View source for Biography:Carl Friedrich Gauss

{{short description|German mathematician and physicist (1777–1855)}}
{{Infobox scientist
| image             = Carl Friedrich Gauss 1840 by Jensen.jpg
| 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       = 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 and sciences
| alma_mater        = {{plainlist|
* 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  = [[Biography:Johann Friedrich Pfaff|Johann Friedrich Pfaff]]
| known_for         = 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)
* [[Physics:Copley Medal|Copley Medal]] (1838)
}}
| doctoral_students = {{collapsible list|title={{nothing}}
| [[Biography:Richard Dedekind|Richard Dedekind]]
| [[Biography:Christian Ludwig Gerling|Christian Ludwig Gerling]]
| [[Biography:Wilhelm Klinkerfues|Wilhelm Klinkerfues]]
| [[Biography:Johann Benedict Listing|Johann Benedict Listing]]
| [[Biography:Bernhard Riemann|Bernhard Riemann]]
| August Ritter
| Karl von Staudt
}}
| notable_students  = {{collapsible list|title={{nothing}}
| [[Biography:Gotthold Eisenstein|Gotthold Eisenstein]]
| Johann Franz Encke
| [[Biography:Carl Wolfgang Benjamin Goldschmidt|Carl Wolfgang Benjamin Goldschmidt]]
| [[Biography:Adolph Theodor Kupffer|Adolph Theodor Kupffer]]
| [[Biography:August Ferdinand Möbius|August Ferdinand Möbius]]
| Moritz Stern
| Georg Frederik Ursin
| Moritz Ludwig Wichmann
}}
| signature         = Carl Friedrich Gauß signature.svg
| footnotes         =
}}
'''Johann Carl Friedrich Gauss''' ({{lang-de|link=no|Gauß}} {{IPA-de|kaʁl ˈfʁiːdʁɪç ˈɡaʊs||De-carlfriedrichgauss.ogg}};<ref>{{cite book|author1=Dudenredaktion|last2=Kleiner|first2=Stefan|last3=Knöbl|first3=Ralf|year=2015|orig-date=1962|title=Das Aussprachewörterbuch|trans-title=The Pronunciation Dictionary| url=https://books.google.com/books?id=T6vWCgAAQBAJ| language=de|edition=7th|location=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=Walter de Gruyter|location=Berlin|isbn=978-3-11-018202-6|pages=402, 520, 529}}</ref> {{lang-la|Carolus Fridericus Gauss}}; 30 April 1777{{spaced ndash}}23 February 1855) was a German [[Biography:Mathematician|mathematician]], [[Geodesy|geodesist]], and [[Physicist|physicist]] who made significant contributions to many fields in mathematics and science. Gauss ranks among history's most influential mathematicians and has been referred to as the "Prince of Mathematicians".<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=T he 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>

Gauss was a [[Philosophy:Child prodigy|child prodigy]] in mathematics. While still a student at the University of Göttingen, he propounded several mathematical [[Theorem|theorem]]s. Gauss completed his masterpieces ''[[Disquisitiones Arithmeticae]]'' and ''Theoria motus corporum coelestium'' as a private scholar. Later he was director of the Göttingen Observatory and professor at the university for nearly half a century, from 1807 until his death in 1855.

Gauss published the second and third complete proofs of the [[Fundamental theorem of algebra|fundamental theorem of algebra]], made contributions to [[Number theory|number theory]] and developed the theories of binary and ternary quadratic forms. He is credited with inventing the [[Fast Fourier transform|fast Fourier transform]] algorithm and was instrumental in the discovery of the dwarf planet [[Astronomy:Ceres (dwarf planet)|Ceres]].<ref>{{Cite journal |last1=Heideman· |first1=Michael T. |last2=Johnson |first2=Don H. |last3=Burrus |first3=C. Sydney |date=1985-09-01 |title=Gauss and the History of the Fast Fourier Transform |url=https://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf |journal=Archive for History of Exact Sciences |volume=34 |issue=3 |pages=265–277|doi=10.1007/BF00348431 }}</ref> His work on the motion of planetoids disturbed by large planets led to the introduction of the [[Physics:Gaussian gravitational constant|Gaussian gravitational constant]] and the [[Least squares|method of least squares]], which he discovered before [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] published on the method,<ref>{{Cite book |last=Schaaf |first=William L. |url=https://archive.org/details/carlfriedrichgau0000unse |title=Carl Friedrich Gauss: Prince of Mathematicians |publisher=Franklin Watts |year=1964 |pages=84 |language=en}}</ref><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}}</ref><ref name=":4">{{Cite journal |last=Stigler |first=Stephen M. |date=1981 |title=Gauss and the Invention of Least Squares |journal=The Annals of Statistics |volume=9 |issue=3 |pages=465–474 |doi=10.1214/aos/1176345451 |jstor=2240811 |issn=0090-5364|doi-access=free }}</ref><ref>{{Cite web |last=Lim |first=Milton |date=2021-03-31 |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=2023-10-14 |website=Actuaries Digital}}</ref> and which is still used in all sciences to minimize measurement error. He also anticipated [[Non-Euclidean geometry|non-Euclidean geometry]], and was the first to analyze it, even coining the term.<ref name=":5">{{Cite book |last=Klein |first=Felix |url=https://archive.org/details/elementarymathem0000klei/mode/1up |title=Elementary Mathematics from an Advanced Standpoint: Geometry |publisher=Dover Publications |year=1924 |isbn=978-0486434810 |pages=176–177 |language=en |author-link=Felix Klein}}</ref><ref name=":0" /><ref name=":1">{{Cite book |last=Schaaf |first=William L. |url=https://archive.org/details/carlfriedrichgau0000unse |title=Carl Friedrich Gauss: Prince of Mathematicians |publisher=Franklin Watts |year=1964 |pages=128–140 |language=en}}</ref> He is considered one of its discoverers alongside [[Biography:Nikolai Lobachevsky|Nikolai Lobachevsky]] and [[Biography:János Bolyai|János Bolyai]].<ref>{{Cite journal |last1=Jenkovszky |first1=László |last2=Lake |first2=Matthew J. |last3=Soloviev |first3=Vladimir |date=2023-03-12 |title=János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword |journal=Symmetry |language=en |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 invented the [[Engineering:Heliotrope (instrument)|heliotrope]] in 1821,<ref>{{Cite book |last=Schaaf |first=William L. |url=https://archive.org/details/carlfriedrichgau0000unse |title=Carl Friedrich Gauss: Prince of Mathematicians |publisher=Franklin Watts |year=1964 |pages=81 |language=en}}</ref> a [[Physics:Magnetometer|magnetometer]] in 1833 and, alongside [[Biography:Wilhelm Eduard Weber|Wilhelm Eduard Weber]], invented the first electromagnetic telegraph in 1833.<ref>{{Cite book |last=Schaaf |first=William L. |url=https://archive.org/details/carlfriedrichgau0000unse |title=Carl Friedrich Gauss: Prince of Mathematicians |publisher=Franklin Watts |year=1964 |pages=115–127 |language=en}}</ref>

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

== Biography ==
=== Youth and education ===
[[File:Braunschweig Brunswick Geburtshaus CF Gauss (1914).jpg|thumb|upright|House of birth in Brunswick (destroyed in World War II)]]
[[File:Carl Friedrich Gauß, Karikatur von Abraham Gotthelf Kästner, 1795.jpg|thumb|Caricature of Abraham Gotthelf Kästner by Gauss (1795)]]
Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to a family of lower social status.<ref>Rudolf Borch (1929) : ''Ahnentafel des Mathematikers Carl Friedrich Gauß'' [Genealogical table]. Ahnentafeln Berühmter Deutscher, Vol. 1, pp. 63–65. Zentralstelle für Deutsche Personen- und Familiengeschichte (ed.)</ref> His father Gebhard Dietrich Gauss (1744–1808) worked in several jobs, as butcher, bricklayer, gardener, and as treasurer of a death-benefit fund. Gauss characterized his father as an honourable and respected man, but rough and dominating at home. He was experienced in writing and calculating, but his wife Dorothea (1743–1839), Carl Friedrich's mother, was nearly illiterate. Carl Friedrich was christened and [[Religion:Confirmation|confirmed]] in a church{{efn|St. Catherine's church}} near the school that he attended as a child.<ref>{{cite web | first1 = Susan | last1 = Chamberless | url = http://www.gausschildren.org/genwiki/index.php?title=Letter:WORTHINGTON,_Helen_to_Carl_F._Gauss_-_1911-07-26 | title = Letter:Worthington, Helen to Carl F. Gauss – 26 July 1911 | publisher = Susan D. Chamberless | date = 11 March 2000 | access-date = 14 September 2011}}</ref> He had one elder brother from his father's first marriage.

Gauss was a [[Philosophy:Child prodigy|child prodigy]] in the field of mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick, who sent him to the local ''Collegium Carolinum'',{{efn|The ''Collegium Carolinum'' was the preceding institution of the ''Technische Hochschule Braunschweig'', now Braunschweig Institute of Technology, but at Gauss' time not equal to a university.}} which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and [[Social:Classics|classical languages]] at the Hanoverian University of Göttingen until 1798.<ref name="scientificmonthly"/> It is not known why Gauss went to Göttingen and not to the University of Helmstedt near his native Brunswick, but it is assumed that the large library of Göttingen, where students were allowed to borrow books and take them home, was the decisive reason.{{sfn|Dunnington|2004|pp=398–404}} One of his professors in mathematics was [[Biography:Abraham Gotthelf Kästner|Abraham Gotthelf Kästner]], whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams.{{sfn|Dunnington|2004|p=24}} Gauss depicted him in a drawing showing a lecture scene where he produced errors in a simple calculation. Astronomy was taught by Karl Felix von Seyffer (1762–1822), with whom Gauss stayed in correspondence after graduation; [[Biography:Heinrich Wilhelm Matthias Olbers|Olbers]] and Gauss mocked him in their correspondence. On the other hand, he thought highly of [[Biography:Georg Christoph Lichtenberg|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 [[Biography:Heinrich Wilhelm Brandes|Heinrich Wilhelm Brandes]].

Though being a registered student at university, it is evident that he was a self-taught student in mathematics, since he independently rediscovered several theorems.<ref>{{MacTutor Biography|id=Gauss}}</ref> He succeeded with a breakthrough in a geometrical problem that had occupied mathematicians since the days of the {{wipe|Ancient Greece}} when he determined in 1796 which regular [[Polygon|polygon]]s can be constructed by compass and straightedge. This discovery was the subject of his first publication and ultimately led Gauss to choose mathematics instead of [[Philology|philology]] as a career.{{sfn|Dunnington|2004|p=28}} Gauss' mathematical diary shows that, in the same year, he was also productive in number theory. He made advanced discoveries in [[Modular arithmetic|modular arithmetic]], found the first proof of the [[Quadratic reciprocity|quadratic reciprocity]] law, and dealt with the [[Prime number theorem|prime number theorem]]. Many ideas for his mathematical magnum opus ''Disquisitiones arithmeticae'', published in 1801, date from this time.

=== Private scholar ===
Gauss graduated as a [[Social:Doctor of Philosophy|Doctor of Philosophy]] in 1799. He did not graduate from Göttingen, as is sometimes stated,{{efn|This error occurs for example in Marsden (1977).<ref name="Marsden" />}}<ref name="Marsden" /> but rather, at the Duke of Brunswick's special request, from the University of Helmstedt, the only state university of the duchy. There, [[Biography:Johann Friedrich Pfaff|Johann Friedrich Pfaff]] assessed his doctoral thesis, and Gauss got the degree ''in absentia'' without the further oral examination that was usually requested. The Duke then granted him his cost of living as a private scholar in Brunswick. Gauss showed his gratitude and loyalty for this bequest when he refused several calls from the Russian Academy of Sciences in St. Peterburg and from Landshut University. Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans:<ref>{{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 mortally wounded in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped. He then followed a call to the University of Göttingen, an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory.

Studying the calculation of asteroid orbits, Gauss established contact with the astronomical community of [[Place:Bremen|Bremen]] and Lilienthal, especially Wilhelm Olbers, [[Biography:Karl Ludwig Harding|Karl Ludwig Harding]] and Friedrich Wilhelm Bessel, as part of the informal group of astronomers known as the [[Astronomy:Celestial police|Celestial police]].{{sfn|Dunnington|2004|pp=50, 54–55, 74–77}} One of their aims was the discovery of further planets, and they assembled data on asteroids and comets as a basis for Gauss's research. Gauss was thereby able to develop new, powerful methods for the determination of orbits, which he later published in his astronomical magnum opus ''Theoria motus corporum coelestium'' (1809).

=== Professor in Göttingen ===
[[File:Goettingen Sternwarte 01.jpeg|thumb|Old Göttingen observatory, circa 1800]]
[[File:Carl Friedrich Gauss on his Deathbed, 1855.jpg|thumb|Gauss on his deathbed (1855)]]
Gauss arrived at Göttingen in November 1807, and in the following years he was confronted with the demand for two thousand francs from the Westphalian government as a war contribution. Without having yet received his salary, he could not raise this enormous amount. Both Olbers and [[Biography: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 [[Religion:Prince-primate|Prince-primate]] Dalberg,{{sfn|Dunnington|2004|pp=86–87}} paid the sum.

Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower,{{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 the building was not finished until October 1816. It contained new up-to-date instruments, for instance two [[Astronomy:Meridian circle|meridian circle]]s from Repsold{{sfn|Brendel|1929|p=84}} and [[Biography:Georg Friedrich von Reichenbach|Reichenbach]],{{sfn|Brendel|1929|p=119}} and a [[Astronomy:Heliometer|heliometer]] from [[Biography:Joseph von Fraunhofer|Fraunhofer]].{{sfn|Brendel|1929|p=56}}

The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: in the first two decades of the 19th century astronomy was the main focus, in the third decade geodesy, and in the fourth decade he occupied himself with physics, mainly magnetism.{{sfn|Klein|1979|p=7}}

Gauss remained mentally active into his old age, even while suffering from [[Medicine:Gout|gout]] and general unhappiness. His last observation was the solar eclipse of July 28, 1851.{{sfn|Brendel|1929|p=144}} On 23 February 1855, Gauss died of a heart attack in Göttingen;{{sfn|Dunnington|2004|p=24}} he is interred in the Albani Cemetery there. Heinrich Ewald, Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer, gave eulogies at his funeral.

=== Gauss'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}}.<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 | location = 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 | location = 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> The cerebral area was determined by Wagner's son Hermann in his doctoral thesis to be {{convert|219588|mm2|abbr=out}}.<ref>{{Cite book | last = Wagner | first = Hermann |  title =  Maassbestimmungen der Oberfläche des grossen Gehirns | publisher = Georg H. Wigand | year = 1864 | location  = 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> Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation for his genius.<ref name=bardi>{{Cite book| last = Bardi | first = Jason | title = The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe | publisher = John Wiley & Sons, Inc. | year = 2008 | page = 189 | isbn = 978-0-470-46736-7}}</ref> After various previous investigations, a [[Physics:Magnetic resonance imaging of the brain|magnetic resonance]] study of 1998, done at the [[Organization:Max Planck Institute for Biophysical Chemistry|Max Planck Institute for Biophysical Chemistry]] in Göttingen, gave no results which could be used to explain his mathematical abilities.<ref>Wolfgang Hänicke, [[Biography:Jens Frahm|Jens Frahm]] und Axel D. Wittmann: [https://web.archive.org/web/20110719065116/http://www.mpibpc.mpg.de/inform/MpiNews/cientif/jahrg5/12.99/scta.html ''Magnetresonanz-Tomografie des Gehirns von Carl Friedrich Gauß.''] In: ''MPI News&nbsp;5'', Heft&nbsp;12, 1999</ref>

In 2013, a neurobiologist at the same institute discovered that Gauss's brain had been mixed up, due to mislabelling, with that of the physician {{interlanguage link|Conrad Heinrich Fuchs|de}}, who died in Göttingen a few months after Gauss.<ref>{{cite journal | last1 = Schweizer | first1 = Renate | last2 = Wittmann | first2 = Axel | last3 = Frahm | first3 = Jens |  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 on 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:Therese Staufenau geb. Gauß, 008.jpg|thumb|upright|Therese Gauss (1834) by Ludwig Becker]]
Gauss married Johanna Osthoff (1780–1809) on 9 October 1805.<ref>{{cite web|url= https://www.ancestry.com/genealogy/records/johanna-osthoff-24-30lwhs|title=Johanna Osthoff 1780–1809 – Ancestry|website=www.ancestry.com|language=en-us|access-date=10 December 2017}}</ref> They had two sons and a daughter: Joseph (1806–1873), Wilhelmina (1808–1840) and Louis (1809–1810). Johanna died on 11 October 1809 one month after the birth of Louis, who himself died a few months later.

Gauss remarried within a year, on 4 August 1810, to Wilhelmine (Minna) Waldeck (1788–1831), a friend of his first wife. They had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879) and {{ill|Therese Staufenau|de}} (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|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 the actor Constantin Staufenau.{{sfn|Dunnington|2004|p=374}} Her sister Wilhelmina married the orientalist Heinrich Ewald.{{sfn|Dunnington|2004|p=206}} Gauss' mother Dorothea lived in his house from 1817 until her death in 1839.<ref name="scientificmonthly"/>

The eldest son Joseph, whilst still a schoolboy, helped his father as an assistant during his survey campaign in summer 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways. In 1836 he studied the railroad system in the US for some months.<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 joined the army for five years. He then worked for the American Fur Company in the Midwest, where he learned the Sioux language. Later, he moved to Missouri and became a successful businessman.<ref name="Gerardy" /> Wilhelm married a niece of the astronomer Friedrich Bessel and also moved to Missouri in 1837,<ref>{{cite journal | author-last = Wolf | author-first = Armin | title = Der Pädagoge und Philosoph Johann Conrad Fallenstein (1731–1813) – Genealogische Beziehungen zwischen [[Biography:Max Weber|Max Weber]], Gauß und Bessel | journal = Genealogie | volume = 7 | year = 1964 | language = de | pages = 266–269}}</ref> starting as a farmer and later becoming wealthy in the shoe business in St. Louis.<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 descendants left in Germany all derive from Joseph, as the Gauss daughters had no children.<ref name="Gerardy" />

=== Personality ===
==== The scholar ====
[[File:GaussSiegel1777-1855.png|thumb|upright|Gauss' [[Seal (emblem)|seal]]]]
At the end of the 18th century, German academic mathematics was in a poor condition: the prolific mathematicians of that time worked in France and other European countries.{{efn|Gauss himself, in a letter to Bolyai, complained about "the shallowness that is so dominating in our contemporary mathematics".<ref>{{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 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>}} The mathematical mainstream mainly dealt with solving practical problems in mechanics, astronomy, geodesy, etc.<ref>{{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-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> In this scientific environment, Gauss can be seen, following [[Biography:Felix Klein|Felix Klein]], as typical of both 18th and 19th-century mathematicians. His interest in practical applicability, for example in geodesy and astronomy, qualified Gauss to be taken as a typical applied mathematician of the [[Philosophy:Age of Enlightenment|century of enlightenment]]. On the other hand, he began research in numerous parts of mathematics without defined links to practical purposes, and thus showed himself as a pioneer of what was later called "[[Pure mathematics|pure mathematics]]". In contrast to earlier mathematicians, such as [[Biography:Leonhard Euler|Leonhard Euler]]—who let their readers take part in their reasoning as they developed new ideas, and included certain erroneous deviations from the correct path—Gauss developed a new style of direct and complete explanation that did not attempt to show the reader the author's train of thought.{{sfn|Klein|1979|p=5-6}}{{sfn|Dunnington|2004|p=217}}

:"Gauss was the first to restore that ''rigor'' of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of preceding period in ''new'' developments."<ref>{{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}} quotation p. 109</ref>

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai on 2 September 1808 as follows:<ref>[https://archive.org/details/briefwechselzwi00gausgoog/page/n124/mode/2up Letter from Gauss to Bolyai from 2 September 1808]</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}} }}

Gauss refused to publish work which he did not consider complete and above criticism. This [[Philosophy:Perfectionism (psychology)|perfectionism]] was in keeping with the motto of his personal [[Seal (emblem)|seal]] {{lang|la|Pauca sed Matura}} ("Few, but Ripe"). His [[Gauss's diary|personal diary]] indicates that he had made several mathematical discoveries years or decades before contemporaries published them. 
He put down new ideas in writing to his colleagues, who encouraged him to publish, and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a publishable 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 his "Collected Works" contain a considerable literary estate, too.{{sfn|Klein|1979|p=29}}{{sfn|Dunnington|2004|p=420–430}} [[Biography:Eric Temple Bell|Eric Temple Bell]] said that if Gauss had published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.<ref name=":2">{{cite book | last = Bell | first = E.T. | chapter = Ch. 14: The Prince of Mathematicians: Gauss | title = Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré | location = New York | publisher = Simon and Schuster | pages = 218–269 | year = 2009 | isbn = 978-0-671-46400-4}}</ref>

On certain occasions, Gauss claimed that a finding published by 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 = [[Biography: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, he was criticized for his negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of the history of science and more time than he wished to spend.<ref name="Biermann" />

[[File:Carl Friedrich Gauß, 003.jpg|thumb|upright|A sketch of Gauss by his friend Johann Benedict Listing, 1830]]
Though Gauss is seen as a master of axiomatic presentation, it became obvious from his posthumously published papers, his diary, and short glosses in his own textbooks, that he worked to a great extent in an empirical way. Gauss was a lifelong busy and enthusiastic calculator. He coped with the enormous workload by using skillful tools.{{sfn|Maennchen|1930|p=4–9}} Gauss used a lot of [[Mathematical table|mathematical table]]s, examined their qualities{{clarify|date=September 2023}}, 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 | 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]]. It has been taken as a curious feature of his working style that he carried out calculations with a high degree of precision, much more than required.{{sfn|Maennchen|1930|p=3}} Very likely, this method gave him a lot of material which he used in finding theorems in number theory.{{sfn|Maennchen|1930|p=4–9}}

It was well known to his close colleagues that Gauss disliked giving academic lectures. He first stated this to Olbers in 1802, so this aversion was not the result of bad experience. Thus he refused to accept any academic position with teaching duties during his years as a private scholar. But from the start of his academic career at Göttingen in 1807, he continuously gave lectures until 1854.{{sfn|Dunnington|2004|p=405–410}} He often complained about the efforts of teaching, feeling that it was a waste of his time, but on the other hand he occasionally described one or other student as talented. In all these 47 years of teaching he gave only three lectures on subjects of pure mathematics, whereas most of his lectures dealt with astronomy, geodesy, and [[Applied mathematics|applied mathematics]]. However, many of Gauss' students went on to become renowned mathematicians, physicists, and astronomers: [[Biography:Moritz Cantor|Moritz Cantor]], [[Biography:Richard Dedekind|Dedekind]], Dirksen, Encke, Gould,{{efn|The index of correspondence shows that Benjamin Gould was presumably the last correspondent who 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" />}} [[Biography:Eduard Heine|Heine]], Klinkerfues, [[Biography:Adolph Theodor Kupffer|Kupffer]], [[Biography:Johann Benedict Listing|Listing]], [[Biography:August Ferdinand Möbius|Möbius]], Nicolai, [[Biography:Bernhard Riemann|Riemann]], Ritter, [[Biography:Ernst Christian Julius Schering|Schering]], Scherk, Schumacher, [[Biography:Ludwig August Seeber|Seeber]], [[Biography:Karl Georg Christian von Staudt|von Staudt]], Stern, Ursin; as geoscientists Sartorius von Waltershausen and Wappäus.<ref>{{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>

Gauss did not write any textbook, and (unlike his friends Bessel, Humboldt, and Olbers) he disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter<ref>{{cite book | last = Gauss | first = C. F. | editor = Königlich Preußische Akademie der Wissenschaften | title = Carl Friedrich Gauss. Werke | volume = VI | pages = 73–86 | url = https://gdz.sub.uni-goettingen.de/id/PPN236007467?tify=%7B%22pages%22%3A%5B77%5D%2C%22pan%22%3A%7B%22x%22%3A0.5%2C%22y%22%3A0.69%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.413%7D | language = de | year = 1874}}</ref> and the essay ''Erdmagnetismus und Magnetometer'' of 1836.<ref name="Biermann" />

Gauss published his papers and books exclusively in [[Social:Latin|Latin]] or in [[Social: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}}}}

[[File:Universitäts-Sternwarte Göttingen 02.jpg|thumb|The new Göttingen Observatory of 1816; Gauss' living rooms were in the western wing (right)]]
At Göttingen University, Gauss was accompanied by a staff of other lecturers in his disciplines, who completed the educational program: for instance the brilliant Thibaut in mathematics, in physics [[Biography:Wilhelm Eduard Weber|Weber]] and [[Biography:Johann Tobias Mayer|Mayer]], well known for his successful textbooks, and [[Biography:Karl Ludwig Harding|Harding]], who took the main part of lectures in astronomy. When the observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. Once they had been on friendly terms with another, but in the course of time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer.{{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' mother-in-law, pleaded with Olbers to try to provide Gauss with another position far from Göttingen.}} The years since 1820 were evaluated as a "period of lower astronomical activity".{{sfn|Brendel|1929|pp=7, 128}} The new, well-equipped observatory did not work as effectively as others; Gauss' astronomical research had the character of a one-man enterprise, and the university established a place for an assistant only after Harding's death in 1834. But nevertheless Gauss twice refused the opportunity to solve the problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy, without no great lecturing duties, as well as from [[Organization:Leipzig University|Leipzig University]] in 1810 and from Vienna University in 1842. Perhaps the reason was the difficult situation of his family.<ref>{{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> In his later years, Gauss was one of the best-paid professors of the university.<ref name="Gerardy" />

When Gauss was asked for help by his friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title,{{efn|name=Bessel|Bessel never had a university education.}} Gauss provided a [[Social:Honorary degree|doctorate ''honoris causa'']] for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for [[Biography:Sophie Germain|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 for the talented mathematician [[Biography:Gotthold Eisenstein|Gotthold Eisenstein]] in Berlin.

After King William IV's death in 1837, the [[Social:Personal union|personal union]] between the kingdoms of [[History:United Kingdom of Great Britain and Ireland|Great Britain and Ireland]] and Hanover ceased. In the same year, the new Hanoverian King Ernest Augustus annulled the constitution given to the state by his brother in 1833. Seven prominent professors, later known as the "Göttingen Seven", protested against this, among them Gauss' friend and collaborator Wilhelm Weber and Gauss' son-in-law Heinrich Ewald. All of them were dismissed, three of them were expelled, but Ewald and Weber could stay in Göttingen. Ewald took a position the [[Organization:University of Tübingen|University of Tübingen]] in 1838, where Gauss' daughter Wilhelmina died soon afterwards in 1840, and with Weber went to Leipzig in 1843; but both of them returned to their Göttingen positions in 1849 as the only ones of the Göttingen Seven. Gauss was deeply affected by this quarrel, but saw no possibility to help them.{{sfn|Dunnington|2004|p=195-200}}

Gauss took part in academic administration: three times he was elected as [[Social:Dean (education)|dean]] of the Philosophy Faculty.{{sfn|Dunnington|2004|p=288}} Being entrusted with the widow's pension fund of the university, he dealt with [[Actuarial science|actuarial science]] and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years, even in his last year of life.{{sfn|Dunnington|2004|p=288}}

==== The private man ====
[[File:Minna Gauß geb. Waldeck, 002.jpg|thumb|upright|Gauss' second wife Wilhelmine Waldeck]]
Soon after Gauss' death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw Gauss as a serene and forward-striving man with childlike modesty,{{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}} He was noted for a sense of justice{{sfn|Sartorius von Waltershausen|1856|p=68}} and religious tolerance.{{sfn|Sartorius von Waltershausen|1856|p=100}} Apart from his closer circle, others regarded him as reserved and unapproachable, "like an 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 grumpy behaviour, but a short time later his mood could change, and he became a charming, open-minded host.<ref name="Biermann" />

Gauss' life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the death of their third child, he plunged into a depression from which he never fully recovered. Soon after her death he wrote a last letter to her in the style of an ancient threnody, the most personal surviving document of Gauss'.<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}} The situation worsened when [[Medicine:Tuberculosis|tuberculosis]] afflicted, and ultimately destroyed the health of, his second wife Minna over 13 years;  both his daughters later suffered from the same disease.{{sfn|Dunnington|2004|p=206, 374}} Both younger sons were educated for some years in Celle far from Göttingen. Gauss himself gave only slight hints of his personal distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings".<ref name="Biermann" />

Gauss grew to dominate his children and eventually had conflicts with his sons, because he did not want any of them to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements. The military career of his elder son Joseph ended after more than two decades with the rank of a poorly paid first lieutenant, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married.<ref name="Gerardy" /> The second son Eugen shared a good measure of Gauss' talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused a scandal in public,<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> he suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken for starting, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties to get an appropriate education, and emigrated as well. Only Gauss' youngest daughter Therese accompanied him in his last years of life.

Collecting numerical data on very different things, useful or useless, became a habit in his later years, for example the number of paths from his home to certain places in Göttingen, or the numbers of living days of persons; he congratulated Humboldt in December 1851, when he had reached the same age as [[Biography:Isaac Newton|Isaac Newton]] at his death, calculated in days.{{sfn|Sartorius von Waltershausen|1856|p=71}}

Gauss had a good knowledge of [[Social:Latin|Latin]] as well as of modern languages. At the age of 62, he began to teach himself [[Social:Russian language|Russian]], very likely to understand scientific writings from Russia, among them those of [[Biography:Nikolai Lobachevsky|Lobachevsky]] on non-Euclidean geometry.<ref>{{cite book | last1 = Lehfeldt | first1 = Werner |  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 | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> Gauss read both classical and modern literature, the English and French in the original languages.{{efn|The first book he loaned from the university library in 1795 was the novel 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" | 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.{{sfn|Wußing|1982|p=80}} Gauss liked singing and went to concerts.{{sfn|Wußing|1982|p=81}} He was a busy newspaper reader, and in his last years he used to 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 ''[[Philosophy:Naturphilosophie|Naturphilosophie]]''.{{sfn|Wußing|1982|p=79}}

Gauss' religious beliefs have been a subject of speculation by some of his biographers. He sometimes said: "God is calculating."{{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 |title=MacTutor |url=https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quotations/ |website=https://mathshistory.st-andrews.ac.uk/}}</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 [[Philosophy:Dogma|dogma]]s or understand the Holy Bible to be true quite literally.{{sfn|Dunnington|2004|p=300}} Sartorius mentioned Gauss' [[Religion:Religious tolerance|religious tolerance]], and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions.{{sfn|Sartorius von Waltershausen|1856|p=100}}

Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, in accordance with the motto "mundus vult decipi". As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors.{{sfn|Sartorius von Waltershausen|1856|p=94}} Gauss was loyal to the House of Hanover, disliked Napoleon and his system, and all kind of violence and revolution caused horror to him. Thus he condemned the methods of the [[Social:Revolutions of 1848|Revolutions of 1848]], though he agreed with some of their aims, such as the idea 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 Prussia and Austria extended widely over the frontiers of the Confederation.}}

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, but he disapproved of the idea of paper money.{{sfn|Sartorius von Waltershausen|1856|p=90}} After his death a great sum of money was found hidden in his rooms.<ref>{{cite book | editor = Niedrsächsische Vermessungs- und Katasterverwaltung | last = Gronwald | first = W. | chapter = Carl Friedrich Gauss – Ein Lebensbild | title = Carl Friedrich Gauss und die Landesvermessung in Niedersachsen | location = Hannover | pages = 1–16 | year = 1955}} p. 12</ref>

== Scientific work ==

=== Algebra and number theory ===
==== Fundamental theorem of algebra ====
[[File:DBP 1977 928 Carl Friedrich Gauß.jpg|thumb|upright|German stamp commemorating Gauss' 200th anniversary: the [[Complex plane|complex plane]]]]
In his doctoral thesis from 1799 Gauss proved the [[Fundamental theorem of algebra|fundamental theorem of algebra]] which states that every non-constant single-variable [[Polynomial|polynomial]] with complex coefficients has at least one complex [[Root of a function|root]]. Mathematicians including [[Biography:Jean le Rond d'Alembert|Jean le Rond d'Alembert]] had produced false proofs before him, and Gauss' dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.<ref>{{Cite arXiv |last1=Basu |first1=Soham |last2=Velleman |first2=Daniel J. |date=2017-04-21 |title=On Gauss's first proof of the fundamental theorem of algebra |class=math.CV |eprint=1704.06585 }}</ref>

==== Disquisitiones Arithmeticae ====
{{Main|Disquisitiones Arithmeticae}}
The entries in Gauss' Mathematical diary indicate that he was busy with the subject of number theory at least since 1796. A detailed study of previous researches showed him that some of his findings had been already done by other scholars. In the years 1798 and 1799 Gauss wrote a voluminous compilation of all these results in the famous ''[[Disquisitiones Arithmeticae]]'', published in 1801, that was fundamental in consolidating number theory as a discipline and covered both elementary and algebraic [[Number theory|number theory]]. Therein he introduces, among other things, the [[Triple bar|triple bar]] symbol ({{math|≡}}) for [[Congruence relation|congruence]] and uses it in a clean presentation of [[Modular arithmetic|modular arithmetic]]. It deals with the unique factorization theorem and [[Primitive root modulo n|primitive roots modulo n]]. In the main chapters, Gauss presents the first two proofs of the law of [[Quadratic reciprocity|quadratic reciprocity]], which allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, and develops the theories of [[Binary quadratic form|binary]] and ternary [[Quadratic form|quadratic form]]s.

Highlights of these theories include the remarkable [[Gauss composition law]] for binary quadratic forms, as well as his enumeration of the number of representations of an integer as 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. From several remarkable analytic results on [[Ideal class group|class numbers]] that Gauss gives without proof towards the end of the fifth chapter, it appears that Gauss already knew the [[Class number formula|class number formula]] in 1801.<ref>{{cite web |title=Did Gauss know Dirichlet's class number formula in 1801? |work=[[MathOverflow]] |date=10 October 2012 |url=https://mathoverflow.net/q/109330 }}</ref>

In the last chapter Gauss gives his proof for the [[Constructible polygon|constructibility]] of a regular [[Heptadecagon|heptadecagon]] (17-sided polygon) with straightedge and compass by reducing this geometrical to an algebraic problem. He shows that a regular polygon is constructible if the number of its sides is a product of distinct [[Fermat number|Fermat primes]] and a [[Exponentiation|power]] of 2. In the same chapter, he gives a result on the number of solutions of certain cubic polynomials with coefficients in [[Finite field|finite field]]s, which amounts to counting integral points on an [[Elliptic curve|elliptic curve]]. Some 150 years later, Andre Weil remarked that this particular result, together with some other unpublished results of Gauss, led him to formulate what is now called [[Weil conjectures]].<ref>{{cite journal | last = Weil | first = André |  title = Numbers of solutions of equations in finite fields | journal = [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] | volume = 55 |  pages = 497–508 | year = 1949 | issue = 5 | doi = 10.1090/S0002-9904-1949-09219-4 | url = https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/S0002-9904-1949-09219-4.pdf}}</ref>{{efn|Especially entry 146 in his [[Gauss's diary|diary]] is now seen as a prelude to the modern developments associated with Weil conjectures and the tools of [[Algebraic geometry|algebraic geometry]].<ref>{{Cite book | title = Carl Friedrich Gauss. Werke | editor = Königlich Preußische Akademie der Wissenschaften | volume = X, 1 | page =  571 | location = Göttingen | language = la | year = 1917 | url =  https://gdz.sub.uni-goettingen.de/id/PPN236018647?tify=%7B%22pages%22%3A%5B599%5D%2C%22pan%22%3A%7B%22x%22%3A0.436%2C%22y%22%3A0.834%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.856%7D}}</ref>}}

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

==== Further investigations ====
In 1831, [[Biography:Ludwig August Seeber|Ludwig August Seeber]] published a book 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 | location = Mannheim  | url = https://books.google.com/books?id=QKJGAAAAcAAJ}}</ref> with accordance with the program outlined in Gauss's ''Disquisitiones''. However, he did not prove a central theorem of his theory, so it remained a mere conjecture. In his review of Seeber's book, Gauss simplified many of Seeber's lengthy arguments, proved this central conjecture, and remarked that this theorem is equivalent to [[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>

Gauss proved [[Fermat's Last Theorem]] for ''n'' = 3 and sketchingly proved it for ''n'' = 5 in his unpublished writings. The particular case of ''n'' = 3 was proved much earlier by [[Biography:Leonhard Euler|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.

Among his published number theoretical works, his two papers on [[Quartic reciprocity|biquadratic residue]]s (published in 1828 and 1832) are considered second in importance only to ''Disquisitions Arithmeticae''. In these papers Gauss introduces the [[Ring theory|ring]] of Gaussian integers <math>\mathbb{Z}[i]</math>, and shows that this ring is a [[Unique factorization domain|unique factorization domain]]. Furthermore, he generalizes into this ring many 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 – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.

In the second paper, he states the general law of biquadratic reciprocity and proves several special cases of it, but proof of the general theorem is lacking, despite Gauss's statements that he found such a proof around 1814. He promised a third paper with a general proof, but this never appeared. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claims the techniques of these proofs ([[Gauss sum]]s) can be applied to prove higher reciprocity laws.

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

=== Analysis ===
One of Gauss's first independent discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers; his systematic investigations on the AGM led him to discover an unusually rich mathematical landscape, and to obtain plenty of new results associated with it. He discovered its relation to elliptic integrals in the years 1798-1799 through the so-called [[Landen's transformation]], and in a diary entry recorded his discovery of the connection of [[Gauss's constant]] to lemniscatic elliptic functions, a result that Gauss stated that "will surely open a new area of analysis". He also made early inroads into the more formal issues of the foundations of [[Complex analysis|complex analysis]], and from a letter to Bessel in 1811 it is clear that he knew the so-called "fundamental theorem of complex analysis" - [[Cauchy's integral theorem]] - and understood the notion of [[Residue (complex analysis)|complex residues]] when integrating around poles.

Another source of inspiration for Gauss's early work in analysis was his acquaintance with [[Pentagonal number theorem|Euler's pentagonal numbers theorem]]. This theorem together with his other researches on the AGM and lemniscatic functions led him to plenty of results on Jacobi theta functions, work which culminated with his 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 |  date = 2021 | edition = 2 | title = Series and Products in the Development of Mathematics | location = Cambridge | pages = 20–22 | publisher = Cambridge University Press | url = https://assets.cambridge.org/97811087/09453/frontmatter/9781108709453_frontmatter.pdf}}</ref> In his publication from 1811 on the determination of the sign of [[Quadratic Gauss sum|quadratic Gauss sum]], Gauss solved the problem by introducing [[Gaussian binomial coefficient]]s and by using a line of reasoning that somehow "hides" its origin in theta function theory, as later mathematicians have shown. All this work was done several decades before the publication of [[Biography:Carl Gustav Jacob Jacobi|Jacobi's]] "[[Fundamenta nova theoriae functionum ellipticarum|Fundamenta nova]]" in 1829; however, Gauss never found the time to systematically write and organize all his thoughts and theorems of this kind, and his contemporaries never knew the scope of his work.

Several mathematical fragments in his Nachlass indicate that he knew quite well parts of the modern theory of modular forms of Felix Klein and [[Biography:Robert Fricke|Robert Fricke]]. In his work on the [[Multivalued function|multivalued]] AGM of two complex numbers, he discovered a very deep connection between the infinitely many values of the AGM to its two "simplest values".<ref>{{cite journal | last = Cox | first = David | 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> His unpublished writings include several drawings that show he was quite aware of the geometric side of the theory; in the context of his work on the complex AGM he recognized and made a sketch of the key concept of [[Fundamental domain|fundamental domain]] for the [[Modular group|modular group]].{{efn|Gauss already dealt implicitly with fundamental domains in his ''Disquisitions Arithmeticae'' (1801), in the context of reduction theory of binary quadratic forms, but this geometric sketch appears for the first time in his analytic work on the AGM.}} One of Gauss's sketches of this kind was his drawing of a [[Tessellation|tessellation]] of the [[Unit disk|unit disk]] by "equilateral" [[Hyperbolic triangle|hyperbolic triangle]]s with all angles equal to <math>\pi/4</math>.

In his lifetime Gauss published almost nothing about those more modern theories of [[Elliptic function|elliptic function]]s, but he did publish most of his results on the related theme of the [[Hypergeometric function|hypergeometric function]]. In his work "Disquisitiones generales circa series infinitam..." (1812), he provided the first systematic treatment of the general hypergeometric function <math>F(\alpha,\beta,\gamma,x)</math>, and showed that many of the functions known to science at the time, such as the [[Elementary function|elementary function]]s and some special functions, are a special case of the hypergeometric function.{{sfn|Schlesinger|1933|p=136–142}} This work was the first one with an exact inquiry of convergence of infinite series in the history of mathematics.{{sfn|Schlesinger|1933|p=142}} Furthermore, it dealt with infinite [[Continued fraction|continued fraction]]s arising as ratios of hypergeometric functions.

In 1822 Gauss published his prize winning essay on conformal mappings, which contains several developments that pertain to the field of complex analysis. In this essay, Gauss made explicit the insight that angle-preserving mappings in the complex plane must be complex [[Analytic function|analytic function]]s, and used the so-called [[Beltrami equation]] to prove the existence of [[Isothermal coordinates|isothermal coordinates]] on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an ellipsoid of revolution. In addition, in unpublished fragments from the years 1834-1839 he investigated and solved the more difficult task of explicitly constructing a conformal mapping from the interior of an ellipse to the unit disk.<ref>{{cite book | last = Monna | first = A. F. | date = 1975 | title = Dirichlet's principle : a mathematical comedy of errors and its influence on the development of analysis | location = Utrecht | pages = 11–12 | url = https://www.gbv.de/dms/tib-ub-hannover/117505722.pdf | publisher = }}</ref> His solution, which combined his early work on elliptic functions and his later ideas on [[Potential theory|potential theory]], reveals his mastery of the theory of logarithmic potential, and his final results corresponded to the formula found by Hermann Schwarz in 1870.

==== Numeric analysis ====
Gauss often deduced theorems [[Philosophy:Inductive reasoning|inductively]] from numerical data he had collected in an empirical way. As such, the use of efficient algorithms to facilitate calculations was vital to his researches, and he made many contributions to numeric analysis. In 1815, he published an article on numeric integration, in which he described his method of [[Gaussian quadrature]], that greatly improved existing methods and inspired much of the work made by later mathematicians.

In a private letter to Gerling from 1823,<ref>[https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22%3A%5B316%5D%2C%22view%22%3A%22info%22%7D Letter Gauss to Gerling from 26 December 1823]</ref> he described a solution of a certain 4X4 system of linear equations by using Gauss-Seidel method – an "indirect" [[Iterative method|iterative method]] for the solution of linear systems, that in some cases converges very rapidly to the exact solution. Gauss recommended it over the usual method (the so-called "direct elimination") for systems of more than 2 equations, stating that it can be done "while half asleep, or while thinking about other things".<ref>{{cite arXiv | author = Yousef Saad |  title = Iterative Methods for Linear Systems of Equations: A Brief Historical Journey | date = 2019-08-02 | class = math.HO | eprint = 1908.01083v1 }}</ref> As such, it was an early contribution to [[Numerical linear algebra|numerical linear algebra]].

Gauss invented an algorithm for calculating what are now called [[Discrete Fourier transform|discrete Fourier transform]]s, sometimes called "the most important numerical algorithm of our lifetime", when calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey published their similar [[Cooley–Tukey FFT 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|trigonometric interpolation]] method, but his paper ''Theoria Interpolationis Methodo Nova Tractata'' was published only posthumously in 1866,<ref>{{cite book | last = Gauss | first = C.F. | date = 1876 | title = Theoria Interpolationis Methodo Nova Tractata | location = 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> preceded by the first presentation by [[Biography:Joseph Fourier|Joseph Fourier]] on the subject in 1807.<ref>{{cite journal | last1 = Heideman | first1 = M. | last2 = Johnson| first2 = D. | last3 = Burrus | first3 = C. | 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 = 2013-03-19 | url-status=live}}</ref>

=== Chronology ===
The first publication following the doctoral thesis dealt with the determination of the [[Religion:Computus#Gauss' Easter algorithm|date of Easter]] (1800), a very elementary matter of mathematics. Gauss aimed to present a most convenient algorithm for people without any knowledge in ecclesiastical or even astronomical chronology, and thus avoided the usually required terms of [[Astronomy:Golden number (time)|golden number]], [[Physics:Epact|epact]], [[Astronomy:Solar cycle (calendar)|solar cycle]], and [[Dominical letter|domenical letter]], and any religious connotations.{{sfn|Maennchen|1930|p=49–63}} Biographers speculated on the reason why Gauss dealt with this matter, but it is likely comprehensible by the historical background. The replacement of the [[Julian calendar]] by the [[Gregorian calendar]] had caused great confusion to the hundreds of states of the [[Religion:Holy Roman Empire|Holy Roman Empire]] since the 16th century, and was finished in Germany not until the year 1700, when the difference of eleven days was deleted, but the difference in calculating the date of Easter remained between Protestant and Catholic territories. A further agreement of 1776 equalized the confessional way of counting, thus in the Protestant states like the Duchy of Brunswick the Easter of 1777, five weeks before Gauss' birth, was the first one calculated in the new manner.<ref name="Olesko" /> The public difficulties of replacement may be the historical background for the confusion on this matter in the Gauss family (see chapter: Anecdotes). For being connected with the Easter regulations, an essay on the date of Pesach followed soon in 1802.

[[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]]

=== Astronomy ===
On 1 January 1801, Italian astronomer [[Biography:Giuseppe Piazzi|Giuseppe Piazzi]] discovered the [[Astronomy:Dwarf planet|dwarf planet]] [[Astronomy:Ceres (dwarf planet)|Ceres]].<ref>{{Cite journal | last = Forbes | first = Eric G. |  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> Piazzi could track Ceres for only somewhat more than a month, following it for three degrees across the night sky, less than 1% of the total orbit, until it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data. Gauss tackled the problem within three months of intense work, and predicted a position for Ceres in December 1801. This turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 7/31 December at Gotha, and independently by Heinrich Olbers on 1/2 January in [[Place:Bremen|Bremen]].<ref>{{cite journal | last1 = Teets | first1 = Donald | first2 = Whitehead | last2 = 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}}</ref>{{efn|The unambigeous identification of a cosmic object as planet among the fixed stars requires at least two observations with interval.}} This confirmation eventually led to the classification of Ceres as [[Astronomy:Minor-planet designation|minor-planet designation]] 1 Ceres; that was taken as the predicted planet between [[Astronomy:Mars|Mars]] and [[Astronomy:Jupiter|Jupiter]] by the most speculative [[Astronomy:Titius–Bode law|Titius–Bode law]].<ref name="Marsden">{{Cite journal| last = Marsden | first = Brian G. | author-link = Brian G. Marsden | date=1977-08-01 | 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>

[[Astronomy:Gauss's method|Gauss's method]] involved determining a [[Conic section|conic section]] in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by [[Kepler's laws of planetary motion|Kepler's Second Law]]). This problem 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}} Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again".

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as ''Theoria motus corporum coelestium in sectionibus conicis solem ambientum''. In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation.<ref>Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.</ref> It introduced the [[Physics:Gaussian gravitational constant|Gaussian gravitational constant]].

Since the new asteroids had been discovered, Gauss occupied himself with the [[Astronomy:Perturbation|perturbation]]s of their [[Astronomy:Orbital elements|orbital elements]]. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was Pallas, because of its great eccentricity and [[Astronomy:Orbital inclination|orbital inclination]], whereby Laplace's method did not work. Gauss used his own tools : the [[Arithmetic–geometric mean|arithmetic–geometric mean]], the [[Hypergeometric function|hypergeometric function]], and his method of interpolation.{{sfn|Brendel|1929|p=194–195}} He found an orbital resonance with [[Astronomy:Jupiter|Jupiter]] in proportion 18 : 7 in 1812; Gauss published this result as [[Cipher|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 = [[Organization:Monthly Notices of the Royal Astronomical Society|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.<ref>{{cite book | last1 = Schroeder | first1 = Manfred R. |  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 | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>}} However, after long years he finished his work in 1816 without a result that seemed sufficient to him. This marked the end of his activities in theoretical astronomy, too.{{sfn|Brendel|1929|p=254}}

One fruit of Gauss's research on Pallas perturbations was his article ''Determinatio Attractionis...'' (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". This method introduced a useful averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time taking the planet to follow the corresponding orbital arcs.{{sfn|Brendel|1929|p=253–254}} Gauss presents his method of evaluating the gravitational attraction of such an elliptic ring, which includes several complicated steps; one such step involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an [[Elliptic integral|elliptic integral]].{{efn|This article is also noteworthy because it was Gauss's only published reference to his (mostly unpublished) work on the AGM algorithm.}} In the late 19th century Gauss's method was adapted by American astronomer George William Hill, who applied it directly to the problem of secular perturbation induced by [[Astronomy:Venus|Venus]] on [[Astronomy:Mercury (planet)|Mercury]] orbit.<ref>{{cite journal | last1 = Hill | first1 = George William |  title = On Gauss's Method of Computing Secular Perturbations, with an Application to the Action of Venus on Mercury | journal = Astronomical Papers of the American Ephemeris | date = 1882 | volume = 1 | pages = 315–361 | bibcode = 1882USNAO...1..315H | url = https://babel.hathitrust.org/cgi/pt?id=uc1.32106020676935&seq=17}}</ref>

=== Theory of errors ===
It is likely that Gauss used the [[Least squares|method of least squares]] for calculating the orbit of Ceres to minimize the impact of [[Observational error|measurement error]].<ref name=":4" /> The method was published first by [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] in 1805, but Gauss claimed in ''Theoria motus'' (1809) that he had been using it since 1794 or 1795.<ref name=":3" /> 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 the method under the assumption of [[Normal distribution|normally distributed]] errors ([[Gauss–Markov theorem]]) in his paper ''Theoria combinationis observationum erroribus minimis obnoxiae'' from 1821.

In this paper, which was relatively little known in the English speaking world in the first century after its publication, he stated and proved [[Gauss's inequality]] (a [[Chebyshev'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 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|variance]] of [[Sample variance|sample variance]].{{efn|In 1947, [[Biography:Andrey Kolmogorov|Andrey Kolmogorov]] corrected a mistake he found in Gauss's formula for the lower bound.}} In a supplement to this paper Gauss described [[Recursive least squares filter|recursive least squares methods]] that went unnoticed until 1950, when his work was rediscovered as a consequence of the growing demand of quick [[Estimation theory|estimation]] for various new technologies. Gauss's work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert, and the Gauss-Helmert theory is considered today as the "classical" theory of errors.

Gauss made several striking contributions to problems in [[Probability theory|probability theory]] that are not directly concerned with the theory of errors, but offer a glimpse into his broad minded view on the applicability of probabilistic thinking. One remarkable example appears as a note in his diary and is concerned with a very unusual problem that came to his mind: to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in ''(0,1)''. He derived this distribution, now known as the Gauss-Kuzmin distribution, as a by-product of his discovery of the [[Ergodicity|ergodicity]] of the Gauss map for continued fractions. Gauss's solution is the first ever result in the metrical theory of continued fractions.

[[File:Georg IV Erlass Landvermessung.jpg|left|thumb|upright|Order of King George IV to the triangulation project]]

=== Arc measurement and geodetic survey ===
Gauss was busy with geodetic problems since 1799, when he helped Karl Ludwig von Lecoq with calculations during his [[Engineering:Surveying|survey]] in Westphalia.{{sfn|Galle|1924|pp=16–18}} Later since 1804, he taught himself some geodetic practise with a sextant in Brunswick,{{sfn|Galle|1924|p=22}} and Göttingen.{{sfn|Galle|1924|p=28}}

Since 1816, his former student Heinrich Christian Schumacher, then professor in [[Place:Copenhagen|Copenhagen]], but living in Altona (Holstein) near [[Place:Hamburg|Hamburg]], made a [[Triangulation|triangulation]] of the Jutland peninsula from Skagen in the north to Lauenburg in the south.{{efn|Lauenburg is in the southernmost place of the Duchy of Holstein, that was held in personal union by the King of {{wipe|Denmark}}.}} The aim was not only the foundation of map production, but also the determination of the geodetic arc of that distance. Schumacher asked Gauss to continue this work further to the south and said he could find support for this project directly from the government of Hanover. Finally in May 1820, King George IV gave the order to Gauss.{{sfn|Galle|1924|p=32}}

Gauss and Schumacher had yet 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 personally, that reached from Thuringia in the south to the river Elbe in the north. The triangel between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum side of {{convert|107|km|1|abbr=in}}. In the thin populated Lüneburg Heath, without significant natural summits or artificial buildings, he had great difficulties to find suitable triangulation points, sometimes cutting lanes through the vegetation was necessary or even the erection of signal towers.{{sfn|Galle|1924|p=75}}

[[File:Gauß Heliotrop@20170430.JPG|thumb|The heliotrope]]
[[File:Vize-Heliotrop Gauß-Ausstellung Bomann-Museum (1).jpg|thumb|Gauss' vice heliotrope, a Troughton sextant with additional mirror]]
For pointing signals, Gauss invented a new instrument with movable mirrors and a small telescope that reflects the sunbeams to the triangulation points, and named it ''[[Engineering:Heliotrope (instrument)|heliotrope]]''. Another suitable construction for the same purpose was a [[Astronomy:Sextant|sextant]] with an additional mirror which he named ''vice heliotrope''.{{sfn|Galle|1924|p=69}} Gauss got assistance by soldiers of the Hanoveran army, among them his eldest son Joseph. Gauss took part in the baseline measurement (Braak Base Line) of Schumacher in the village Braak near Hamburg in 1820, and used the result for the evaluation of his triangulation.{{sfn|Dunnington|2004|p=121}}

The arc measurement needed a precise astronomical determination of two points in the network. Gauss and Schumacher used the favourite occasion that both observatories in Göttingen and in Altona, in the garden of Schumacher's house, laid nearly in the same [[Earth:Longitude|longitude]]. The [[Earth:Latitude|latitude]] was measured with both their own instruments and a zenith sector of 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 [[Earth:Principal Triangulation of Great Britain|Principal Triangulation of Great Britain]].{{sfn|Galle|1924|p=61}}}}

An additional result was a better value of [[Flattening|flattening]] of the approximative [[Earth:Earth ellipsoid|earth ellipsoid]].{{sfn|Galle|1924|pp=37–38, 49–50}}{{efn|The new value of about 1/298.39 what was a significant improvement against the former value of 1/302.78 from Walbeck (1820). The calculation was done by Eduard Schmidt, private lecturer at Göttingen University.{{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 intended the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover. The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Carl Friedrich Gauss, who applied his mathematical inventions as the method of least squares and his [[Gaussian elimination|elimination method]] to it. The project was finished in 1844, but Gauss did not publish a final report of the project and his method of projection; this work was not done 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 | location = Hannover | language = de}}</ref>

In 1828, when studying differences in [[Earth:Latitude|latitude]], Gauss first defined a physical approximation for the [[Earth:Figure of the Earth|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 | url=https://books.google.com/books?id=tIg_AAAAcAAJ&pg=PA73 | language=de | access-date=2021-07-06 | page=73}}</ref> later his doctoral student [[Biography:Johann Benedict Listing|Johann Benedict Listing]] called this the ''[[Geoid|geoid]]''.<ref>{{cite book | last = Listing | first = J.B. | title = Ueber unsere jetzige Kenntniss der Gestalt und Grösse der Erde | publisher = Dieterich | year = 1872 | location = Göttingen | url=https://books.google.com/books?id=yQ9TAAAAcAAJ | language = de | page = 9}}</ref>

=== Differential geometry ===
{{Main|Theorema Egregium}}
The geodetic survey of Hanover fueled Gauss' interest in [[Differential geometry|differential geometry]] and [[Topology|topology]], fields of mathematics dealing with [[Curve|curve]]s and [[Surface (topology)|surfaces]]. This led him in 1828 to the publication of a memoir that marks the birth of modern [[Differential geometry of surfaces|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 instead pioneered a revolutionary approach that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. Its crowning 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|angle]]s and [[Distance|distance]]s on the surface. That is, curvature does not depend on how the surface might be [[Embedding|embedded]] in 3-dimensional space or 2-dimensional space.

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended [[Manifold|manifold]]s - it makes clear the distinction between the intrinsic properties of the manifold (the [[Physics:Metric tensor|metric]]) and its physical realization (the embedding) in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a [[Sphere|sphere]] or an [[Ellipsoid|ellipsoid]] cannot be transformed to a plane without distortion, what causes a fundamental problem in designing [[Map projection|projection]]s for geographical maps.

An additional significant portion of his essay is dedicated to a profound study of [[Geodesic|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.

One key differential geometric conception was lacking from Gauss's memoir, that of [[Geodesic curvature|geodesic curvature]]. However, his posthumous papers show that this notion did not escape his mind, and in the years of composing his memoir he also wrote up a manuscript in which he introduced it and referred to it as "side curvature" (in German: "Seitenkrümmung"). More importantly, he proved its [[Invariant (mathematics)|invariance]] under isometric transformations, a result later obtained by [[Biography:Ferdinand Minding|Ferdinand Minding]]. Based on this evidence and the announcement in his memoir of further investigations on the curvature integral, it is very likely that he knew the more general version of the Gauss-Bonnet theorem proved by [[Biography:Pierre Ossian Bonnet|Pierre Ossian Bonnet]] in 1848, which is closer in spirit to the global version of this theorem.<ref>Minimal surfaces (2010), p.50</ref>{{efn|In pp.73-74 of his essay on Gauss's contributions to calculus of variations, [[Biography:Oskar Bolza|Oskar Bolza]] mentions a transformation for the expression of geodesic curvature, that Gauss gives at the last section of this manuscript, which is "the actual core not only of Gauss's theorem about total curvature, but also of Bonnet's later generalization (1848)".}}

=== Non-Euclidean geometries===
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|upright|Lithography by Siegfried Bendixen (1828)]]
{{Main|Non-Euclidean geometry}}
In the lifetime of Gauss a vivid discussion on Euclid‘s parallel axiom was going on. Numerous mathematicians made efforts to prove it, whereas some of them discussed the possibility of geometrical systems without it.{{sfn|Bühler|1981|p=100–102}} Gauss himself was only interested in the geometrical aspects of the physical space, but did not care about the philosophical aspects of an enlarged geometry. In 1816, he gave his first short public comment on this matter in a book review, and in the follwing time he occasionally made some remarks in letters to his correspondents.{{sfn|Klein|1979|p=57–60}} He is the one who coined the term "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>

Not until Lobachevsky (1829) and Janos Bolyai (1832) had published their ideas of a non-Euclidean geometry – for the first time in history of mathematics – ,{{sfn|Bühler|1981|p=100–102}} Gauss himself put down his ideas, but avoided any influence to the contemporary scientific discussion, because he did not publish about it.{{sfn|Klein|1979|p=57–60}} Gauss commended the ideas of Janos Bolyai in a letter to his father,<ref>[https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up Letter from Gauss to Bolyai from 6 March 1832]</ref> claiming that these were congruent to his own thoughts since some decades.{{sfn|Klein|1979|p=57–60}}<ref>{{cite book | last = Krantz | first = Steven G. | 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> But it is not clear to what extent he preceded Lobachevsky and Bolyai, for he gave only vague and obscure remarks on it in his letters.{{sfn|Bühler|1981|p=100–102}}

Sartorius mentioned it first in 1856, but only the edition of left papers in Volume VIII of his Collected Works (1900) showed Gauss's own progress on that matter, at a time when Non-Euclidean geometry had yet grown out of controversial discussion.{{sfn|Klein|1979|p=57–60}}

In 1854, Gauss selected the topic for [[Biography:Bernhard Riemann|Bernhard Riemann]]'s inaugural lecture {{lang|de|Über die Hypothesen, welche der Geometrie zu Grunde liegen}} from three proposals.<ref name=Monastyrsky>{{Cite book|title=Riemann, Topology, and Physics|first=Michael|last=Monastyrsky|publisher=Birkhäuser|year=1987|isbn=978-0-8176-3262-5|pages=21–22}}</ref><ref>{{Cite book | last = Riemann | first = Bernhard | title = On the Hypotheses which lie at the Bases of Geometry | translator = William Kingdon Clifford | url = http://www.emis.de/classics/Riemann/WKCGeom.pdf}}</ref> On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.{{sfn|Bühler|1981|p=154}}

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

His earliest "serious" encounter with topological notions occurred to him in the course of his astronomical work, and in a small article from 1804 he determined the limits of the region on the [[Celestial sphere|celestial sphere]] in which comets and asteroids might appear, region which he termed "Zodiacus". He determined this region, and observed that if the Earth's and comet's orbits are [[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 [[Astronomy:7 Iris|7 Iris]], he  published another short article in which he further elaborated the 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>

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

In a fragment from 1833, Gauss defined the [[Linking number|linking number]] of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. In the same note, he lamented on the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as [[Braid theory|braid]]s and [[Tangle (mathematics)|tangle]]s.<ref name="Epple 1998 45–52"/>

In his later years Gauss held the emerging field of topology in a very high esteem and expected great future developments for it, but since there is so few written material by Gauss from this period, his influence was made mainly through occasional remarks and oral communications.{{efn|As a result, there are only indirect reports by his colleagues and students Mobius, Listing and Riemann as testimony to his influence.}} For example, an indirect report by  Mobius referred to a surface constructed by Gauss, which Gauss called "double ring" and sayed something about its [[Connected space|connectivity]] properties.{{sfn|Stäckel|1917|p=51-53}} This report is consistent with a fragment of Gauss, written around 1840, which sketched a theory of the order of connectivity of surfaces.<ref>{{cite web |title = Who first defined "simply connected"? |work=[[MathOverflow]] |date=22 March 2014 |url=https://mathoverflow.net/questions/161121/who-first-defined-simply-connected-reference}}</ref> Finally, it is worth mentioning that in the introduction to Listing's book "''Vorstudien zur Topologie''" (1847), Listing expressed his indebtness to Gauss's influence.

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

For example, he was a vivid spirit in applying complex numbers to various problems, and used them in his work on [[Perspective (graphical)|perspective]] and [[Projective geometry|projective geometry]]: in a short 1836 note on "Projections of the Cube", 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| last = Penrose | first = Roger | title = Drawing with Complex Numbers  | journal = Mathematical Intelligencer | year = 2000 | volume = 22 | issue = 4 | pages = 8–13 | doi = 10.1007/BF03026760 | arxiv=math/0001097| s2cid = 119136586 }}</ref> In an unpublished 1819 note entitled "the Sphere", he conceived of the complex plane extended by a [[Point at infinity|point at infinity]] as the [[Stereographic projection|stereographic projection]] of a sphere (the [[Riemann sphere]]), and described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane.{{sfn|Stäckel|1917|p=77-78}}

Under the context of Gauss's heading of extended algebraic systems, it must be mentioned that there is solid evidence that he had in his foresight the algebraic system of quaternions, the discovery of the great [[Biography:William Rowan Hamilton|William Rowan Hamilton]]. In 1819, Gauss drafted an unpublished short treatise on "Rotations of Space", in which he elaborated on the use of quadruples of real numbers (of which he called "scales") to describe 3D rotations.<ref>{{cite journal | last = Pujol | first = J. | title = Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: a Historical Reassessment | journal = Commun. Math. Anal. | volume = 13 |  pages = 1–14 | year = 2012 | issue = 2 | url = https://projecteuclid.org/journals/communications-in-mathematical-analysis/volume-13/issue-2/Hamilton-Rodrigues-Gauss-Quaternions-and-Rotations-a-Historical-Reassessment/cma/1349803591.full}}</ref>

In elementary geometry, he contributed his solution to the problem of constructing the largest-area ellipse that can be inscribed in a given quadrilateral, which was published in 1810 as an addition to Schumacher's translation of [[Biography:Lazare Carnot|Lazare Carnot]]'s treatise ''Géométrie de position''.{{sfn|Stäckel|1917|p=71-74 }} He discovered a surprising result about the computation of area of [[Pentagon|pentagon]]s. He made many contributions to [[Spherical geometry|spherical geometry]], and in this context solved some practical problems about [[Astronomy:Celestial navigation|navigation by stars]].{{efn|These navigational tasks appear as problems no. 78,79 in Heinrich Dörrie's book "100 Great Problems Of Elementary Mathematics". Their solutions by Gauss were published in 1812 as part of Bode's Astronomisches Jahrbuch.}}

One of his most remarkable investigations was concerned with [[Biography:John Napier|John Napier]]'s "[[Pentagramma mirificum]]" - a certain spherical [[Pentagram|pentagram]] whose properties intrigued and occupied Gauss's mind for several decades. In his studies of the Pentagramma he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons and Poncelet pentagons in the plane.<ref>{{cite journal | last = Schechtman | first = Vadim | 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>

=== Magnetism and telegraphy ===
==== Geomagnetism ====
[[File:Göttingen-Gauß-Weber-Monument.01.JPG|thumb|upright|Gauss-Weber monument in Göttingen]]
[[File:A magnetometer used by Carl Friedrich Gauss, from Gerlach und F. Traumüller, 1899.png|thumb|The Gauss-Weber magnetometer]]
Gauss' interest in magnetism is obvious since the first decennium of the 19th century. Since 1826, when [[Biography:Alexander von Humboldt|Alexander von Humboldt]] visited him in Göttingen, both scientists began intensive research on [[Physics:Geomagnetism|geomagnetism]], partly independent, 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 | url = https://www.hin-online.de/index.php/hin/article/view/154/280}}</ref> In 1828, Gauss was Humboldt's personal guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquaintance with the physicist [[Biography:Wilhelm Eduard Weber|Wilhelm Weber]].{{sfn|Dunnington|2004|p=136}}

When Weber got the chair for physics in Göttingen as successor of [[Biography:Johann Tobias Mayer|Johann Tobias Mayer]] by Gauss' recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of [[Physics:Magnetism|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 supported measurements of [[Earth:Earth's magnetic field|Earth's magnetic field]] in many regions of the world with equal methods at arranged dates in the years 1836 to 1841. In 1836, Humboldt was helpful to organize the worldwide spread of observatories including the [[Place:British Empire|British dominions]] with a letter to the Duke of Sussex, then president of the Royal Society, wherein he asked for support for a program of global research based on Gauss' methods.<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 = [[Physics:Philosophical Magazine|Philosophical Magazine]] | issue = 9 | pages = 42–53 | year = 1836 | url = https://archive.org/details/londonedinburghp09lond/page/42/mode/2up}}</ref> Together with other instigators, this led to a global programm known as "[[Biography:Edward Sabine#Magnetical crusade|Magnetical crusade]]" under directory of [[Biography:Edward Sabine|Edward Sabine]]. The dates, times, and intervalls of observations were determined in advance, the ''Göttingen mean time'' was used as standard.<ref name="Rupke">{{cite book | last1 = Rupke | first1 = Nicolaas |  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 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> Finally 61 stations participated in this global program. Gauss and Weber founded a series for the publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to [[Organization:Leipzig University|Leipzig]] in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Assiciation activity.<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 | url = https://www.hin-online.de/index.php/hin/article/view/357/719}}</ref>

Following Humboldt's example, Gauss ordered a magnetic [[Astronomy:Observatory|observatory]] to be built in the garden of his observatory, but both scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity, but discriminated Humboldt's concept of magnetic intensity to the terms of "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of intensity of the magnetic field, and constructed a suitable [[Physics:Magnetometer|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" /> The precision of the magnetometer was about ten times higher than of previous instruments. With this work, Gauss was the first one who derived a non-mechanical quantity by basic mechanical quantities.<ref name="Rupke" />

Gauss carried out a "General Theory of Terrestrial Magnetism" (1839), in what he believed to describe the nature of magnetic Force; following Felix Klein, this work is actually a presentation of observations by use of [[Physics:Spherical harmonics|spherical harmonics]] rather than a physical theory.{{sfn|Klein|1979|pp=21–23}} The theory predicted the existence of exactly two [[Astronomy:Poles of astronomical bodies#Magnetic poles|magnetic poles]] on the earth, thus 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 | 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}} In his "General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances" (1840) Gauss gave the baseline of a theory of the magnetic [[Physics:Magnetic vector potential|potential]], based on Lagrange, Laplace, and Poisson;{{sfn|Klein|1979|pp=21–23}} it seems rather unlikely that he had knowledge of the previous works of [[Biography: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" />

Gauss got a remarkable influence on the begin of geophysics in Russia, when [[Biography:Adolph Theodor Kupffer|Adolph Theodor Kupffer]], one of his former students, founded a magnetic observatory in St. Petersburg, following the eample of the observatory in Götttingen, and similar 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 [[Biography:Hans Christian Ørsted|Hans Christian Ørsted]] on [[Physics:Electromagnetism|electromagnetism]] and [[Biography:Michael Faraday|Michael Faraday]] on [[Physics:Electromagnetic induction|electromagnetic induction]] drew Gauss' attention to these matters.{{sfn|Schaefer|1929|p=6}} Gauss and Weber found the rules for branched [[Physics:Electricity|electric]] circuits, later benamed as [[Kirchhoff's circuit laws]],{{sfn|Schaefer|1929|p=108}} and made inquiries on electromagnetism. They constructed the first electromechanical telegraph in 1833, and Weber himself connected the observatory with the institute for physics in the town centre of Göttingen,{{efn|A thunderstorm damaged the cable in 1845.<ref name="Timm" />}} but they did not care for any further development of this invention with regard to commercial purposes.<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 | 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=García |first2=Gonzalo Barrio |last3=Lires |first3=María Álvarez |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=23592931 }}</ref><ref>{{Cite web |last=Laboratory |first=National High Magnetic Field |title=Gauss-Weber Telegraph – 1833 - Magnet Academy |url=https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/museum/gauss-weber-telegraph-1833/ |access-date=2023-03-31 |website=nationalmaglab.org |language=en}}</ref>

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

In the same year Gauss had an insightful speculative thought, according to which electromagnetic interaction between two [[Physics:Electric charge|electric charge]]s propagates in space in finite speed, in a manner similar to light, and that the magnitude of this interaction might depend on their relative velocity. In this way, he refuted the notion of immediate [[Physics:Action at a distance|action at a distance]]. In unpublished fragments and in an 1845 letter to Weber, Gauss attempted to unite electricity and magnetism by forming a single expression for the interaction between two charges in relative motion, from which both [[Physics:Coulomb's law|Coulomb's law]] and the effects of magnetism could be derived.

His unpublished insights in these directions eventually merged into the so-called [[Physics:Weber electrodynamics|Weber electrodynamics]], a theory that became obsolete today due to some essential  difficulties to reconcile it with the undisputed [[Biography:James Clerk Maxwell|Maxwell]]'s theory. In retrospect, despite its incorrectness, the Gauss-Weber theory contained some of the germs of later ideas, such as the existence of an [[Physics:Electromagnetic field|electromagnetic field]] that is in some sense independent of its point sources (Faraday's view), as well as the notion of [[Physics:Retarded potential|retarded potential]].

=== Optics ===
Instrument maker Johann Georg Repsold in [[Place:Hamburg|Hamburg]] asked Gauss in 1807 for help to construct an achromatic lens system. Based on Gauss' calculations, Repsold succeeded with a new objective in 1810. A main problem, among other difficulties, was the non precise knowledge of the [[Physics:Refractive index|refractive index]] and [[Physics:Dispersion (optics)|dispersion]] of the used glass types. In a short article from 1817 Gauss dealt with the problem of removal of [[Physics:Chromatic aberration|chromatic aberration]] in [[Physics:Gauss lens|double lenses]], and made calculations about adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician [[Biography:Carl August von Steinheil|Carl August von Steinheil]], who in 1860 indroduced the achromatic [[Physics:Achromatic lens|Steinheil doublet]], based in part on Gauss's calculations.{{sfn|Schaefer|1929|p=159–165}} Many results in [[Physics:Geometrical optics|geometrical optics]] are scattered in Gauss's correspondences and handnotes.
  
In his influential ''Dioptrical Investigations'' (1840), Gauss gave the first systematic analysis on the formation of images under a [[Physics:Paraxial approximation|paraxial approximation]] ([[Physics:Gaussian optics|Gaussian optics]]).<ref name=Hecht>{{Cite book | title = Optics | first = Eugene | last = Hecht |  publisher = Addison Wesley | year = 1987 | isbn = 978-0-201-11609-0 | page = 134}}</ref> Gauss demonstrated, that under a paraxial approximation an optical system can be characterized by its [[Physics: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 [[Physics:Lens|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}}

[[File:Carl Friedrich Gauß, Büste von Heinrich Hesemann, 1855.jpg|thumb|left|upright|Gauss bust by Heinrich Hesemann (1855)]]

=== Mechanics ===
Gauss' first and last business in mechanics concerned the [[Earth's rotation|earth's rotation]]. When his university friend [[Biography:Johann Benzenberg|Benzenberg]] carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as an effect of the [[Physics:Coriolis force|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 his results correspondent sufficiently with Benzenberg's data, who published Gauss' considerations as appendix to his book on falling experiments.{{sfn|Geppert|1933|p=3-11}}

After [[Biography:Léon Foucault|Foucault]] had demonstrated his [[Physics:Foucault pendulum|pendulum]] in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum; the time of oscillation was 3.1 seconds. It is described in the Gauss–Gerling correspondence, and Weber made some experiments with this obviously working apparatus in 1853, but no data were published.{{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>

[[Physics:Gauss's principle of least constraint|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 [[Physics:D'Alembert's principle|D'Alembert's principle]] with [[Biography:Joseph-Louis Lagrange|Lagrange]]'s principle of Virtual Work, and showing analogies to the method of [[Least squares|least squares]].{{sfn|Geppert|1933|p=16-26}}

=== Metrology ===
In 1828, Gauss was appointed to head of a Board for weights and measures of the Kingdom of Hanover. He provided the creation of [[Standard (metrology)|standards]] of length and measures. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical preparation.<ref name="Olesko">{{cite book | last1 = Olesko | first1 = Kathryn |  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 = 236–253 | chapter = Der praktische Gauß – Präzisionsmessung für den Alltag | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> In his correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision.{{sfn|Geppert|1933|p=59-60}} He gave his final reports on the Hanoveran [[Foot (unit)|foot]] and [[Pound (mass)|pound]] to the government in 1841. This work got more than regional importance by the order of a law of 1836, that connected the Hanoveran measures with the English ones.<ref name="Olesko" />

== Anecdotes ==
[[File:Gauss keresztlevele.jpg|thumb|[[Social:Parish register|Parochial registration]] of Gauss' christening on 4 May 1777 with later added birth date]]
Several stories of his early genius have been reported. Carl Friedrich Gauss' mother had never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the [[Unsolved:Ascension of Jesus|Feast of the Ascension]], which occurs 39 days after Easter.{{sfn|Dunnington|2004|p=69}} Gauss later solved this puzzle about his birthdate in the context of [[Religion:Computus|finding the date of Easter]], deriving methods to compute the date in both past and future years.<ref>{{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}}</ref> Gauss felt sorry for his new born daughter Wilhelmine, because she was born on the [[Physics:Leap year|leap day]] in 1808 and thus would celebrate her birthday only every four years.{{sfn|Wußing|1982|p=43}}

In his memorial on Gauss, Wolfgang Sartorius von Waltershausen tells a story about the three-years-aged Gauss, who corrected a math error his father made. The most popular story, also told by Sartorius, tells of a school exercise: the teacher, J.G. Büttner, and his assistant, Martin Bartels, ordered students to add an arithmetic series. Out of about a hundred pupils, Gauss was the first to solve the problem correctly by a significant margin.{{sfn|Sartorius von Waltershausen|1856|p=11-13}} Although (or because) Sartorius gave no details, in the course of time many versions of this story have been created, with more and more details regarding the nature of the series{{snd}}the most frequent being the classical problem of adding together all the integers from 1 to 100{{snd}}and the circumstances in the classroom.<ref name="hayesreckoning">{{cite journal | author = Hayes | first = Brian | date = 2006 | title = Gauss's Day of Reckoning | url = https://www.americanscientist.org/article/gausss-day-of-reckoning | url-status = live | journal = American Scientist | volume = 94 | issue = 3 | page = 200 | doi = 10.1511/2006.59.200 |archive-url = https://web.archive.org/web/20120112140951/http://www.americanscientist.org/issues/id.3483,y.0,no.,content.true,page.1,css.print/issue.aspx|archive-date=12 January 2012|access-date=30 October 2012}}</ref>{{efn|Some authors, such as [[Biography:Joseph J. Rotman|Joseph J. Rotman]], question whether it ever happened.<ref>{{Cite book |last=Rotman |first=Joseph J. |title=A first course in abstract algebra: with applications |date=2006 |publisher=Pearson Prentice Hall |isbn=0-13-186267-7 |edition=3rd |location=Upper Saddle River, N.J. |pages=7–8 |oclc=61309485}}</ref>}}

Gauss' favorite English author was Walter Scott, but when he sometimes read the words "the moon rises broad in the nord west", he was very amused.{{sfn|Sartorius von Waltershausen|1856|p=93-94}}

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|title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics|year=2003|publisher=Joseph Henry Press|location= Washington, DC|isbn=978-0-309-08549-6|page=[https://archive.org/details/primeobsessionbe00derb_0/page/202 202]|url=https://archive.org/details/primeobsessionbe00derb_0|url-access=registration}}</ref>

== Honours and awards ==
[[File:Carl Friedrich Gauß, Copley-Medaille, 1838.jpg|thumb|upright|Copley Medal for Gauss (1838)]]
The first membership of a scientific society was given to Gauss in 1802 by the Russian Academy of Sciences. Further memberships (corresponding, foreign or full) were from the Academy of Sciences in Göttingen (1802/ 1807),<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/Overview.aspx?src=CalmView.Persons&r=(((Surname=%27Gauss%27))) | publisher = The Royal Society | access-date = 7 April 2023}}</ref> the 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 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 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 in Copenhagen (1821), 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), 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), 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/ Academy members]</ref> the [[Organization:Royal Society of Sciences in Uppsala|Royal Society of Sciences in Uppsala]] (1843), the Royal Irish Academy in Dublin (1843), the 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 [[Organization:Russian Geographical Society|Russian Geographical Society]] (1851), the Imperial Academy of Sciences in Vienna (1848), 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=2021-04-16|website=search.amphilsoc.org}}</ref> the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem.{{sfn|Dunnington|2004|p=359}}

Gauss was an honorary member of the University of Kazan and of the Philosophy Faculty of the [[Organization:Charles University|University of Prague]] since 1849.

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 | location = Paris | url = https://books.google.com/books?id=biIgAQAAMAAJ&pg=PA460}}</ref> the Danish Academy of Science prize in 1823 for "his study of angle-preserving maps", and the [[Physics:Copley Medal|Copley Medal]] from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".{{sfn|Dunnington|2004|p=359}}

Gauss was appointed Knight of the French Legion of Honour<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> in 1837 and was one of the first members of the Prussian 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> Furthermore, he received the Order of the Crown of Westphalia (1810), the Danish Order of the Dannebrog (1817), the Hanoverian Royal Guelphic Order (1815), the Swedish Order of the Polar Star (1844), the Order of Henry the Lion (1849), and the Bavarian Maximilian Order for Science and Art (1853).{{sfn|Dunnington|2004|p=359}}

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). On the occasion of his golden doctor degree jubilee he got the honorary citizenship of both towns of Brunswick and Göttingen in 1849.{{sfn|Dunnington|2004|p=359}} Soon after his death a medal was issued by order of King George V of Hanover with the back side inscription : ''GEORGIVS V REX HANNOVERAE MATHEMATICORVM PRINCIPI'' and the circumscription : ''ACADEMIAE SVAE GEORGIAE AVGVSTAE DECORI AETERNO''.<ref>[https://www.bing.com/images/search?view=detailV2&ccid=wIlk7Itq&id=E1F48105357FA67A9040B8576772536CDC2DCD8F&thid=OIP.wIlk7ItqdyDpnOxhaU26HgHaDu&mediaurl=https%3A%2F%2Fimg.ma-shops.de%2Fsauer%2Fpic%2F560_ma525_g.jpg&exph=494&expw=981&q=gauss+medaille&simid=608054884459764795&form=IRPRST&ck=23EBCB22E7F8AF16024961E62AFDC380&selectedindex=26&qpvt=gauss+medaille&ajaxhist=0&ajaxserp=0&vt=0&sim=11&cdnurl=https%3A%2F%2Fth.bing.com%2Fth%2Fid%2FR.c08964ec8b6a7720e99cec61694dba1e%3Frik%3Dj80t3GxTcmdXuA%26pid%3DImgRaw%26r%3D0 Image of the Royal medal].</ref>

The ″Gauss-Gesellschaft Göttingen″ (''Gauss Society'') was founded in 1964 for researches on life and work of Carl Friedrich Gauss and related persons and edits the ″Mitteilungen der Gauss-Gesellschaft″ (''Communications of the Gauss Society'').<ref>Website of [http://www.gauss-gesellschaft-goettingen.de/gaussges.html Gauss Society Göttingen].</ref>

== Writings ==
=== Mathematics and astronomy ===
[[File:Gauß-Denkmal - Braunschweig, Germany - DSC04436.JPG|thumb|upright|Statue of Gauss in 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  | location = 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/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B268%2C269%5D%2C%22view%22%3A%22info%22%7D}}
* 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/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B296%5D%2C%22view%22%3A%22info%22%7D}}
* 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/PPN250442582_0004?tify=%7B%22pages%22%3A%5B287%5D%2C%22view%22%3A%22info%22%7D}}
** {{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 = Netto | trans-title = The four Gaussian proofs of the fundamental theorem of algebra | publisher = Wilhelm Engelmann | location = 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://zs.thulb.uni-jena.de/receive/jportal_jpvolume_00201970}} (Calculation of the date of Easter)
* 1801: {{Cite book | title = Disquisitiones Arithmeticae | publisher = Gerh. Fleischer jun. | location = 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 = [[Company:Springer Science+Business Media|Springer]] | location = 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}}
* 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://zs.thulb.uni-jena.de/rsc/viewer/jportal_derivate_00237503/Monatlich_Correspondenz_130168688_5_1802_0444%20.tif}} (Calculation of the date of Pesach)
* 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/PPN35283028X_0016_1NS?tify=%7B%22pages%22%3A%5B498%2C499%5D%2C%22view%22%3A%22info%22%7D}} (Introduces [[Gauss's lemma (number theory)|Gauss's lemma]], uses it in the third proof of quadratic reciprocity)
* 1809: {{Cite book | last = Gauss | first = Carl Friedrich | display-authors = 0 | title = Theoria motus corporum coelestium in sectionibus conicis solem ambientium |language=la | publisher = Friedrich Perthes & Johann Heinrich Besser | location = 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 | last = Gauss | first = Carl Friedrich | display-authors = 0 | year = 1857 | title = Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections | translator-last = Davis | translator-first=Charles Henry | publisher=Little, Brown & Co. | url = https://archive.org/details/theoryofmotionof00gausrich/}}
* 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/PPN35283028X_0001_2NS?tify=%7B%22pages%22%3A%5B170%2C171%5D%2C%22view%22%3A%22info%22%7D}} (Orbit of [[Astronomy: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/PPN35283028X_0001_2NS?tify=%7B%22pages%22%3A%5B194%2C195%5D%2C%22view%22%3A%22info%22%7D}} (Determination of the sign of the [[Quadratic Gauss sum|quadratic Gauss sum]], uses this to give the fourth proof of quadratic reciprocity)
* 1812: {{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/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}}
* 1815: {{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/PPN35283028X_0003_2NS?tify=%7B%22pages%22%3A%5B200%2C201%5D%2C%22view%22%3A%22info%22%7D}}
* 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/PPN35283028X_0004_2NS?tify=%7B%22pages%22%3A%5B262%2C263%5D%2C%22view%22%3A%22info%22%7D}} (Fifth and sixth proofs of quadratic reciprocity)
* 1818: {{cite journal | title = Demonstratio 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/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}}
* 1821: {{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/PPN35283028X_0005_2NS?tify=%7B%22pages%22%3A%5B276%5D%2C%22view%22%3A%22info%22%7D}}
* 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/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}}
* 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 | bibcode = 1828stco.book.....G | url = https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0006_2NS?tify=%7B%22pages%22%3A%5B271%5D%2C%22view%22%3A%22info%22%7D | last1 = Gauss | first1 = Carl Friedrich | date = 1828 | publisher = Dieterich }} (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
** {{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 | location = Philadelphia | year = 1995 | doi=10.1137/1.9781611971248 | url = https://epubs.siam.org/doi/book/10.1137/1.9781611971248 | last1 = Gauss | first1 = Carl Friedrich | last2 = Stewart | first2 = G. W. | isbn = 978-0-89871-347-3}}
* 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/PPN35283028X_0006_2NS?tify=%7B%22pages%22%3A%5B312%2C313%5D%2C%22view%22%3A%22info%22%7D}}
** {{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/PPN35283028X_0006_2NS?tify=%7B%22pages%22%3A%5B241%5D%2C%22view%22%3A%22info%22%7D}}
* 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/PPN35283028X_0007_2NS?tify=%7B%22pages%22%3A%5B273%5D%2C%22view%22%3A%22info%22%7D}} (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 | pages = 3–46 | url = https://gdz.sub.uni-goettingen.de/id/PPN250442582_0002}}
* 1847: {{cite journal | title = Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen | volume = Dritter Band | pages = 3–44 | url = https://gdz.sub.uni-goettingen.de/id/PPN250442582_0003}}
* {{Citation |  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]
** {{cite journal | title = A commentary on Gauss's mathematical diary, 1796–1814 | author = [[Biography:Jeremy Gray|Jeremy Gray]] | journal = Expositiones Mathematicae | volume = 2 | pages = 97–130 | year = 1984 }}

=== Physics ===
* 1804: {{cite book | last1 = Gauss | first1 = Carl Friedrich | editor-last1 = Benzenberg | editor-first1 = Johann Friedrich |  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 | location = Dortmund | pages = 363–371 | chapter = Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde | chapter-url = https://www.digitale-sammlungen.de/de/view/bsb10060427?page=392,393}}
* 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/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}}
* 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 (Crelle's Journal) | volume = 1829 | issue = 4| pages = 232–235 | doi = 10.1515/crll.1829.4.232 | s2cid = 199545985 | url = https://zenodo.org/record/1448816}}
* 1832: {{cite journal | title = Principia generalia theoriae figurae fluidorum fluidorum in statu aequilibrii | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores | volume = 7 | pages = 39–88 | url = https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0007_2NS?tify=%7B%22pages%22%3A%5B223%5D%2C%22view%22%3A%22info%22%7D}}
* 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/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}}
** [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: {{cite journal | title = Erdmagnetismus und Magnetometer | language = de | journal = Jahrbuch für 1836 | volume = 1836 | pages = 1–47 | editor = H.C. Schumacher | publisher = J.G.Cotta'sche Buchhandlung | location = Tübingen | url = https://www.digitale-sammlungen.de/de/view/bsb10538569?page=21}}
* 1840: {{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 | location = 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/PPN250442582_0001?tify=%7B%22pages%22%3A%5B548%2C549%5D%2C%22view%22%3A%22info%22%7D}}

==== together with Wilhelm Weber ====
* 1837–1839: {{Cite book | title = Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1836–1838 | publisher = Dieterichsche Buchhandlung| location = 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 | location = 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 | location = 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) | location = Göttingen| language = la, de | year = 1863–1933 | url = https://gdz.sub.uni-goettingen.de/id/PPN235957348?tify=%7B%22view%22:%22toc%22%7D}}

=== Correspondence ===
* {{Cite book| title = Briefwechsel zwischen Gauss und Bessel | editor = Königlich Preußische Akademie der Wissenschaften |publisher = Wilhelm Engelmann| location = 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 | location = Munich | language = de}} (letters to 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 | location = 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| location = 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 | location = 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 | location = 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 | location = 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 |  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 | location = Vienna | language = de}} (letters from 1835 to 1843)
* {{cite book | title = Briefwechsel zwischen Carl Friedrich Gauß und 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 | location = 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 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 | location = 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 | location = 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 | location = 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 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 | location = Göttingen | language = de | isbn = 978-3525821169}} (letters from 1795 to 1815)

<gallery>
File:Disqvisitiones-800.jpg|Title page of Gauss' magnum opus, ''[[Disquisitiones Arithmeticae]]'' (1801)
File:Gauss-12.jpg|Title of Theorema arithmetici demonstratio nova (1808)
File:Gauss-5.jpg|alt=Title page of an 1890 copy of Carl Gauss' "Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium."|Title page of ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium'' (1809)
File:Gauss-9.jpg|alt=Title page of an 1857 copy of "Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections: A Translation of Gauss' "Theoria Motus," translated to English by Charles Henry Davis|Title page to the English Translation of ''Theoria Motus'' by Charles Henry Davis (1857)
File:Gauss-2.jpg|alt=Title page of an 1833 copy of "Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata."|Title page of ''Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata'' (1833)
File:Gauss-16.jpg|Volume II of "''Carl Friedrich Gauss Werke''," 1876
</gallery>

The Göttingen Academy of Sciences and Humanities provides a complete collection of the yet known letters from and to Carl Friedrich Gauss that is accessible online.<ref name="Correspondence">{{cite web | url = https://gauss.adw-goe.de/?locale-attribute=de | title = Der komplette Briefwechsel von Carl Friedrich Gauss | publisher = Akademie der Wissenschaften zu Göttingen | access-date = 10 March 2023}}</ref> Written estate 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>

== Names and Commemorations ==
* List of things named after Carl Friedrich Gauss

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

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

=== Sources ===
{{refbegin|indent=yes}}
* {{cite book | last = Brendel | first = Martin |  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 |  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 | location = New York}}
** {{cite book| first1 = Jeremy | last1 = 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 | location = 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 | 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 | 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 |  series = Grundlehren der mathematischen Wissenschaften 24 | publisher = Springer-Verlag | orig-date = 1926 | date = 1979 | location = 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 book | last = Sartorius von Waltershausen | first = Wolfgang |  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 = Helen Worthington Gauss | location = Colorado Springs | year = 1966 | url = https://ia600405.us.archive.org/20/items/gaussmemorial00walt/gaussmemorial00walt.pdf}}
* {{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 |  editor = Königlich Preußische Akademie der Wissenschaften | title = Carl Friedrich Gauss. Werke | volume = X, 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 = Wußing | first = Hans | title = Carl Friedrich Gauß | year = 1982 | location = Leipzig | edition = 4 | publisher = BSB B. G. Teubner | language =de}}
* {{cite book | last = Stäckel | first = Paul |  editor = Königlich Preußische Akademie der Wissenschaften | title = Carl Friedrich Gauss. Werke | volume = X, 2 | 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}}
{{Refend}}

=== Further reading ===
{{refbegin|indent=yes}}
* {{Cite book |  last2 = Boyer | first2 = Carl B. |  last1 = Merzbach | first1 = Uta C. | title = A History of Mathematics | url = https://books.google.com/books?id=bR9HAAAAQBAJ | publisher = John Wiley & Sons | location = New Jersey | edition = 3rd | year = 2011 | isbn = 978-0470630563}}
* {{cite book|last=[[Biography:Tord Hall|Hall]]|first=Tord|title=Carl Friedrich Gauss: A Biography|location=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. | 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 | location = 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 |  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 = Gauss | author-first = Carl Friedrich | translator-last = Janeway | translator-first = Carol Brown | translator-link = Carol Brown Janeway | location = | year = 2006}}
{{Refend}}

== External links ==
{{commons|Johann Carl Friedrich Gauß}}
{{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 [[Physics:Astrophysics Data System|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}}
* {{MacTutor Biography|id=Gauss}}
* {{cite web|url=http://www.bbc.co.uk/programmes/b00ss0lf |title=Carl Friedrich Gauss |series=A Brief History of Mathematics|website=BBC 4}}
* {{cite web|last=Grimes|first=James|title=5050 And a Gauss Trick|url=http://www.numberphile.com/videos/one_to_million.html|work=Numberphile|publisher=[[Biography:Brady Haran|Brady Haran]]|archive-url=https://archive.today/20130411122756/http://www.numberphile.com/videos/one_to_million.html|archive-date=11 April 2013}}
* {{cite web|url=https://www.uni-goettingen.de/en/62596.html |title=Carl Friedrich Gauß|location=Göttingen University}}
* {{cite web|url=http://www.gausschildren.org/cfgauss/cfgauss.htm |title=Gauss|website=www.gausschildren.org}}
* [https://denkmalatlas.niedersachsen.de/viewer/themen/Gauss-Steine/ ''Carl Friedrich Gauss – Spuren seines Lebens''] (Places used as points for triangulation)
{{refend}}

{{Age of Enlightenment}}
{{Scientists whose names are used as non SI units}}

{{DEFAULTSORT:Gauss, Carl Friedrich}}
[[Category:Ceres (dwarf planet)]]
[[Category:Differential geometers]]
[[Category:Number theorists]]
[[Category:Linear algebraists]]
[[Category:Optical physicists]]

{{Sourceattribution|Carl Friedrich Gauss}}