Editing Carl Friedrich Gauss

{{short description|German mathematician, astronomer, geodesist, and physicist (1777–1855)}}
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{{Infobox scientist
| image             = Carl Friedrich Gauss 1840 by Jensen.jpg
| alt               = Portrait of arl Friedrich Gauss 1840 by Jensen
| caption           = Portrait by [[Christian Albrecht Jensen]], 1840 (copy from Gottlieb Biermann, 1887)<ref>{{cite journal | author-last = Axel D. Wittmann | author-first = Inna V. Oreshina | title = On Jensen's Paintings of C. F. Gauss | journal = Mitteilungen der Gauss-Gesellschaft | issue = 46 | pages = 57–61 | year = 2009 | url = http://www.gauss-gesellschaft-goettingen.de/mitteil.html#2009}}</ref>
| birth_name        = Johann Carl Friedrich Gauss
| birth_date        = {{birth date|df=yes|1777|4|30}}
| birth_place       = [[Braunschweig|Brunswick]], Principality of Brunswick-Wolfenbüttel, {{awrap|Holy Roman Empire}}
| death_date        = {{death date and age|df=yes|1855|2|23|1777|4|30}}
| death_place       = [[Göttingen]], Kingdom of Hanover, {{awrap|German Confederation}}
| fields            = Mathematics, Astronomy, Geodesy, Magnetism
| alma_mater        = {{plainlist|
* [[Braunschweig University of Technology|Collegium Carolinum]]
* [[University of Göttingen]]
* [[University of Helmstedt]] (PhD)
}}
| workplaces        = University of Göttingen
| thesis_title      = Demonstratio nova...
| thesis_url        = http://www.e-rara.ch/zut/content/titleinfo/1336299
| thesis_year       = 1799
| doctoral_advisor  = [[Johann Friedrich Pfaff]]
| known_for         = [[List of things named after Carl Friedrich Gauss|Full list]]
| spouse            = {{plainlist|
* {{marriage|Johanna Osthoff|1805|1809|reason=died}}
* {{marriage|Minna Waldeck|1810|1831|reason=died}}
}}
| children          = 6
| awards            = {{plainlist|
* [[Lalande Prize]] (1809)
* [[Copley Medal]] (1838)
}}
| doctoral_students = {{collapsible list|title={{nothing}}
| [[Richard Dedekind]]
| [[Christian Ludwig Gerling]]
| [[Wilhelm Klinkerfues]]
| [[Johann Benedict Listing]]
| [[Bernhard Riemann]]
| [[August Ritter (civil engineer)|August Ritter]]
| [[Karl von Staudt]]
}}
| notable_students  = {{collapsible list|title={{nothing}}
| [[Gotthold Eisenstein]]
| [[Johann Franz Encke]]
| [[Carl Wolfgang Benjamin Goldschmidt]]
| [[Adolph Theodor Kupffer]]
| [[August Ferdinand Möbius]]
| [[Moritz Stern]]
| [[Georg Frederik Ursin]]
| [[Moritz Ludwig George Wichmann|Moritz Ludwig Wichmann]]
}}
| signature         = Carl Friedrich Gauß signature.svg
}}
'''Johann Carl Friedrich Gauss''' ({{IPAc-en|g|aʊ|s|audio=LL-Q1860 (eng)-Flame, not lame-Carl Friedrich Gauss.wav}};<ref>[http://www.dictionary.com/browse/gauss "Gauss"]. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> {{langx|de|link=no|Gauß}} {{IPA|de|kaʁl ˈfʁiːdʁɪç ˈɡaʊs||De-carlfriedrichgauss.ogg}};<ref>{{cite book | year = 2015 | orig-date = 1962 | title = Duden – Das Aussprachewörterbuch | trans-title = The Pronunciation Dictionary | url = https://books.google.com/books?id=T6vWCgAAQBAJ | language = de | edition = 7th | place = Berlin | publisher = Dudenverlag | isbn = 978-3-411-04067-4 | pages = 246, 381, 391}}</ref><ref>{{cite book | last1 = Krech | first1 = Eva-Maria | last2 = Stock | first2 = Eberhard | last3 = Hirschfeld | first3 = Ursula | last4 = Anders | first4 = Lutz-Christian | title = Deutsches Aussprachewörterbuch | trans-title = German Pronunciation Dictionary | url = https://books.google.com/books?id=E-1tr_oVkW4C&q=deutsches+ausspracheworterbuch | language = de | year = 2009 | publisher =W alter de Gruyter | place = Berlin | isbn = 978-3-11-018202-6 | pages = 402, 520, 529}}</ref> {{langx|la|Carolus Fridericus Gauss}}; 30 April 1777{{spaced ndash}}23 February 1855) was a German [[mathematician]], [[astronomer]], [[Geodesy|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 [[theorem]]s. As an independent scholar, he wrote the [[masterpiece]]s ''[[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 [[Gauss composition law|composition law]], the [[Quadratic reciprocity|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 function|hypergeometric series]]. Due to Gauss' extensive and fundamental contributions to science and mathematics, more than [[List of things named after Carl Friedrich Gauss|100 mathematical and scientific concepts]] are named after him.

Gauss was instrumental in the identification of [[Ceres (dwarf planet)|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 [[Least squares|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 (instrument)|heliotrope]] in 1821, a [[magnetometer]] in 1833 and – with [[Wilhelm Eduard Weber]] – the first electromagnetic [[Telegraphy|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 [[Posthumous publication|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 [[Richard Dedekind|Dedekind]] and [[Bernhard Riemann|Riemann]], became well-known and influential mathematicians in their own right.

== Biography ==
=== Youth and education ===
[[File:Braunschweig Brunswick Geburtshaus CF Gauss (1914).jpg|thumb|upright|Birthplace in Brunswick (destroyed in World War II)]]
[[File:Goe.Kurze.Geismarstr.Gauss.Wohnhaus.JPG|thumb|upright|Gauss's home as student in Göttingen]]
Gauss was born on 30 April 1777 in [[Braunschweig|Brunswick]] in the [[Duchy of Brunswick-Wolfenbüttel]] (now in the German state of [[Lower Saxony]]). His family was of relatively low social status.<ref>{{cite book | last = Borch | first = Rudolf | title = Ahnentafel des Mathematikers Carl Friedrich Gauß | trans-title = Ancestors' Tabel of the mathematician Carl Friedrich Gauss | series = Ahnentafeln Berühmter Deutscher | volume = 1 | publisher = Zentralstelle für Deutsche Personen- und Familiengeschichte | date = 1929 | pages = 63–65 | language = de}}</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.{{sfn|Dunnington|2004|p=8}} He had one elder brother from his father's first marriage.{{sfn|Dunnington|2004|pp=8–9}}

Gauss was a [[child prodigy]] in mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the [[Charles William Ferdinand, Duke of Brunswick|Duke of Brunswick]] who sent him to the local ''Collegium Carolinum'',{{efn|The ''Collegium Carolinum'' was a preceding institution of the [[Technical University of Braunschweig]], but at Gauss's time not equal to a university.{{sfn|Dunnington|2004|p=17}}
}} which he attended from 1792 to 1795 with [[Eberhard August Wilhelm von Zimmermann]] as one of his teachers.{{sfn|Schlesinger|1933|p=10}}{{sfn|Dunnington|2004|p=14}}<ref name="Ullrich">{{cite book | last1 = Ullrich | first1 = Peter | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 17–29 | chapter = Herkunft, Schul- und Studienzeit von Carl Friedrich Gauß | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> Thereafter the Duke granted him the resources for studies of mathematics, sciences, and [[Classics|classical languages]] at the [[University of Göttingen]] until 1798.<ref name="scientificmonthly">{{cite journal |last1=Dunnington |first1=Waldo |year=1927 |title=The Sesquicentennial of the Birth of Gauss |url=http://www.mathsong.com/cfgauss/Dunnington/1927/ |journal=[[The Scientific Monthly]] |volume=24 |issue=5 |pages=402–414 |bibcode=1927SciMo..24..402D |jstor=7912 |archive-url=https://web.archive.org/web/20080226020629/http://www.mathsong.com/cfgauss/Dunnington/1927/ |archive-date=26 February 2008}}  Also available at {{cite web |title=The Sesquicentennial of the Birth of Gauss |url=http://gausschildren.org/genwiki/index.php?title=The_Sesquicentennial_of_the_Birth_of_Gauss}} 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 [[epigram]]s.{{sfn|Dunnington|2004|p=24}}{{efn|Once Gauss drew a lecture scene with professor Kästner producing errors in a simple calculation.<ref name="Ullrich" />}} Astronomy was taught by [[Karl Felix Seyffer]], with whom Gauss stayed in correspondence after graduation;{{sfn|Dunnington|2004|p=26}} [[Heinrich Wilhelm Matthias Olbers|Olbers]] and Gauss mocked him in their correspondence.<ref>{{cite book | last = Wattenberg | first = Diedrich | author-link = Diedrich Wattenberg | title = Wilhelm Olbers im Briefwechsel mit Astronomen seiner Zeit | publisher = GNT – Verlag für Geschichte der Naturwissenschaften und der Technik | date = 1994 | place = Stuttgart | page = 41 | isbn = 3-928186-19-1 | language = de}}</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.{{sfn|Dunnington|2004|p=26}} Fellow students of this time were [[Johann Friedrich Benzenberg]], [[Farkas Bolyai]], and [[Heinrich Wilhelm Brandes]].{{sfn|Dunnington|2004|p=26}}

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 Greece|Ancient Greeks]] when he determined in 1796 which regular [[polygon]]s can be constructed by [[Compass and straightedge constructions|compass and straightedge]]. This discovery ultimately led Gauss to choose mathematics instead of [[philology]] as a career.{{sfn|Dunnington|2004|p=28}} 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.{{sfn|Dunnington|2004|p=37}}

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>{{Cite web |date=2017-02-06 |title=Gauss's Day of Reckoning |url=https://www.americanscientist.org/article/gausss-day-of-reckoning |access-date=2025-03-27 |website=American Scientist |language=en}}</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>{{Cite book |last=Posamentier |first=Alfred S. |title=Math Makers: The Lives and Works of 50 Famous Mathematicians|pages=242–243|publisher=Prometheus Books|year=2019}}</ref> Other accounts state that he computed the sum as 100 sets of 101 and divided by 2.<ref>{{Cite web |title=The Story of Gauss - National Council of Teachers of Mathematics |url=https://www.nctm.org/Publications/TCM-blog/Blog/The-Story-of-Gauss/ |access-date=2025-03-27 |website=www.nctm.org}}</ref>

=== Private scholar ===
Gauss graduated as a [[Doctor of Philosophy]] in 1799, not in Göttingen, as is sometimes stated,{{efn|This error occurs for example in Marsden (1977).<ref name="Marsden">{{Cite journal| last = Marsden | first = Brian G. | author-link = Brian G. Marsden | date=1 August 1977 | title = Carl Friedrich Gauss, Astronomer | url = http://adsabs.harvard.edu/abs/1977JRASC..71..309M | journal = [[Journal of the Royal Astronomical Society of Canada]] | volume = 71 | pages = 309–323 | bibcode=1977JRASC..71..309M | issn = 0035-872X}}</ref>}}<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 ''[[Graduation|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 [[Ludwig Maximilian University of Munich|Landshut University]].<ref name="Reich2000" /><ref name="Beuermann">{{cite book | last = Beuermann | first = Klaus | editor-last = Beuermann | editor-first = Klaus | title = Grundsätze über die Anlage neuer Sternwarten unter Beziehung auf die Sternwarte der Universität Göttingen von Georg Heinrich Borheck | publisher = Universitätsverlag Göttingen | year = 2005 | place = Göttingen | pages = 37–45 | chapter = Carl Friedrich Gauß und die Göttinger Sternwarte | isbn = 3-938616-02-4 | chapter-url = https://library.oapen.org/bitstream/handle/20.500.12657/32489/610361.pdf?sequence=1&isAllowed=y}}</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 [[War of the Fourth Coalition|Napoleon's wars]] cancelled those plans:<ref>{{Cite journal | last = Michling | first = Horst | title = Zum Projekt einer Gauß-Sternwarte in Braunschweig | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 3 | page = 24 | year = 1966 | language = de}}</ref> the Duke was killed in the [[Battle of Jena–Auerstedt|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, Lower Saxony|Lilienthal]], especially [[Wilhelm Olbers]], [[Karl Ludwig Harding]], and [[Friedrich Wilhelm Bessel]], forming part of the informal group of astronomers known as the [[Celestial police]].{{sfn|Dunnington|2004|pp=50, 54–55, 74–77}} 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).{{sfn|Dunnington|2004|pp=91–92}}

=== Professor in Göttingen ===
[[File:Goettingen Sternwarte 01.jpeg|thumb|Old Göttingen observatory, {{Circa|1800}}]]
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 [[Göttingen Observatory|astronomical observatory]],{{sfn|Dunnington|2004|pp=85–87}} and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand [[franc]]s from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and [[Pierre-Simon Laplace|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]] [[Karl Theodor Anton Maria von Dalberg|Dalberg]],{{sfn|Dunnington|2004|pp=86–87}} paid the sum.{{sfn|Dunnington|2004|pp=85–87}}

Gauss took on the directorship of the 60-year-old observatory, founded in 1748 by [[Prince-elector]] [[George II of Great Britain|George II]] and built on a converted fortification tower,{{sfn|Brendel|1929|pp=81–82}} with usable, but partly out-of-date instruments.{{sfn|Brendel|1929|p=49}} 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,{{sfn|Brendel|1929|p=83}} 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 circle]]s from [[Johann Georg Repsold|Repsold]]{{sfn|Brendel|1929|p=84}} and [[Georg Friedrich von Reichenbach|Reichenbach]],{{sfn|Brendel|1929|p=119}} and a [[heliometer]] from [[Joseph von Fraunhofer|Fraunhofer]].{{sfn|Brendel|1929|p=56}}

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.{{sfn|Klein|1979|p=7}}

Gauss made no secret of his aversion to giving academic lectures.<ref name="Reich2000">{{Cite journal | last = Reich | first = Karin | title = Gauß' Schüler | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 37 | pages = 33–62 | year = 2000 | language = de}}</ref><ref name="Beuermann" /> But from the start of his academic career at Göttingen, he continuously gave lectures until 1854.{{sfn|Dunnington|2004|pp=405–410}} 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">{{cite book | last1 = Wittmann | first1 = Axel | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedrsächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 131–149 | chapter = Carl Friedrich Gauß und sein Wirken als Astronom | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> and only three lectures on subjects of pure mathematics.<ref name="Reich2000" />{{efn|Gauss announced 195 lectures, 70 per cent of them on astronomical, 15 per cent on mathematical, 9 per cent on geodetical, and 6 per cent on physical subjects.<ref name="Wittmann" />}} Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: [[Moritz Cantor]], [[Richard Dedekind|Dedekind]], [[Enno Dirksen|Dirksen]], [[Johann Franz Encke|Encke]], [[Benjamin Apthorp Gould|Gould]],{{efn|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.<ref name="Correspondence" />}} [[Eduard Heine|Heine]], [[Ernst Klinkerfues|Klinkerfues]], [[Adolph Theodor Kupffer|Kupffer]], [[Johann Benedict Listing|Listing]], [[August Ferdinand Möbius|Möbius]], [[Friedrich Bernhard Gottfried Nicolai|Nicolai]], [[Bernhard Riemann|Riemann]], [[August Ritter (civil engineer)|Ritter]], [[Ernst Christian Julius Schering|Schering]], [[Heinrich Scherk|Scherk]], [[Heinrich Christian Schumacher|Schumacher]], [[Karl Georg Christian von Staudt|von Staudt]], [[Moritz Abraham Stern|Stern]], [[Georg Frederik Ursin|Ursin]]; as geoscientists [[Wolfgang Sartorius von Waltershausen|Sartorius von Waltershausen]], and [[Johann Eduard Wappäus|Wappäus]].<ref name="Reich2000" />

Gauss did not write any textbook and disliked the [[Popular science|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 language|German]].{{efn|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.{{sfn|Brendel|1929|page=211}}}}{{efn|The ''Theoria motus...'' was completed in the German language in 1806, but on request of the editor [[Friedrich Christoph Perthes]] Gauss translated it into Latin.{{sfn|Dunnington|2004|p=90}}}} He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians.{{sfn|Dunnington|2004|pp=37–38}}

[[File:Universitäts-Sternwarte Göttingen 02.jpg|thumb|The new Göttingen observatory of 1816; Gauss's living rooms were in the western wing (right)]]
[[File:Die Göttinger Sieben von Eduard Ritmüller.jpg|thumb|upright|[[Wilhelm Eduard Weber|Wilhelm Weber]] and [[Heinrich Ewald]] (first row) as members of the [[Göttingen Seven]]]]
[[File:Carl Friedrich Gauss on his Deathbed, 1855.jpg|thumb|Gauss on his deathbed (1855) (daguerreotype from Philipp Petri){{sfn|Dunnington|2004|p=324}}]]
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>{{cite book | author-last = Cantor | author-first = Moritz | author-link = Moritz Cantor | title = Thibaut, Bernhard Friedrich | publisher = Duncker & Humblot | date = 1894 | series = [[Allgemeine Deutsche Biographie]] | volume = 37 | pages = 745–746 | location = Leipzig | language = de | url = https://de.wikisource.org/wiki/ADB:Thibaut,_Bernhard_Friedrich}}</ref> the physicist [[Johann Tobias Mayer|Mayer]], known for his textbooks,<ref>{{cite book | author-last = Folkerts | author-first = Menso | author-link = Menso Folkerts | title = Mayer, Johann Tobias | publisher = Duncker & Humblot | date = 1990 | series = [[Neue Deutsche Biographie]] | volume = 16 | page = 530 | location = | language = de | url = https://www.deutsche-biographie.de/gnd100373267.html#ndbcontent}}</ref> his successor [[Wilhelm Eduard Weber|Weber]] since 1831, and in the observatory [[Karl Ludwig Harding|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" />{{efn|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's mother-in-law, pleaded with Olbers to try to provide Gauss with another position far from Göttingen.<ref name="Küssner" />}} Gauss used the new [[meridian circle]]s nearly exclusively, and kept them away from Harding, except for some very seldom joint observations.{{sfn|Brendel|1929|pp=106–108}}

[[Martin Brendel|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".{{sfn|Brendel|1929|pp=7, 128}} 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">{{Cite journal | last = Küssner | first = Martha | title = Friedrich Wilhelm Bessels Beziehungen zu Göttingen und Erinnerungen an ihn | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 15 | pages = 3–19 | year = 1978 | language = de}}</ref>{{sfn|Brendel|1929|pp=106–108}}{{efn|Gauss's first assistant was [[Carl Wolfgang Benjamin Goldschmidt|Benjamin Goldschmidt]], and his second [[Wilhelm Klinkerfues]], who later became one of his successors.<ref name="Wittmann" />}}

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 [[Honorary degree|doctorate ''honoris causa'']] for Bessel from the Philosophy Faculty of Göttingen in March 1811.{{efn|name=Bessel|Bessel never got a university education.<ref>{{cite book | last = Hamel | first = Jürgen | author-link = Jürgen Hamel | title = Friedrich Wilhelm Bessel | publisher = BSB B.G.Teubner Verlagsgesellschaft | place = Leipzig | page = 29 | year = 1984}}</ref>{{sfn|Dunnington|2004|p=76}}}} Gauss gave another recommendation for an honorary degree for [[Sophie Germain]] but only shortly before her death, so she never received it.<ref>{{cite journal | last1 = Mackinnon | first1 = Nick | year = 1990 | title = Sophie Germain, or, Was Gauss a feminist? | journal = [[The Mathematical Gazette]] | volume = 74 | issue = 470 | pages = 346–351 | publisher = The Mathematical Association | doi = 10.2307/3618130 | jstor = 3618130 | s2cid=126102577}}</ref> He also gave successful support to the mathematician [[Gotthold Eisenstein]] in Berlin.<ref>{{Cite journal | last = Biermann | first = Kurt-R. | title = Gotthold Eisenstein | journal = [[Journal für die reine und angewandte Mathematik]] | volume = 214 | pages = 19–30 | year = 1964 | doi = 10.1515/crll.1964.214-215.19 | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN243919689_0214_0215?tify=%7B%22pages%22%3A%5B25%5D%2C%22pan%22%3A%7B%22x%22%3A0.434%2C%22y%22%3A0.442%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.661%7D}}</ref>

Gauss was loyal to the [[House of Hanover]]. After King [[William IV]] died in 1837, the new Hanoverian King [[Ernest Augustus, King of Hanover|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.{{sfn|Dunnington|2004|pp=195&ndash;200}}

Gauss took part in academic administration: three times he was elected as [[Dean (education)|dean]] of the Faculty of Philosophy.{{sfn|Dunnington|2004|p=288}} 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.{{sfn|Dunnington|2004|p=288}}

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;{{sfn|Dunnington|2004|p=24}} and was interred in the [[Albanifriedhof|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.{{sfn|Sartorius von Waltershausen|1856|p=104}}

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.{{sfn|Dunnington|2004|p=237}}

=== 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 {{convert|1492|g|lb|2}}.<ref>{{Cite book | last = Wagner | first = Rudolf | title = 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 | publisher = Dieterich | year = 1860 | place = Göttingen | url = https://books.google.com/books?id=YqTH2HDXgqkC&q=%C3%9Cber+die+typischen+Verschiedenheiten+der+Windungen+der+Hemisph%C3%A4ren+und+%C3%BCber+die+Lehre+vom+Hirngewicht,+mit+besondrer+R%C3%BCcksicht+auf+die+Hirnbildung+intelligenter+M%C3%A4nner.+Vorstudie}}</ref><ref>{{Cite book | last = Wagner | first = Rudolf | title = 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 | publisher = Dieterich | year = 1862 | place = Göttingen | url = https://books.google.com/books?id=czvfLAp94VsC&q=%C3%9Cber+den+Hirnbau+der+Mikrocephalen+mit+vergleichender+R%C3%BCcksicht+auf+den+Bau+des+Gehirns+der+normalen+Menschen+und+der+Quadrumanen.+Vorstudien+zu+einer+wissenschaftlichen+Morphologie+und+Physiologie+des+menschlichen+Gehirns+als+S}}</ref> Wagner's son [[Hermann Wagner (geographer)|Hermann]], a geographer, estimated the cerebral area to be {{convert|219588|mm2|abbr=out}} in his doctoral thesis.<ref>{{Cite book | last = Wagner | first = Hermann | author-link = Hermann Wagner (geographer) | title =  Maassbestimmungen der Oberfläche des grossen Gehirns | publisher = Georg H. Wigand | year = 1864 | place = Cassel & Göttingen | url = https://books.google.com/books?id=8ToAAAAAQAAJ&q=Maassbestimmungen+der+Oberfl%C3%A4che+des+grossen+Gehirns|language=de|trans-title=Measurements of the surface of the large brain}}</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>{{cite journal | last1 = Schweizer | first1 = Renate | last2 = Wittmann | first2 = Axel | last3 = Frahm | first3 = Jens | author-link3 = Jens Frahm | title = A rare anatomical variation newly identifies the brains of C.F. Gauss and C.H. Fuchs in a collection at the University of Göttingen | journal = Brain | volume = 137 | issue = 4 | page = e269 | year = 2014 | doi = 10.1093/brain/awt296 | pmid = 24163274 | doi-access = free | hdl = 11858/00-001M-0000-0014-C6F0-6 | hdl-access = free }} (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>{{cite web|website=Max Planck Society |url=https://www.mpg.de/7589532/Carl_Friedrich_Gauss_brain |title=Unravelling the true identity of the brain of Carl Friedrich Gauss}}</ref>

=== Family ===
[[File:Minna Gauß geb. Waldeck, 002.jpg|thumb|upright|Gauss's second wife Wilhelmine Waldeck]]
Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick.{{sfn|Dunnington|2004|p=66}} 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.{{sfn|Wußing|1982|p=44}} 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.{{sfn|Dunnington|2004|pp=77, 88, 93}}

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>{{cite journal|first1=Florian |last1=Cajori|author-link1=Florian Cajori |url=https://www.jstor.org/stable/1626244 |title=Carl Friedrich Gauss and his children|journal=Science|volume= 9|issue=229|pages=697–704|date=19 May 1899 |publisher=American Association for the Advancement of Science|jstor= 1626244 |series=New Series |doi=10.1126/science.9.229.697 |pmid=17817224 |bibcode=1899Sci.....9..697C }}</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.{{sfn|Dunnington|2004|p=374}} Her sister Wilhelmina married the orientalist [[Heinrich Ewald]].{{sfn|Dunnington|2004|p=206}} 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">{{Cite journal | last = Gerardy | first = Theo | title = C. F. Gauß und seine Söhne | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 3 | pages = 25–35 | year = 1966 | language = de}}</ref>{{efn|On this journey he met the geodesist [[Ferdinand Rudolph Hassler]], who was a scientific correspondent of Carl Friedrich Gauss.<ref>{{Cite journal | last = Gerardy | first = Theo | title = Geodäten als Korrespondenten von Carl Friedrich Gaus | journal = Allgemeine Vermessungs-Nachrichten | issue =84 | pages = 150–160 | year = 1977 | language = de}} p. 157</ref>{{sfn|Dunnington|2004|p=286}} }}

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 [[Friedrich Bessel|Bessel]];<ref>{{cite journal | author-last = Wolf | author-first = Armin | title = Der Pädagoge und Philosoph Johann Conrad Fallenstein (1731–1813) – Genealogische Beziehungen zwischen [[Max Weber]], Gauß und Bessel | journal = Genealogie | volume = 7 | year = 1964 | language = de | pages = 266–269}}</ref> he then moved to Missouri, started as a farmer and became wealthy in the shoe business in [[St. Louis, Missouri|St. Louis]] in later years.<ref>{{cite journal | author-last = Weinberger | author-first = Joseph | title = Carl Friedrich Gauß 1777–1855 und seine Nachkommen | journal = Archiv für Sippenforschung und alle verwandten Gebiete | volume = 43/44 | issue = 66 | year = 1977 | language = de | pages = 73–98}}</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" />

<gallery>
File:Joseph Gauß, 001.jpg|Joseph Gauss
File:Joseph Gauß, 003.jpg|Sophie Gauss née Erythropel<br>Joseph's wife
File:Minna Ewald geb. Gauß, 003.jpg|Wilhelmina Gauss
File:Ewald, Georg Heinrich August (1803-1875).jpg|[[Heinrich Ewald]]<br>Wilhelmina's husband
File:Eugen Gauß, 001.jpg|Eugen (Eugene) Gauss
File:Eugen Gauß, 003.jpg|Henrietta Gauss née Fawcett<br>Eugene's wife
File:Wilhelm Gauß, 002.jpg|Wilhelm (Charles William) Gauss
File:Wilhelm Gauß, 001.jpg|Louisa Aletta Gauss née Fallenstein<br>William's wife
File:Therese Staufenau geb. Gauß, 008.jpg|Therese Gauss
File:Therese Staufenau geb. Gauß, 010.jpg|Constantin Staufenau<br>Therese's husband
</gallery>

=== Personality ===
==== Scholar ====
[[File:Carl Friedrich Gauß, Karikatur von Abraham Gotthelf Kästner, 1795.jpg|thumb|A student draws his professor of mathematics: Caricature of [[Abraham Gotthelf Kästner]] by Gauss (1795){{efn|Following Bolyai's handwritten Hungarian text at the bottom, Gauss intentionally characterized Kästner with the added the wrong addition.}}]]
[[File:Carl Friedrich Gauß, 003.jpg|thumb|upright|A student draws his professor of mathematics: Gauss sketched by his student [[Johann Benedict Listing]], 1830]]
In the first two decades of the 19th century, Gauss was the only important mathematician in Germany comparable to the leading French mathematicians.<ref name="Schubring">{{cite book | last1 = Schubring | first1 = Gert | editor-last1 = Fauvel | editor-first1 = John | editor-last2 = Flood | editor-first2 = Raymond | editor-last3 = Wilson | editor-first3 = Robin | editor-link1 = John Fauvel | editor-link2 = Raymond Flood (mathematician)| editor-link3 = Robin Wilson (mathematician) | title = Möbius and his band: Mathematics and Astronomy in Nineteenth-century Germany  | publisher = Oxford University Press | date = 1993 | pages = 21–33 | chapter = The German mathematical community}}</ref>  His ''Disquisitiones Arithmeticae'' was the first mathematical book from Germany to be translated into the French language.<ref>{{cite book | author-last = Schubring | author-first = Gert | title = Geschichte der Mathematik in ihren Kontexten | publisher = Birkhäuser| date = 2021 | pages = 133–134 | language = de}}</ref>

Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration.{{sfn|Klein|1894|pp=100–101}} 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,{{sfn|Klein|1979|pp=5–6}} Gauss introduced a new style of direct and complete exposition that did not attempt to show the reader the author's train of thought.{{sfn|Dunnington|2004|p=217}}

{{blockquote|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.|source={{harvnb|Klein|1894|p=101}} }}

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows:<ref>{{Cite web|url=https://archive.org/details/briefwechselzwi00gausgoog/page/n124/mode/2up|title=Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai|first=Carl Friedrich Gauss|last=Farkas Bólyai|date=22 April 1899|publisher=B. G. Teubner|via=Internet Archive}}</ref>
{{blockquote|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.|source={{harvnb|Dunnington|2004|p=416}} }}

His posthumous papers, his scientific [[Gauss's diary|diary]],<ref>{{cite journal | author-link1 = Felix Klein | editor-last = Klein | editor-first = Felix | title = Gauß' wissenschaftliches Tagebuch 1796–1814 | doi=10.1007/BF01449013 | year = 1903 | journal = [[Mathematische Annalen]] | volume = 57 | pages = 1–34 | s2cid = 119641638 | language = la, de | url = https://gdz.sub.uni-goettingen.de/id/PPN235181684_0057?tify=%7B%22pages%22%3A%5B8%2C9%5D%2C%22view%22%3A%22info%22%7D}} p. 2</ref> and short glosses in his own textbooks show that he empirically worked to a great extent.{{sfn|Bachmann|1922|pp=4–6}}{{sfn|Schlesinger|1933|p=18}} 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.{{sfn|Maennchen|1930|pp=64–65}} He coped with the enormous workload by using skillful tools.{{sfn|Maennchen|1930|pp=4–9}} Gauss used numerous [[mathematical table]]s, examined their exactness, and constructed new tables on various matters for personal use.<ref>{{cite book | last1 = Reich | first1 = Karin | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedrsächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 73–86 | chapter = Logarithmentafeln – Gauß' "tägliches Arbeitsgeräth" | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> He developed new tools for effective calculation, for example the [[Gaussian elimination]].<ref>{{Citation | last1=Althoen | first1=Steven C. | last2=McLaughlin | first2=Renate | title=Gauss–Jordan reduction: a brief history | doi=10.2307/2322413 | year=1987 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=94 | issue=2 | pages=130–142 | jstor=2322413 | publisher=Mathematical Association of America}}</ref> Gauss's calculations and the tables he prepared were often more precise than practically necessary.{{sfn|Maennchen|1930|p=3}} Very likely, this method gave him additional material for his theoretical work.{{sfn|Maennchen|1930|pp=4–9}}{{sfn|Bachmann|1922|p=5}}

[[File:GaussSiegel1777-1855.png|thumb|upright|Gauss's [[Seal (emblem)|seal]] with his motto ''Pauca sed Matura'']]
Gauss was only willing to publish work when he considered it complete and above criticism. This [[perfectionism (psychology)|perfectionism]] was in keeping with the motto of his personal [[Seal (emblem)|seal]] {{lang|la|Pauca sed Matura}} ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself 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">{{Cite journal | last = Biermann | first = Kurt-R. | title = Über die Beziehungen zwischen C. F. Gauß und F. W. Bessel | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 3 | pages = 7–20 | year = 1966 | language = de}}</ref> Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too.{{sfn|Klein|1979|p=29}}{{sfn|Dunnington|2004|pp=420–430}} Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics",{{sfn|Sartorius von Waltershausen|1856|p=79}} 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">{{cite book |last=Derbyshire |first=John |url=https://archive.org/details/primeobsessionbe00derb_0/page/202 |title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics |publisher=Joseph Henry Press |year=2003 |isbn=978-0-309-08549-6 |place=Washington, DC |page=202 |url-access=registration}}</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">{{Cite journal | last = [[Stephen Stigler|Stigler]] | first = Stephen M. | title = Gauss and the Invention of Least Squares | journal = [[Annals of Statistics]] | volume = 9 | issue = 3 | pages = 465–474 | year = 1981| doi = 10.1214/aos/1176345451 | doi-access = free }}</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 man ====
Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty,{{sfn|Sartorius von Waltershausen|1856|p=102}} but also of "iron character"{{sfn|Sartorius von Waltershausen|1856|p=95}} with an unshakeable strength of mind.{{sfn|Sartorius von Waltershausen|1856|p=8}} Apart from his closer circle, others regarded him as reserved and unapproachable "like an [[Twelve Olympians|Olympian]] sitting enthroned on the summit of science".{{sfn|Wußing|1982|p=41}} 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 [[Johann Friedrich Ludwig Hausmann|Hausmann]] he opposed to a call for [[Justus Liebig]] on a university chair in Göttingen, "because he was always involved in some polemic."{{sfn|Dunnington|2004|p=253}}

[[File:Göttingen, Kurze Straße 15, 001.jpg|thumb|Gauss's residence from 1808 to 1816 in the first floor]]
Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient [[threnody]], the most personal of his surviving documents.<ref>{{cite web | url = https://gauss.adw-goe.de/handle/gauss/2086 | title = Letter from Carl Friedrich Gauss to Johanna Gauss, 23. October 1809| website = Der komplette Briefwechsel von Carl Friedrich Gauss | date = 23 October 1809| publisher = Akademie der Wissenschaften zu Göttingen | access-date = 26 March 2023}}</ref>{{sfn| Dunnington|2004|pp=94–95}} His second wife and his two daughters suffered from [[tuberculosis]].{{sfn|Dunnington|2004|p=206, 374}} In a letter to [[Friedrich Wilhelm Bessel|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">{{cite web | url = https://homepages.rootsweb.com/~schmblss/home/Letters/Gauss/1898-12-21.htm | title = Letter: Charles Henry Gauss to Florian Cajori – 21 December 1898 | access-date = 25 March 2023}}</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.{{sfn|Dunnington|2004|p=374}}

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 [[Alexander von Humboldt|Humboldt]] in December 1851 for having reached the same age as [[Isaac Newton]] at his death, calculated in days.{{sfn|Sartorius von Waltershausen|1856|p=71}}

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.{{sfn|Dunnington|2004|p=241}}{{efn|The first book he loaned from the university library in 1795 was the novel ''[[Clarissa; or, The History of a Young Lady|Clarissa]]'' from [[Samuel Richardson]].<ref>{{cite book | last1 = Reich | first1 = Karin | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedrsächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 105–115 | chapter = Gauß' geistige Väter: nicht nur "summus Newton", sondern auch "summus Euler" | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>}} His favorite English author was [[Walter Scott]], his favorite German [[Jean Paul]]. At the age of 62, he began to teach himself [[Russian language|Russian]], very likely to understand scientific writings from Russia, among them those of [[Nikolai Lobachevsky|Lobachevsky]] on non-Euclidean geometry.<ref>{{cite book | last1 = Lehfeldt | first1 = Werner | author-link  = Werner Lehfeldt | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedrsächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 302–310 | chapter = Carl Friedrich Gauß' Beschäftigung mit der russischen Sprache | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>{{sfn|Wußing|1982|p=80}} Gauss liked singing and went to concerts.{{sfn|Wußing|1982|p=81}} He was a busy newspaper reader; in his last years, he would visit an academic press salon of the university every noon.{{sfn|Sartorius von Waltershausen|1856|p=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]]''.{{sfn|Wußing|1982|p=79}}

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, ergo decipiatur|mundus vult decipi]]".{{sfn|Sartorius von Waltershausen|1856|p=94}} 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.{{sfn|Sartorius von Waltershausen|1856|p=95}}{{efn|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.}} He had a low estimation of the constitutional system and he criticized parliamentarians of his time for their perceived ignorance and logical errors.{{sfn|Sartorius von Waltershausen|1856|p=94}}

Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes"{{sfn|Sartorius von Waltershausen|1856|p=97}} and "I succeeded – not on account of my hard efforts, but by the grace of the Lord."<ref>{{cite web | url = https://gauss.adw-goe.de/handle/gauss/731 | title = Letter from Carl Friedrich Gauss to Wilhelm Olbers, 3 September 1805| website = Der komplette Briefwechsel von Carl Friedrich Gauss | date = 23 October 1809| publisher = Akademie der Wissenschaften zu Göttingen | access-date = 26 March 2023}}</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.{{sfn|Dunnington|2004|p=300}} According to Sartorius, Gauss' [[religious tolerance]],  "insatiable thirst for truth" and sense of justice were motivated by his religious convictions.{{sfn|Sartorius von Waltershausen|1856|p=100}}

== Mathematics ==
=== Algebra and number theory ===
==== Fundamental theorem of algebra ====
[[File:DBP 1977 928 Carl Friedrich Gauß.jpg|thumb|upright|German stamp commemorating Gauss's 200th anniversary: the [[complex plane]] or ''Gauss plane'']]
In his doctoral thesis from 1799, Gauss proved the [[fundamental theorem of algebra]] which states that every non-constant single-variable [[polynomial]] with complex coefficients has at least one complex [[root of a function|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>{{Cite arXiv | last1 = Basu | first1 = Soham | last2 = Velleman | first2 = Daniel J. | date = 21 April 2017 | title = On Gauss's first proof of the fundamental theorem of algebra | class = math.CV | eprint = 1704.06585}}</ref>

==== ''Disquisitiones Arithmeticae'' ====
{{Main|Disquisitiones Arithmeticae}}
In the preface to the ''Disquisitiones'', Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered.{{sfn|Bachmann|1922|p=8}} 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 ({{math|≡}}) for [[Congruence relation|congruence]] and uses it for a clean presentation of [[modular arithmetic]].{{sfn|Bachmann|1922|pp=8–9}} It deals with the [[unique factorization theorem]] and [[primitive root modulo n|primitive roots modulo n]]. In the main sections, Gauss presents the first two proofs of the law of [[quadratic reciprocity]]{{sfn|Bachmann|1922|pp=16–25}} and develops the theories of [[Binary quadratic form|binary]]{{sfn|Bachmann|1922|pp=14–16, 25}} and ternary [[quadratic form]]s.{{sfn|Bachmann|1922|pp=25–28}}

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 [[Legendre's three-square theorem|theorem on three squares]], he proves the triangular case of the [[Fermat polygonal number theorem]] for ''n'' = 3.{{sfn|Bachmann|1922|p=29}} From several analytic results on [[ideal class group|class numbers]] that Gauss gives without proof towards the end of the fifth section,{{sfn|Bachmann|1922|pp=22&ndash;23}} it appears that Gauss already knew the [[class number formula]] in 1801.{{sfn|Bachmann|1922|pp=66&ndash;69}}

In the last section, Gauss gives proof for the [[Constructible polygon|constructibility]] of a regular [[heptadecagon]] (17-sided polygon) with [[Compass and straightedge constructions|straightedge and compass]] by reducing this geometrical problem to an algebraic one.<ref name="Denker">{{cite book | last1 = Denker | first1 = Manfred | last2 = Patterson | first2 = Samuel James | author-link2  = Samuel James Patterson | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 53–62 | chapter = Gauß – der geniale Mathematiker | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</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 prime]]s. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in [[finite field]]s, which amounts to counting integral points on an [[elliptic curve]].<ref name="Stuhler">{{cite book | last1 = Stuhler | first1 = Ulrich | author-link  = Ulrich Stuhler | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 62–72 | chapter = Arithmetisch-geometrisches Mittel und elliptische Integrale: Gauß und die komplexe Analysis | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.{{sfn|Dunnington|2004|p=44}}<ref name="Frei">{{cite book |last=Frei |first=Günther |author-link=Günther Frei |chapter-url=https://books.google.com/books?id=IUFTcOsMTysC&pg=159 |title=The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae |date=2007 |publisher=Springer |isbn=978-3-540-20441-1 |editor-last1=Goldstein |editor-first1=Catherine |editor-link1=Catherine Goldstein |place=Berlin, Heidelberg, New York |pages=159–198 |chapter=The Unpublished Section Eight: On the Way for Function Fields over a Finite Field |doi=10.1007/978-3-540-34720-0 |editor-last2=Schappacher |editor-first2=Norbert |editor-link2=Norbert Schappacher |editor-last3=Schwermer |editor-first3=Joachim |editor-link3=Joachim Schwermer}}</ref>

==== Further investigations ====
One of Gauss's first results was the empirically found conjecture of 1792 – the later called [[prime number theorem]] – giving an estimation of the number of prime numbers by using the [[Logarithmic integral function|integral logarithm]].<ref>{{cite book | last1 = Koch | first1 = H. | author-link1 = Herbert Koch | last2 = Pieper | first2 = H. | title = Zahlentheorie | publisher = VEB Deutscher Verlag der Wissenschaften | place = Berlin | date = 1976 | pages = 6, 124}}</ref>{{efn|Gauss told the story later in detail in a letter to [[Johann Franz Encke|Encke]].{{sfn|Bachmann|1922|p=4}}}}

In 1816, [[Heinrich Wilhelm Matthias Olbers|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>{{cite journal | last1 = Kleiner | first1 = I. | year = 2000 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem | journal = [[Elemente der Mathematik]] | volume = 55 | pages = 19–37 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079
 | s2cid = 53319514 | url-status = dead | archive-url = https://web.archive.org/web/20110608052614/http://math.stanford.edu/~lekheng/flt/kleiner.pdf | archive-date = 8 June 2011}}</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.{{sfn|Bachmann|1922|pp= 60&ndash;61}}

Gauss contributed to solving the [[Kepler conjecture]] in 1831 with the proof that a [[Close-packing of equal spheres|greatest packing density]] of spheres in the three-dimensional space is given when the centres of the spheres form a [[Cubic crystal system|cubic face-centred]] arrangement,<ref>{{Cite journal | last = Hales | first = Thomas C. | author-link = Thomas Callister Hales | title = Historical overview of the Kepler conjecture | doi = 10.1007/s00454-005-1210-2 | mr = 2229657 | year = 2006 | journal = [[Discrete & Computational Geometry]] | issn = 0179-5376 | volume=36 | issue = 1 | pages = 5–20| doi-access = free}}</ref> when he reviewed a book of [[Ludwig August Seeber]] on the theory of reduction of positive ternary quadratic forms.<ref>{{Cite book | last = Seeber | first = Ludwig August | year = 1831 | title = Untersuchungen über die Eigenschaften der positiven ternaeren quadratischen Formen | place = Mannheim  | url = https://books.google.com/books?id=QKJGAAAAcAAJ}}</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>{{cite journal | date = July 1831 | title = Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber | url = https://babel.hathitrust.org/cgi/pt?id=mdp.39015064427944&seq=387 | journal = [[Göttingische gelehrte Anzeigen]] | issue = 108 | pages = 1065–1077}}</ref>

In  two papers on [[Quartic reciprocity|biquadratic residue]]s (1828, 1832) Gauss introduced the [[ring theory|ring]] of [[Gaussian integers]] <math>\mathbb{Z}[i]</math>, showed that it is a [[unique factorization domain]],<ref name="Kleiner1998">{{cite journal | url = https://ems.press/journals/em/articles/664 | title = From Numbers to Rings: The Early History of Ring Theory | first1 = Israel | last1 = Kleiner | author-link = Israel Kleiner (mathematician) | journal = [[Elemente der Mathematik]] | volume = 53 | number = 1 | doi = 10.1007/s000170050029 | year = 1998 | pages = 18–35 | zbl = 0908.16001 | doi-access = free}}</ref> and generalized some key arithmetic concepts, such as [[Fermat's little theorem]] and [[Gauss's lemma (number theory)|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>{{cite book | last1 = Lemmermeyer | first1 = Franz | title = Reciprocity Laws: from Euler to Eisenstein | series = Springer Monographs in Mathematics | publisher = Springer | place = Berlin | year = 2000 | page = 15 | isbn = 3-540-66957-4 | doi= 10.1007/978-3-662-12893-0}}</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 sum]]s) can be applied to prove higher reciprocity laws.{{sfn|Bachmann|1922|pp= 52, 57&ndash;59}}

=== Analysis ===
One of Gauss's first discoveries was the notion of the [[arithmetic-geometric mean]] (AGM) of two positive real numbers.{{sfn|Schlesinger|1933|p=41–57}} 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">{{cite journal | last = Cox | first = David A. | author-link = David A. Cox | date = January 1984 | title = The Arithmetic-Geometric Mean of Gauss | url = https://www.researchgate.net/publication/248675540 | journal = [[L'Enseignement mathématique]] | volume = 30 | issue = 2 | pages = 275–330}}</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 [[residue (complex analysis)|complex residues]] when integrating around [[pole (complex analysis)|poles]].<ref name="Stuhler" /><ref>Letter Gauss to Bessel from 18 December 1811, partly printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B96%5D%2C%22pan%22%3A%7B%22x%22%3A0.553%2C%22y%22%3A0.327%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.893%7D ''Collected Works'', Volume 8, pp. 90&ndash;92].</ref>

[[Pentagonal number theorem|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>{{cite book | last = Roy | first = Ranjan | author-link = Ranjan Roy | date = 2021 | edition = 2 | title = Series and Products in the Development of Mathematics | place = Cambridge | pages = 20–22 |volume=2 | publisher = Cambridge University Press | url = https://assets.cambridge.org/97811087/09453/frontmatter/9781108709453_frontmatter.pdf |isbn=9781108709378}}</ref> His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.{{sfn|Schlesinger|1933|pp=185-186}}{{efn|Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order), [[Ludwig Adolf Sohncke|Sohncke]] in 1837 (7th and other orders).}}{{efn|In a letter to Bessel from 1828, Gauss commented: "Mr. Abel has [...] anticipated me, and relieves me of the effort [of publishing] in respect to one third of these matters ..."{{sfn|Schlesinger|1933|p=41}}}}

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 function|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]].{{sfn|Schlesinger|1933|pp=101–106}}<ref name="Houzel">{{cite book | last = Houzel | first = Christian | editor-last1 = Goldstrein | editor-first1 = Catherine | editor-last2 = Schappacher | editor-first2 = Norbert | editor-last3 = Schwermer | editor-first3 = Joachim | editor-link1 = Catherine Goldstein | editor-link2 = Norbert Schappacher | editor-link3 = Joachim Schwermer | title = The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae | publisher = Springer | place = Berlin, Heidelberg, New York| date = 2007 | page = 293 | chapter = Elliptic Functions and Arithmetic | doi = 10.1007/978-3-540-34720-0 | isbn = 978-3-540-20441-1 | url = https://link.springer.com/book/10.1007/978-3-540-34720-0}}</ref> One of Gauss's sketches of this kind was a drawing of a [[tessellation]] of the [[unit disk]] by "equilateral" [[hyperbolic triangle]]s with all angles equal to <math>\pi/4</math>.<ref>Printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B110%5D%2C%22view%22%3A%22info%22%7D ''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.{{efn|This remark appears at article 335 of chapter 7 of ''Disquisitiones Arithmeticae'' (1801).}} 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 [[Gaussian binomial coefficient|q-analogs of binomial coefficient]]s 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 [[Carl Gustav Jacob Jacobi|Jacobi]] and [[Hermite]] has exposed the crux of his argument.{{sfn|Schlesinger|1933|pp=122&ndash;123}}

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.{{sfn|Schlesinger|1933|pp=136–142}} This work is the first exact inquiry into [[Convergent series|convergence]] of infinite series in the history of mathematics.{{sfn|Schlesinger|1933|p=142}} Furthermore, it deals with infinite [[continued fraction]]s arising as ratios of hypergeometric functions, which are now called [[Gauss continued fraction]]s.{{sfn|Schlesinger|1933|pp=136–154}}

In 1823, Gauss won the prize of the Danish Society with an essay on [[conformal mapping]]s, which contains several developments that pertain to the field of complex analysis.{{sfn|Stäckel|1917|pp=90&ndash;91}} Gauss stated that angle-preserving mappings in the complex plane must be complex [[analytic function]]s, 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]].{{sfn|Bühler|1981|p=103}}

==== Numerical analysis ====
Gauss often deduced theorems [[Inductive reasoning|inductively]] from numerical data he had collected empirically.{{sfn|Schlesinger|1933|p=18}} 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>{{cite book | last = Gautschi | first = Walter | author-link = Walter Gautschi | edition = 1 | title = E.B. Christoffel. The Influence of his Work on Mathematics and the Physical Science | editor-last1 = Butzer | editor-first1 = Paul B. | editor-last2 = Fehér | editor-first2 = Franziska | editor-link1 = Paul Butzer | chapter = A Survey of Gauss-Christoffel Quadrature Formulae | place = Birkhäuser, Basel | year = 1981 | pages = 72–147 | publisher = Springer | doi = 10.1007/978-3-0348-5452-8_6 | isbn = 978-3-0348-5452-8 | chapter-url = https://link.springer.com/chapter/10.1007/978-3-0348-5452-8_6}}</ref>

In a private letter to [[Christian Ludwig Gerling|Gerling]] from 1823,<ref>{{Cite book|url=https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22:%5B316%5D,%22view%22:%22info%22%7D|title=Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling|publisher=Elsner}}</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>{{cite arXiv | author = Yousef Saad | author-link = Yousef Saad | title = Iterative Methods for Linear Systems of Equations: A Brief Historical Journey | date = 2 August 2019 | class = math.HO | eprint = 1908.01083v1}}</ref>

Gauss invented an algorithm for calculating what is now called [[discrete Fourier transform]]s when calculating the orbits of Pallas and Juno in 1805, 160 years before [[James Cooley|Cooley]] and [[John Tukey|Tukey]] found their similar [[Cooley-Tukey FFT algorithm|Cooley–Tukey algorithm]].<ref>{{cite journal | last1 = Cooley | first1 = James W. | first2 = John W. | last2 = Tukey | title = An algorithm for the machine calculation of complex Fourier series | journal = [[Mathematics of Computation]] | volume = 19 | issue = 90 | pages = 297–301 | year = 1965 | doi = 10.2307/2003354 | jstor = 2003354 | doi-access=free }}</ref> He developed it as a [[trigonometric interpolation]] method, but the paper ''Theoria Interpolationis Methodo Nova Tractata'' was published only posthumously in 1876,<ref>{{cite book | last = Gauss | first = C.F. | date = 1876 | title = Theoria Interpolationis Methodo Nova Tractata | place = Göttingen | language = la | pages = 265–327 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B273%5D%2C%22pan%22%3A%7B%22x%22%3A0.524%2C%22y%22%3A0.333%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.856%7D | publisher = K. Gesellschaft der Wissenschaften zu Göttingen}}</ref> well after [[Joseph Fourier]]'s introduction of the subject in 1807.<ref>{{cite journal | last1 = Heideman | first1 = Michael T. | last2 = Johnson| first2 = Don H. | last3 = Burrus | first3 = C. Sidney | author-link3 = C. Sidney Burrus | title = Gauss and the history of the fast Fourier transform | journal = IEEE ASSP Magazine | year = 1984 | volume = 1 | issue = 4 | pages = 14–21 | doi = 10.1109/MASSP.1984.1162257|s2cid=10032502 | url = http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-url=https://web.archive.org/web/20130319053449/http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-date = 19 March 2013 | url-status=live}}</ref>

=== Geometry ===
==== Differential geometry ====
{{Main|Theorema Egregium}}
The geodetic survey of [[Hanover]] fuelled Gauss's interest in [[differential geometry]] and [[topology]], fields of mathematics dealing with [[curve]]s and [[Surface (topology)|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 coordinate system|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 [[angle]]s and [[distance]]s on the surface, regardless of the [[embedding]] of the surface in three-dimensional or two-dimensional space.{{sfn|Stäckel|1917|pp=110–119}}

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended [[manifold]]s; it clarifies the distinction between the intrinsic properties of the manifold (the [[metric tensor|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 [[map projection|projection]]s for geographical maps.{{sfn|Stäckel|1917|pp=110–119}} A portion of this essay is dedicated to a profound study of [[geodesic]]s. 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.{{sfn|Stäckel|1917|pp=105–106}}

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 [[invariant (mathematics)|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]].{{sfn|Bolza|1921|pp=70–74}}

==== Non-Euclidean geometry ====
{{Main|Non-Euclidean geometry}}
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|upright|Lithography by [[Siegfried Bendixen]] (1828)]]
During Gauss' lifetime, the [[Parallel postulate]] of [[Euclidean geometry]] was heavily discussed.{{sfn|Stäckel|1917|pp=19–20}} Numerous efforts were made to prove it in the frame of the Euclidean [[axiom]]s, whereas some mathematicians discussed the possibility of geometrical systems without it.{{sfn|Bühler|1981|pp=100–102}} 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.{{sfn|Klein|1979|pp=57–60}}{{sfn|Stäckel|1917|pp=19–20}} 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">{{Cite journal | last = Winger | first = R. M. | date = 1925 | title = Gauss and non-Euclidean geometry | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-31/issue-7/Gauss-and-non-euclidean-geometry/bams/1183486559.full | journal = Bulletin of the American Mathematical Society | volume = 31 | issue = 7 | pages = 356–358 | doi = 10.1090/S0002-9904-1925-04054-9 | issn = 0002-9904 | doi-access = free }}</ref> but he strongly forbade Taurinus to make any use of it.{{sfn|Klein|1979|pp=57–60}} Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.<ref>{{Cite book |last=Bonola |first=Roberto |url=https://archive.org/details/noneuclideangeom00bono/page/64 |title=Non-Euclidean Geometry: A Critical and Historical Study of its Development |publisher=The Open Court Publishing Company |year=1912 |pages=64–67 |language=en}}</ref><ref name=":0" /><ref>{{Cite book |last=Klein |first=Felix |url=https://archive.org/details/elementarymathem0000klei/page/176 |title=Elementary Mathematics from an Advanced Standpoint: Geometry |publisher=Dover Publications |year=1939 |pages=176–177 |language=en}}</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.{{sfn|Bühler|1981|pp=100–102}} In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.{{sfn|Klein|1979|pp=57–60}}<ref>{{Cite journal | last1 = Jenkovszky | first1 = László | last2 = Lake | first2 = Matthew J. | last3 = Soloviev | first3 = Vladimir | date = 12 March 2023 | title = János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword | journal = Symmetry| volume = 15 | issue = 3 | pages = 707 | doi = 10.3390/sym15030707 | arxiv = 2303.17011 | bibcode=2023Symm...15..707J | issn = 2073-8994 | doi-access = free}}</ref> Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai<ref>{{Cite web|url=https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up|title=Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai|first=Carl Friedrich Gauss|last=Farkas Bólyai|date=22 April 1899|publisher=B. G. Teubner|via=Internet Archive}}</ref> claiming that these were congruent to his own thoughts of some decades.{{sfn|Klein|1979|pp=57–60}}<ref>{{cite book | last = Krantz | first = Steven G. | author-link = Steven G. Krantz | title = An Episodic History of Mathematics: Mathematical Culture through Problem Solving|url=https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA171 | access-date = 9 February 2013 | date = 2010 | publisher = [[The Mathematical Association of America]] | isbn = 978-0-88385-766-3| pages = 171f}}</ref> However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.{{sfn|Bühler|1981|pp=100–102}}

[[Wolfgang Sartorius von Waltershausen|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.{{sfn|Klein|1979|pp=57–60}}

==== Early topology ====
Gauss was also an early pioneer of [[topology]] or ''Geometria Situs'', as it was called in his lifetime. The first proof of the [[fundamental theorem of algebra]] in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.{{sfn|Ostrowski|1920|pp=1–18}}

[[File:Carl Friedrich Gauß, Büste von Heinrich Hesemann, 1855.jpg|thumb|left|upright|Gauss bust by Heinrich Hesemann (1855){{efn|Hesemann also took a death mask from Gauss.{{sfn|Dunnington|2004|p=324}}}}]]
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 [[Link (knot theory)|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">{{cite journal | last = Epple | first = Moritz | title = Orbits of asteroids, a braid, and the first link invariant | journal =  [[The Mathematical Intelligencer]] | volume = 20 | pages = 45–52 | year = 1998 | issue = 1 | doi = 10.1007/BF03024400 | s2cid = 124104367 | url = https://link.springer.com/article/10.1007/BF03024400}}</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 [[Knot theory|knot]]s.<ref>{{Cite book | last = Epple | first = Moritz | author-link = Moritz Epple | title = History of Topology | chapter = Geometric Aspects in the Development of Knot Theory | date = 1999 | editor-last = James | editor-first = I.M. | place = Amsterdam | publisher = Elseviwer | pages = 301–357 | chapter-url = https://www.maths.ed.ac.uk/~v1ranick/papers/epple3.pdf}}</ref> To do so he devised a symbolical scheme, the [[Gauss code]], that in a sense captured the characteristic features of tract figures.<ref>{{Cite book | last1 = Lisitsa | first1 = Alexei | last2 = Potapov | first2 = Igor | last3 = Saleh | first3 = Rafiq | title = Language and Automata Theory and Applications | chapter = Automata on Gauss Words | date = 2009 | editor-last = Dediu | editor-first = Adrian Horia | editor2-last = Ionescu | editor2-first = Armand Mihai | editor3-last = Martín-Vide | editor3-first = Carlos | series = Lecture Notes in Computer Science | volume = 5457 | place = Berlin, Heidelberg | publisher = Springer | pages = 505–517 | doi = 10.1007/978-3-642-00982-2_43 | isbn = 978-3-642-00982-2 | chapter-url = https://cgi.csc.liv.ac.uk/~igor/papers/lata2009.pdf}}</ref>{{sfn|Stäckel|1917|pp=50–51}}

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 [[Braid theory|braid]]s and [[Tangle (mathematics)|tangle]]s.<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.{{sfn|Stäckel|1917|pp=51–55}}

=== Minor mathematical accomplishments ===
Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example, in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography,<ref>Printed in ''Collected Works '' Volume 2, pp. 305–310</ref> he stated the [[Axonometry|fundamental theorem of axonometry]], which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers.<ref>{{cite journal | last1 = Eastwood | first1 = Michael | last2 = Penrose | first2 = Roger | author-link2 = Michael Eastwood | author-link = Roger Penrose | title = Drawing with Complex Numbers  | journal = The Mathematical Intelligencer | year = 2000 | volume = 22 | issue = 4 | pages = 8–13 | doi = 10.1007/BF03026760 | arxiv=math/0001097| s2cid = 119136586 }}</ref> He described rotations of this sphere as the action of certain [[Mobius transformation|linear fractional transformations]] on the extended complex plane,{{sfn|Schlesinger|1933|p=198}} and gave a proof for the geometric theorem that the [[Altitude (triangle)|altitudes]] of a triangle always meet in a single [[orthocenter]].<ref>Carl Friedrich Gauss: Zusätze.II. In: {{cite book | last = Carnot | first = Lazare | author-link = Lazare Carnot | translator =  H.C. Schumacher | title = Geometrie der Stellung | pages = 363–364 | year = 1810 | publisher =  Hammerich | place = Altona | language=de}} (Text by Schumacher, algorithm by Gauss), republished in [https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B397%5D%2C%22pan%22%3A%7B%22x%22%3A0.529%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.657%7D ''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>{{cite journal | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter | title = Frieze patterns | journal = [[Acta Arithmetica]] | volume = 18 | pages = 297–310 | year = 1971 | url = http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf | doi = 10.4064/aa-18-1-297-310 | doi-access = free}}</ref> he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects.<ref>[https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B489%5D%2C%22pan%22%3A%7B%22x%22%3A0.547%2C%22y%22%3A0.533%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.548%7D ''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>{{cite journal | last = Schechtman | first = Vadim | author-link = Vadim Schechtman | title = Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...) | journal = Annales de la faculté des sciences de Toulouse Mathématiques | volume = 22 |  pages = 353–375 | year = 2013 | issue = 2  | doi = 10.5802/afst.1375 | url = https://eudml.org/doc/275393| doi-access = free }}</ref>

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given [[quadrilateral]],<ref>[https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B389%5D%2C%22pan%22%3A%7B%22x%22%3A0.549%2C%22y%22%3A0.675%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D ''Bestimmung der größten Ellipse, welche die vier Ebenen eines gegebenen Vierecks berührt''], printed in ''Collected Works'' Volume 4, pp. 385–392; [https://zs.thulb.uni-jena.de/rsc/viewer/jportal_derivate_00237703/Monatliche_Correspondenz_130168688_22_1810_0115.tif?logicalDiv=jportal_jpvolume_00203050&q=August%201810 original] in ''Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde'', Volume 22, 1810, pp. 112–121</ref>{{sfn|Stäckel|1917|p=71-72}} and discovered a surprising result about the computation of area of [[pentagon]]s.<ref>Printed in ''Collected Works '' Volume 4, pp. 406–407</ref>{{sfn|Stäckel|1917|p=76}}

== Sciences ==
=== Astronomy ===
{{Main|Discovery of Ceres}}
[[File:Carl Friedrich Gauß, Pastellgemälde von Johann Christian August Schwartz, 1803, ohne Rahmen.jpg|thumb|left|upright|Carl Friedrich Gauss 1803 by Johann Christian August Schwartz]]
On 1 January 1801, Italian astronomer [[Giuseppe Piazzi]] discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called [[Titius–Bode law]], and named it [[Ceres (dwarf planet)|Ceres]].<ref>{{Cite journal | last = Forbes | first = Eric G. | author-link = Eric G. Forbes | year = 1971 | title = Gauss and the Discovery of Ceres | url=http://adsabs.harvard.edu/full/1971JHA.....2..195F | url-status=live | journal = [[Journal for the History of Astronomy]] | volume = 2 | issue = 3 | pages = 195–199 | bibcode=1971JHA.....2..195F | doi=10.1177/002182867100200305 |archive-url=https://web.archive.org/web/20210718200510/http://adsabs.harvard.edu/full/1971JHA.....2..195F | archive-date=18 July 2021 |s2cid=125888612}}</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 Observatory|Gotha]], and independently [[Heinrich Wilhelm Matthäus Olbers|Heinrich Olbers]] on 1 and 2 January in [[Bremen]], identified the object near the predicted position.<ref>{{cite journal | last1 = Teets | first1 = Donald | last2 = Whitehead | first2 = Karen | title = The discovery of Ceres. How Gauss became famous | journal = [[Mathematics Magazine]] | volume = 19 | issue = 90 | pages = 83–91 | year = 1965 | url = https://www.maa.org/programs/maa-awards/writing-awards/the-discovery-of-ceres-how-gauss-became-famous | access-date = 22 March 2023 | archive-date = 3 April 2023 | archive-url = https://web.archive.org/web/20230403074017/https://www.maa.org/programs/maa-awards/writing-awards/the-discovery-of-ceres-how-gauss-became-famous | url-status = dead }}</ref>{{efn|The unambiguous identification of a cosmic object as planet among the fixed stars requires at least two observations with interval.}}

[[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.{{sfn|Klein|1979|p=8}}

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 [[Perturbation (astronomy)|perturbation]]s of their [[orbital elements]]. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was [[(2) Pallas|Pallas]], because of its great [[Eccentricity (astronomy)|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.{{sfn|Brendel|1929|pp=194–195}} He found an [[orbital resonance#Coincidental 'near' ratios of mean motion|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.{{sfn|Brendel|1929|p=206}}<ref>{{cite journal | last = Taylor | first = D. B. | year = 1982 | title = The secular motion of Pallas | journal = [[Monthly Notices of the Royal Astronomical Society]] | volume = 199 | issue = 2 | pages = 255–265 | bibcode=1982MNRAS.199..255T |doi=10.1093/mnras/199.2.255 | doi-access=free}}</ref>{{efn|Brendel (1929) thought this cipher to be insoluble, but actually decoding was very easy.{{sfn|Brendel|1929|p=206}}<ref>{{cite book | last1 = Schroeder | first1 = Manfred R. | author-link  = Manfred R. Schroeder | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 259–260 | chapter = Gauß, die Konzertsaalakustik und der Asteroid Palls | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>}} 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.{{sfn|Brendel|1929|p=254}}

[[File:Goettingen Sternwarte Besemann.png|thumb|Göttingen observatory seen from the North-west (by Friedrich Besemann, {{Circa|1835}})]]
One fruit of Gauss's research on Pallas perturbations was the ''Determinatio Attractionis...'' (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". It introduced an averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time the planet takes to follow the corresponding orbital arcs.{{sfn|Brendel|1929|pp=253–254}} 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]].{{sfn|Schlesinger|1933|pp=169–170}}

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.{{sfn|Brendel|1929|pp=8–9}} After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables of [[Astronomical nutation|nutation]] and [[Aberration (astronomy)|aberration]], solar coordinates, and refraction.{{sfn|Brendel|1929|p=3}} He made many contributions to [[spherical geometry]], and in this context solved some practical problems about [[Celestial navigation|navigation by stars]].{{sfn|Brendel|1929|p=54}} He published a great number of observations, mainly on minor planets and comets; his last observation was the [[solar eclipse of 28 July 1851]].{{sfn|Brendel|1929|p=144}}

=== Chronology ===
Gauss's first publication following his doctoral thesis dealt with the determination of the [[Date of Easter#Gauss's Easter algorithm|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 (time)|golden number]], [[epact]], [[Solar cycle (calendar)|solar cycle]], [[Dominical letter|domenical letter]], and any religious connotations.{{sfn|Maennchen|1930|pp=49–63}} 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 theory ===
Gauss likely used the [[Least squares|method of least squares]] to minimize the impact of [[Observational error|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.{{sfn|Schaaf|1964|p=84}}<ref name=":3">{{Cite journal | last = Plackett | first = R.L. | author-link = R. L. Plackett | date = 1972 | title = The discovery of the method of least squares | url = https://hedibert.org/wp-content/uploads/2016/08/plackett1972-thediscoveryofthemethodofleastsquares.pdf | journal = Biometrika | volume = 59 | issue = 2 | pages =239–251| doi = 10.2307/2334569 | jstor = 2334569 }}</ref><ref>{{Cite web | last = Lim | first = Milton | date = 31 March 2021 | title = Gauss, Least Squares, and the Missing Planet | url = https://www.actuaries.digital/2021/03/31/gauss-least-squares-and-the-missing-planet/ | access-date = 14 October 2023 | website = Actuaries Digital}}</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 [[normal distribution|normally distributed]] errors ([[Gauss–Markov theorem]]), in the two-part paper ''Theoria combinationis observationum erroribus minimis obnoxiae'' (1823).<ref>{{cite journal |first=R. L. |last=Plackett |author-link=Robin Plackett |title=A Historical Note on the Method of Least Squares |journal=[[Biometrika]] |volume=36 |issue=3/4 |year=1949 |pages=458–460 |doi=10.2307/2332682 |jstor=2332682 |pmid=15409359 }}</ref>

In the first paper he proved [[Gauss's inequality]] (a [[Chebyshev's inequality|Chebyshev-type inequality]]) for [[Unimodality|unimodal distributions]], and stated without proof another inequality for [[Moment (mathematics)|moments]] of the fourth order (a special case of the Gauss-Winckler inequality).<ref>{{Cite journal | last = Avkhadiev | first = F. G. | title = A Simple Proof of the Gauss-Winckler Inequality | journal = [[The American Mathematical Monthly]] | volume = 112 | issue = 5 | pages = 459–462 | year = 2005| doi = 10.2307/30037497 | jstor = 30037497 | doi-access =  }}</ref> He derived lower and upper bounds for the [[variance]] of the [[sample variance]]. In the second paper, Gauss described [[Recursive least squares filter|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>{{cite journal | last1=Schaffrin | first1=Burkhard | last2=Snow | first2=Kyle | title=Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth | journal=Linear Algebra and Its Applications | publisher=Elsevier BV | volume=432 | issue=8 | year=2010 | issn=0024-3795 | doi=10.1016/j.laa.2009.09.014 | pages=2061–2076| doi-access=free }}</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-Kuzmin-Wirsing operator|Gauss map for continued fractions]]. Gauss's solution is the first-ever result in the metrical theory of continued fractions.<ref>{{Cite journal | last = Sheynin | first = O. B. | title = C. F. Gauss and the Theory of Errors | journal = [[Archive for History of Exact Sciences]] | volume = 20 | issue = 1 | pages = 21–72 | year = 1979| doi = 10.1007/BF00776066 | jstor = 41133536 | doi-access =  }}</ref>

=== Geodesy ===
[[File:Georg IV Erlass Landvermessung.jpg|thumb|upright|Order of King [[George IV]] from 9 May 1820 to the triangulation project (with the additional signature of Count [[Ernst zu Münster]] below)]]
[[File:Gauß Heliotrop@20170430.JPG|thumb|The [[Heliotrope (instrument)|heliotrope]]]]
[[File:Vize-Heliotrop Gauß-Ausstellung Bomann-Museum (1).jpg|thumb|Gauss's vice heliotrope, a [[Edward Troughton|Troughton]] sextant with additional mirror]]
Gauss was busy with geodetic problems since 1799 when he helped [[Karl Ludwig von Lecoq]] with calculations during his [[Surveying|survey]] in [[Westphalia]].{{sfn|Galle|1924|pp=16–18}} Beginning in 1804, he taught himself some practical geodesy in Brunswick{{sfn|Galle|1924|p=22}} and Göttingen.{{sfn|Galle|1924|p=28}}

Since 1816, Gauss's former student [[Heinrich Christian Schumacher]], then professor in [[Copenhagen]], but living in [[Altona, Hamburg|Altona]] ([[Duchy of Holstein|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.{{efn|Lauenburg was the southernmost town of the [[Duchy of Holstein]], that was held in personal union by the King of [[Denmark]].}} 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.{{sfn|Galle|1924|p=32}}

An [[arc measurement]] needs a precise astronomical determination of at least two points in the [[Triangulation network|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 [[Jesse Ramsden|Ramsden]] that was transported to both observatories.{{sfn|Galle|1924|p=61}}{{efn|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]].{{sfn|Galle|1924|p=61}}}}

Gauss and Schumacher had already determined some angles between [[Lüneburg]], Hamburg, and Lauenburg for the geodetic connection in October 1818.{{sfn|Galle|1924|p=60}} 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 (Dransfeld)|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 {{convert|107|km|1|abbr=in}}. 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" />{{sfn|Galle|1924|pp=75–80}}

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 (instrument)|heliotrope]]''.{{sfn|Schaaf|1964|p=81}} Another suitable construction for the same purpose was a [[sextant]] with an additional mirror which he named ''vice heliotrope''.{{sfn|Galle|1924|p=69}} Gauss was assisted by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in the [[Baseline (surveying)|baseline]] measurement ([[Braak Base Line]]) of Schumacher in the village of [[Braak, Schleswig-Holstein|Braak]] near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.{{sfn|Dunnington|2004|p=121}}

An additional result was a better value for the [[flattening]] of the approximative [[Earth ellipsoid]].{{sfn|Galle|1924|pp=37–38, 49–50}}{{efn|The value from [[Henrik Johan Walbeck|Walbeck]] (1820) of 1/302,78 was improved to 1/298.39; the calculation was done by Eduard Schmidt, private lecturer at Göttingen University.{{sfn|Galle|1924|p=49-50}}}} Gauss developed the [[Transverse Mercator projection#Ellipsoidal transverse Mercator|universal transverse Mercator projection]] of the ellipsoidal shaped Earth (what he named ''conform projection''){{sfn|Dunnington|2004|p=164}} 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.{{sfn|Dunnington|2004|p=135}} 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 [[Gaussian elimination|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.{{sfn|Galle|1924|p=129}}<ref>{{cite book | last = Schreiber | first = Oscar | title = Theorie der Projectionsmethode der Hannoverschen Landesvermessung | publisher = Hahnsche Buchhandlung | year = 1866 | place = Hannover | language = de}}</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">{{cite book | last = Gauß | first = C.F. | title=Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector | publisher=Vandenhoeck und Ruprecht | year = 1828 | language=de | page=73}}</ref> later his doctoral student [[Johann Benedict Listing]] called this the ''[[geoid]]''.<ref>{{cite book | last = Listing | first = J.B. | title = Ueber unsere jetzige Kenntniss der Gestalt und Grösse der Erde | publisher = Dieterich | year = 1872 | place = Göttingen | url=https://books.google.com/books?id=yQ9TAAAAcAAJ | language = de | page = 9}}</ref>

=== Magnetism and telegraphy ===
==== Geomagnetism ====
[[File:Göttingen-Gauß-Weber-Monument.01.JPG|thumb|upright|Gauss-Weber monument in Göttingen by Ferdinand Hartzer (1899)]]
[[File:A magnetometer used by Carl Friedrich Gauss, from Gerlach und F. Traumüller, 1899.png|thumb|The Gauss–Weber magnetometer]]
Gauss had been interested in magnetism since 1803.{{sfn|Dunnington|2004|p=153}} After [[Alexander von Humboldt]] visited Göttingen in 1826, both scientists began intensive research on [[geomagnetism]], partly independently, partly in productive cooperation.<ref>{{cite journal | author-last = Reich | author-first = Karin  | title = Alexander von Humboldt und Carl Friedrich Gauss als Wegbereiter der neuen Disziplin Erdmagnetismus | journal = Humboldt Im Netz | volume = 12 | issue = 22 | pages = 33–55 | year = 2011 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/154/280}}</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 Eduard Weber|Wilhelm Weber]].{{sfn|Dunnington|2004|p=136}}

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.{{sfn|Dunnington|2004|p=161}} 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">{{cite journal | author-last = Reich | author-first = Karin  | title = Der Humboldt'sche Magnetische Verein im historischen Kontext | journal = Humboldt Im Netz | volume = 24 | issue = 46 | pages = 53–74 | year = 2023 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/357/719}}</ref>

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the [[British Empire|British dominions]] with a letter to the [[Prince Augustus Frederick, Duke of Sussex|Duke of Sussex]], then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods.<ref>{{cite journal | author-last = Biermann | author-first = Kurt-R. | title = Aus der Vorgeschichte der Aufforderung Alexander von Humboldt von 1836 an den Präsidenten der Royal Society zur Errichtung geomagnetischer Stationen (Dokumente zu den Beziehungen zwischen A.v. Humboldt und C. F. Gauß) | journal = Humboldt Im Netz | volume = 6 | issue = 11 | year = 2005 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/154/280}}</ref><ref>{{cite journal | author-last = Humboldt | author-first = Alexander von | title = 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 | journal = [[Philosophical Magazine]] | issue = 9 | pages = 42–53 | year = 1836 | volume = Bd. 6 | doi = 10.18443/70 | url = https://www.hin-online.de/index.php/hin/article/view/70/112}}</ref> Together with other instigators, this led to a global program known as "[[Edward Sabine#Magnetical crusade|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">{{cite book | last1 = Rupke | first1 = Nicolaas | author-link  = Nicolaas Adrianus Rupke | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 188–201 | chapter = Carl Friedrich Gauß und der Erdmagnetismus | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</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 University|Leipzig]] in 1843 as late effect of the [[Göttingen Seven|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 [[Magnetic declination|declination]], [[Magnetic dip|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" />{{sfn|Schaaf|1964|pp=115–127}} 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.{{sfn|Klein|1979|pp=21–23}} The theory predicted the existence of exactly two [[Poles of astronomical bodies#Magnetic poles|magnetic poles]] on the Earth, thus [[Christopher Hansteen|Hansteen]]'s idea of four magnetic poles became obsolete,<ref name="Roussanova">{{cite journal | author-last = Roussanova | author-first = Elena  | title = Russland ist seit jeher das gelobte Land für Magnetismus gewesen: Alexander von Humboldt, Carl Friedrich Gauß und die Erforschjung des Erdmagnetismus in Russland | journal = Humboldt Im Netz | volume = 12 | issue = 22 | pages = 56–83 | year = 2011 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/155/282}}</ref> and the data allowed to determine their location with rather good precision.{{sfn|Schaefer|1929|p=87}}

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"/>

==== Electromagnetism ====
[[File:Gauss-Weber-Telegraf Paulinerkirche 02.jpg|thumb|upright|Town plan of Göttingen with course of the telegraphic connection]]
The discoveries of [[Hans Christian Ørsted]] on [[electromagnetism]] and [[Michael Faraday]] on [[electromagnetic induction]] drew Gauss's attention to these matters.{{sfn|Schaefer|1929|p=6}} Gauss and Weber found rules for branched [[Electricity|electric]] circuits, which were later found independently and first published by [[Gustav Kirchhoff]] and named after him as [[Kirchhoff's circuit laws]],{{sfn|Schaefer|1929|p=108}} and made inquiries into electromagnetism. They constructed the first [[Electrical telegraph|electromechanical telegraph]] in 1833, and Weber himself connected the observatory with the institute for physics in the town centre of Göttingen,{{efn|A thunderstorm damaged the cable in 1845.<ref name="Timm" />}} but they made no further commercial use of this invention.<ref name="Timm">{{cite book | last1 = Timm | first1 = Arnulf | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 169–183 | chapter = Der elektrische Telegraph von Gauß und Weber | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref><ref>{{Cite book | last1 = Martín-Rodríguez | first1 = Fernando | last2  = Barrio García | first2 = Gonzalo | last3 = Álvarez Lires | first3 = María | title = 2010 Second Region 8 IEEE Conference on the History of Communications | chapter = Technological archaeology: Technical description of the Gauss-Weber telegraph | date = 2010 | chapter-url = https://ieeexplore.ieee.org/document/5735309 | pages = 1–4 | doi = 10.1109/HISTELCON.2010.5735309|hdl=11093/1859 | isbn = 978-1-4244-7450-9 | s2cid = 2359293}}</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 of induction|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>{{cite book | last = Roche | first = John J. | chapter = A critical study of the vector potential | editor-last = Roche | editor-first = John | date = 1990 | title = Physicists Look Back: Studies in the History of Physics | publisher = Adam Hilger | place = Bristol, New York | pages = 147–149 | isbn = 0-85274-001-8}}</ref>

Gauss tried to find a unifying law for long-distance effects of [[electrostatics]], [[electrodynamics]], electromagnetism, and [[electric Induction|induction]], comparable to Newton's law of gravitation,{{sfn|Schaefer|1929|pp=148–152}} but his attempt ended in a "tragic failure".<ref name="Rupke" />

===Potential theory===
Since Isaac Newton had shown theoretically that the Earth and rotating stars assume non-spherical shapes, the problem of attraction of ellipsoids gained importance in mathematical astronomy. In his first publication on potential theory, the "Theoria attractionis..." (1813), Gauss provided a [[closed-form expression]] to the gravitational attraction of a homogeneous [[triaxial ellipsoid]] at every point in space.{{sfn|Geppert|1933|p=32}} In contrast to previous research of [[Colin Maclaurin|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 [[divergence theorem|Gauss's theorem]] in [[vector analysis]].{{sfn|Geppert|1933|pp=32&ndash;40}}

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 vector potential|magnetic potential]], based on Lagrange, Laplace, and Poisson;{{sfn|Klein|1979|pp=21–23}} it seems rather unlikely that he knew the previous works of [[George Green (mathematician)|George Green]] on this subject.{{sfn|Schaefer|1929|p=6}} 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" />

=== Optics ===
Gauss's calculations enabled instrument maker [[Johann Georg Repsold]] in [[Hamburg]] to construct a new [[achromatic lens]] system in 1810. A main problem, among other difficulties, was that the [[refractive index]] and [[Dispersion (optics)|dispersion]] of the glass used were not precisely known.{{sfn|Schaefer|1929|pp=153–154}} In a short article from 1817 Gauss dealt with the problem of removal of [[chromatic aberration]] in [[Gauss lens|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 [[Achromatic lens|Steinheil doublet]], partly based on Gauss's calculations.{{sfn|Schaefer|1929|pp=159–165}} Many results in [[geometrical optics]] are scattered in Gauss's correspondences and hand notes.{{sfn|Dunnington|2004|p=170}}
  
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>{{Cite book | title = Optics | first = Eugene | last = Hecht | author-link = Eugene Hecht | publisher = Addison Wesley | year = 1987 | isbn = 978-0-201-11609-0 | page = 134}}</ref> He characterized optical systems under a paraxial approximation only by its [[Cardinal point (optics)|cardinal points]],<ref name=Bass>{{Cite book|title=Handbook of Optics| first1=Michael|last1=Bass|first2=Casimer|last2=DeCusatis| first3=Jay|last3=Enoch|first4=Vasudevan|last4=Lakshminarayanan|publisher=McGraw Hill Professional|year=2009|isbn=978-0-07-149889-0|page=17.7}}</ref> and he derived the Gaussian [[lens]] formula, applicable without restrictions in respect to the thickness of the lenses.<ref name=Ostdiek>{{Cite book | title = Inquiry into Physics | first1 = Vern J. | last1 = Ostdiek | first2 = Donald J. | last2 = Bord | publisher = Cengage Learning | year = 2007 | isbn = 978-0-495-11943-2 | page = 381}}</ref>{{sfn|Schaefer|1929|p=189–208}}

=== Mechanics ===
Gauss's first work in mechanics concerned the [[earth's rotation]]. When his university friend [[Johann Benzenberg|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.{{sfn|Geppert|1933|pp=3–11}}

After [[Léon Foucault|Foucault]] had demonstrated the earth's rotation by his [[Foucault pendulum|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.{{sfn|Geppert|1933|p=12-16}}<ref>{{Cite journal | last = Siebert | first = Manfred | title = Das Foucault-Pendel von C. F. Gauß| journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 35 | pages = 49–52 | year = 1998 | volume = 35 | bibcode = 1998GGMit..35...49S | language = de}}</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 [[Joseph-Louis Lagrange|Lagrange]]'s [[principle of virtual work]], and showing analogies to the method of [[least squares]].{{sfn|Geppert|1933|p=16-26}}

=== Metrology ===
In 1828, Gauss was appointed as head of the board for weights and measures of the Kingdom of Hanover. He created [[Standard (metrology)|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">{{cite book | last1 = Olesko | first1 = Kathryn | author-link  = Kathryn Olesko | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 236–253 | chapter = Der praktische Gauß – Präzisionsmessung für den Alltag | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> In the correspondence with Schumacher, who was also working on this matter, he described new ideas for high-precision scales.{{sfn|Geppert|1933|pp=59–60}} He submitted the final reports on the Hanoverian [[Foot (unit)|foot]] and [[Pound (mass)|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 awards ==
[[File:Carl Friedrich Gauß, Copley-Medaille, 1838.jpg|thumb|upright|[[Copley Medal]] for Gauss (1838)]]
Gauss first became member of a scientific society, the [[Russian Academy of Sciences]], in 1802.{{sfn|Dunnington|2004|p=351}} Further memberships (corresponding, foreign or full) were awarded by the [[Göttingen Academy of Sciences and Humanities|Academy of Sciences]] in Göttingen (1802/ 1807),<ref>{{cite web | title = Carl Friedrich Gauss | url = https://adw-goe.de/mitglieder/?tx_find_find%5B__referrer%5D%5B%40extension%5D=Find&tx_find_find%5B__referrer%5D%5B%40controller%5D=Search&tx_find_find%5B__referrer%5D%5B%40action%5D=index&tx_find_find%5B__referrer%5D%5Barguments%5D=YToxOntzOjEwOiJjb250cm9sbGVyIjtzOjY6IlNlYXJjaCI7fQ%3D%3Db53126a3c4f75d5722ed8fcf2a020b56f6bfcf62&tx_find_find%5B__referrer%5D%5B%40request%5D=%7B%22%40extension%22%3A%22Find%22%2C%22%40controller%22%3A%22Search%22%2C%22%40action%22%3A%22index%22%7Da191c3ea4436cf01fca51b5985f59eeefe0305c1&tx_find_find%5B__trustedProperties%5D=%7B%22q%22%3A%7B%22default%22%3A1%7D%7Db6f34fb16399247fb46d6dfdf0edb66a94783c82&tx_find_find%5Bq%5D%5Bdefault%5D=Gauss#members-paginator-top| publisher = Niedersächsische Akademie der Wissenschaften zu Göttingen | access-date = 8 April 2023}}</ref> the [[French Academy of Sciences]] (1804/ 1820),<ref>{{cite web | title = Les membres du passé | url = https://www.academie-sciences.fr/en/Liste-des-membres-depuis-la-creation-de-l-Academie-des-sciences/les-membres-du-passe-dont-le-nom-commence-par-g.html | publisher = Académie des Sciences – Institut de France | access-date = 7 April 2023}}</ref> the [[Royal Society]] of London (1804),<ref>{{cite web | title = Fellows | url = https://catalogues.royalsociety.org/CalmView/Record.aspx?src=CalmView.Persons&id=NA8223&pos=1 | publisher = The Royal Society | access-date = 7 April 2023}}</ref> the [[Prussian Academy of Sciences|Royal Prussian Academy]] in Berlin (1810),<ref>{{cite web | title = Karl Friedrich Gauss | url = https://www.bbaw.de/die-akademie/akademie-historische-aspekte/mitglieder-historisch/historisches-mitglied-karl-friedrich-gauss-843 | publisher = Berlin-Brandenburgische Akademie der Wissenschaften | access-date = 7 April 2023}}</ref> the [[Accademia nazionale delle scienze|National Academy of Science]] in Verona (1810),<ref>{{cite web | title = Elenco Cronologico soci stranieri | url = https://www.accademiaxl.it/accademia/elenco-cronologico-soci-stranieri/ | publisher = Accademia nazionale delle scienze | access-date = 7 April 2023}}</ref> the [[Royal Society of Edinburgh]] (1820),<ref>{{cite web | title = Past Fellows | url = https://rse.org.uk/fellowship/past-fellows/ | publisher = The Royal Society of Edinburgh | page = 345 | access-date = 7 April 2023}}</ref> the [[Bavarian Academy of Sciences and Humanities|Bavarian Academy of Sciences]] of Munich (1820),<ref>{{cite web | title = Verstorbene Mitglieder: Prof. Dr. Carl Friedrich Gauss | url = https://badw.de/gelehrtengemeinschaft/verstorbene.html?tx_badwdb_badwperson%5Baction%5D=show&tx_badwdb_badwperson%5Bcontroller%5D=BADWPerson&tx_badwdb_badwperson%5BpartialType%5D=BADWPersonDetailsPartial&tx_badwdb_badwperson%5Bper_id%5D=951&cHash=053ab8d7aab2b0405adb8d951f239ed0 | publisher = Bayerische Akademie der Wissenschaften | access-date = 7 April 2023}}</ref> the [[Royal Danish Academy of Sciences and Letters|Royal Danish Academy]] in Copenhagen (1821),{{sfn|Dunnington|2004|p=352}} the [[Royal Astronomical Society]] in London (1821),<ref>{{cite web | title = Carl Frederick Gauss | date = 30 April 1777 | url = https://www.ras.ac.uk/obituaries/Carl_Frederick/Gauss | publisher = The Royal Astronomical Society | access-date = 7 April 2023}}</ref> the [[Royal Swedish Academy of Sciences]] (1821),{{sfn|Dunnington|2004|p=352}} the [[American Academy of Arts and Sciences]] in Boston (1822),<ref name=AAAS>{{cite web | title = Book of Members, 1780–2010: Chapter G| date = 9 February 2023 | url = https://www.amacad.org/person/carl-friedrich-gauss | publisher = American Academy of Arts and Sciences | access-date = 7 April 2023}}</ref>  the [[Royal Bohemian Society of Sciences]] in Prague (1833),{{sfn|Dunnington|2004|p=353}} the [[Royal Academy of Science, Letters and Fine Arts of Belgium]] (1841/1845),<ref>[https://academieroyale.be/fr/who-who-detail/relations/hans-friedrich-karl-gauss/ Académie Royale de Belgique: Academy members]</ref> the [[Royal Society of Sciences in Uppsala]] (1843),{{sfn|Dunnington|2004|p=353}} the [[Royal Irish Academy]] in Dublin (1843),{{sfn|Dunnington|2004|p=353}} the [[Royal Netherlands Academy of Arts and Sciences|Royal Institute of the Netherlands]] (1845/ 1851),<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00000342 |title=C.F. Gauss (1797–1855) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=19 July 2015}}</ref> the [[Spanish Royal Academy of Sciences]] in Madrid (1850),<ref>{{cite web | title = Extranjeros | url = https://rac.es/sobre-nosotros/miembros/academicos-historicos/correspondiente-internacional/ | publisher = Real Academia de Ciencias | access-date = 7 April 2023}}</ref> the [[Russian Geographical Society]] (1851),{{sfn|Dunnington|2004|p=355}} the [[Austrian Academy of Sciences|Imperial Academy of Sciences]] in Vienna (1848),{{sfn|Dunnington|2004|p=355}} the [[American Philosophical Society]] (1853),<ref>{{Cite web|title=APS Member History|url=https://search.amphilsoc.org/memhist/search?year=1853;year-max=1853;smode=advanced;startDoc=1|access-date=16 April 2021|website=search.amphilsoc.org}}</ref> the [[Cambridge Philosophical Society]],{{sfn|Dunnington|2004|p=355}} and the [[Koninklijke Hollandsche Maatschappij der Wetenschappen|Royal Hollandish Society of Sciences]] in Haarlem.{{sfn|Dunnington|2004|pp=351–355}}{{sfn|Dunnington|2004|p=359}}

Both the [[University of Kazan]] and the Philosophy Faculty of the [[Charles University|University of Prague]] appointed him honorary member in 1848.{{sfn|Dunnington|2004|pp=351–355}}

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>{{cite journal | author-last = Maidron | author-first = M. E. | title = Le prix d'astronomie fondé par Lalande | journal = Revue Scientifique | volume = XIV | page = 460 | year = 1887 | place = Paris | url = https://books.google.com/books?id=biIgAQAAMAAJ&pg=PA460}}</ref> the Danish Academy of Science prize in 1823 for his memoir on conformal projection,{{sfn|Dunnington|2004|p=353}} and the [[Copley Medal]] from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".{{sfn|Dunnington|2004|p=359}}<ref>{{cite web | title = Copley Medal: Past winners | date = 30 November 2023 | url = https://royalsociety.org/medals-and-prizes/copley-medal/ | publisher = The Royal Society | access-date = 3 June 2024}}</ref><ref name="Wittmann" />

Gauss was appointed Knight of the French [[Legion of Honour]] in 1837,<ref>{{cite web | title = Gauss, Charles Frédéric | url = https://www.leonore.archives-nationales.culture.gouv.fr/ui/notice/157088 | publisher = Archives Nationales | access-date = 7 April 2023}}</ref> and became one of the first members of the Prussian [[Order Pour le Merite#Civil class|Order Pour le Merite]] (Civil class) when it was established in 1842.<ref>{{cite web | title = Carl Friedrich Gauss | url = https://www.orden-pourlemerite.de/mitglieder/carl-friedrich-gau%C3%9F?m=2&u=4 | publisher = Orden pour le Mérite für Wissenschaften und Künste | access-date = 7 April 2023}}</ref> He received the [[Order of the Crown of Westphalia]] (1810),{{sfn|Dunnington|2004|p=359}} the Danish [[Order of the Dannebrog]] (1817),{{sfn|Dunnington|2004|p=359}} the Hanoverian [[Royal Guelphic Order]] (1815),{{sfn|Dunnington|2004|p=359}} the Swedish [[Order of the Polar Star]] (1844),{{sfn|Dunnington|2004|p=354}} the [[Order of Henry the Lion]] (1849),{{sfn|Dunnington|2004|p=354}} and the [[Bavarian Maximilian Order for Science and Art]] (1853).{{sfn|Dunnington|2004|p=355}}

The Kings of Hanover appointed him the honorary titles "[[Hofrath]]" (1816){{sfn|Dunnington|2004|p=288}}  and "Geheimer Hofrath"{{efn|literally translation: ''Secrete Councillor of the Court''}} (1845). In 1949, on the occasion of his golden doctor degree jubilee, he received [[honorary citizenship]] of both Brunswick and Göttingen.{{sfn|Dunnington|2004|p=355}} 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".{{sfn|Wußing|1982|p=86}}

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>{{cite web | title = Gauss-Gesellschaft e.V. (Gauss Society) Göttingen | url = https://www.gauss-gesellschaft-goettingen.de/gauss-e.htm | access-date = 4 April 2024}}</ref>

== Names and commemorations ==
* [[List of things named after Carl Friedrich Gauss]]

== Selected writings ==
=== Mathematics and astronomy ===
[[File:Gauß-Denkmal - Braunschweig, Germany - DSC04436.JPG|thumb|upright|Statue of Gauss in [[Braunschweig|Brunswick]] (1880), made by [[Hermann Heinrich Howaldt]], designed by [[Fritz Schaper]]]]
* 1799: {{Cite book | title = Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse | trans-title = New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors of the first or second degree  | publisher = C. G. Fleckeisen  | place = Helmstedt | url =  https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B5%5D%2C%22pan%22%3A%7B%22x%22%3A0.469%2C%22y%22%3A0.692%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.422%7D   }} (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) [https://edoc.hu-berlin.de/handle/18452/732 Original book]
* 1816: {{cite journal | title = Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 3 | pages = 107–134 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B37%5D%2C%22pan%22%3A%7B%22x%22%3A0.498%2C%22y%22%3A0.444%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.657%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B268%2C269%5D%2C%22view%22%3A%22info%22%7D Original]
* 1816: {{cite journal | title = Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 3 | pages = 135–142 | url =  https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B63%5D%2C%22pan%22%3A%7B%22x%22%3A0.471%2C%22y%22%3A0.444%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.657%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B296%5D%2C%22view%22%3A%22info%22%7D Original]
* 1850: {{cite journal | title = Beiträge zur Theorie der algebraischen Gleichungen | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen | volume = 4 | pages = 34–35 | url =    
https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B77%5D%2C%22pan%22%3A%7B%22x%22%3A0.452%2C%22y%22%3A1.019%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.456%7D}}    [https://gdz.sub.uni-goettingen.de/id/PPN250442582_0004?tify=%7B%22pages%22%3A%5B287%5D%2C%22view%22%3A%22info%22%7D Original] (Lecture from 1849)
** {{Cite book | title = Die vier Gauss'schen Beweise für die Zerlegung ganzer algebraischer Funktionen in reelle Faktoren ersten und zweiten Grades. (1799–1849) | translator-last = [[Eugen Netto|Netto]] | trans-title = The four Gaussian proofs of the fundamental theorem of algebra | publisher = Wilhelm Engelmann | place = Leipzig | year = 1890 | url = https://archive.org/details/bub_gb_LoW4AAAAIAAJ/mode/2up}} (German)
* 1800: {{cite journal | title = Berechnung des Osterfestes | trans-title = Calculation of Easter | journal = Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde | volume = 2 | pages = 121–130 | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B77%5D%2C%22pan%22%3A%7B%22x%22%3A0.406%2C%22y%22%3A0.752%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.381%7D}} [https://zs.thulb.uni-jena.de/receive/jportal_jpvolume_00201970 Original]
* 1801: {{Cite book | title = Disquisitiones Arithmeticae | publisher = Gerh. Fleischer jun. | place = Leipzig | url=https://gdz.sub.uni-goettingen.de/id/PPN235993352?tify=%7B%22pages%22%3A%5B5%5D%2C%22view%22%3A%22info%22%7D}}
** {{Cite book | title = Disquisitiones Arithmeticae & other papers on number theory | translator-last = Clarke | translator-first = Arthur A. | publisher = [[Springer Science+Business Media|Springer]] | place = New York | year = 1986 | isbn = 978-0-387-96254-2 | doi = 10.1007/978-1-4939-7560-0 | edition = 2nd, corrected | url = https://link.springer.com/book/10.1007/978-1-4939-7560-0#about-this-book | last1 = Gauss | first1 = Carl Friedrich }} (translated from the [https://www.science.org/doi/10.1126/science.154.3749.642.b second German edition, Göttingen 1860])
* 1802: {{cite journal | title = Berechnung des jüdischen Osterfestes | trans-title = Calculation of Jewish Easter | journal = Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde | volume = 5 | pages = 435–437 | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B84%5D%2C%22pan%22%3A%7B%22x%22%3A0.406%2C%22y%22%3A0.752%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.381%7D}} [https://zs.thulb.uni-jena.de/rsc/viewer/jportal_derivate_00237503/Monatlich_Correspondenz_130168688_5_1802_0444%20.tif Original]
* 1804: {{cite journal | title = Über die Grenzen der geocentrischen Oerter der Planeten | trans-title = On the limits of the geocentric places of the planets | journal = Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde | volume = 10 | pages = 171–193 | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B110%5D%2C%22view%22%3A%22info%22%7D}} [https://zs.thulb.uni-jena.de/rsc/viewer/jportal_derivate_00240854/Monatliche_Correspondenz_130168688_10_0175.tif Original] (on the Zodiacus)
* 1808: {{cite journal | title = Theorematis arithmetici demonstratio nova | journal = Commentationes Societatis Regiae Scientiarum Gottingensis. Comm. Math. | volume = 16  | pages = 69–74 | url = https://gdz.sub.uni-goettingen.de/id/PPN23599524X?tify=%7B%22pages%22%3A%5B5%5D%2C%22pan%22%3A%7B%22x%22%3A0.545%2C%22y%22%3A0.469%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.622%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0016_1NS?tify=%7B%22pages%22%3A%5B498%2C499%5D%2C%22view%22%3A%22info%22%7D Original] (Introduces [[Gauss's lemma (number theory)|Gauss's lemma]], uses it in the third proof of quadratic reciprocity)
* 1808: {{Cite book | title = Methodus peculiaris elevationem poli determinandi |language=la | place = Göttingen | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B41%5D%2C%22pan%22%3A%7B%22x%22%3A0.509%2C%22y%22%3A0.541%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.561%7D}}
* 1809: {{Cite book | title = Theoria motus corporum coelestium in sectionibus conicis solem ambientium |language=la | publisher = Friedrich Perthes & Johann Heinrich Besser | place = Hamburg | url = https://gdz.sub.uni-goettingen.de/id/PPN236008730?tify=%7B%22pages%22%3A%5B7%5D%2C%22pan%22%3A%7B%22x%22%3A0.453%2C%22y%22%3A0.658%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.507%7D}} [https://gutenberg.beic.it/webclient/DeliveryManager?pid=12217180 Original book]
** {{cite book | year = 1857 | title = Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections | translator-last = Davis | translator-first = Charles Henry | translator-link = Charles Henry Davis | publisher = Little, Brown & Co. | url = https://archive.org/details/theoryofmotionof00gausrich/}}
** {{cite book | title=Theory of the motion of the celestial bodies moving around the Sun in conic sections. Reprint of the 1809 original. (Theoria motus corporum coelestium in sectionibus conicis solem ambientium.) (Latin) | url= | publisher=[[Cambridge University Press]]  | series=
Cambridge Library Collection - Mathematics | isbn=978-1-108-14311-0| zbl=1234.01016 | year=2011}}
* 1811: {{cite journal | title = Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1806, 1807, 1808, 1809 | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Math. | volume = 1 | pages = 1–26 | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B5%5D%2C%22pan%22%3A%7B%22x%22%3A0.547%2C%22y%22%3A0.399%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.731%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0001_2NS?tify=%7B%22pages%22%3A%5B170%2C171%5D%2C%22view%22%3A%22info%22%7D Original] (from 1810) (Orbit of [[2 Pallas|Pallas]])
* 1811: {{cite journal | title = Summatio quarundam serierum singularium | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 1 | pages = 1–40 | url = https://gdz.sub.uni-goettingen.de/id/PPN23599524X?tify=%7B%22pages%22%3A%5B13%5D%2C%22pan%22%3A%7B%22x%22%3A0.425%2C%22y%22%3A0.693%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.896%7D}}  [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0001_2NS?tify=%7B%22pages%22%3A%5B194%2C195%5D%2C%22view%22%3A%22info%22%7D Original] (from 1808) (Determination of the sign of the [[quadratic Gauss sum]], uses this to give the fourth proof of quadratic reciprocity)
* 1813: {{cite journal | title = Disquisitiones generales circa seriem infinitam <math>1+\frac{\alpha\beta}{\gamma.1}+\mbox{etc.}</math> | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 2 | pages = 1–42 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B131%5D%2C%22pan%22%3A%7B%22x%22%3A0.603%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0002_2NS?tify=%7B%22pages%22%3A%5B227%5D%2C%22pan%22%3A%7B%22x%22%3A0.559%2C%22y%22%3A0.496%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.456%7D Original] (from 1812, contains the [[Gauss's continued fraction]])
* 1816: {{cite journal | title = Methodus nova integralium valores per approximationem inveniendi | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 3 | pages = 39–76 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B171%5D%2C%22pan%22%3A%7B%22x%22%3A0.516%2C%22y%22%3A0.761%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B200%2C201%5D%2C%22view%22%3A%22info%22%7D Original] (from 1814)
* 1818: {{cite journal | title = Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 4 | pages = 3–20 | url = https://gdz.sub.uni-goettingen.de/id/PPN23599524X?tify=%7B%22pages%22%3A%5B51%5D%2C%22pan%22%3A%7B%22x%22%3A0.462%2C%22y%22%3A0.326%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.896%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0004_2NS?tify=%7B%22pages%22%3A%5B262%2C263%5D%2C%22view%22%3A%22info%22%7D Original] (from 1817) (Fifth and sixth proofs of quadratic reciprocity)
* 1818: {{cite journal | title = Determinatio attractionis, quam in punctum positionis datae exerceret planeta, si eius massa per totamorbitam, ratione temporis, quo singulae partes describuntur, uniformiter esset dispertita | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 4 | pages = 21–48 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B339%5D%2C%22pan%22%3A%7B%22x%22%3A0.531%2C%22y%22%3A0.592%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0004_2NS?tify=%7B%22pages%22%3A%5B281%5D%2C%22pan%22%3A%7B%22x%22%3A0.499%2C%22y%22%3A0.677%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.434%7D Original] (Only reference to the – mostly unpublished – work on the algorithm of the [[arithmetic-geometric mean]].)
* 1823: {{cite journal | title = Theoria combinationis observationum erroribus minimis obnoxiae. Pars Prior | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 5 | pages = 33–62 | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B5%5D%2C%22pan%22%3A%7B%22x%22%3A0.47%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}}  [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0005_2NS?tify=%7B%22pages%22%3A%5B276%5D%2C%22view%22%3A%22info%22%7D Original] (from 1821)
* 1823: {{cite journal | title = Theoria combinationis observationum erroribus minimis obnoxiae. Pars Posterior | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 5 | pages = 63–90 | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B31%5D%2C%22pan%22%3A%7B%22x%22%3A0.433%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0005_2NS?tify=%7B%22pages%22%3A%5B306%5D%2C%22pan%22%3A%7B%22x%22%3A0.413%2C%22y%22%3A0.66%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.413%7D Original] 
* 1825: {{cite journal | title = Allgemeine Auflösung der Aufgabe die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird | journal = Astronomische Abhandlungen | volume = 3 | place = Altona | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B193%5D%2C%22pan%22%3A%7B%22x%22%3A0.303%2C%22y%22%3A0.717%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.507%7D}} (Prize winning essay from 1822 on conformal mapping)
* 1828: {{cite book | title = Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector | trans-title = Determination of the Difference in Latitude between the Observatories of Göttingen and Altona by Observations with Ramsden's Zenith sector | publisher = Vandenhoeck und Ruprecht | year = 1828 | place = Göttingen | url = https://gdz.sub.uni-goettingen.de/id/PPN23601515X?tify=%7B%22pages%22%3A%5B7%5D%2C%22pan%22%3A%7B%22x%22%3A0.229%2C%22y%22%3A0.617%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D | language=de}} [https://books.google.com/books?id=tIg_AAAAcAAJ&pg=PA04 Original book]
* 1828: {{cite journal | title = Supplementum theoriae combinationis observationum erroribus minimis obnoxiae | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 6 | pages = 57–98 | date = 1828 | bibcode = 1828stco.book.....G | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B59%5D%2C%22pan%22%3A%7B%22x%22%3A0.433%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D | last1 = Gauss | first1 = Carl Friedrich }} (from 1826)
** {{Cite book | title = Theory of the Combination of Observations Least Subject to Errors. Part One, Part Two, Supplement (Classics in Applied Mathematics) | translator-last = G. W. Stewart | publisher = Society for Industrial and Applied Mathematics | place = Philadelphia | year = 1995 | doi=10.1137/1.9781611971248 | url = https://epubs.siam.org/doi/book/10.1137/1.9781611971248 | isbn = 978-0-89871-347-3 | last1 = Gauss | first1 = Carl Friedrich | last2 = Stewart | first2 = G. W. }} (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
* 1828: {{cite journal | title = Disquisitiones generales circa superficies curvas | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 6 | pages = 99–146 | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B221%5D%2C%22pan%22%3A%7B%22x%22%3A0.398%2C%22y%22%3A0.479%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D}}    [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0006_2NS?tify=%7B%22pages%22%3A%5B312%2C313%5D%2C%22view%22%3A%22info%22%7D Original] (from 1827)
** {{Cite book | title = General Investigations of Curved Surfaces | translator-last = J. C. Morehead and A. M. Hiltebeitel | publisher = The Princeton University Library | year = 1902 | url = https://www.gutenberg.org/files/36856/36856-pdf.pdf}}
* 1828: {{cite journal | title = Theoria residuorum biquadraticorum, Commentatio prima | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 6 | pages = 27–56 | url = https://gdz.sub.uni-goettingen.de/id/PPN23599524X?tify=%7B%22pages%22%3A%5B69%5D%2C%22pan%22%3A%7B%22x%22%3A0.365%2C%22y%22%3A0.646%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.747%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0006_2NS?tify=%7B%22pages%22%3A%5B241%5D%2C%22view%22%3A%22info%22%7D Original] (from 1825)
* 1832: {{cite journal | title = Theoria residuorum biquadraticorum, Commentatio secunda | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 7 | pages = 89–148 | url = https://gdz.sub.uni-goettingen.de/id/PPN23599524X?tify=%7B%22pages%22%3A%5B97%5D%2C%22pan%22%3A%7B%22x%22%3A0.402%2C%22y%22%3A0.326%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.896%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0007_2NS?tify=%7B%22pages%22%3A%5B273%5D%2C%22view%22%3A%22info%22%7D Original] (from 1831) (Introduces the [[Gaussian integers]], states (without proof) the law of [[biquadratic reciprocity]], proves the supplementary law for 1 + ''i'')
* 1845: {{cite journal | title = Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen | volume = Zweiter Band, von den Jahren 1842–1844 | pages = 3–46 | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B263%5D%2C%22pan%22%3A%7B%22x%22%3A0.464%2C%22y%22%3A0.808%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN250442582_0002 Original] (from 1843)
* 1847: {{cite journal | title = Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen | volume = Dritter Band, von den Jahren 1845–1847 | pages = 3–44 | url = https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22%3A%5B305%5D%2C%22pan%22%3A%7B%22x%22%3A0.384%2C%22y%22%3A0.733%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN250442582_0003 Original] (from 1846)
* 1848: {{cite journal | title = Schreiben des Herrn Geheimen Hofrathes Gauss an den Herausgeber | trans-title = Letter of Mr. Secret Councillor of the Court Gauss to the editor | journal = [[Astronomische Nachrichten]] | volume = 27 | pages = 1–3 | doi = 10.1002/asna.18480270102 | bibcode = 1848AN.....27....1G | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B198%5D%2C%22view%22%3A%22info%22%7D | date = 1848 | last1 = Gauss }} [https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?journal=AN...&year=1848&volume=..27&letter=.&db_key=GEN&page_ind=4&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES Original]
* 1903: [https://gdz.sub.uni-goettingen.de/id/PPN236018647?tify=%7B%22pages%22%3A%5B516%5D%2C%22pan%22%3A%7B%22x%22%3A0.452%2C%22y%22%3A0.567%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.946%7D Wissenschaftliches Tagebuch]  ({{cite journal | author-link1 = Felix Klein | editor-last = Klein | editor-first = Felix | title = Gauß' wissenschaftliches Tagebuch 1796–1814 | doi=10.1007/BF01449013 | year = 1903 | journal = [[Mathematische Annalen]] | volume = 57 | pages = 1–34 | s2cid = 119641638 | language = la, de | url = https://gdz.sub.uni-goettingen.de/id/PPN235181684_0057?tify=%7B%22pages%22%3A%5B8%2C9%5D%2C%22view%22%3A%22info%22%7D}}) [https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323 Original book] (from 1847, on the Zodiacus)
** {{cite journal | title = A commentary on Gauss's mathematical diary, 1796–1814 | author = [[Jeremy Gray]] | journal = Expositiones Mathematicae | volume = 2 | pages = 97–130 | year = 1984 }}

=== Physics ===
* 1804: [https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B503%5D%2C%22pan%22%3A%7B%22x%22%3A0.468%2C%22y%22%3A0.479%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.609%7D Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde] ( in original book: {{cite book | last1 = Benzenberg | first1 = Johann Friedrich | author-link = Johann Benzenberg | title = Versuche über das Gesetz des Falls, über den Widerstand der Luft und über die Umdrehung der Erde | trans-title = Experiments on the Law of falling Bodies, on the Resistance of Air, and of the Rotation of the Earth | publisher = Gebrüder Mallinckrodt | place = Dortmund | pages = 363–371}} [https://www.digitale-sammlungen.de/de/view/bsb10060427?page=392,393 Original])
* 1813: {{cite journal | title = Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. | volume = 2 | pages = 1–24 | url = https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B4%5D%2C%22pan%22%3A%7B%22x%22%3A0.405%2C%22y%22%3A0.528%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.877%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0002_2NS?tify=%7B%22pages%22%3A%5B318%5D%2C%22pan%22%3A%7B%22x%22%3A0.539%2C%22y%22%3A0.64%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.456%7D Original] (contains [[Gauss's theorem]] of vector analysis)
* 1817: {{cite journal | title = Ueber die achromatischen Doppelobjective besonders in Rücksicht der vollkommnern Aufhebung der Farbenzerstreuung | trans-title = On achromatic double lenses with special regard to a more complete dispersion of colours | journal = Zeitschrift für Astronomie und verwandte Wissenschaften | volume = IV | pages = 345–351 | language = de | url = https://babel.hathitrust.org/cgi/pt?id=hvd.hxigq8&seq=349}}
* 1829: {{cite journal | year = 1829 | title = Über ein neues allgemeines Grundgesetz der Mechanik | trans-title = On a new General Fundamental Law of Mechanics | journal = Journal für die reine und angewandte Methematik | volume = 1829 | issue = 4| pages = 232–235 | doi = 10.1515/crll.1829.4.232 | s2cid = 199545985 | url = https://zenodo.org/record/1448816}}
* 1830: {{cite journal | title = Principia generalia theoriae figurae fluidorum in statu aequilibrii | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores | volume = 7 | pages = 39–88 | url = https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B32%5D%2C%22pan%22%3A%7B%22x%22%3A0.497%2C%22y%22%3A0.694%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.424%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0007_2NS?tify=%7B%22pages%22%3A%5B223%5D%2C%22view%22%3A%22info%22%7D Original] (from 1829)
* 1841: {{cite journal | title = Intensitas vis magneticae terrestris ad mensuram absolutam revocata | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores | volume = 8 | pages = 3–44 | url = https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B82%5D%2C%22pan%22%3A%7B%22x%22%3A0.398%2C%22y%22%3A0.669%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D}}    [https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0008_2NS?tify=%7B%22pages%22%3A%5B198%5D%2C%22pan%22%3A%7B%22x%22%3A0.569%2C%22y%22%3A0.65%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.449%7D Original] (from 1832){{efn|Gauss presented the text to the Göttingen Academy in December 1832, a preprint in Latin with a small number of copies appeared in 1833. It was soon translated and published in German and French. The complete text in Latin was published in 1841.<ref name="Reich" />}}
** [http://21sci-tech.com/translations/gaussMagnetic.pdf The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement.] Translated by Susan P. Johnson.
* 1836: [https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B320%5D%2C%22pan%22%3A%7B%22x%22%3A0.412%2C%22y%22%3A0.459%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D Erdmagnetismus und Magnetometer] (Original book: {{cite book | language = de | title = Jahrbuch für 1836 | volume = 1836 | pages = 1–47 | editor = H.C. Schumacher | publisher = J.G.Cotta'sche Buchhandlung | place = Tübingen | url = https://www.digitale-sammlungen.de/de/view/bsb10538569?page=21}})
* 1840: [https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B200%5D%2C%22pan%22%3A%7B%22x%22%3A0.332%2C%22y%22%3A0.772%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte] (Original book: {{Cite book | title = Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte | trans-title = General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances | publisher = Weidmannsche Buchhandlung | place = Leipzig | year = 1840 | language = de | url = https://www.deutschestextarchiv.de/book/show/gauss_lehrsaetze_1840}}
* 1843: {{cite journal | title = Dioptrische Untersuchungen | trans-title = Dioptrical Investigations | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen | volume = Erster Band | pages = 1–34 | language = de | url = https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B248%5D%2C%22pan%22%3A%7B%22x%22%3A0.417%2C%22y%22%3A0.824%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.731%7D}} [https://gdz.sub.uni-goettingen.de/id/PPN250442582_0001?tify=%7B%22pages%22%3A%5B548%2C549%5D%2C%22view%22%3A%22info%22%7D Original] (from 1840)

==== Together with Wilhelm Weber ====
* 1837–1839: {{Cite book | title = Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1836–1838 | publisher = Dieterichsche Buchhandlung| place = Göttingen | language = de | url = https://babel.hathitrust.org/cgi/pt?id=njp.32101058086446&view=1up&seq=9}}
* 1840–1843: {{Cite book | title = Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839–1841 | publisher = Weidmannsche Verlagsbuchhandlung | place = Leipzig | language = de | url = https://babel.hathitrust.org/cgi/pt?id=njp.32101058086396&view=1up&seq=199}}
* 1840: {{Cite book | title = Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins | publisher = Weidmannsche Verlagsbuchhandlung | place = Leipzig | language = de | url = https://babel.hathitrust.org/cgi/pt?id=njp.32101058086388&view=1up&seq=5}}

=== Collected works ===
* {{Cite book| title = Carl Friedrich Gauss. Werke | editor = Königlich Preußische Akademie der Wissenschaften | volume = 1–12 | publisher = (diverse publishers) | place = Göttingen| language = la, de | year = 1863–1933 | url = https://gdz.sub.uni-goettingen.de/id/PPN235957348?tify=%7B%22view%22:%22toc%22%7D}} (includes unpublished literary estate)

=== Correspondence ===
* {{Cite book| title = Briefwechsel zwischen Gauss und Bessel | editor = Königlich Preußische Akademie der Wissenschaften |publisher = Wilhelm Engelmann| place = Leipzig | year = 1880 | language=de | url=https://archive.org/details/briefwechselzwi00berlgoog/page/n5/mode/2up}} (letters from December 1804 to August 1844)
* {{cite book | last1 = Schoenberg | first1 = Erich | last2 = Perlick | first2 = Alfons | title = Unbekannte Briefe von C. F. Gauß und Fr. W. Bessel | date = 1955 | series = Abhandlungen der Bayerischen Akademie der Wissenschaften, Math.-nat. Klasse, Neue Folge, No. 71 | pages = 5–21 | publisher = Verlag der Bayerischen Akademie der Wissenschaften | place = Munich | language = de}} (letters to [[Palm Heinrich Ludwig von Boguslawski|Boguslawski]] from February 1835 to January 1848)
* {{cite book | title = Der Briefwechsel zwischen Carl Friedrich Gauß und [[Johann Elert Bode]] | date = 2014 | editor-last = Schwemin | editor-first = Friedhelm | series = Acta Historica Astronomica | volume = 53 | publisher = Akademische Verlaganstalt | place = Leipzig | language = de | isbn = 978-3-944913-43-8}} (letters from February 1802 to October 1826)
* {{Cite book| title = Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai | editor-last = Franz Schmidt | editor-first = Paul Stäckel |publisher = B.G. Teubner| place = Leipzig | year = 1899 | language=de | url=https://archive.org/details/briefwechselzwi00gausgoog/page/n4/mode/2up}} (letters from September 1797 to February 1853; added letters of other correspondents)
* {{Cite book| title = Obgleich und indeßen. Der Briefwechsel zwischen Carl Friedrich Gauss und [[Johann Franz Encke]] | editor = Axel Wittmann |publisher = Verlag Kessel | place = Remagen | year = 2018 | language=de | isbn = 978-3945941379}} (letters from June 1810 to June 1854)
* {{Cite book| title = Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling | editor = Clemens Schaefer |publisher = Otto Elsner | place = Berlin | year = 1927 | language=de | url=https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22%3A%5B5%5D%2C%22view%22%3A%22info%22%7D}} (letters from June 1810 to June 1854)
* {{Cite book| title = Briefe zwischen A. v. Humboldt und Gauss | editor = [[Karl Christian Bruhns]] |publisher = Wilhelm Engelmann | place = Leipzig | year = 1877 | language=de | url=https://archive.org/details/briefezwischena00gausgoog/page/n5/mode/2up}} (letters from July 1807 to December 1854; added letters of other correspondents)
* {{cite book | last1 = Reich | first1 = Karin | last2 = Roussanova | first2 = Elena | author-link1 = Karin Reich | title = [[Karl Kreil]] und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext | date = 2018 | series = Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68| publisher = Verlag der Österreichischen Akademie der Wissenschaften | place = Vienna | language = de}} (letters from 1835 to 1843)
* {{cite book | title = Briefwechsel zwischen Carl Friedrich Gauß und [[Karl Ludwig von Lecoq|Carl Ludwig von Lecoq]] | date = 1959 | editor-last = Gerardy | editor-first = Theo | series = Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 | pages = 37–63 | publisher = Vandenhoeck & Ruprecht | place = Göttingen | language = de}} (letters from February 1799 to September 1800)
* {{cite journal | last = Forbes | first = Eric G. | year = 1971 | title = The Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802–05) | journal = Annals of Science | volume = 27 | issue = 3 | pages = 213–237 | doi = 10.1080/00033797100203767 | url = https://www.tandfonline.com/doi/abs/10.1080/00033797100203767}}
* {{cite journal | last = Cunningham | first = Clifford | author-link = Clifford Cunningham | year = 2004 | title = Discovery of the Missing Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802–05) | journal = Annals of Science | volume = 61 | issue = 4 | pages = 469–481 | doi = 10.1080/00033790310001660164 | url = https://www.tandfonline.com/doi/abs/10.1080/00033790310001660164}}
* {{cite book | date = 1900 | editor = Carl Schilling | series = [[Wilhelm Olbers]]. Sein Leben und seine Werke. Zweiter Band | title = Briefwechsel zwischen Olbers und Gauss: Erste Abtheilung | publisher = Julius Springer | place = Berlin  | language = de| url = https://archive.org/details/p1wilhelmolberss02olbeuoft/page/n5/mode/2up}} (letters from January 1802 to October 1819)
* {{cite book | date = 1909 | editor = Carl Schilling | series = Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band | title = Briefwechsel zwischen Olbers und Gauss: Zweite Abtheilung | publisher = Julius Springer | place = Berlin  | language = de| url = https://archive.org/details/p2wilhelmolberss02olbeuoft/page/n7/mode/2up}} (letters from January 1820 to May 1839; added letters of other correspondents)
* {{Cite book| title = Briefwechsel zwischen C. F. Gauss und [[H. C. Schumacher]] | editor = [[Christian August Friedrich Peters]] |publisher = Gustav Esch | place = Altona | year = 1860–1865 | language = de}}
** [https://books.google.com/books?id=aIRtAAAAMAAJ Volumes 1+2] (letters from April 1808 to March 1836)
** [https://books.google.com/books?id=NDcDAAAAQAAJ Volumes 3+4] (letters from March 1836 to April 1845)
** [https://books.google.com/books?id=3jEDAAAAQAAJ Volumes 5+6] (letters from April 1845 to November 1850)
* {{cite book | title = Briefwechsel zwischen Carl Friedrich Gauß und [[Eberhard August Wilhelm Zimmermann|Eberhard August Zimmermann]] | date = 1987 | editor-last = Poser | editor-first = Hans | series = Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 | publisher = Vandenhoeck & Ruprecht | place = Göttingen | language = de | isbn = 978-3525821169}} (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">{{cite web | url = https://gauss.adw-goe.de/?locale-attribute=en | title = The complete correspondence of Carl Friedrich Gauss | publisher = Akademie der Wissenschaften zu Göttingen | access-date = 10 March 2023}}</ref> The literary estate is kept and provided by the [[Göttingen State and University Library]].<ref>{{Cite journal | last = Rohlfing | first = Helmut | title = Das Erbe des Genies. Der Nachlass Carl Friedrich Gauß an der Niedersächsischen Staats- und Universitätsbibliothek Göttingen | journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 40 | pages = 7–23 | year = 2003 | volume = 35 | bibcode = 1998GGMit..35...49S | language = de}}</ref> Written materials from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.<ref>{{cite web | url = https://www.stadtarchiv-braunschweig.findbuch.net/php/main.php#4720495820303231 | title = Familienarchiv Gauß | website = Stadtarchiv Braunschweig | department = Signature: G IX 021 | access-date = 25 March 2023}}</ref>

== References ==
=== Notes ===
{{notelist}}

===Citations===
{{Reflist}}

=== Sources ===
{{refbegin|indent=yes}}
* {{cite book |last=Bachmann |first=Paul |author-link=Paul Gustav Heinrich Bachmann |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 1) |chapter=Über Gauss' zahlentheoretische Arbeiten |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B7%5D%2C%22pan%22%3A%7B%22x%22%3A0.127%2C%22y%22%3A0.73%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.39%7D |language=de |year=1922}}
* {{cite book |last=Bolza |first=Oskar |author-link=Oskar Bolza |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 5) |chapter=Gauss und die Variationsrechnung |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B457%5D%2C%22view%22%3A%22toc%22%7D |language=de |year=1921}}
* {{cite book |last=Brendel |first=Martin |author-link=Martin Brendel |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=XI, 2 (Abhandlung 3) |chapter=Über die astronomischen Arbeiten von Gauss |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236059505?tify=%7B%22pages%22%3A%5B396%2C397%5D%2C%22view%22%3A%22info%22%7D |language=de |year=1929}}
* {{Cite book |title=Gauss: A Biographical Study |first=Walter Kaufmann |last=Bühler |publisher=Springer-Verlag |year=1981 |isbn=978-0-387-10662-5 |url=https://books.google.com/books?id=ktruAAAAMAAJ}}
* {{cite book |last=Dunnington |first=G. Waldo |author-link=G. Waldo Dunnington |title=Carl Friedrich Gauss: Titan of Science |url=https://books.google.com/books?id=4mwSrfxBSzkC |year=2004 |publisher=The Mathematical Association of America |isbn=978-0-88385-547-8 |oclc=53933110}} First edition: {{cite book |title=Carl Friedrich Gauss: Titan of Science. A Study of his Life and Work |year=1955 |publisher=Exposition Press |place=New York}}
** {{cite book |first1=Jeremy |last1=Gray |author-link=Jeremy Gray |chapter=Introduction to Dunnington's "Gauss" |title=Carl Friedrich Gauss: Titan of Science. A Study of his Life and Work |pages=xix–xxvi |year=1955 |publisher=Exposition Press |place=New York}} With a critical view on Dunnington's style and appraisals
* {{cite book |last=Galle |first=Andreas |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=XI, 2 (Abhandlung 1) |chapter=Über die geodätischen Arbeiten von Gauss |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236059505?tify=%7B%22pages%22%3A%5B7%5D%2C%22view%22%3A%22info%22%7D |language=de |year=1924}}
* {{cite book |last=Geppert |first=Harald |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 7) |chapter=Über Gauss' Arbeiten zur Mechanik und Potentialtheorie |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B635%5D%2C%22pan%22%3A%7B%22x%22%3A0.506%2C%22y%22%3A0.4%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.73%7D |language=de |year=1933}}
* {{cite book |last=Klein |first=Felix |chapter=The Development of Mathematics at the German Universities |title=Lectures on Mathematics |publisher=Macmillan and Co. |place=New York, London |pages=99–101 |year=1894 |chapter-url=https://archive.org/details/lecturesonmathem00klei/page/n108/mode/2up}}
* {{Cite book |title=Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Teil 1 |trans-title=Lectures on the Development of Mathematics in the 19th Century |first=Felix |last=Klein |author-link=Felix Klein |series=Grundlehren der mathematischen Wissenschaften 24 |publisher=Springer-Verlag |orig-date=1926 |date=1979 |place=Berlin, Heidelberg, New York |language=de |isbn=3-540-09234-X |url=https://gdz.sub.uni-goettingen.de/id/PPN375425993}}
* {{cite book |last=Maennchen |first=Philipp |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 6) |chapter=Gauss als Zahlenrechner |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B555%5D%2C%22pan%22%3A%7B%22x%22%3A0.127%2C%22y%22%3A0.73%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.38%7D |language=de |year=1930}}
* {{cite journal |author-last=O'Hara |author-first=James Gabriel |title=Gauss and the Royal Society: The Reception of his Ideas on Magnetism in Britain |journal=[[Notes and Records of the Royal Society]] |volume=38 |issue=1 |pages=17–78 |year=1983 |doi=10.1098/rsnr.1983.0002 |url=https://royalsocietypublishing.org/doi/10.1098/rsnr.1983.0002}}
* {{cite book |last=Ostrowski |first=Alexander |author-link=Alexander Ostrowski |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 3) |chapter=Über den ersten und vierten Gaussschen Beweis des Fundamentalsatzes der Algebra |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B309%5D%2C%22pan%22%3A%7B%22x%22%3A0.203%2C%22y%22%3A0.554%7D%2C%22view%22%3A%22toc%22%2C%22zoom%22%3A0.462%7D |language=de |year=1920}}
* {{cite book |last=Sartorius von Waltershausen |first=Wolfgang |author-link=Wolfgang Sartorius von Waltershausen |title=Gauss zum Gedächtniss |language=de |url=https://archive.org/details/bub_gb_h_Q5AAAAcAAJ/page/n4/mode/1up |publisher=S. Hirzel |year=1856}}
** {{Cite book |title=Carl Friedrich Gauss. A Memorial |translator-last=Worthington Gauss |translator-first=Helen |location=Colorado Springs |publisher=Colorado College |year=1966 |url=https://ia600405.us.archive.org/20/items/gaussmemorial00walt/gaussmemorial00walt.pdf}}
* {{Cite book |last=Schaaf |first=William L. |title=Carl Friedrich Gauss: Prince of Mathematicians |publisher=Franklin Watts |place=New York |year=1964 |url=https://archive.org/details/carlfriedrichgau0000unse}}
* {{cite book |last=Schaefer |first=Clemens |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=XI, 2 (Abhandlung 2) |chapter=Über Gauss' physikalische Arbeiten |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236059505?tify=%7B%22pages%22%3A%5B176%2C177%5D%2C%22view%22%3A%22info%22%7D |language=de |year=1929}}
* {{cite book |last=Schlesinger |first=Ludwig |author-link=Ludwig Schlesinger |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 2) |chapter=Über Gauss' Arbeiten zur Funktionentheorie |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B85%5D%2C%22view%22%3A%22info%22%7D |language=de |year=1933}}
* {{cite book |last=Stäckel |first=Paul |author-link=Paul Stäckel |editor=Königlich Preußische Akademie der Wissenschaften |title=Carl Friedrich Gauss. Werke |volume=X, 2 (Abhandlung 4) |chapter=Gauss als Geometer |chapter-url=https://gdz.sub.uni-goettingen.de/id/PPN236019856?tify=%7B%22pages%22%3A%5B331%5D%2C%22view%22%3A%22toc%22%7D |language=de |year=1917}}
* {{cite book |last1=Stuloff |first1=Nikolai |date=1964 |editor=Historische Kommmission der Bayerischen Akademie der Wissenschaften |editor-link=Bavarian Academy of Sciences and Humanities |series=[[Neue Deutsche Biographie]] |volume=6 |pages=101–107 |title=Gauß, Carl Friedrich |publisher=Duncker & Humblot |place=Berlin |language=de |url=https://www.deutsche-biographie.de/gnd104234644.html#ndbcontent |isbn=}}
* {{cite book |last=Wußing |first=Hans |author-link=Hans Wußing |title=Carl Friedrich Gauß |year=1982 |place=Leipzig |edition=4 |publisher=BSB B. G. Teubner |language=de}}
{{Refend}}

== Further reading ==
{{refbegin|indent=yes}}
* {{Cite book |author2-link=Carl Benjamin Boyer |last2=Boyer |first2=Carl B. |author1-link=Uta Merzbach |last1=Merzbach |first1=Uta C. |title=A History of Mathematics |url=https://books.google.com/books?id=bR9HAAAAQBAJ |publisher=John Wiley & Sons |place=New Jersey |edition=3rd |year=2011 |isbn=978-0470630563}}
* {{cite book |last=Hall |first=Tord |author-link=Tord Hall |title=Carl Friedrich Gauss: A Biography |place=Cambridge, MA |publisher=[[MIT Press]] |year=1970 |isbn=978-0-262-08040-8 |oclc=185662235 |url=https://archive.org/details/carlfriedrichgau00tord}}
* {{cite book |last=Nahin |first=Paul J. |author-link=Paul J. Nahin |title=An Imaginary Tale: The Story of √-1 |url=https://books.google.com/books?id=OPyPwaElDvUC&pg=PA82 |date=2010 |publisher=Princeton University Press |isbn=9 78-1-4008-3389-4}}
* {{cite book |last=Simmons |first=J. |title=The Giant Book of Scientists: The 100 Greatest Minds of All Time |publisher=The Book Company |place=Sydney |year=1996}}
* {{cite book |last=Tent |first=Margaret |title=The Prince of Mathematics: Carl Friedrich Gauss |publisher=A. K. Peters |isbn=978-1-56881-455-1 |year=2006}}

=== Fictional ===
* {{cite book |last=Kehlmann |first=Daniel |author-link=Daniel Kehlmann |title=Die Vermessung der Welt |publisher=Rowohlt |year=2005 |isbn=978-3-498-03528-0 |oclc=144590801 |title-link=Measuring the World |language=de}}
** {{Cite book |title=[[Measuring the World]] |author-last=Kehlmann |author-first=Daniel |translator-last=Janeway |translator-first=Carol Brown |translator-link=Carol Brown Janeway |place= |year=2006}}
{{Refend}}

== External links ==
{{commons}}
{{wikisource|Author:Carl Friedrich Gauss|Carl Friedrich Gauss}}
{{wikiquote}}
{{Library resources box|onlinebooks=yes|about=yes|by=yes}}
{{refbegin|indent=yes}}
* {{Internet Archive author |sname=Carl Friedrich Gauss}}
* [http://adsabs.harvard.edu/cgi-bin/nph-abs_connect?db_key=AST&db_key=PRE&qform=AST&arxiv_sel=astro-ph&arxiv_sel=cond-mat&arxiv_sel=cs&arxiv_sel=gr-qc&arxiv_sel=hep-ex&arxiv_sel=hep-lat&arxiv_sel=hep-ph&arxiv_sel=hep-th&arxiv_sel=math&arxiv_sel=math-ph&arxiv_sel=nlin&arxiv_sel=nucl-ex&arxiv_sel=nucl-th&arxiv_sel=physics&arxiv_sel=quant-ph&arxiv_sel=q-bio&sim_query=YES&ned_query=YES&adsobj_query=YES&aut_logic=OR&obj_logic=OR&author=Gauss%0D%0AGau%C3%9F&object=&start_mon=&start_year=&end_mon=12&end_year=1965&ttl_logic=OR&title=&txt_logic=OR&text=&nr_to_return=200&start_nr=1&jou_pick=ALL&ref_stems=&data_and=ALL&group_and=ALL&start_entry_day=&start_entry_mon=&start_entry_year=&end_entry_day=&end_entry_mon=&end_entry_year=&min_score=&sort=SCORE&data_type=SHORT&aut_syn=YES&ttl_syn=YES&txt_syn=YES&aut_wt=1.0&obj_wt=1.0&ttl_wt=0.3&txt_wt=3.0&aut_wgt=YES&obj_wgt=YES&ttl_wgt=YES&txt_wgt=YES&ttl_sco=YES&txt_sco=YES&version=1 Publications of  C.&nbsp;F.&nbsp;Gauss] in [[Astrophysics Data System]]
* {{cite journal|url=http://adsabs.harvard.edu//full/seri/MNRAS/0016//0000080.000.html |journal=Monthly Notices of the Royal Astronomical Society |volume=16 |year=1856|issue=80|title= Obituary| access-date = 8 April 2024}}
* {{cite web|url=http://www.bbc.co.uk/programmes/b00ss0lf |title=Carl Friedrich Gauss |series=A Brief History of Mathematics|website=BBC 4 | access-date = 8 April 2024}}
* {{Cite web|url=http://american_almanac.tripod.com/gauss.htm|title=Mind Over Mathematics: How Gauss Determined The Date of His Birth|website=american_almanac.tripod.com | access-date = 8 April 2024}}
* {{cite web|url = https://mathoverflow.net/questions/370190/how-did-gauss-characterize-the-metrical-relations-in-the-uniform-4-4-4-tiling|title=On Gauss' tesselation of the unit disk | access-date = 8 April 2024}}
* {{cite web|url=https://www.uni-goettingen.de/en/62596.html |title=Carl Friedrich Gauß|place=Göttingen University | access-date = 8 April 2024}}
* {{cite web|url = http://www.coloradocollege.edu/library/SpecialCollections/CenturyChest/transcription24.html
 |title= ''Descendants of Gauss'' |website=www.coloradocollege.edu |archive-url= https://web.archive.org/web/20100528042818/http://www.coloradocollege.edu/library/SpecialCollections/CenturyChest/transcription24.html | access-date = 4 June 2024|archive-date= 28 May 2010 }}
* {{cite web|url=http://www.gausschildren.org/cfgauss/cfgauss.htm |title=Gauss|website=www.gausschildren.org | access-date = 8 April 2024}}
* [https://denkmalatlas.niedersachsen.de/viewer/themen/Gauss-Steine/ Carl Friedrich Gauss – Spuren seines Lebens] (Places used as points for triangulation)
{{refend}}

{{Carl Friedrich Gauss}}
{{Copley Medallists 1801–1850}}
{{Scientists whose names are used as non SI units}}
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{{DEFAULTSORT:Gauss, Carl Friedrich}}
[[Category:Carl Friedrich Gauss| ]]
[[Category:1777 births]]
[[Category:1855 deaths]]
[[Category:18th-century German astronomers]]
[[Category:18th-century German mathematicians]]
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[[Category:Differential geometers]]
[[Category:Hyperbolic geometers]]
[[Category:German optical physicists]]
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[[Category:People from the Duchy of Brunswick]]
[[Category:Scientists from Braunschweig]]
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