Editing Carl Friedrich Gauss (section)

=== Geometry ===
==== Differential geometry ====
{{Main|Theorema Egregium}}
The geodetic survey of [[Hanover]] fuelled Gauss's interest in [[differential geometry]] and [[topology]], fields of mathematics dealing with [[curve]]s and [[Surface (topology)|surfaces]]. This led him in 1828 to the publication of a work that marks the birth of modern [[differential geometry of surfaces]], as it departed from the traditional ways of treating surfaces as [[Cartesian coordinate system|cartesian graphs]] of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the [[Theorema Egregium]] (''remarkable theorem''), established a property of the notion of [[Gaussian curvature]]. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring [[angle]]s and [[distance]]s on the surface, regardless of the [[embedding]] of the surface in three-dimensional or two-dimensional space.{{sfn|Stäckel|1917|pp=110–119}}

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended [[manifold]]s; it clarifies the distinction between the intrinsic properties of the manifold (the [[metric tensor|metric]]) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a [[sphere]] or an [[ellipsoid]] cannot be transformed to a plane without distortion, which causes a fundamental problem in designing [[map projection|projection]]s for geographical maps.{{sfn|Stäckel|1917|pp=110–119}} A portion of this essay is dedicated to a profound study of [[geodesic]]s. In particular, Gauss proves the local [[Gauss–Bonnet theorem]] on geodesic triangles, and generalizes [[Legendre's theorem on spherical triangles]] to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.{{sfn|Stäckel|1917|pp=105–106}}

Gauss's memoir from 1828 lacks the conception of [[geodesic curvature]]. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its [[invariant (mathematics)|invariance]] under isometric transformations, a result that was later obtained by [[Ferdinand Minding]] and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by [[Pierre Ossian Bonnet]] in 1848 and known as the [[Gauss–Bonnet theorem]].{{sfn|Bolza|1921|pp=70–74}}

==== Non-Euclidean geometry ====
{{Main|Non-Euclidean geometry}}
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|upright|Lithography by [[Siegfried Bendixen]] (1828)]]
During Gauss' lifetime, the [[Parallel postulate]] of [[Euclidean geometry]] was heavily discussed.{{sfn|Stäckel|1917|pp=19–20}} Numerous efforts were made to prove it in the frame of the Euclidean [[axiom]]s, whereas some mathematicians discussed the possibility of geometrical systems without it.{{sfn|Bühler|1981|pp=100–102}} Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s  that a non-Euclidean geometry without the parallel postulate could solve the problem.{{sfn|Klein|1979|pp=57–60}}{{sfn|Stäckel|1917|pp=19–20}} In a letter to [[Franz Taurinus]] of 1824, he presented a short comprehensible outline of what he named a "[[non-Euclidean geometry]]",<ref name=":0">{{Cite journal | last = Winger | first = R. M. | date = 1925 | title = Gauss and non-Euclidean geometry | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-31/issue-7/Gauss-and-non-euclidean-geometry/bams/1183486559.full | journal = Bulletin of the American Mathematical Society | volume = 31 | issue = 7 | pages = 356–358 | doi = 10.1090/S0002-9904-1925-04054-9 | issn = 0002-9904 | doi-access = free }}</ref> but he strongly forbade Taurinus to make any use of it.{{sfn|Klein|1979|pp=57–60}} Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.<ref>{{Cite book |last=Bonola |first=Roberto |url=https://archive.org/details/noneuclideangeom00bono/page/64 |title=Non-Euclidean Geometry: A Critical and Historical Study of its Development |publisher=The Open Court Publishing Company |year=1912 |pages=64–67 |language=en}}</ref><ref name=":0" /><ref>{{Cite book |last=Klein |first=Felix |url=https://archive.org/details/elementarymathem0000klei/page/176 |title=Elementary Mathematics from an Advanced Standpoint: Geometry |publisher=Dover Publications |year=1939 |pages=176–177 |language=en}}</ref>

The first publications on non-Euclidean geometry in the history of mathematics were authored by [[Nikolai Lobachevsky]] in 1829 and [[Janos Bolyai]] in 1832.{{sfn|Bühler|1981|pp=100–102}} In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.{{sfn|Klein|1979|pp=57–60}}<ref>{{Cite journal | last1 = Jenkovszky | first1 = László | last2 = Lake | first2 = Matthew J. | last3 = Soloviev | first3 = Vladimir | date = 12 March 2023 | title = János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword | journal = Symmetry| volume = 15 | issue = 3 | pages = 707 | doi = 10.3390/sym15030707 | arxiv = 2303.17011 | bibcode=2023Symm...15..707J | issn = 2073-8994 | doi-access = free}}</ref> Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai<ref>{{Cite web|url=https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up|title=Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai|first=Carl Friedrich Gauss|last=Farkas Bólyai|date=22 April 1899|publisher=B. G. Teubner|via=Internet Archive}}</ref> claiming that these were congruent to his own thoughts of some decades.{{sfn|Klein|1979|pp=57–60}}<ref>{{cite book | last = Krantz | first = Steven G. | author-link = Steven G. Krantz | title = An Episodic History of Mathematics: Mathematical Culture through Problem Solving|url=https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA171 | access-date = 9 February 2013 | date = 2010 | publisher = [[The Mathematical Association of America]] | isbn = 978-0-88385-766-3| pages = 171f}}</ref> However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.{{sfn|Bühler|1981|pp=100–102}}

[[Wolfgang Sartorius von Waltershausen|Sartorius]] first mentioned Gauss's work on non-Euclidean geometry in 1856, but only the publication of Gauss's [[Nachlass]] in Volume VIII of the Collected Works (1900) showed Gauss's ideas on the matter, at a time when non-Euclidean geometry was still an object of some controversy.{{sfn|Klein|1979|pp=57–60}}

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

[[File:Carl Friedrich Gauß, Büste von Heinrich Hesemann, 1855.jpg|thumb|left|upright|Gauss bust by Heinrich Hesemann (1855){{efn|Hesemann also took a death mask from Gauss.{{sfn|Dunnington|2004|p=324}}}}]]
Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the [[celestial sphere]] in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are [[Link (knot theory)|linked]], then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid [[7 Iris]], he  published a further qualitative discussion of the Zodiacus.<ref name="Epple 1998 45–52">{{cite journal | last = Epple | first = Moritz | title = Orbits of asteroids, a braid, and the first link invariant | journal =  [[The Mathematical Intelligencer]] | volume = 20 | pages = 45–52 | year = 1998 | issue = 1 | doi = 10.1007/BF03024400 | s2cid = 124104367 | url = https://link.springer.com/article/10.1007/BF03024400}}</ref>

In Gauss's letters of 1820–1830, he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of [[Knot theory|knot]]s.<ref>{{Cite book | last = Epple | first = Moritz | author-link = Moritz Epple | title = History of Topology | chapter = Geometric Aspects in the Development of Knot Theory | date = 1999 | editor-last = James | editor-first = I.M. | place = Amsterdam | publisher = Elseviwer | pages = 301–357 | chapter-url = https://www.maths.ed.ac.uk/~v1ranick/papers/epple3.pdf}}</ref> To do so he devised a symbolical scheme, the [[Gauss code]], that in a sense captured the characteristic features of tract figures.<ref>{{Cite book | last1 = Lisitsa | first1 = Alexei | last2 = Potapov | first2 = Igor | last3 = Saleh | first3 = Rafiq | title = Language and Automata Theory and Applications | chapter = Automata on Gauss Words | date = 2009 | editor-last = Dediu | editor-first = Adrian Horia | editor2-last = Ionescu | editor2-first = Armand Mihai | editor3-last = Martín-Vide | editor3-first = Carlos | series = Lecture Notes in Computer Science | volume = 5457 | place = Berlin, Heidelberg | publisher = Springer | pages = 505–517 | doi = 10.1007/978-3-642-00982-2_43 | isbn = 978-3-642-00982-2 | chapter-url = https://cgi.csc.liv.ac.uk/~igor/papers/lata2009.pdf}}</ref>{{sfn|Stäckel|1917|pp=50–51}}

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

Gauss's influence in later years to the emerging field of topology, which he held in high esteem, was through occasional remarks and oral communications to Mobius and Listing.{{sfn|Stäckel|1917|pp=51–55}}