Editing Carl Friedrich Gauss (section)

== Sciences ==
=== Astronomy ===
{{Main|Discovery of Ceres}}
[[File:Carl Friedrich Gauß, Pastellgemälde von Johann Christian August Schwartz, 1803, ohne Rahmen.jpg|thumb|left|upright|Carl Friedrich Gauss 1803 by Johann Christian August Schwartz]]
On 1 January 1801, Italian astronomer [[Giuseppe Piazzi]] discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called [[Titius–Bode law]], and named it [[Ceres (dwarf planet)|Ceres]].<ref>{{Cite journal | last = Forbes | first = Eric G. | author-link = Eric G. Forbes | year = 1971 | title = Gauss and the Discovery of Ceres | url=http://adsabs.harvard.edu/full/1971JHA.....2..195F | url-status=live | journal = [[Journal for the History of Astronomy]] | volume = 2 | issue = 3 | pages = 195–199 | bibcode=1971JHA.....2..195F | doi=10.1177/002182867100200305 |archive-url=https://web.archive.org/web/20210718200510/http://adsabs.harvard.edu/full/1971JHA.....2..195F | archive-date=18 July 2021 |s2cid=125888612}}</ref> He could track it only for a short time until it disappeared behind the glare of the Sun. The mathematical tools of the time were not sufficient to predict the location of its reappearance from the few data available. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree when [[Franz Xaver von Zach]] on 7 and 31 December at [[Gotha Observatory|Gotha]], and independently [[Heinrich Wilhelm Matthäus Olbers|Heinrich Olbers]] on 1 and 2 January in [[Bremen]], identified the object near the predicted position.<ref>{{cite journal | last1 = Teets | first1 = Donald | last2 = Whitehead | first2 = Karen | title = The discovery of Ceres. How Gauss became famous | journal = [[Mathematics Magazine]] | volume = 19 | issue = 90 | pages = 83–91 | year = 1965 | url = https://www.maa.org/programs/maa-awards/writing-awards/the-discovery-of-ceres-how-gauss-became-famous | access-date = 22 March 2023 | archive-date = 3 April 2023 | archive-url = https://web.archive.org/web/20230403074017/https://www.maa.org/programs/maa-awards/writing-awards/the-discovery-of-ceres-how-gauss-became-famous | url-status = dead }}</ref>{{efn|The unambiguous identification of a cosmic object as planet among the fixed stars requires at least two observations with interval.}}

[[Gauss's method]] leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.{{sfn|Klein|1979|p=8}}

The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as ''Theoria motus corporum coelestium in sectionibus conicis solem ambientum''.<ref>Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.</ref> It introduced the [[Gaussian gravitational constant]].<ref name="Wittmann" />

Since the new asteroids had been discovered, Gauss occupied himself with the [[Perturbation (astronomy)|perturbation]]s of their [[orbital elements]]. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was [[(2) Pallas|Pallas]], because of its great [[Eccentricity (astronomy)|eccentricity]] and [[orbital inclination]], whereby Laplace's method did not work. Gauss used his own tools: the [[arithmetic–geometric mean]], the [[hypergeometric function]], and his method of interpolation.{{sfn|Brendel|1929|pp=194–195}} He found an [[orbital resonance#Coincidental 'near' ratios of mean motion|orbital resonance]] with [[Jupiter]] in proportion 18:7 in 1812; Gauss gave this result as [[cipher]], and gave the explicit meaning only in letters to Olbers and Bessel.{{sfn|Brendel|1929|p=206}}<ref>{{cite journal | last = Taylor | first = D. B. | year = 1982 | title = The secular motion of Pallas | journal = [[Monthly Notices of the Royal Astronomical Society]] | volume = 199 | issue = 2 | pages = 255–265 | bibcode=1982MNRAS.199..255T |doi=10.1093/mnras/199.2.255 | doi-access=free}}</ref>{{efn|Brendel (1929) thought this cipher to be insoluble, but actually decoding was very easy.{{sfn|Brendel|1929|p=206}}<ref>{{cite book | last1 = Schroeder | first1 = Manfred R. | author-link  = Manfred R. Schroeder | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 259–260 | chapter = Gauß, die Konzertsaalakustik und der Asteroid Palls | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref>}} After long years of work, he finished it in 1816 without a result that seemed sufficient to him. This marked the end of his activities in theoretical astronomy.{{sfn|Brendel|1929|p=254}}

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

Even after Gauss's contributions to theoretical astronomy came to an end, more practical activities in [[observational astronomy]] continued and occupied him during his entire career. As early as 1799, Gauss dealt with the determination of longitude by use of the lunar parallax, for which he developed more convenient formulas than those were in common use.{{sfn|Brendel|1929|pp=8–9}} After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables of [[Astronomical nutation|nutation]] and [[Aberration (astronomy)|aberration]], solar coordinates, and refraction.{{sfn|Brendel|1929|p=3}} He made many contributions to [[spherical geometry]], and in this context solved some practical problems about [[Celestial navigation|navigation by stars]].{{sfn|Brendel|1929|p=54}} He published a great number of observations, mainly on minor planets and comets; his last observation was the [[solar eclipse of 28 July 1851]].{{sfn|Brendel|1929|p=144}}

=== Chronology ===
Gauss's first publication following his doctoral thesis dealt with the determination of the [[Date of Easter#Gauss's Easter algorithm|date of Easter]] (1800), an elementary mathematical topic. Gauss aimed to present a convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usual terms of [[Golden number (time)|golden number]], [[epact]], [[Solar cycle (calendar)|solar cycle]], [[Dominical letter|domenical letter]], and any religious connotations.{{sfn|Maennchen|1930|pp=49–63}} This choice of topic likely had historical grounds. The replacement of the [[Julian calendar]] by the [[Gregorian calendar]] had caused confusion in the [[Holy Roman Empire]] since the 16th century and was not finished in Germany until 1700, when the difference of eleven days was deleted. Even after this, Easter fell on different dates in Protestant and Catholic territories, until this difference was abolished by agreement in 1776. In the Protestant states, such as the Duchy of Brunswick, the Easter of 1777, five weeks before Gauss's birth, was the first one calculated in the new manner.<ref name="Olesko" />

=== Error theory ===
Gauss likely used the [[Least squares|method of least squares]] to minimize the impact of [[Observational error|measurement error]] when calculating the orbit of Ceres.<ref name="Stigler" /> The method was published first by [[Adrien-Marie Legendre]] in 1805, but Gauss claimed in ''Theoria motus'' (1809) that he had been using it since 1794 or 1795.{{sfn|Schaaf|1964|p=84}}<ref name=":3">{{Cite journal | last = Plackett | first = R.L. | author-link = R. L. Plackett | date = 1972 | title = The discovery of the method of least squares | url = https://hedibert.org/wp-content/uploads/2016/08/plackett1972-thediscoveryofthemethodofleastsquares.pdf | journal = Biometrika | volume = 59 | issue = 2 | pages =239–251| doi = 10.2307/2334569 | jstor = 2334569 }}</ref><ref>{{Cite web | last = Lim | first = Milton | date = 31 March 2021 | title = Gauss, Least Squares, and the Missing Planet | url = https://www.actuaries.digital/2021/03/31/gauss-least-squares-and-the-missing-planet/ | access-date = 14 October 2023 | website = Actuaries Digital}}</ref> In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares".<ref name="Stigler" /> Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of [[normal distribution|normally distributed]] errors ([[Gauss–Markov theorem]]), in the two-part paper ''Theoria combinationis observationum erroribus minimis obnoxiae'' (1823).<ref>{{cite journal |first=R. L. |last=Plackett |author-link=Robin Plackett |title=A Historical Note on the Method of Least Squares |journal=[[Biometrika]] |volume=36 |issue=3/4 |year=1949 |pages=458–460 |doi=10.2307/2332682 |jstor=2332682 |pmid=15409359 }}</ref>

In the first paper he proved [[Gauss's inequality]] (a [[Chebyshev's inequality|Chebyshev-type inequality]]) for [[Unimodality|unimodal distributions]], and stated without proof another inequality for [[Moment (mathematics)|moments]] of the fourth order (a special case of the Gauss-Winckler inequality).<ref>{{Cite journal | last = Avkhadiev | first = F. G. | title = A Simple Proof of the Gauss-Winckler Inequality | journal = [[The American Mathematical Monthly]] | volume = 112 | issue = 5 | pages = 459–462 | year = 2005| doi = 10.2307/30037497 | jstor = 30037497 | doi-access =  }}</ref> He derived lower and upper bounds for the [[variance]] of the [[sample variance]]. In the second paper, Gauss described [[Recursive least squares filter|recursive least squares methods]]. His work on the theory of errors was extended in several directions by the geodesist [[Friedrich Robert Helmert]] to the [[Gauss-Helmert model]].<ref>{{cite journal | last1=Schaffrin | first1=Burkhard | last2=Snow | first2=Kyle | title=Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth | journal=Linear Algebra and Its Applications | publisher=Elsevier BV | volume=432 | issue=8 | year=2010 | issn=0024-3795 | doi=10.1016/j.laa.2009.09.014 | pages=2061–2076| doi-access=free }}</ref>

Gauss also contributed to problems in [[probability theory]] that are not directly concerned with the theory of errors. One example appears as a diary note where he tried to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in ''(0,1)''. He derived this distribution, now known as the [[Gauss-Kuzmin distribution]], as a by-product of the discovery of the [[ergodicity]] of the [[Gauss-Kuzmin-Wirsing operator|Gauss map for continued fractions]]. Gauss's solution is the first-ever result in the metrical theory of continued fractions.<ref>{{Cite journal | last = Sheynin | first = O. B. | title = C. F. Gauss and the Theory of Errors | journal = [[Archive for History of Exact Sciences]] | volume = 20 | issue = 1 | pages = 21–72 | year = 1979| doi = 10.1007/BF00776066 | jstor = 41133536 | doi-access =  }}</ref>

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

Since 1816, Gauss's former student [[Heinrich Christian Schumacher]], then professor in [[Copenhagen]], but living in [[Altona, Hamburg|Altona]] ([[Duchy of Holstein|Holstein]]) near [[Hamburg]] as head of an observatory, carried out a [[triangulation]] of the [[Jutland]] peninsula from [[Skagen]] in the north to [[Lauenburg]] in the south.{{efn|Lauenburg was the southernmost town of the [[Duchy of Holstein]], that was held in personal union by the King of [[Denmark]].}} This project was the basis for map production but also aimed at determining the geodetic arc between the terminal sites. Data from geodetic arcs were used to determine the dimensions of the earth [[geoid]], and long arc distances brought more precise results. Schumacher asked Gauss to continue this work further to the south in the Kingdom of Hanover; Gauss agreed after a short time of hesitation. Finally, in May 1820, King [[George IV]] gave the order to Gauss.{{sfn|Galle|1924|p=32}}

An [[arc measurement]] needs a precise astronomical determination of at least two points in the [[Triangulation network|network]]. Gauss and Schumacher used the coincidence that both observatories in Göttingen and Altona, in the garden of Schumacher's house, laid nearly in the same [[longitude]]. The [[latitude]] was measured with both their instruments and a [[zenith sector]] of [[Jesse Ramsden|Ramsden]] that was transported to both observatories.{{sfn|Galle|1924|p=61}}{{efn|This Ramsden sector was loaned by the [[Board of Ordnance]], and had earlier been used by [[William Mudge]] in the [[Principal Triangulation of Great Britain]].{{sfn|Galle|1924|p=61}}}}

Gauss and Schumacher had already determined some angles between [[Lüneburg]], Hamburg, and Lauenburg for the geodetic connection in October 1818.{{sfn|Galle|1924|p=60}} During the summers of 1821 until 1825 Gauss directed the triangulation work personally, from [[Thuringia]] in the south to the river [[Elbe]] in the north. The [[triangle]] between [[Hoher Hagen (Dransfeld)|Hoher Hagen]], [[Großer Inselsberg]] in the [[Thuringian Forest]], and [[Brocken]] in the [[Harz]] mountains was the largest one Gauss had ever measured with a maximum size of {{convert|107|km|1|abbr=in}}. In the thinly populated [[Lüneburg Heath]] without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.<ref name="Olesko" />{{sfn|Galle|1924|pp=75–80}}

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

An additional result was a better value for the [[flattening]] of the approximative [[Earth ellipsoid]].{{sfn|Galle|1924|pp=37–38, 49–50}}{{efn|The value from [[Henrik Johan Walbeck|Walbeck]] (1820) of 1/302,78 was improved to 1/298.39; the calculation was done by Eduard Schmidt, private lecturer at Göttingen University.{{sfn|Galle|1924|p=49-50}}}} Gauss developed the [[Transverse Mercator projection#Ellipsoidal transverse Mercator|universal transverse Mercator projection]] of the ellipsoidal shaped Earth (what he named ''conform projection''){{sfn|Dunnington|2004|p=164}} for representing geodetical data in plane charts.

When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the whole [[Kingdom of Hanover]] with a Royal decree from 25 March 1828.{{sfn|Dunnington|2004|p=135}} The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Gauss, who applied his mathematical inventions such as the [[method of least squares]] and the [[Gaussian elimination|elimination method]] to it. The project was finished in 1844, and Gauss sent a final report of the project to the government; his method of projection was not edited until 1866.{{sfn|Galle|1924|p=129}}<ref>{{cite book | last = Schreiber | first = Oscar | title = Theorie der Projectionsmethode der Hannoverschen Landesvermessung | publisher = Hahnsche Buchhandlung | year = 1866 | place = Hannover | language = de}}</ref>

In 1828, when studying differences in [[latitude]], Gauss first defined a physical approximation for the [[figure of the Earth]] as the surface everywhere perpendicular to the direction of gravity;<ref name="Gauß1828">{{cite book | last = Gauß | first = C.F. | title=Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector | publisher=Vandenhoeck und Ruprecht | year = 1828 | language=de | page=73}}</ref> later his doctoral student [[Johann Benedict Listing]] called this the ''[[geoid]]''.<ref>{{cite book | last = Listing | first = J.B. | title = Ueber unsere jetzige Kenntniss der Gestalt und Grösse der Erde | publisher = Dieterich | year = 1872 | place = Göttingen | url=https://books.google.com/books?id=yQ9TAAAAcAAJ | language = de | page = 9}}</ref>

=== Magnetism and telegraphy ===
==== Geomagnetism ====
[[File:Göttingen-Gauß-Weber-Monument.01.JPG|thumb|upright|Gauss-Weber monument in Göttingen by Ferdinand Hartzer (1899)]]
[[File:A magnetometer used by Carl Friedrich Gauss, from Gerlach und F. Traumüller, 1899.png|thumb|The Gauss–Weber magnetometer]]
Gauss had been interested in magnetism since 1803.{{sfn|Dunnington|2004|p=153}} After [[Alexander von Humboldt]] visited Göttingen in 1826, both scientists began intensive research on [[geomagnetism]], partly independently, partly in productive cooperation.<ref>{{cite journal | author-last = Reich | author-first = Karin  | title = Alexander von Humboldt und Carl Friedrich Gauss als Wegbereiter der neuen Disziplin Erdmagnetismus | journal = Humboldt Im Netz | volume = 12 | issue = 22 | pages = 33–55 | year = 2011 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/154/280}}</ref> In 1828, Gauss was Humboldt's guest during the conference of the [[Society of German Natural Scientists and Physicians]] in Berlin, where he got acquainted with the physicist [[Wilhelm Eduard Weber|Wilhelm Weber]].{{sfn|Dunnington|2004|p=136}}

When Weber got the chair for physics in Göttingen as successor of [[Johann Tobias Mayer]] by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of [[magnetism]] with a representation for the unit of magnetism in terms of mass, charge, and time.{{sfn|Dunnington|2004|p=161}} They founded the ''Magnetic Association'' (German: ''Magnetischer Verein''), an international working group of several observatories, which carried out measurements of [[Earth's magnetic field]] in many regions of the world using equivalent methods at arranged dates in the years 1836 to 1841.<ref name="Reich">{{cite journal | author-last = Reich | author-first = Karin  | title = Der Humboldt'sche Magnetische Verein im historischen Kontext | journal = Humboldt Im Netz | volume = 24 | issue = 46 | pages = 53–74 | year = 2023 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/357/719}}</ref>

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the [[British Empire|British dominions]] with a letter to the [[Prince Augustus Frederick, Duke of Sussex|Duke of Sussex]], then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods.<ref>{{cite journal | author-last = Biermann | author-first = Kurt-R. | title = Aus der Vorgeschichte der Aufforderung Alexander von Humboldt von 1836 an den Präsidenten der Royal Society zur Errichtung geomagnetischer Stationen (Dokumente zu den Beziehungen zwischen A.v. Humboldt und C. F. Gauß) | journal = Humboldt Im Netz | volume = 6 | issue = 11 | year = 2005 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/154/280}}</ref><ref>{{cite journal | author-last = Humboldt | author-first = Alexander von | title = Letter of the Baron von Humboldt to His Royal Highness the Duke of Sussex,..., on the Advancement of the Knowledge of Terrestrial Magnetism, by the Establishment of Magnetic Stations and correspionding Observations | journal = [[Philosophical Magazine]] | issue = 9 | pages = 42–53 | year = 1836 | volume = Bd. 6 | doi = 10.18443/70 | url = https://www.hin-online.de/index.php/hin/article/view/70/112}}</ref> Together with other instigators, this led to a global program known as "[[Edward Sabine#Magnetical crusade|Magnetical crusade]]" under the direction of [[Edward Sabine]]. The dates, times, and intervals of observations were determined in advance, the ''Göttingen mean time'' was used as the standard.<ref name="Rupke">{{cite book | last1 = Rupke | first1 = Nicolaas | author-link  = Nicolaas Adrianus Rupke | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 188–201 | chapter = Carl Friedrich Gauß und der Erdmagnetismus | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> 61 stations on all five continents participated in this global program. Gauss and Weber founded a series for publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to [[Leipzig University|Leipzig]] in 1843 as late effect of the [[Göttingen Seven|Göttingen Seven affair]] marked the end of Magnetic Association activity.<ref name="Reich" />

Following Humboldt's example, Gauss ordered a magnetic [[observatory]] to be built in the garden of the observatory, but the scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic [[Magnetic declination|declination]], [[Magnetic dip|inclination]], and intensity and differentiated, unlike Humboldt, between "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of the intensity of the magnetic field and constructed a suitable [[magnetometer]] to measure ''absolute values'' of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus.<ref name="Reich" />{{sfn|Schaaf|1964|pp=115–127}} The precision of the magnetometer was about ten times higher than that of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities.<ref name="Rupke" />

Gauss carried out a ''General Theory of Terrestrial Magnetism'' (1839), in what he believed to describe the nature of magnetic force; according to Felix Klein, this work is a presentation of observations by use of [[spherical harmonics]] rather than a physical theory.{{sfn|Klein|1979|pp=21–23}} The theory predicted the existence of exactly two [[Poles of astronomical bodies#Magnetic poles|magnetic poles]] on the Earth, thus [[Christopher Hansteen|Hansteen]]'s idea of four magnetic poles became obsolete,<ref name="Roussanova">{{cite journal | author-last = Roussanova | author-first = Elena  | title = Russland ist seit jeher das gelobte Land für Magnetismus gewesen: Alexander von Humboldt, Carl Friedrich Gauß und die Erforschjung des Erdmagnetismus in Russland | journal = Humboldt Im Netz | volume = 12 | issue = 22 | pages = 56–83 | year = 2011 | language = de | url = https://www.hin-online.de/index.php/hin/article/view/155/282}}</ref> and the data allowed to determine their location with rather good precision.{{sfn|Schaefer|1929|p=87}}

Gauss influenced the beginning of geophysics in Russia, when [[Adolph Theodor Kupffer]], one of his former students, founded a magnetic observatory in [[St. Petersburg]], following the example of the observatory in Göttingen, and similarly, [[Ivan Simonov]] in [[Kazan]].<ref name="Roussanova"/>

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

Gauss's main theoretical interests in electromagnetism were reflected in his attempts to formulate quantitive laws governing electromagnetic induction. In notebooks from these years, he recorded several innovative formulations; he discovered the [[vector potential]] function, independently rediscovered by [[Franz Ernst Neumann]] in 1845, and in January 1835 he wrote down an "induction law" equivalent to [[Faraday's law of induction|Faraday's law]], which stated that the [[electromotive force]] at a given point in space is equal to the [[instantaneous rate of change]] (with respect to time) of this function.<ref>Printed in the ''Collected Works'', Volume 5, pp. 609–610.</ref><ref>{{cite book | last = Roche | first = John J. | chapter = A critical study of the vector potential | editor-last = Roche | editor-first = John | date = 1990 | title = Physicists Look Back: Studies in the History of Physics | publisher = Adam Hilger | place = Bristol, New York | pages = 147–149 | isbn = 0-85274-001-8}}</ref>

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

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

In the ''General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances'' (1840) Gauss gave a basic theory of [[Magnetic vector potential|magnetic potential]], based on Lagrange, Laplace, and Poisson;{{sfn|Klein|1979|pp=21–23}} it seems rather unlikely that he knew the previous works of [[George Green (mathematician)|George Green]] on this subject.{{sfn|Schaefer|1929|p=6}} However, Gauss could never give any reasons for magnetism, nor a theory of magnetism similar to Newton's work on gravitation, that enabled scientists to predict geomagnetic effects in the future.<ref name="Rupke" />

=== Optics ===
Gauss's calculations enabled instrument maker [[Johann Georg Repsold]] in [[Hamburg]] to construct a new [[achromatic lens]] system in 1810. A main problem, among other difficulties, was that the [[refractive index]] and [[Dispersion (optics)|dispersion]] of the glass used were not precisely known.{{sfn|Schaefer|1929|pp=153–154}} In a short article from 1817 Gauss dealt with the problem of removal of [[chromatic aberration]] in [[Gauss lens|double lenses]], and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician [[Carl August von Steinheil]], who in 1860 introduced the achromatic [[Achromatic lens|Steinheil doublet]], partly based on Gauss's calculations.{{sfn|Schaefer|1929|pp=159–165}} Many results in [[geometrical optics]] are scattered in Gauss's correspondences and hand notes.{{sfn|Dunnington|2004|p=170}}
  
In the ''Dioptrical Investigations'' (1840), Gauss gave the first systematic analysis of the formation of images under a [[paraxial approximation]] ([[Gaussian optics]]).<ref name=Hecht>{{Cite book | title = Optics | first = Eugene | last = Hecht | author-link = Eugene Hecht | publisher = Addison Wesley | year = 1987 | isbn = 978-0-201-11609-0 | page = 134}}</ref> He characterized optical systems under a paraxial approximation only by its [[Cardinal point (optics)|cardinal points]],<ref name=Bass>{{Cite book|title=Handbook of Optics| first1=Michael|last1=Bass|first2=Casimer|last2=DeCusatis| first3=Jay|last3=Enoch|first4=Vasudevan|last4=Lakshminarayanan|publisher=McGraw Hill Professional|year=2009|isbn=978-0-07-149889-0|page=17.7}}</ref> and he derived the Gaussian [[lens]] formula, applicable without restrictions in respect to the thickness of the lenses.<ref name=Ostdiek>{{Cite book | title = Inquiry into Physics | first1 = Vern J. | last1 = Ostdiek | first2 = Donald J. | last2 = Bord | publisher = Cengage Learning | year = 2007 | isbn = 978-0-495-11943-2 | page = 381}}</ref>{{sfn|Schaefer|1929|p=189–208}}

=== Mechanics ===
Gauss's first work in mechanics concerned the [[earth's rotation]]. When his university friend [[Johann Benzenberg|Benzenberg]] carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as the [[Coriolis force]], he asked Gauss for a theory-based calculation of the values for comparison with the experimental ones. Gauss elaborated a system of fundamental equations for the motion, and the results corresponded sufficiently with Benzenberg's data, who added Gauss's considerations as an appendix to his book on falling experiments.{{sfn|Geppert|1933|pp=3–11}}

After [[Léon Foucault|Foucault]] had demonstrated the earth's rotation by his [[Foucault pendulum|pendulum]] experiment in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum. It is described in the Gauss–Gerling correspondence and Weber made some experiments with this apparatus in 1853, but no data were published.{{sfn|Geppert|1933|p=12-16}}<ref>{{Cite journal | last = Siebert | first = Manfred | title = Das Foucault-Pendel von C. F. Gauß| journal = Mitteilungen der Gauß-Gesellschaft Göttingen | issue = 35 | pages = 49–52 | year = 1998 | volume = 35 | bibcode = 1998GGMit..35...49S | language = de}}</ref>

[[Gauss's principle of least constraint]] of 1829 was established as a general concept to overcome the division of mechanics into statics and dynamics, combining [[D'Alembert's principle]] with [[Joseph-Louis Lagrange|Lagrange]]'s [[principle of virtual work]], and showing analogies to the method of [[least squares]].{{sfn|Geppert|1933|p=16-26}}

=== Metrology ===
In 1828, Gauss was appointed as head of the board for weights and measures of the Kingdom of Hanover. He created [[Standard (metrology)|standards]] for length and measure. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical construction.<ref name="Olesko">{{cite book | last1 = Olesko | first1 = Kathryn | author-link  = Kathryn Olesko | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 236–253 | chapter = Der praktische Gauß – Präzisionsmessung für den Alltag | language = de | isbn = 3-930457-72-5 | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> In the correspondence with Schumacher, who was also working on this matter, he described new ideas for high-precision scales.{{sfn|Geppert|1933|pp=59–60}} He submitted the final reports on the Hanoverian [[Foot (unit)|foot]] and [[Pound (mass)|pound]] to the government in 1841. This work achieved international importance due to an 1836 law that connected the Hanoverian measures with the English ones.<ref name="Olesko" />