Editing Carl Friedrich Gauss (section)

=== 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}}