Gauss was instrumental in the identification ofCeres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of theGaussian gravitational constant and themethod of least squares, which he had discovered beforeAdrien-Marie Legendre published it. Gauss was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders ofgeophysics and formulated the fundamental principles ofmagnetism. Fruits of his practical work were the inventions of theheliotrope in 1821, amagnetometer in 1833 and – alongsideWilhelm Eduard Weber – the first electromagnetictelegraph in 1833.
Gauss refused to publish incomplete work and left several works to be editedposthumously. He believed that the act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to disliking teaching, but some of his students became influential mathematicians, such asRichard Dedekind andBernhard Riemann.
Birthplace in Brunswick (destroyed in World War II)Gauss's home as student in Göttingen
Gauss was born on 30 April 1777 inBrunswick in theDuchy of Brunswick-Wolfenbüttel (now in the German state ofLower Saxony). His family was of relatively low social status.[5] His father Gebhard Dietrich Gauss (1744–1808) worked variously as a butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate.[6] He had one elder brother from his father's first marriage.[7]
He was likely a self-taught student in mathematics since he independently rediscovered several theorems.[11] He solved a geometrical problem that had occupied mathematicians since theAncient Greeks when he determined in 1796 which regularpolygons can be constructed bycompass and straightedge. This discovery ultimately led Gauss to choose mathematics instead ofphilology as a career.[16] Gauss's mathematical diary, a collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opusDisquisitiones Arithmeticae (1801) date from this time.[17]
Gauss graduated as aDoctor of Philosophy in 1799, not in Göttingen, as is sometimes stated,[c][18] but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of the duchy.Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degreein absentia without further oral examination.[11] The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from theRussian Academy of Sciences inSt. Peterburg andLandshut University.[19][20] Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. ArchitectPeter Joseph Krahe made preliminary designs, but one ofNapoleon's wars cancelled those plans:[21] the Duke was killed in thebattle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped.
When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community ofBremen andLilienthal, especiallyWilhelm Olbers,Karl Ludwig Harding, andFriedrich Wilhelm Bessel, as part of the informal group of astronomers known as theCelestial police.[22] One of their aims was the discovery of further planets. They assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opusTheoria motus corporum coelestium (1809).[23]
In November 1807, Gauss followed a call to theUniversity of Göttingen, then an institution of the newly foundedKingdom of Westphalia underJérôme Bonaparte, as full professor and director of theastronomical observatory,[24] and kept the chair until his death in 1855. He was soon confronted with the demand for two thousandfrancs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers andLaplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person fromFrankfurt, later discovered to bePrince-primateDalberg,[25] paid the sum.[24]
Gauss took on the directorate of the 60-year-old observatory, founded in 1748 byPrince-electorGeorge II and built on a converted fortification tower,[26] with usable, but partly out-of-date instruments.[27] The construction of a new observatory had been approved by Prince-electorGeorge III in principle since 1802, and the Westphalian government continued the planning,[28] but Gauss could not move to his new place of work until September 1816.[20] He got new up-to-date instruments, including twomeridian circles fromRepsold[29] andReichenbach,[30] and aheliometer fromFraunhofer.[31]
The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade.[32]
Gauss did not write any textbook and disliked thepopularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essayErdmagnetismus und Magnetometer of 1836.[36] Gauss published his papers and books exclusively inLatin orGerman.[f][g] He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians.[39]
The new Göttingen observatory of 1816; Gauss's living rooms were in the western wing (right)Wilhelm Weber andHeinrich Ewald (first row) as members of theGöttingen SevenGauss on his deathbed (1855) (daguerreotype from Philipp Petri)[40]
In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by a strong calculus as the sole tasks of astronomy.[34] At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures,[41] the physicistMayer, known for his textbooks,[42] his successorWeber since 1831, and in the observatoryHarding, who took the main part of lectures in practical 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.[20] They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer.[20][h] Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations.[44]
Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which the years since 1820 are taken as a "period of lower astronomical activity".[45] The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834.[43][44][i]
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 burdening lecturing duties, as well as fromLeipzig University in 1810 and fromVienna University in 1842, perhaps because of the family's difficult situation.[43] Gauss's salary was raised from 1000Reichsthaler in 1810 to 2500 Reichsthaler in 1824,[20] and in his later years he was one of the best-paid professors of the university.[46]
When Gauss was asked for help by his colleague and friendFriedrich Wilhelm Bessel in 1810, who was in trouble atKönigsberg University because of his lack of an academic title, Gauss provided adoctoratehonoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811.[j] Gauss gave another recommendation for an honorary degree forSophie Germain but only shortly before her death, so she never received it.[49] He also gave successful support to the mathematicianGotthold Eisenstein in Berlin.[50]
Gauss was loyal to theHouse of Hanover. After KingWilliam IV died in 1837, the new Hanoverian KingErnest Augustus annulled the 1833 constitution. Seven professors, later known as the "Göttingen Seven", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss was deeply affected by this quarrel but saw no possibility to help them.[51]
Gauss took part in academic administration: three times he was elected asdean of the Faculty of Philosophy.[52] Being entrusted with the widow'spension fund of the university, he dealt withactuarial 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.[52]
Gauss remained mentally active into his old age, even while suffering fromgout and general unhappiness. On 23 February 1855, he died of a heart attack in Göttingen;[13] and was interred in theAlbani Cemetery there.Heinrich Ewald, Gauss's son-in-law, andWolfgang Sartorius von Waltershausen, Gauss's close friend and biographer, gave eulogies at his funeral.[53]
Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150,000 Thaler; after his death, about 18,000 Thaler were found hidden in his rooms.[54]
The day after Gauss's death his brain was removed, preserved, and studied byRudolf Wagner, who found its mass to be slightly above average, at 1,492 grams (3.29 lb).[55][56] Wagner's sonHermann, a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis.[57] In 2013, a neurobiologist at theMax Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physicianConrad Heinrich Fuchs, who died in Göttingen a few months after Gauss.[58] A further investigation showed no remarkable anomalies in the brains of both persons. 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.[59]
Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick.[60] They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after the birth of Louis, who himself died a few months later.[61] Gauss chose the first names of his children in honour ofGiuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids.[62]
On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than a decade.[63] Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau.[64] Her sister Wilhelmina married the orientalistHeinrich Ewald.[65] Gauss's mother Dorothea lived in his house from 1817 until she died in 1839.[12]
The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in the summer of 1821. After a short time at university, in 1824 Joseph joined theHanoverian 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 theRoyal Hanoverian State Railways. In 1836 he studied the railroad system in the US for some months.[46][k]
Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for theAmerican Fur Company in the Midwest. Later, he moved toMissouri and became a successful businessman.[46] Wilhelm married a niece of the astronomerBessel;[68] he then moved to Missouri, started as a farmer and became wealthy in the shoe business inSt. Louis in later years.[69] Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children.[46]
A student draws his professor of mathematics: Caricature ofAbraham Gotthelf Kästner by Gauss (1795)[l]A student draws his professor of mathematics: Gauss sketched by his studentJohann Benedict Listing, 1830
In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones;[70] hisDisquisitiones Arithmeticae was the first mathematical book from Germany to be translated into the French language.[71]
Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration.[72] Whereas previous mathematicians likeLeonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path,[73] Gauss however introduced a new style of direct and complete explanation that did not attempt to show the reader the author's train of thought.[74]
Gauss was the first to restore thatrigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period innew developments.
But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows:[75]
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.
The posthumous papers, his scientificdiary,[76] and short glosses in his own textbooks show that he empirically worked to a great extent.[77][78] He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked the results by masterly estimation. Nevertheless, his calculations were not always free from mistakes.[79] He coped with the enormous workload by using skillful tools.[80] Gauss used a lot ofmathematical tables, examined their exactness, and constructed new tables on various matters for personal use.[81] He developed new tools for effective calculation, for example theGaussian elimination.[82] 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, and prepared tables with more decimal places than ever requested for practical purposes.[83] Very likely, this method gave him a lot of material which he used in finding theorems in number theory.[80][84]
Gauss refused to publish work that he did not consider complete and above criticism. Thisperfectionism was in keeping with the motto of his personalsealPauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind".[36] Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too.[85][86] Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics",[87] and supposedly once espoused a belief in the necessity of immediately understandingEuler's identity as a benchmark pursuant to becoming a first-class mathematician.[88]
On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not the first to publish" differed from that of his scientific contemporaries.[89] In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a 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.[36]
Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty,[90] but also of "iron character"[91] with an unshakeable strength of mind.[92] Apart from his closer circle, others regarded him as reserved and unapproachable "like anOlympian sitting enthroned on the summit of science".[93] His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host.[36] Gauss abominated polemic natures; together with his colleagueHausmann he opposed to a call forJustus Liebig on a university chair in Göttingen, "because he was always involved in some polemic."[94]
Gauss's residence from 1808 to 1816 in the first floor
Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancientthrenody, the most personal surviving document of Gauss.[95][96] The situation worsened whentuberculosis ultimately destroyed the health of his second wife Minna over 13 years; both his daughters later suffered from the same disease.[97] Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings".[36]
Because of his wife's illness, both younger sons were educated for some years inCelle, far from Göttingen. The military career of his elder son Joseph ended after more than two decades with the rank of a poorly paidfirst lieutenant, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married.[46] The second son Eugen shared a good measure of his father's 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,[98] Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support.[46] The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life.[64]
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 number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age asIsaac Newton at his death, calculated in days.[99]
Similar to his excellent knowledge ofLatin he was also acquainted with modern languages. At the age of 62, he began to teach himselfRussian, very likely to understand scientific writings from Russia, among them those ofLobachevsky on non-Euclidean geometry.[100] Gauss read both classical and modern literature, and English and French works in the original languages.[101][m] His favorite English author wasWalter Scott, his favorite GermanJean Paul.[103] Gauss liked singing and went to concerts.[104] He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon.[105] 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 ofNaturphilosophie.[106]
Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto "mundus vult decipi".[105] He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of theRevolutions of 1848, though he agreed with some of their aims, such as the idea of a unified Germany.[91][n] 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.[105]
Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes"[107] and "I succeeded – not on account of my hard efforts, but by the grace of the Lord."[108] Gauss was a member of theLutheran church, like most of the population in northern Germany. It seems that he did not believe alldogmas or understand the Holy Bible quite literally.[109] Sartorius mentioned Gauss'sreligious tolerance, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions.[110]
German stamp commemorating Gauss's 200th anniversary: thecomplex plane orGauss plane
In his doctoral thesis from 1799, Gauss proved thefundamental theorem of algebra which states that every non-constant single-variablepolynomial with complex coefficients has at least one complexroot. Mathematicians includingJean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.[111]
In the preface to theDisquisitiones, Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself.[112] TheDisquisitiones Arithmeticae, written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraicnumber theory. Therein he introduces thetriple bar symbol (≡) forcongruence and uses it for a clean presentation ofmodular arithmetic.[113] It deals with theunique factorization theorem andprimitive roots modulo n. In the main sections, Gauss presents the first two proofs of the law ofquadratic reciprocity[114] and develops the theories ofbinary[115] and ternaryquadratic forms.[116]
TheDisquisitiones include theGauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of histheorem on three squares, he proves the triangular case of theFermat polygonal number theorem forn = 3.[117] From several analytic results onclass numbers that Gauss gives without proof towards the end of the fifth section,[118] it appears that Gauss already knew theclass number formula in 1801.[119]
In the last section, Gauss gives proof for theconstructibility of a regularheptadecagon (17-sided polygon) withstraightedge and compass by reducing this geometrical problem to an algebraic one.[120] He shows that a regular polygon is constructible if the number of its sides is either apower of 2 or the product of a power of 2 and any number of distinctFermat primes. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients infinite fields, which amounts to counting integral points on anelliptic curve.[121] An unfinished eighth chapter was found among left papers only after his death, consisting of work done during 1797–1799.[122][123]
One of Gauss's first results was the empirically found conjecture of 1792 – the later calledprime number theorem – giving an estimation of the number of prime numbers by using theintegral logarithm.[124][o]
When Olbers encouraged Gauss in 1816 to compete for a prize from the French Academy on the proof forFermat's Last Theorem (FLT), he refused because of his low esteem on this matter. However, among his left works a short undated paper was found with proofs of FLT for the casesn = 3 andn = 5.[126] The particular case ofn = 3 was proved much earlier byLeonhard Euler, but Gauss developed a more streamlined proof which made use ofEisenstein integers; though more general, the proof was simpler than in the real integers case.[127]
Gauss contributed to solving theKepler conjecture in 1831 with the proof that agreatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form acubic face-centred arrangement,[128] when he reviewed a book ofLudwig August Seeber on the theory of reduction of positive ternary quadratic forms.[129] Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.[130]
In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.[133]
One of Gauss's first discoveries was the notion of thearithmetic-geometric mean (AGM) of two positive real numbers.[134] He discovered its relation to elliptic integrals in the years 1798–1799 through theLanden's transformation, and a diary entry recorded the discovery of the connection ofGauss's constant tolemniscatic elliptic functions, a result that Gauss stated that "will surely open an entirely new field of analysis".[135] He also made early inroads into the more formal issues of the foundations ofcomplex analysis, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" –Cauchy's integral theorem – and understood the notion ofcomplex residues when integrating aroundpoles.[121][136]
Several mathematical fragments in hisNachlass indicate that he knew parts of the modern theory ofmodular forms.[121] In his work on themultivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM to its two "simplest values".[135] In his unpublished writings he recognized and made a sketch of the key concept offundamental domain for themodular group.[140][141] One of Gauss's sketches of this kind was a drawing of atessellation of theunit disk by "equilateral"hyperbolic triangles with all angles equal to.[142]
An example of Gauss's insight in the fields of analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of thelemniscate curve, which inspired Abel's theorem on lemniscate division.[r] Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign ofquadratic Gauss sum, in which he solved the main problem by introducingq-analogs of binomial coefficients and manipulating them by several original identities that seem to stem out of his work on elliptic functions theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic functions theory, and only the later work of mathematicians such as Jacobi andHermite has exposed the crux of his argument.[143]
In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the generalhypergeometric function, and shows that many of the functions known at the time are special cases of the hypergeometric function.[144] This work is the first one with an exact inquiry ofconvergence of infinite series in the history of mathematics.[145] Furthermore, it deals with infinitecontinued fractions arising as ratios of hypergeometric functions which are now calledGauss continued fractions.[146]
In 1823, Gauss won the prize of the Danish Society with an essay onconformal mappings, which contains several developments that pertain to the field of complex analysis.[147] Gauss stated that angle-preserving mappings in the complex plane must be complexanalytic functions, and used the later calledBeltrami equation to prove the existence ofisothermal coordinates on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and anellipsoid of revolution.[148]
Gauss often deduced theoremsinductively from numerical data he had collected empirically.[78] As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions tonumerical analysis, as the method ofGaussian quadrature published in 1816.[149]
In a private letter to Gerling from 1823,[150] he described a solution of a 4X4 system of linear equations by usingGauss-Seidel method – an "indirect"iterative method for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.[151]
The geodetic survey of Hanover fuelled Gauss's interest indifferential geometry andtopology, fields of mathematics dealing withcurves andsurfaces. This led him in 1828 to the publication of a memoir that marks the birth of moderndifferential geometry of surfaces, as it departed from the traditional ways of treating surfaces ascartesian 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, theTheorema Egregium (remarkable theorem), established a property of the notion ofGaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuringangles anddistances on the surface, regardless of theembedding of the surface in three-dimensional or two-dimensional space.[155]
The Theorema Egregium leads to the abstraction of surfaces as doubly-extendedmanifolds; it clarifies the distinction between the intrinsic properties of the manifold (themetric) 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 asphere or anellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designingprojections for geographical maps.[155] A portion of this essay is dedicated to a profound study ofgeodesics. In particular, Gauss proves the localGauss–Bonnet theorem on geodesic triangles, and generalizesLegendre'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.[156]
Gauss's memoir from 1828 lacks the conception ofgeodesic curvature. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved itsinvariance under isometric transformations, a result that was later obtained byFerdinand 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 byPierre Ossian Bonnet in 1848 and known asGauss–Bonnet theorem.[157]
In the lifetime of Gauss, a vivid discussion on theParallel postulate inEuclidean geometry was going on.[158] Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it.[159] Gauss thought about the basics of geometry since the 1790s years, but in the 1810s he realized that a non-Euclidean geometry without the parallel postulate could solve the problem.[160][158] In a letter toFranz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry",[161] but he strongly forbade Taurinus to make any use of it.[160] Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.[162][161][163]
The first publications on non-Euclidean geometry in the history of mathematics were authored byNikolai Lobachevsky in 1829 andJanos Bolyai in 1832.[159] In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.[160][164] Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai[165] claiming that these were congruent to his own thoughts of some decades.[160][166] However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his letter remarks are only vague and obscure.[159]
Sartorius mentioned Gauss's work on non-Euclidean geometry firstly in 1856, but only the edition of left papers in Volume VIII of the Collected Works (1900) showed Gauss's ideas on that matter, at a time when non-Euclidean geometry had yet grown out of controversial discussion.[160]
Gauss was also an early pioneer oftopology orGeometria Situs, as it was called in his lifetime. The first proof of thefundamental 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.[167]
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 thecelestial sphere in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits arelinked, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid7 Iris, he published a further qualitative discussion of the Zodiacus.[168]
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 ofknots.[169] To do so he devised a symbolical scheme, theGauss code, that in a sense captured the characteristic features of tract figures.[170][171]
In a fragment from 1833, Gauss defined thelinking 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 asbraids andtangles.[168]
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.[172]
Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example, in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography,[173] he stated thefundamental theorem of axonometry, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers.[174] He described rotations of this sphere as the action of certainlinear fractional transformations on the extended complex plane,[175] and gave a proof for the geometric theorem that thealtitudes of a triangle always meet in a singleorthocenter.[176]
Gauss was concerned withJohn Napier's "Pentagramma mirificum" – a certain sphericalpentagram – for several decades;[177] he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects.[178] In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.[179]
Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a givenquadrilateral,[180][181] and discovered a surprising result about the computation of area ofpentagons.[182][183]
Carl Friedrich Gauss 1803 by Johann Christian August Schwartz
On 1 January 1801, Italian astronomerGiuseppe Piazzi discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-calledTitius–Bode law, and named itCeres.[184] 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 extrapolate a position from the few data for its reappearance. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree whenFranz Xaver von Zach on 7 and 31 December atGotha, and independentlyHeinrich Olbers on 1 and 2 January inBremen, identified the object near the predicted position.[185][t]
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.[186]
The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 asTheoria motus corporum coelestium in sectionibus conicis solem ambientum.[187] It introduced theGaussian gravitational constant.[34]
Since the new asteroids had been discovered, Gauss occupied himself with theperturbations of theirorbital elements. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object wasPallas, because of its greateccentricity andorbital inclination, whereby Laplace's method did not work. Gauss used his own tools: thearithmetic–geometric mean, thehypergeometric function, and his method of interpolation.[188] He found anorbital resonance withJupiter in proportion 18:7 in 1812; Gauss gave this result ascipher, and gave the explicit meaning only in letters to Olbers and Bessel.[189][190][u] 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.[192]
Göttingen observatory seen from the North-west (by Friedrich Besemann,c. 1835)
One fruit of Gauss's research on Pallas perturbations was theDeterminatio 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 taking the planet to follow the corresponding orbital arcs.[193] 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 anelliptic integral.[194]
While Gauss's contributions to theoretical astronomy came to an end, more practical activities inobservational astronomy continued and occupied him during his entire career. Even early in 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.[195] After appointment as director of observatory he attached importance to the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables for nutation and aberration, the solar coordinates, and refraction.[196] He made many contributions tospherical geometry, and in this context solved some practical problems aboutnavigation by stars.[197] He published a great number of observations, mainly on minor planets and comets; his last observation was thesolar eclipse of 28 July 1851.[198]
The first publication following the doctoral thesis dealt with the determination of thedate of Easter (1800), an elementary matter of mathematics. Gauss aimed to present a most convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usually required terms ofgolden number,epact,solar cycle,domenical letter, and any religious connotations.[199] Biographers speculated on the reason why Gauss dealt with this matter, but it is likely comprehensible by the historical background. The replacement of theJulian calendar by theGregorian calendar had caused confusion in theHoly Roman Empire since the 16th century and was not finished in Germany until 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's birth, was the first one calculated in the new manner.[200]
Gauss likely used themethod of least squares for calculating the orbit of Ceres to minimize the impact ofmeasurement error.[89] The method was published first byAdrien-Marie Legendre in 1805, but Gauss claimed inTheoria motus (1809) that he had been using it since 1794 or 1795.[201][202][203] In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares".[89] Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption ofnormally distributed errors (Gauss–Markov theorem), in the two-part paperTheoria combinationis observationum erroribus minimis obnoxiae (1823).[204]
Gauss also contributed to problems inprobability 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 theGauss-Kuzmin distribution, as a by-product of the discovery of theergodicity of theGauss map for continued fractions. Gauss's solution is the first-ever result in the metrical theory of continued fractions.[207]
Order of KingGeorge IV from 9 May 1820 to the triangulation project (with the additional signature of CountErnst zu Münster below)TheheliotropeGauss's vice heliotrope, aTroughton sextant with additional mirror
Gauss was busy with geodetic problems since 1799 when he helpedKarl Ludwig von Lecoq with calculations during hissurvey inWestphalia.[208] Beginning in 1804, he taught himself some geodetic practice with a sextant in Brunswick,[209] and Göttingen.[210]
Since 1816, Gauss's former studentHeinrich Christian Schumacher, then professor inCopenhagen, but living inAltona (Holstein) nearHamburg as head of an observatory, carried out atriangulation of theJutland peninsula fromSkagen in the north toLauenburg in the south.[v] 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 earthgeoid, 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, KingGeorge IV gave the order to Gauss.[211]
Anarc measurement needs a precise astronomical determination of at least two points in thenetwork. Gauss and Schumacher used the favourite occasion that both observatories in Göttingen and Altona, in the garden of Schumacher's house, laid nearly in the samelongitude. Thelatitude was measured with both their instruments and azenith sector ofRamsden that was transported to both observatories.[212][w]
Gauss and Schumacher had already determined some angles betweenLüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818.[213] During the summers of 1821 until 1825 Gauss directed the triangulation work personally, fromThuringia in the south to the riverElbe in the north. Thetriangle betweenHoher Hagen,Großer Inselsberg in theThuringian Forest, andBrocken in theHarz mountains was the largest one Gauss had ever measured with a maximum size of 107 km (66.5 miles). In the thinly populatedLüneburg Heath without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.[200][214]
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 itheliotrope.[215] Another suitable construction for the same purpose was asextant with an additional mirror which he namedvice heliotrope.[216] Gauss got assistance by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in thebaseline measurement (Braak Base Line) of Schumacher in the village ofBraak near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.[217]
When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the wholeKingdom of Hanover with a Royal decree from 25 March 1828.[221] 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 themethod of least squares and theelimination 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.[222][223]
In 1828, when studying differences inlatitude, Gauss first defined a physical approximation for thefigure of the Earth as the surface everywhere perpendicular to the direction of gravity;[224] later his doctoral studentJohann Benedict Listing called this thegeoid.[225]
When Weber got the chair for physics in Göttingen as successor ofJohann Tobias Mayer by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge ofmagnetism with a representation for the unit of magnetism in terms of mass, charge, and time.[229] They founded theMagnetic Association (German:Magnetischer Verein), an international working group of several observatories, which supported measurements ofEarth's magnetic field in many regions of the world with equal methods at arranged dates in the years 1836 to 1841.[230]
In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in theBritish dominions with a letter to theDuke of Sussex, then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods.[231][232] Together with other instigators, this led to a global program known as "Magnetical crusade" under the direction ofEdward Sabine. The dates, times, and intervals of observations were determined in advance, theGöttingen mean time was used as the standard.[233] 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 toLeipzig in 1843 as late effect of theGöttingen Seven affair marked the end of Magnetic Association activity.[230]
Following Humboldt's example, Gauss ordered a magneticobservatory 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 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 the intensity of the magnetic field and constructed a suitablemagnetometer to measureabsolute values of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus.[230][234] The precision of the magnetometer was about ten times higher than of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities.[233]
Gauss carried out aGeneral 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 ofspherical harmonics rather than a physical theory.[235] The theory predicted the existence of exactly twomagnetic poles on the Earth, thusHansteen's idea of four magnetic poles became obsolete,[236] and the data allowed to determine their location with rather good precision.[237]
Gauss influenced the beginning of geophysics in Russia, whenAdolph Theodor Kupffer, one of his former students, founded a magnetic observatory inSt. Petersburg, following the example of the observatory in Göttingen, and similarly,Ivan Simonov inKazan.[236]
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 idea ofvector potential function (independently rediscovered byFranz Ernst Neumann in 1845), and in January 1835 he wrote down an "induction law" equivalent toFaraday's law, which stated that theelectromotive force at a given point in space is equal to theinstantaneous rate of change (with respect to time) of this function.[242][243]
Gauss tried to find a unifying law for long-distance effects ofelectrostatics,electrodynamics, electromagnetism, andinduction, comparable to Newton's law of gravitation,[244] but his attempt ended in a "tragic failure".[233]
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 aclosed-form expression to the gravitational attraction of a homogeneoustriaxial ellipsoid at every point in space.[245] In contrast to previous research ofMaclaurin, 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-calledGauss's theorem invector analysis.[246]
In theGeneral theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances (1840) Gauss gave the baseline of a theory of the magneticpotential, based on Lagrange, Laplace, and Poisson;[235] it seems rather unlikely that he knew the previous works ofGeorge Green on this subject.[238] 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.[233]
Gauss's calculations enabled instrument makerJohann Georg Repsold inHamburg to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was the nonprecise knowledge of therefractive index anddispersion of the used glass types.[247] In a short article from 1817 Gauss dealt with the problem of removal ofchromatic aberration indouble lenses, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the opticianCarl August von Steinheil, who in 1860 introduced the achromaticSteinheil doublet, partly based on Gauss's calculations.[248] Many results ingeometrical optics are only scattered in Gauss's correspondences and hand notes.[249]
In theDioptrical Investigations (1840), Gauss gave the first systematic analysis on the formation of images under aparaxial approximation (Gaussian optics).[250] He characterized optical systems under a paraxial approximation only by itscardinal points,[251] and he derived the Gaussianlens formula, applicable without restrictions in respect to the thickness of the lenses.[252][253]
Gauss's first business in mechanics concerned theearth's rotation. When his university friendBenzenberg carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as an effect of theCoriolis 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.[254]
AfterFoucault had demonstrated the earth's rotation by hispendulum 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.[255][256]
In 1828, Gauss was appointed to head of a Board for weights and measures of the Kingdom of Hanover. He provided the creation ofstandards of length and measures. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical preparation.[200] In the correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision.[258] He submitted the final reports on the Hanoverianfoot andpound to the government in 1841. This work got more than regional importance by the order of a law of 1836 that connected the Hanoverian measures with the English ones.[200]
Gauss received theLalande 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,[278] the Danish Academy of Science prize in 1823 for his memoir on conformal projection,[270] and theCopley Medal from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".[277][279][34]
The Kings of Hanover appointed him the honorary titles "Hofrath" (1816)[52] and "Geheimer Hofrath"[z] (1845). In 1949, on the occasion of his golden doctor degree jubilee, he got thehonorary citizenship of both towns of Brunswick and Göttingen.[274] Soon after his death a medal was issued by order of KingGeorge V of Hanover with the back inscription dedicated "to the Prince of Mathematicians".[283]
The "Gauss-Gesellschaft Göttingen" ("Göttingen Gauss Society") was founded in 1964 for research on life and work of Carl Friedrich Gauss and related persons and edits theMitteilungen der Gauss-Gesellschaft (Communications of the Gauss Society).[284]
1800:"Berechnung des Osterfestes" [Calculation of Easter].Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde (in German).2:121–130.Original
1811:"Summatio quarundam serierum singularium".Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math.1:1–40.Original (from 1808) (Determination of the sign of thequadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity)
1843:"Dioptrische Untersuchungen" [Dioptrical Investigations].Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen (in German). Erster Band:1–34.Original (from 1840)
Königlich Preußische Akademie der Wissenschaften, ed. (1863–1933).Carl Friedrich Gauss. Werke (in Latin and German). Vol. 1–12. Göttingen: (diverse publishers). (includes unpublished literary estate)
Königlich Preußische Akademie der Wissenschaften, ed. (1880).Briefwechsel zwischen Gauss und Bessel (in German). Leipzig: Wilhelm Engelmann. (letters from December 1804 to August 1844)
Schoenberg, Erich; Perlick, Alfons (1955).Unbekannte Briefe von C. F. Gauß und Fr. W. Bessel. Abhandlungen der Bayerischen Akademie der Wissenschaften, Math.-nat. Klasse, Neue Folge, No. 71 (in German). Munich: Verlag der Bayerischen Akademie der Wissenschaften. pp. 5–21. (letters toBoguslawski from February 1835 to January 1848)
Schwemin, Friedhelm, ed. (2014).Der Briefwechsel zwischen Carl Friedrich Gauß undJohann Elert Bode. Acta Historica Astronomica (in German). Vol. 53. Leipzig: Akademische Verlaganstalt.ISBN978-3-944913-43-8. (letters from February 1802 to October 1826)
Axel Wittmann, ed. (2018).Obgleich und indeßen. Der Briefwechsel zwischen Carl Friedrich Gauss undJohann Franz Encke (in German). Remagen: Verlag Kessel.ISBN978-3945941379. (letters from June 1810 to June 1854)
Reich, Karin; Roussanova, Elena (2018).Karl Kreil und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext. Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68 (in German). Vienna: Verlag der Österreichischen Akademie der Wissenschaften. (letters from 1835 to 1843)
Gerardy, Theo, ed. (1959).Briefwechsel zwischen Carl Friedrich Gauß undCarl Ludwig von Lecoq. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 (in German). Göttingen: Vandenhoeck & Ruprecht. pp. 37–63. (letters from February 1799 to September 1800)
Carl Schilling, ed. (1909).Briefwechsel zwischen Olbers und Gauss: Zweite Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. (letters from January 1820 to May 1839; added letters of other correspondents)
Volumes 1+2 (letters from April 1808 to March 1836)
Volumes 3+4 (letters from March 1836 to April 1845)
Volumes 5+6 (letters from April 1845 to November 1850)
Poser, Hans, ed. (1987).Briefwechsel zwischen Carl Friedrich Gauß undEberhard August Zimmermann. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 (in German). Göttingen: Vandenhoeck & Ruprecht.ISBN978-3525821169. (letters from 1795 to 1815)
^Once Gauss drew a lecture scene with professor Kästner producing errors in a simple calculation.[11]
^This error occurs for example in Marsden (1977).[18]
^Gauss announced 195 lectures, 70 per cent of them on astronomical, 15 per cent on mathematical, 9 per cent on geodetical, and 6 per cent on physical subjects.[34]
^The index of correspondence shows that Benjamin Gould was presumably the last correspondent who, on 13 February 1855, sent a letter to Gauss in his lifetime. It was an actual letter of farewell, but it is uncertain whether it reached the addressee just in time.[35]
^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.[37]
^TheTheoria motus... was completed in the German language in 1806, but on request of the editorFriedrich Christoph Perthes Gauss translated it into Latin.[38]
^Both Gauss and Harding dropped only veiled hints on this personal problem in their correspondence. A letter to Schumacher indicates that Gauss tried to get rid of his colleague and searched for a new position for him outside of Göttingen, but without result. Apart from that, Charlotte Waldeck, Gauss's mother-in-law, pleaded with Olbers to try to provide Gauss with another position far from Göttingen.[43]
^The political background was the confusing situation of theGerman Confederation with 39 nearly independent states, the sovereigns of three of them being Kings of other countries (Netherlands, Danmark, United Kingdom), whereas theKingdom of Prussia and theAustrian Empire extended widely over the frontiers of the Confederation.
^Gauss told the story later in detail in a letter toEncke.[125]
^Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order),Sohncke in 1837 (7th and other orders).
^In a letter to Bessel from 1828, Gauss commented: "Mr. Abel has [...] anticipated me, and relieves me of the effort [of publishing] in respect to one third of these matters ..."[139]
^This remark appears at article 335 of chapter 7 ofDisquisitiones Arithmeticae (1801).
^The value fromWalbeck (1820) of 1/302,78 was improved to 1/298.39; the calculation was done by Eduard Schmidt, private lecturer at Göttingen University.[219]
^literally translation:Secrete Councillor of the Court
^Gauss presented the text to the Göttingen Academy in December 1832, a preprint in Latin with a small number of copies appeared in 1833. It was soon translated and published in German and French. The complete text in Latin was published in 1841.[230]
^Krech, Eva-Maria; Stock, Eberhard; Hirschfeld, Ursula; Anders, Lutz-Christian (2009).Deutsches Aussprachewörterbuch [German Pronunciation Dictionary] (in German). Berlin: W alter de Gruyter. pp. 402, 520, 529.ISBN978-3-11-018202-6.
^Borch, Rudolf (1929).Ahnentafel des Mathematikers Carl Friedrich Gauß [Ancestors' Tabel of the mathematician Carl Friedrich Gauss]. Ahnentafeln Berühmter Deutscher (in German). Vol. 1. Zentralstelle für Deutsche Personen- und Familiengeschichte. pp. 63–65.
^Wattenberg, Diedrich (1994).Wilhelm Olbers im Briefwechsel mit Astronomen seiner Zeit (in German). Stuttgart: GNT – Verlag für Geschichte der Naturwissenschaften und der Technik. p. 41.ISBN3-928186-19-1.
^abcdeReich, Karin (2000). "Gauß' Schüler".Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (37):33–62.
^abcdefBeuermann, Klaus (2005)."Carl Friedrich Gauß und die Göttinger Sternwarte"(PDF). In Beuermann, Klaus (ed.).Grundsätze über die Anlage neuer Sternwarten unter Beziehung auf die Sternwarte der Universität Göttingen von Georg Heinrich Borheck. Göttingen: Universitätsverlag Göttingen. pp. 37–45.ISBN3-938616-02-4.
^Michling, Horst (1966). "Zum Projekt einer Gauß-Sternwarte in Braunschweig".Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 24.
^abcdeBiermann, Kurt-R. (1966). "Über die Beziehungen zwischen C. F. Gauß und F. W. Bessel".Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3):7–20.
^abcKüssner, Martha (1978). "Friedrich Wilhelm Bessels Beziehungen zu Göttingen und Erinnerungen an ihn".Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (15):3–19.
^Wolf, Armin (1964). "Der Pädagoge und Philosoph Johann Conrad Fallenstein (1731–1813) – Genealogische Beziehungen zwischenMax Weber, Gauß und Bessel".Genealogie (in German).7:266–269.
^Weinberger, Joseph (1977). "Carl Friedrich Gauß 1777–1855 und seine Nachkommen".Archiv für Sippenforschung und alle verwandten Gebiete (in German). 43/44 (66):73–98.
^Schubring, Gert (1993). "The German mathematical community". InFauvel, John;Flood, Raymond;Wilson, Robin (eds.).Möbius and his band: Mathematics and Astronomy in Nineteenth-century Germany. Oxford University Press. pp. 21–33.
^Schubring, Gert (2021).Geschichte der Mathematik in ihren Kontexten (in German). Birkhäuser. pp. 133–134.
^Lisitsa, Alexei; Potapov, Igor; Saleh, Rafiq (2009)."Automata on Gauss Words"(PDF). In Dediu, Adrian Horia; Ionescu, Armand Mihai; Martín-Vide, Carlos (eds.).Language and Automata Theory and Applications. Lecture Notes in Computer Science. Vol. 5457. Berlin, Heidelberg: Springer. pp. 505–517.doi:10.1007/978-3-642-00982-2_43.ISBN978-3-642-00982-2.
^Carl Friedrich Gauss: Zusätze.II. In:Carnot, Lazare (1810).Geometrie der Stellung (in German). Translated by H.C. Schumacher. Altona: Hammerich. pp. 363–364. (Text by Schumacher, algorithm by Gauss), republished inCollected Works Volume 4, p. 396-398
^Schreiber, Oscar (1866).Theorie der Projectionsmethode der Hannoverschen Landesvermessung (in German). Hannover: Hahnsche Buchhandlung.
^Gauß, C.F. (1828).Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector (in German). Vandenhoeck und Ruprecht. p. 73.
^Printed in theCollected Works, Volume 5, pp. 609–610.
^Roche, John J. (1990). "A critical study of the vector potential". In Roche, John (ed.).Physicists Look Back: Studies in the History of Physics. Bristol, New York: Adam Hilger. pp. 147–149.ISBN0-85274-001-8.
^Siebert, Manfred (1998). "Das Foucault-Pendel von C. F. Gauß".Mitteilungen der Gauß-Gesellschaft Göttingen (in German).35 (35):49–52.Bibcode:1998GGMit..35...49S.
^Rohlfing, Helmut (2003). "Das Erbe des Genies. Der Nachlass Carl Friedrich Gauß an der Niedersächsischen Staats- und Universitätsbibliothek Göttingen".Mitteilungen der Gauß-Gesellschaft Göttingen (in German).35 (40):7–23.Bibcode:1998GGMit..35...49S.
^"Familienarchiv Gauß". Signature: G IX 021.Stadtarchiv Braunschweig. Retrieved25 March 2023.
Gray, Jeremy (1955). "Introduction to Dunnington's "Gauss"".Carl Friedrich Gauss: Titan of Science. A Study of his Life and Work. New York: Exposition Press. pp. xix–xxvi. With a critical view on Dunnington's style and appraisals
Galle, Andreas (1924)."Über die geodätischen Arbeiten von Gauss". In Königlich Preußische Akademie der Wissenschaften (ed.).Carl Friedrich Gauss. Werke (in German). Vol. XI, 2 (Abhandlung 1).
Maennchen, Philipp (1930)."Gauss als Zahlenrechner". In Königlich Preußische Akademie der Wissenschaften (ed.).Carl Friedrich Gauss. Werke (in German). Vol. X, 2 (Abhandlung 6).
Schaefer, Clemens (1929)."Über Gauss' physikalische Arbeiten". In Königlich Preußische Akademie der Wissenschaften (ed.).Carl Friedrich Gauss. Werke (in German). Vol. XI, 2 (Abhandlung 2).
Stäckel, Paul (1917)."Gauss als Geometer". In Königlich Preußische Akademie der Wissenschaften (ed.).Carl Friedrich Gauss. Werke (in German). Vol. X, 2 (Abhandlung 4).
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