left Göttingen in September 1830 and emigrated to the United States, where he spent five years with the army. He then worked for the American Fur Company in the Midwest. He later moved to Missouri and became a successful businessman.<ref name="Gerardy" /> Wilhelm married a niece of the astronomer Bessel;<ref>Template:Cite journal</ref> he then moved to Missouri, started as a farmer and became wealthy in the shoe business in St. Louis in later years.<ref>Template:Cite journal</ref> Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children.<ref name="Gerardy" />
- Joseph Gauß, 001.jpg
Joseph Gauss
- Joseph Gauß, 003.jpg
Sophie Gauss née Erythropel
Joseph's wife - Minna Ewald geb. Gauß, 003.jpg
Wilhelmina Gauss
- Ewald, Georg Heinrich August (1803-1875).jpg
Heinrich Ewald
Wilhelmina's husband - Eugen Gauß, 001.jpg
Eugen (Eugene) Gauss
- Eugen Gauß, 003.jpg
Henrietta Gauss née Fawcett
Eugene's wife - Wilhelm Gauß, 002.jpg
Wilhelm (Charles William) Gauss
- Wilhelm Gauß, 001.jpg
Louisa Aletta Gauss née Fallenstein
William's wife - Therese Staufenau geb. Gauß, 008.jpg
Therese Gauss
- Therese Staufenau geb. Gauß, 010.jpg
Constantin Staufenau
Therese's husband
PersonalityEdit
ScholarEdit
In the first two decades of the 19th century, Gauss was the only important mathematician in Germany comparable to the leading French mathematicians.<ref name="Schubring">Template:Cite book</ref> His Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into the French language.<ref>Template:Cite book</ref>
Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration.Template:Sfn In contrast to previous mathematicians like Leonhard Euler, who let their readers take part in their reasoning, including certain erroneous deviations from the correct path,Template:Sfn Gauss introduced a new style of direct and complete exposition that did not attempt to show the reader the author's train of thought.Template:Sfn
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Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments.{{#if:Template:Harvnb|{{#if:|}}
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But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
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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.{{#if:Template:Harvnb|{{#if:|}}
— {{#if:|, in }}Template:Comma separated entries}}
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His posthumous papers, his scientific diary,<ref>Template:Cite journal p. 2</ref> and short glosses in his own textbooks show that he empirically worked to a great extent.Template:SfnTemplate:Sfn He was a lifelong busy and enthusiastic calculator, working extraordinarily quickly and checking his results through estimation. Nevertheless, his calculations were not always free from mistakes.Template:Sfn He coped with the enormous workload by using skillful tools.Template:Sfn Gauss used numerous mathematical tables, examined their exactness, and constructed new tables on various matters for personal use.<ref>Template:Cite book</ref> He developed new tools for effective calculation, for example the Gaussian elimination.<ref>Template:Citation</ref> Gauss's calculations and the tables he prepared were often more precise than practically necessary.Template:Sfn Very likely, this method gave him additional material for his theoretical work.Template:SfnTemplate:Sfn
Gauss was only willing to publish work when he considered it complete and above criticism. This perfectionism was in keeping with the motto of his personal seal {{#invoke:Lang|lang}} ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself by 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".<ref name="Biermann">Template:Cite journal</ref> Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too.Template:SfnTemplate:Sfn Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics",Template:Sfn 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.<ref name="First-Class">Template:Cite book</ref>
On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not the first to publish" differed from that of his scientific contemporaries.<ref name="Stigler">Template:Cite journal</ref> In contrast to his perfectionism in presenting mathematical ideas, his citations were criticized as negligent. He justified himself with an unusual view of correct citation practice: he would only give complete references, with respect to the previous authors of importance, which no one should ignore, but citing in this way would require knowledge of the history of science and more time than he wished to spend.<ref name="Biermann" />
Private manEdit
Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty,Template:Sfn but also of "iron character"Template:Sfn with an unshakeable strength of mind.Template:Sfn Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on the summit of science".Template:Sfn His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host.<ref name="Biermann" /> Gauss disliked polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on a university chair in Göttingen, "because he was always involved in some polemic."Template:Sfn
Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody, the most personal of his surviving documents.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Sfn His second wife and his two daughters suffered from tuberculosis.Template:Sfn In a letter to Bessel, dated December 1831, Gauss hinted at his distress, describing himself as "the victim of the worst domestic sufferings".<ref name="Biermann" />
Because of his wife's illness, both younger sons were educated for some years in Celle, far from Göttingen. The military career of his elder son Joseph ended after more than two decades at the poorly paid rank of first lieutenant, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married.<ref name="Gerardy" /> The second son Eugen shared a good measure of his father's talent in computation and languages but had a lively 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,<ref name="gausschildren">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support.<ref name="Gerardy" /> The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life.Template:Sfn
In his later years Gauss habitually collected various types of useful or useless numerical data, such as the number of paths from his home to certain places in Göttingen or peoples' ages in days; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days.Template:Sfn
Beyond his excellent knowledge of Latin, he was also acquainted with modern languages. Gauss read both classical and modern literature, and English and French works in the original languages.Template:SfnTemplate:Efn His favorite English author was Walter Scott, his favorite German Jean Paul. 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.<ref>Template:Cite book</ref>Template:Sfn Gauss liked singing and went to concerts.Template:Sfn He was a busy newspaper reader; in his last years, he would visit an academic press salon of the university every noon.Template:Sfn 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.Template:Sfn
Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto "mundus vult decipi".Template:Sfn He disliked Napoleon and his system and was horrified by violence and revolution of all kinds. Thus he condemned the methods of the Revolutions of 1848, though he agreed with some of their aims, such as that of a unified Germany.Template:SfnTemplate:Efn He had a low estimation of the constitutional system and he criticized parliamentarians of his time for their perceived ignorance and logical errors.Template:Sfn
Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes"Template:Sfn and "I succeeded – not on account of my hard efforts, but by the grace of the Lord."<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Gauss was a member of the Lutheran church, like most of the population in northern Germany, but it seems that he did not believe all Lutheran dogma or understand the Bible fully literally.Template:Sfn According to Sartorius, Gauss' religious tolerance, "insatiable thirst for truth" and sense of justice were motivated by his religious convictions.Template:Sfn
MathematicsEdit
Algebra and number theoryEdit
Fundamental theorem of algebraEdit
In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts led to considerable clarification of the concept of complex numbers.<ref>Template:Cite arXiv</ref>
Disquisitiones ArithmeticaeEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In the preface to the Disquisitiones, Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered.Template:Sfn The Disquisitiones Arithmeticae, written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces the triple bar symbol (Template:Math) for congruence and uses it for a clean presentation of modular arithmetic.Template:Sfn It deals with the unique factorization theorem and primitive roots modulo n. In the main sections, Gauss presents the first two proofs of the law of quadratic reciprocityTemplate:Sfn and develops the theories of binaryTemplate:Sfn and ternary quadratic forms.Template:Sfn
The Disquisitiones include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares, he proves the triangular case of the Fermat polygonal number theorem for n = 3.Template:Sfn From several analytic results on class numbers that Gauss gives without proof towards the end of the fifth section,Template:Sfn it appears that Gauss already knew the class number formula in 1801.Template:Sfn
In the last section, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one.<ref name="Denker">Template:Cite book</ref> He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or the product of a power of 2 and any number of distinct Fermat primes. In the same section, 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.<ref name="Stuhler">Template:Cite book</ref> An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.Template:Sfn<ref name="Frei">Template:Cite book</ref>
Further investigationsEdit
One of Gauss's first results was the empirically found conjecture of 1792 – the later called prime number theorem – giving an estimation of the number of prime numbers by using the integral logarithm.<ref>Template:Cite book</ref>Template:Efn
In 1816, Olbers encouraged Gauss to compete for a prize from the French Academy for a proof for Fermat's Last Theorem; he refused, considering the topic uninteresting. However, after his death a short undated paper was found with proofs of the theorem for the cases n = 3 and n = 5.<ref>Template:Cite journal</ref> 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.Template:Sfn
Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form a cubic face-centred arrangement,<ref>Template:Cite journal</ref> when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms.<ref>Template:Cite book</ref> Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.<ref>Template:Cite journal</ref>
In two papers on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers <math>\mathbb{Z}[i]</math>, showed that it is a unique factorization domain,<ref name="Kleiner1998">Template:Cite journal</ref> and generalized some key arithmetic concepts, such as Fermat's little theorem and Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity<ref name="Kleiner1998" /> – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.<ref>Template:Cite book</ref>
In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.Template:Sfn
AnalysisEdit
One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers.Template:Sfn He discovered its relation to elliptic integrals in the years 1798–1799 through 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 "will surely open an entirely new field of analysis".<ref name="Cox">Template:Cite journal</ref> 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.<ref name="Stuhler" /><ref>Letter Gauss to Bessel from 18 December 1811, partly printed in the Collected Works, Volume 8, pp. 90–92.</ref>
Euler's pentagonal numbers theorem, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on Jacobi theta functions,<ref name="Stuhler" /> culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case.<ref>Template:Cite book</ref> His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.Template:SfnTemplate:EfnTemplate:Efn
Several mathematical fragments in his Nachlass indicate that he knew parts of the modern theory of modular forms.<ref name="Stuhler" /> In his work on the multivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM and its two "simplest values".<ref name="Cox" /> In his unpublished writings he recognized and made a sketch of the key concept of fundamental domain for the modular group.Template:Sfn<ref name="Houzel">Template:Cite book</ref> 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 <math>\pi/4</math>.<ref>Printed in the Collected Works, Volume 8, p. 104.</ref>
An example of Gauss's insight in analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the lemniscate curve, which inspired Abel's theorem on lemniscate division.Template:Efn Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of quadratic Gauss sums, in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem from his work on elliptic function theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic function theory, and only the later work of mathematicians such as Jacobi and Hermite has exposed the crux of his argument.Template:Sfn
In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general hypergeometric function <math>F(\alpha,\beta,\gamma,x)</math>, and shows that many of the functions known at the time are special cases of the hypergeometric function.Template:Sfn This work is the first exact inquiry into convergence of infinite series in the history of mathematics.Template:Sfn Furthermore, it deals with infinite continued fractions arising as ratios of hypergeometric functions, which are now called Gauss continued fractions.Template:Sfn
In 1823, Gauss won the prize of the Danish Society with an essay on conformal mappings, which contains several developments that pertain to the field of complex analysis.Template:Sfn Gauss stated that angle-preserving mappings in the complex plane must be complex analytic functions, and used the later-named 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.Template:Sfn
Numerical analysisEdit
Gauss often deduced theorems inductively from numerical data he had collected empirically.Template:Sfn As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to numerical analysis, such as the method of Gaussian quadrature, published in 1816.<ref>Template:Cite book</ref>
In a private letter to Gerling from 1823,<ref>Template:Cite book</ref> he described a solution of a 4x4 system of linear equations with the Gauss-Seidel method – an "indirect" iterative method for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.<ref>Template:Cite arXiv</ref>
Gauss invented an algorithm for calculating what is now called discrete Fourier transforms when calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey found their similar Cooley–Tukey algorithm.<ref>Template:Cite journal</ref> He developed it as a trigonometric interpolation method, but the paper Theoria Interpolationis Methodo Nova Tractata was published only posthumously in 1876,<ref>Template:Cite book</ref> well after Joseph Fourier's introduction of the subject in 1807.<ref>Template:Cite journal</ref>
GeometryEdit
Differential geometryEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The geodetic survey of Hanover fuelled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led him in 1828 to the publication of a work that marks the birth of modern differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as cartesian graphs of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, regardless of the embedding of the surface in three-dimensional or two-dimensional space.Template:Sfn
The Theorema Egregium leads to the abstraction of surfaces as doubly-extended manifolds; it clarifies the distinction between the intrinsic properties of the manifold (the metric) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing projections for geographical maps.Template:Sfn A portion of this essay is dedicated to a profound study of geodesics. In particular, Gauss proves the local Gauss–Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.Template:Sfn
Gauss's memoir from 1828 lacks the conception of geodesic curvature. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its invariance under isometric transformations, a result that was later obtained by Ferdinand Minding and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by Pierre Ossian Bonnet in 1848 and known as the Gauss–Bonnet theorem.Template:Sfn
Non-Euclidean geometryEdit
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During Gauss' lifetime, the Parallel postulate of Euclidean geometry was heavily discussed.Template:Sfn Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it.Template:Sfn Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s that a non-Euclidean geometry without the parallel postulate could solve the problem.Template:SfnTemplate:Sfn In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry",<ref name=":0">Template:Cite journal</ref> but he strongly forbade Taurinus to make any use of it.Template:Sfn Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.<ref>Template:Cite book</ref><ref name=":0" /><ref>Template:Cite book</ref>
The first publications on non-Euclidean geometry in the history of mathematics were authored by Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832.Template:Sfn In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.Template:Sfn<ref>Template:Cite journal</ref> Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> claiming that these were congruent to his own thoughts of some decades.Template:Sfn<ref>Template:Cite book</ref> However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.Template:Sfn
Sartorius first mentioned Gauss's work on non-Euclidean geometry in 1856, but only the publication of Gauss's Nachlass in Volume VIII of the Collected Works (1900) showed Gauss's ideas on the matter, at a time when non-Euclidean geometry was still an object of some controversy.Template:Sfn
Early topologyEdit
Gauss was also an early pioneer of topology or Geometria Situs, as it was called in his lifetime. The first proof of the fundamental theorem of algebra in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.Template:Sfn
Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the celestial sphere in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are linked, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid 7 Iris, he published a further qualitative discussion of the Zodiacus.<ref name="Epple 1998 45–52">Template:Cite journal</ref>
In Gauss's letters of 1820–1830, he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of knots.<ref>Template:Cite book</ref> To do so he devised a symbolical scheme, the Gauss code, that in a sense captured the characteristic features of tract figures.<ref>Template:Cite book</ref>Template:Sfn
In a fragment from 1833, Gauss defined the linking number of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. On the same note, he lamented the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as braids and tangles.<ref name="Epple 1998 45–52"/>
Gauss's influence in later years to the emerging field of topology, which he held in high esteem, was through occasional remarks and oral communications to Mobius and Listing.Template:Sfn
Minor mathematical accomplishmentsEdit
Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example, in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography,<ref>Printed in Collected Works Volume 2, pp. 305–310</ref> 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.<ref>Template:Cite journal</ref> He described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane,Template:Sfn and gave a proof for the geometric theorem that the altitudes of a triangle always meet in a single orthocenter.<ref>Carl Friedrich Gauss: Zusätze.II. In: Template:Cite book (Text by Schumacher, algorithm by Gauss), republished in Collected Works Volume 4, p. 396-398</ref>
Gauss was concerned with John Napier's "Pentagramma mirificum" – a certain spherical pentagram – for several decades;<ref>Template:Cite journal</ref> he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects.<ref>Pentagramma mirificum, printed in Collected Works Volume III, pp. 481–490</ref> In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.<ref>Template:Cite journal</ref>
Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given quadrilateral,<ref>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</ref>Template:Sfn and discovered a surprising result about the computation of area of pentagons.<ref>Printed in Collected Works Volume 4, pp. 406–407</ref>Template:Sfn
SciencesEdit
AstronomyEdit
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On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called Titius–Bode law, and named it Ceres.<ref>Template:Cite journal</ref> He could track it only for a short time until it disappeared behind the glare of the Sun. The mathematical tools of the time were not sufficient to predict the location of its reappearance from the few data available. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree when Franz Xaver von Zach on 7 and 31 December at Gotha, and independently Heinrich Olbers on 1 and 2 January in Bremen, identified the object near the predicted position.<ref>Template:Cite journal</ref>Template:Efn
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.Template:Sfn
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.<ref>Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.</ref> It introduced the Gaussian gravitational constant.<ref name="Wittmann" />
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.Template:Sfn 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.Template:Sfn<ref>Template:Cite journal</ref>Template:Efn 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.Template:Sfn
One fruit of Gauss's research on Pallas perturbations was the Determinatio Attractionis... (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". It introduced an averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time the planet takes to follow the corresponding orbital arcs.Template:Sfn Gauss presents the method of evaluating the gravitational attraction of such an elliptic ring, which includes several steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral.Template:Sfn
Even after Gauss's contributions to theoretical astronomy came to an end, more practical activities in observational astronomy continued and occupied him during his entire career. As early as 1799, Gauss dealt with the determination of longitude by use of the lunar parallax, for which he developed more convenient formulas than those were in common use.Template:Sfn After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables of nutation and aberration, solar coordinates, and refraction.Template:Sfn He made many contributions to spherical geometry, and in this context solved some practical problems about navigation by stars.Template:Sfn He published a great number of observations, mainly on minor planets and comets; his last observation was the solar eclipse of 28 July 1851.Template:Sfn
ChronologyEdit
Gauss's first publication following his doctoral thesis dealt with the determination of the date of Easter (1800), an elementary mathematical topic. Gauss aimed to present a convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usual terms of golden number, epact, solar cycle, domenical letter, and any religious connotations.Template:Sfn This choice of topic likely had historical grounds. The replacement of the Julian calendar by the Gregorian calendar had caused confusion in the Holy Roman Empire since the 16th century and was not finished in Germany until 1700, when the difference of eleven days was deleted. Even after this, Easter fell on different dates in Protestant and Catholic territories, until this difference was abolished by agreement in 1776. In the Protestant states, such as the Duchy of Brunswick, the Easter of 1777, five weeks before Gauss's birth, was the first one calculated in the new manner.<ref name="Olesko" />
Error theoryEdit
Gauss likely used the method of least squares to minimize the impact of measurement error when calculating the orbit of Ceres.<ref name="Stigler" /> 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.Template:Sfn<ref name=":3">Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares".<ref name="Stigler" /> Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of normally distributed errors (Gauss–Markov theorem), in the two-part paper Theoria combinationis observationum erroribus minimis obnoxiae (1823).<ref>Template:Cite journal</ref>
In the first paper he proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of the Gauss-Winckler inequality).<ref>Template:Cite journal</ref> He derived lower and upper bounds for the variance of the sample variance. In the second paper, Gauss described recursive least squares methods. His work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert to the Gauss-Helmert model.<ref>Template:Cite journal</ref>
Gauss also contributed to problems in probability theory that are not directly concerned with the theory of errors. One example appears as a diary note where he tried to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in (0,1). He derived this distribution, now known as the Gauss-Kuzmin distribution, as a by-product of the discovery of the ergodicity of the Gauss map for continued fractions. Gauss's solution is the first-ever result in the metrical theory of continued fractions.<ref>Template:Cite journal</ref>
GeodesyEdit
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Gauss was busy with geodetic problems since 1799 when he helped Karl Ludwig von Lecoq with calculations during his survey in Westphalia.Template:Sfn Beginning in 1804, he taught himself some practical geodesy in BrunswickTemplate:Sfn and Göttingen.Template:Sfn
Since 1816, Gauss's former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg as head of an observatory, carried out a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south.Template:Efn This project was the basis for map production but also aimed at determining the geodetic arc between the terminal sites. Data from geodetic arcs were used to determine the dimensions of the earth geoid, and long arc distances brought more precise results. Schumacher asked Gauss to continue this work further to the south in the Kingdom of Hanover; Gauss agreed after a short time of hesitation. Finally, in May 1820, King George IV gave the order to Gauss.Template:Sfn
An arc measurement needs a precise astronomical determination of at least two points in the network. Gauss and Schumacher used the coincidence that both observatories in Göttingen and Altona, in the garden of Schumacher's house, laid nearly in the same longitude. The latitude was measured with both their instruments and a zenith sector of Ramsden that was transported to both observatories.Template:SfnTemplate:Efn
Gauss and Schumacher had already determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818.Template:Sfn During the summers of 1821 until 1825 Gauss directed the triangulation work personally, from Thuringia in the south to the river Elbe in the north. The triangle between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum size of Template:Convert. In the thinly populated Lüneburg Heath without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.<ref name="Olesko" />Template:Sfn
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.Template:Sfn Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope.Template:Sfn Gauss was assisted 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.Template:Sfn
An additional result was a better value for the flattening of the approximative Earth ellipsoid.Template:SfnTemplate:Efn Gauss developed the universal transverse Mercator projection of the ellipsoidal shaped Earth (what he named conform projection)Template:Sfn for representing geodetical data in plane charts.
When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover with a Royal decree from 25 March 1828.Template:Sfn The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Gauss, who applied his mathematical inventions such as the method of least squares and the elimination method to it. The project was finished in 1844, and Gauss sent a final report of the project to the government; his method of projection was not edited until 1866.Template:Sfn<ref>Template:Cite book</ref>
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;<ref name="Gauß1828">Template:Cite book</ref> later his doctoral student Johann Benedict Listing called this the geoid.<ref>Template:Cite book</ref>
Magnetism and telegraphyEdit
GeomagnetismEdit
Gauss had been interested in magnetism since 1803.Template:Sfn After Alexander von Humboldt visited Göttingen in 1826, both scientists began intensive research on geomagnetism, partly independently, partly in productive cooperation.<ref>Template:Cite journal</ref> In 1828, Gauss was Humboldt's guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquainted with the physicist Wilhelm Weber.Template:Sfn
When Weber got the chair for physics in Göttingen as successor of Johann Tobias Mayer by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of magnetism with a representation for the unit of magnetism in terms of mass, charge, and time.Template:Sfn They founded the Magnetic Association (German: Magnetischer Verein), an international working group of several observatories, which carried out measurements of Earth's magnetic field in many regions of the world using equivalent methods at arranged dates in the years 1836 to 1841.<ref name="Reich">Template:Cite journal</ref>
In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the British dominions with a letter to the Duke of Sussex, then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> Together with other instigators, this led to a global program known as "Magnetical crusade" under the direction of Edward Sabine. The dates, times, and intervals of observations were determined in advance, the Göttingen mean time was used as the standard.<ref name="Rupke">Template:Cite book</ref> 61 stations on all five continents participated in this global program. Gauss and Weber founded a series for publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to Leipzig in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Association activity.<ref name="Reich" />
Following Humboldt's example, Gauss ordered a magnetic observatory to be built in the garden of the observatory, but the scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity and differentiated, unlike Humboldt, between "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of the 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.<ref name="Reich" />Template:Sfn The precision of the magnetometer was about ten times higher than that of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities.<ref name="Rupke" />
Gauss carried out a General Theory of Terrestrial Magnetism (1839), in what he believed to describe the nature of magnetic force; according to Felix Klein, this work is a presentation of observations by use of spherical harmonics rather than a physical theory.Template:Sfn The theory predicted the existence of exactly two magnetic poles on the Earth, thus Hansteen's idea of four magnetic poles became obsolete,<ref name="Roussanova">Template:Cite journal</ref> and the data allowed to determine their location with rather good precision.Template:Sfn
Gauss influenced the beginning of geophysics in Russia, when Adolph Theodor Kupffer, one of his former students, founded a magnetic observatory in St. Petersburg, following the example of the observatory in Göttingen, and similarly, Ivan Simonov in Kazan.<ref name="Roussanova"/>
ElectromagnetismEdit
The discoveries of Hans Christian Ørsted on electromagnetism and Michael Faraday on electromagnetic induction drew Gauss's attention to these matters.Template:Sfn Gauss and Weber found rules for branched electric circuits, which were later found independently and first published by Gustav Kirchhoff and named after him as Kirchhoff's circuit laws,Template:Sfn and made inquiries into 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,Template:Efn but they made no further commercial use of this invention.<ref name="Timm">Template:Cite book</ref><ref>Template:Cite book</ref>
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 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.<ref>Printed in the Collected Works, Volume 5, pp. 609–610.</ref><ref>Template:Cite book</ref>
Gauss tried to find a unifying law for long-distance effects of electrostatics, electrodynamics, electromagnetism, and induction, comparable to Newton's law of gravitation,Template:Sfn but his attempt ended in a "tragic failure".<ref name="Rupke" />
Potential theoryEdit
Since Isaac Newton had shown theoretically that the Earth and rotating stars assume non-spherical shapes, the problem of attraction of ellipsoids gained importance in mathematical astronomy. In his first publication on potential theory, the "Theoria attractionis..." (1813), Gauss provided a closed-form expression to the gravitational attraction of a homogeneous triaxial ellipsoid at every point in space.Template:Sfn In contrast to previous research of Maclaurin, Laplace and Lagrange, Gauss's new solution treated the attraction more directly in the form of an elliptic integral. In the process, he also proved and applied some special cases of the so-called Gauss's theorem in vector analysis.Template:Sfn
In the General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances (1840) Gauss gave a basic theory of magnetic potential, based on Lagrange, Laplace, and Poisson;Template:Sfn it seems rather unlikely that he knew the previous works of George Green on this subject.Template:Sfn 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.<ref name="Rupke" />
OpticsEdit
Gauss's calculations enabled instrument maker Johann Georg Repsold in Hamburg to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was that the refractive index and dispersion of the glass used were not precisely known.Template:Sfn In a short article from 1817 Gauss dealt with the problem of removal of chromatic aberration in double lenses, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician Carl August von Steinheil, who in 1860 introduced the achromatic Steinheil doublet, partly based on Gauss's calculations.Template:Sfn Many results in geometrical optics are scattered in Gauss's correspondences and hand notes.Template:Sfn
In the Dioptrical Investigations (1840), Gauss gave the first systematic analysis of the formation of images under a paraxial approximation (Gaussian optics).<ref name=Hecht>Template:Cite book</ref> He characterized optical systems under a paraxial approximation only by its cardinal points,<ref name=Bass>Template:Cite book</ref> and he derived the Gaussian lens formula, applicable without restrictions in respect to the thickness of the lenses.<ref name=Ostdiek>Template:Cite book</ref>Template:Sfn
MechanicsEdit
Gauss's first work 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 the Coriolis force, he asked Gauss for a theory-based calculation of the values for comparison with the experimental ones. Gauss elaborated a system of fundamental equations for the motion, and the results corresponded sufficiently with Benzenberg's data, who added Gauss's considerations as an appendix to his book on falling experiments.Template:Sfn
After Foucault had demonstrated the earth's rotation by his pendulum experiment in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum. It is described in the Gauss–Gerling correspondence and Weber made some experiments with this apparatus in 1853, but no data were published.Template:Sfn<ref>Template:Cite journal</ref>
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.Template:Sfn
MetrologyEdit
In 1828, Gauss was appointed as head of the board for weights and measures of the Kingdom of Hanover. He created standards for length and measure. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical construction.<ref name="Olesko">Template:Cite book</ref> In the correspondence with Schumacher, who was also working on this matter, he described new ideas for high-precision scales.Template:Sfn He submitted the final reports on the Hanoverian foot and pound to the government in 1841. This work achieved international importance due to an 1836 law that connected the Hanoverian measures with the English ones.<ref name="Olesko" />
Honours and awardsEdit
Gauss first became member of a scientific society, the Russian Academy of Sciences, in 1802.Template:Sfn Further memberships (corresponding, foreign or full) were awarded by the Academy of Sciences in Göttingen (1802/ 1807),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the French Academy of Sciences (1804/ 1820),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Society of London (1804),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Prussian Academy in Berlin (1810),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the National Academy of Science in Verona (1810),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Society of Edinburgh (1820),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Bavarian Academy of Sciences of Munich (1820),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Danish Academy in Copenhagen (1821),Template:Sfn the Royal Astronomical Society in London (1821),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Swedish Academy of Sciences (1821),Template:Sfn the American Academy of Arts and Sciences in Boston (1822),<ref name=AAAS>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Royal Bohemian Society of Sciences in Prague (1833),Template:Sfn the Royal Academy of Science, Letters and Fine Arts of Belgium (1841/1845),<ref>Académie Royale de Belgique: Academy members</ref> the Royal Society of Sciences in Uppsala (1843),Template:Sfn the Royal Irish Academy in Dublin (1843),Template:Sfn the Royal Institute of the Netherlands (1845/ 1851),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Spanish Royal Academy of Sciences in Madrid (1850),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Russian Geographical Society (1851),Template:Sfn the Imperial Academy of Sciences in Vienna (1848),Template:Sfn the American Philosophical Society (1853),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Cambridge Philosophical Society,Template:Sfn and the Royal Hollandish Society of Sciences in Haarlem.Template:SfnTemplate:Sfn
Both the University of Kazan and the Philosophy Faculty of the University of Prague appointed him honorary member in 1848.Template:Sfn
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,<ref>Template:Cite journal</ref> the Danish Academy of Science prize in 1823 for his memoir on conformal projection,Template:Sfn and the Copley Medal from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".Template:Sfn<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="Wittmann" />
Gauss was appointed Knight of the French Legion of Honour in 1837,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and became one of the first members of the Prussian Order Pour le Merite (Civil class) when it was established in 1842.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> He received the Order of the Crown of Westphalia (1810),Template:Sfn the Danish Order of the Dannebrog (1817),Template:Sfn the Hanoverian Royal Guelphic Order (1815),Template:Sfn the Swedish Order of the Polar Star (1844),Template:Sfn the Order of Henry the Lion (1849),Template:Sfn and the Bavarian Maximilian Order for Science and Art (1853).Template:Sfn
The Kings of Hanover appointed him the honorary titles "Hofrath" (1816)Template:Sfn and "Geheimer Hofrath"Template:Efn (1845). In 1949, on the occasion of his golden doctor degree jubilee, he received honorary citizenship of both Brunswick and Göttingen.Template:Sfn Soon after his death a medal was issued by order of King George V of Hanover with the back inscription dedicated "to the Prince of Mathematicians".Template:Sfn
The "Gauss-Gesellschaft Göttingen" ("Göttingen Gauss Society") was founded in 1964 for research on the life and work of Carl Friedrich Gauss and related persons. It publishes the Mitteilungen der Gauss-Gesellschaft (Communications of the Gauss Society).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Names and commemorationsEdit
Selected writingsEdit
Mathematics and astronomyEdit
- 1799: Template:Cite book (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) Original book
- 1816: Template:Cite journal Original
- 1816: Template:Cite journal Original
- 1850: Template:Cite journal Original (Lecture from 1849)
- Template:Cite book (German)
- 1800: Template:Cite journal Original
- 1801: Template:Cite book
- Template:Cite book (translated from the second German edition, Göttingen 1860)
- 1802: Template:Cite journal Original
- 1804: Template:Cite journal Original (on the Zodiacus)
- 1808: Template:Cite journal Original (Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity)
- 1808: Template:Cite book
- 1809: Template:Cite book Original book
- 1811: Template:Cite journal Original (from 1810) (Orbit of Pallas)
- 1811: Template:Cite journal Original (from 1808) (Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity)
- 1813: Template:Cite journal Original (from 1812, contains the Gauss's continued fraction)
- 1816: Template:Cite journal Original (from 1814)
- 1818: Template:Cite journal Original (from 1817) (Fifth and sixth proofs of quadratic reciprocity)
- 1818: Template:Cite journal Original (Only reference to the – mostly unpublished – work on the algorithm of the arithmetic-geometric mean.)
- 1823: Template:Cite journal Original (from 1821)
- 1823: Template:Cite journal Original
- 1825: Template:Cite journal (Prize winning essay from 1822 on conformal mapping)
- 1828: Template:Cite book Original book
- 1828: Template:Cite journal (from 1826)
- Template:Cite book (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
- 1828: Template:Cite journal Original (from 1827)
- 1828: Template:Cite journal Original (from 1825)
- 1832: Template:Cite journal Original (from 1831) (Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i)
- 1845: Template:Cite journal Original (from 1843)
- 1847: Template:Cite journal Original (from 1846)
- 1848: Template:Cite journal Original
- 1903: Wissenschaftliches Tagebuch (Template:Cite journal) Original book (from 1847, on the Zodiacus)
PhysicsEdit
- 1804: Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde ( in original book: Template:Cite book Original)
- 1813: Template:Cite journal Original (contains Gauss's theorem of vector analysis)
- 1817: Template:Cite journal
- 1829: Template:Cite journal
- 1830: Template:Cite journal Original (from 1829)
- 1841: Template:Cite journal Original (from 1832)Template:Efn
- The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement. Translated by Susan P. Johnson.
- 1836: Erdmagnetismus und Magnetometer (Original book: Template:Cite book)
- 1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte (Original book: Template:Cite book
- 1843: Template:Cite journal Original (from 1840)
Together with Wilhelm WeberEdit
- 1837–1839: Template:Cite book
- 1840–1843: Template:Cite book
- 1840: Template:Cite book
Collected worksEdit
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CorrespondenceEdit
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- Template:Cite book (letters to Boguslawski from February 1835 to January 1848)
- Template:Cite book (letters from February 1802 to October 1826)
- Template:Cite book (letters from September 1797 to February 1853; added letters of other correspondents)
- Template:Cite book (letters from June 1810 to June 1854)
- Template:Cite book (letters from June 1810 to June 1854)
- Template:Cite book (letters from July 1807 to December 1854; added letters of other correspondents)
- Template:Cite book (letters from 1835 to 1843)
- Template:Cite book (letters from February 1799 to September 1800)
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- Template:Cite book (letters from January 1802 to October 1819)
- Template:Cite book (letters from January 1820 to May 1839; added letters of other correspondents)
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- Volumes 1+2 (letters from April 1808 to March 1836)
- Volumes 3+4 (letters from March 1836 to April 1845)
- Volumes 5+6 (letters from April 1845 to November 1850)
- Template:Cite book (letters from 1795 to 1815)
The Göttingen Academy of Sciences and Humanities provides a complete collection of the known letters from and to Carl Friedrich Gauss that is accessible online.<ref name="Correspondence">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The literary estate is kept and provided by the Göttingen State and University Library.<ref>Template:Cite journal</ref> Written materials from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
ReferencesEdit
NotesEdit
CitationsEdit
SourcesEdit
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Further readingEdit
FictionalEdit
External linksEdit
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- Publications of C. F. Gauss in Astrophysics Data System
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- Carl Friedrich Gauss – Spuren seines Lebens (Places used as points for triangulation)
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