Carl Friedrich Gauss

12168: 12177: 2320: 1598:, although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. 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, 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. 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. 3150: 1415: 10972: 3027: 13508: 2597: 1536:("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". Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding 3823: 631: 1312: 3611: 1039: 1392: 1031: 83: 1057: 2959: 1380: 3142: 3292: 474: 1681: 13496: 1300: 1077:, 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. 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. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations. 446: 639: 1216: 1366: 2438:(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. 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 1286: 1427: 1340: 2586: 2570: 1354: 2420: 1328: 1274: 830: 658:). His family was of relatively low social status. 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. He had one elder brother from his father's first marriage. 13568: 12446: 1509: 3567: 3602:. Out of about a hundred pupils, Gauss was the first to solve the problem correctly by a significant margin. Although (or because) Sartorius gave no details, over time many versions of this story have been created, with more and more details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100 – and the circumstances in the classroom. 1497:, and short glosses in his own textbooks show that he worked to a great extent in an empirical way. 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. He coped with the enormous workload by using skillful tools. Gauss used a lot of 13544: 13520: 13556: 13532: 1570: 1544:

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.

13484: 2339:. 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 when 1505:. 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. Very likely, this method gave him a lot of material which he used in finding theorems in number theory. 1629:. Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of the contemporary school of 11104: 6677: 3503:, 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. 3117:– for several decades; he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane. 2486:(1809) that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of 3236:

variation of magnetic declination, inclination, and intensity, but discriminated Humboldt's concept of magnetic intensity to the terms of "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of intensity of the magnetic field, and constructed a suitable

1145:", 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. 2998:

in 1832. In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion. Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai claiming that these were congruent to

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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. After appointment as director of observatory he attached importance to

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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

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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, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle,

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On 4 August 1810, the widower 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

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Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about the burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described

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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

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subdivides Gauss's astronomic activity chronologically into seven periods, of which the years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man

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of length and measures. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical preparation. In the correspondence with Schumacher, who was also working on this matter, he described new ideas for scales of high precision. He submitted the final reports on

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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

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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. In contrast to his perfectionism in presenting

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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. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program;

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of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus. 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.

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ultimately destroyed the health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. 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".

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Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. 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.

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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

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was going on. 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. Gauss thought about the basics of geometry since the 1790s years, but in the 1810s he realized that a non-Euclidean

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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,

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sitting enthroned on the summit of science". 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

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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,

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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

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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. The public difficulties of replacement may be the historical background for the confusion on this matter in the Gauss family (see chapter:

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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

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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

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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 and

6662:Über die typischen Verschiedenheiten der Windungen der Hemisphären und über die Lehre vom Hirngewicht, mit besondrer Rücksicht auf die Hirnbildung intelligenter Männer. Vorstudien zu einer wissenschaftlichen Morphologie und Physiologie des menschlichen Gehirns als Seelenorgan, Vol. 1 2361:

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.

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and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the

1207:, who died in Göttingen a few months after Gauss. 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. 3002:

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.

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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.

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on the theory of reduction of positive ternary quadratic forms. 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.

1646:, though he agreed with some of their aims, such as the idea of a unified Germany. 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. 1449:

let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, 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.

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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.

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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

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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

1732:, 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. The 3335:

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 of

6679:Über den Hirnbau der Mikrocephalen mit vergleichender Rücksicht auf den Bau des Gehirns der normalen Menschen und der Quadrumanen. Vorstudien zu einer wissenschaftlichen Morphologie und Physiologie des menschlichen Gehirns als Seelenorgan, Vol. 2 2793:

with a Royal decree from 25 March 1828. 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

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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

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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

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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 ancient

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from the French Academy of Science in 1809 for the theory of planets and the means of determining their orbits from only three observations, the Danish Academy of Science prize in 1823 for his memoir on conformal projection, and the

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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, he stated the

2416:, and gave the explicit meaning only in letters to Olbers and Bessel. 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. 1479:

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

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was used as standard. 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

2268:(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 of 2664:, 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 10159: 10105: 8626: 4873: 4686: 4616: 4454: 4359: 4293: 1182:

Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms.

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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 from

11769: 10314:"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)" 8283: 4853: 4836: 4787: 4561: 4544: 4511: 4490: 4272: 4251: 4148: 4082: 4016: 3945: 3903: 3883: 3863: 11726: 10050: 4808: 4712: 4377: 4123: 4093: 4057: 3844: 2037:

AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM to its two "simplest values". In his unpublished writings he recognized and made a sketch of the key concept of

11672: 4173: 475: 11539: 10337:"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" 4396: 4339: 4318: 3415:

on this subject. 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.

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Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded – not on account of my hard efforts, but by the grace of the Lord." Gauss was a member of the

3207:, then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods. Together with other instigators, this led to a global program known as " 3331:

in 1833, and Weber himself connected the observatory with the institute for physics in the town centre of Göttingen, but they did not care for any further development of this invention for commercial purposes.

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in the south. 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

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than a decade. 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. Her sister Wilhelmina married the orientalist

10169: 4225: 609:. 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 as 817:. 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 opus 3577:

Several stories of his early genius have been reported. Carl Friedrich Gauss's mother had never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the

1960:. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws. 3593:

tells a story about the three-year-old Gauss, who corrected a math error his father made. The most popular story, also told by Sartorius, tells of a school exercise: the teacher Büttner and his assistant

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in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two

459: 13663: 3392:, 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 2450:

the fundamental astronomical constants in correspondence with Bessel. Gauss himself provided tables for nutation and aberration, the solar coordinates, and refraction. He made many contributions to

2896:) 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 4294:"Determinatio attractionis, quam in punctum positionis datae exerceret planeta, si eius massa per totamorbitam, ratione temporis, quo singulae partes describuntur, uniformiter esset dispertita" 13673: 1239:

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 the

13683: 2284:, and any religious connotations. Biographers speculated on the reason why Gauss dealt with this matter, but it is likely comprehensible by the historical background. The replacement of the 13668: 4360:"Allgemeine Auflösung der Aufgabe die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" 2089:

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 the

10994: 9632: 2157: 4971:. 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. 2987:", but he strongly forbade Taurinus to make any use of it. Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well. 5076:. Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68 (in German). Vienna: Verlag der Österreichischen Akademie der Wissenschaften. 12901: 559:

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 of

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without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.

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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.

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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

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He was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved a geometrical problem that had occupied mathematicians since the

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from numerical data he had collected empirically. As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to

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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.

2093:, which inspired Abel's theorem on lemniscate division. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of 1445:

Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like

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as a career. 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 opus

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in later years. 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.

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in 1799, not in Göttingen, as is sometimes stated, but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of the duchy.

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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 the

1946: 1789:

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 his

12281: 5089:. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 (in German). Göttingen: Vandenhoeck & Ruprecht. pp. 37–63. 888:

and built on a converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector

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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.

3674: 2928:. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its 13733: 2999:

his own thoughts of some decades. However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his letter remarks are only vague and obscure.

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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

2178:, which contains several developments that pertain to the field of complex analysis. Gauss stated that angle-preserving mappings in the complex plane must be complex 13688: 12484: 11038: 2084: 1992:, a result that Gauss stated that "will surely open an entirely new field of analysis". He also made early inroads into the more formal issues of the foundations of 13643: 5218:. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 (in German). Göttingen: Vandenhoeck & Ruprecht. 1621:

on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in the original languages. His favorite English author was

10436:"Russland ist seit jeher das gelobte Land für Magnetismus gewesen: Alexander von Humboldt, Carl Friedrich Gauß und die Erforschjung des Erdmagnetismus in Russland" 3558:

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.

3090:, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers. He described rotations of this sphere as the action of certain 2702:, Hamburg, and Lauenburg for the geodetic connection in October 1818. During the summers of 1821 until 1825 Gauss directed the triangulation work personally, from 3864:"Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse" 2679:. 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 same 5238:

provides a complete collection of the known letters from and to Carl Friedrich Gauss that is accessible online. The literary estate is kept and provided by the

4467: 4285: 4264: 4161: 4136: 4070: 3876: 1199:, a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the 12894: 3710: 2159:, and shows that many of the functions known at the time are special cases of the hypergeometric function. This work is the first one with an exact inquiry of 3034:

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

1501:, examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the 5465:

In a letter to Bessel from 1828, Gauss commented: "Mr. Abel has anticipated me, and relieves me of the effort in respect to one third of these matters ..."

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in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem.

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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.

4882: 2022:, which includes Euler's theorem as a special case. His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808. 924:, and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: 766:

without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the

13678: 1863:(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 cases 1458:

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 in

12250: 5298:

Gauss announced 195 lectures, 70 percent of them on astronomical, 15 percent on mathematical, 9 percent on geodetical, and 6 percent on physical subjects.

4800: 4524: 4503: 4331: 3896: 5972: 4029: 3846:

Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse

3166: 3066:

intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as

1160:

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.

11478:

Rohlfing, Helmut (2003). "Das Erbe des Genies. Der Nachlass Carl Friedrich Gauß an der Niedersächsischen Staats- und Universitätsbibliothek Göttingen".

7068: 4622: 2319: 1638:

Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto "

11836: 11571: 6579: 5422:

with 39 nearly independent states, the sovereigns of three of them being Kings of other countries (Netherlands, Danmark, United Kingdom), whereas the

3916: 3252:(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 12887: 8906: 13455: 4049: 2936:

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

1824:

by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a

1203:

in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physician

8546: 8415: 6697: 4579: 4036: 3806: 178: 11868: 11852: 11555: 5235: 3626: 1438:

In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his

13129: 12356: 11814: 11659: 5040: 3965: 3851:

New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors of the first or second degree

3686: 2779: 1200: 4921:

Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins

12477: 5709:]. Ahnentafeln Berühmter Deutscher (in German). Vol. 1. Zentralstelle für Deutsche Personen- und Familiengeschichte. pp. 63–65. 3768: 3499:

carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as an effect of the

2225:

for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.

1084:

enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834.

1642:". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the 13463: 3654: 2917: 1391: 1311: 1251:

Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the

666: 4384:

Determination of the Difference in Latitude between the Observatories of Göttingen and Altona by Observations with Ramsden's Zenith sector

13738: 13623: 13608: 11881: 9076: 5392:

Following Bolyai's handwritten Hungarian text at the bottom, Gauss intentionally characterized Kästner with the added the wrong addition.

4860:

Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte

4854:

Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte

3650: 3149: 11016: 8830: 8678: 4177: 3925:

Die vier Gauss'schen Beweise für die Zerlegung ganzer algebraischer Funktionen in reelle Faktoren ersten und zweiten Grades. (1799–1849)

1365: 793:

When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community of

2405: 2365:

The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as

13633: 9506:

Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector

6043: 4379:

Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector

6723:"A rare anatomical variation newly identifies the brains of C.F. Gauss and C.H. Fuchs in a collection at the University of Göttingen" 3204: 3038:

in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are

6051:

Grundsätze über die Anlage neuer Sternwarten unter Beziehung auf die Sternwarte der Universität Göttingen von Georg Heinrich Borheck

5239: 13728: 13618: 13603: 12578: 12470: 5531: 5487:

The unambiguous identification of a cosmic object as planet among the fixed stars requires at least two observations with interval.

4424:

Theory of the Combination of Observations Least Subject to Errors. Part One, Part Two, Supplement (Classics in Applied Mathematics)

3518: 3448:, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician 1594:, far from Göttingen. The military career of his elder son Joseph ended after more than two decades with the rank of a poorly paid 5547:(1820) of 1/302,78 was improved to 1/298.39; the calculation was done by Eduard Schmidt, private lecturer at Göttingen University. 2331:

discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called

13723: 13638: 13628: 13613: 13268: 9786: 3432:

to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was the nonprecise knowledge of the

2789:

When the arc measurement was finished, Gauss began the enlargement of the triangulation to the west to get a survey of the whole

1738:, written since 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic 11280: 8801: 7870:(2005). "Arithmetisch-geometrisches Mittel und elliptische Integrale: Gauß und die komplexe Analysis". In Mittler, Elmar (ed.). 3545:

In 1828, Gauss was appointed to head of a Board for weights and measures of the Kingdom of Hanover. He provided the creation of

1118:

for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for

13718: 13409: 12008: 11744: 7993: 2734:

mountains was the largest one Gauss had ever measured with a maximum size of 107 km (66.5 miles). In the thinly populated

1821: 775: 733: 674: 12293: 11237: 7564:

Reich, Karin (2005). "Gauß' geistige Väter: nicht nur "summus Newton", sondern auch "summus Euler"". In Mittler, Elmar (ed.).

5861: 5824: 2109:

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general

1414: 1285: 17: 13743: 13658: 12067: 12036: 11976: 11618: 11594: 10754: 10729: 10704: 10558: 9922: 9794: 9271: 8531: 8399: 7946: 7314: 5685: 5654: 4993: 4440: 3998: 3666: 3264:'s idea of four magnetic poles became obsolete, and the data allowed to determine their location with rather good precision. 1339: 154: 11781: 8768: 8064: 6940:

Wolf, Armin (1964). "Der Pädagoge und Philosoph Johann Conrad Fallenstein (1731–1813) – Genealogische Beziehungen zwischen

5268: 4974: 3678: 3590: 3165:, partly independently, partly in productive cooperation. In 1828, Gauss was Humboldt's guest during the conference of the 1840:. An unfinished eighth chapter was found among left papers only after his death, consisting of work done during 1797–1799. 1176: 1001: 965: 875: 5242:. Written materials from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick. 3257: 3042:, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid 1015:

of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay

665:

in mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the

1639: 10508:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 169–183. 10383:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 188–201. 8955:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 259–260. 8726:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 236–253. 8303: 3521:

of 1829 was established as a general concept to overcome the division of mechanics into statics and dynamics, combining

2348: 1980:(AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through the 12774: 12349: 11948: 11795: 9118: 7573:. Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 105–115. 7531:. Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 302–310. 6256:. Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 131–149. 5354: 5223: 5214: 5028: 4865:

General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances

3670: 3405:

General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances

1790: 1100:

in 1810 to 2400 Reichsthaler in 1824, and in his later years he was one of the best-paid professors of the university.

400: 42: 8863:

Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.

7879:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 62–72. 7839:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 53–62. 5799:. Göttinger Bibliotheksschriften 30 (in German). Niedersächsische Staats- und Universitätsbibliothek. pp. 17–29. 5671: 1832:. In the same chapter, he gives a result on the number of solutions of certain cubic polynomials with coefficients in 689:, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his 651: 11899: 11715: 10936: 10604: 10513: 10388: 9193: 8960: 8731: 8316: 8212: 7884: 7844: 7578: 7536: 7177: 7172:. Göttinger Bibliotheksschriften 30 (in German). Niedrsächsische Staats- und Universitätsbibliothek. pp. 73–86. 6261: 6058: 5920: 5804: 5176: 4908: 3690: 3662: 3642: 2854: 12786: 11710:]. Grundlehren der mathematischen Wissenschaften 24 (in German). Berlin, Heidelberg, New York: Springer-Verlag. 11408: 11215: 8264: 8131: 4937: 2755:. Gauss got assistance by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in the 13698: 11203: 4919: 4897: 4074: 3982: 3748: 3026: 1957: 1559: 1138: 698: 13507: 13248: 11505: 7709:

Basu, Soham; Velleman, Daniel J. (21 April 2017). "On Gauss's first proof of the fundamental theorem of algebra".

4738:"Ueber die achromatischen Doppelobjective besonders in Rücksicht der vollkommnern Aufhebung der Farbenzerstreuung" 3778:" (1816) and "Geheimer Hofrath" (1845). In 1949, on the occasion of his golden doctor degree jubilee, he got the 1662:, and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions. 1122:

but only shortly before her death, so she never received it. He also gave successful support to the mathematician

547:

as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the

13708: 12109: 4124:"Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1806, 1807, 1808, 1809" 3706: 2370: 2115: 1794: 1353: 1245: 989: 977: 901: 548: 12157: 12133: 4447:(Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation) 3180:

by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of

2458:. He published a great number of observations, mainly on minor planets and comets; his last observation was the 1027:. He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians. 677:

as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and

13258: 10065: 9838: 8149: 6387: 3702: 3020: 2526: 2241: 2217:

In a private letter to Gerling from 1823, he described a solution of a 4X4 system of linear equations by using

2019: 1996:, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – 1989: 1891: 1696: 973: 533: 362: 3120:

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given

1379: 1167:

and general unhappiness. On 23 February 1855, he died of a heart attack in Göttingen; and was interred in the

13648: 13588: 13287: 12722: 12450: 12342: 11535: 7486: 7195: 6983: 5125:"Discovery of the Missing Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802–05)" 4243: 3345: 2624: 2558: 2517:

of the fourth order (a special case of Gauss-Winckler inequality). He derived lower and upper bounds for the

2397: 2384:. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was 2098: 1853: 985: 425: 12147: 12123: 8127:"Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber" 5790:

Ullrich, Peter (2005). "Herkunft, Schul- und Studienzeit von Carl Friedrich Gauß". In Mittler, Elmar (ed.).

5344:

Gauss's mother-in-law, pleaded with Olbers to try to provide Gauss with another position far from Göttingen.

2265: 1805:

that Gauss gives without proof towards the end of the fifth chapter, it appears that Gauss already knew the

12674: 12431: 11386: 9764: 7055: 6987: 3976: 3698: 3694: 3638: 3522: 3412: 3353: 1997: 1759: 1299: 867:

wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from

787: 11060: 5165: 5149: 3884:"Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia" 3582:, which occurs 39 days after Easter. Gauss later solved this puzzle about his birthdate in the context of 2620:. Beginning in 1804, he taught himself some geodetic practise with a sextant in Brunswick, and Göttingen. 13713: 13703: 13693: 13188: 12973: 12594: 6693: 5915:(in German). Stuttgart: GNT – Verlag für Geschichte der Naturwissenschaften und der Technik. p. 41. 5331: 5055: 5005: 4956: 3622: 3324: 2858: 2596: 2245: 1949: 1734: 1723: 1196: 863:

from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and

783: 767: 710: 702: 524: 11685: 8126: 1419: 686: 13474: 13138: 12662: 12522: 12421: 8588: 3822: 3744: 3658: 3646: 3630: 3411:, based on Lagrange, Laplace, and Poisson; it seems rather unlikely that he knew the previous works of 2873:. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring 2229: 2014:, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on 2001: 1981: 1953: 1708: 1524: 1475:

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows:

885: 12750: 9673: 6430:

Küssner, Martha (1978). "Friedrich Wilhelm Bessels Beziehungen zu Göttingen und Erinnerungen an ihn".

4704: 1038: 961: 859:, and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand 844: 682: 512: 410: 159: 13748: 13298: 9909:. Lecture Notes in Computer Science. Vol. 5457. Berlin, Heidelberg: Springer. pp. 505–517. 7163:

Reich, Karin (2005). "Logarithmentafeln – Gauß' "tägliches Arbeitsgeräth"". In Mittler, Elmar (ed.).

5452:

Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order),

5380: 4858: 4584: 4310: 4273:"Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae" 3530: 3449: 3408: 3312: 3193: 3184:

with a representation for the unit of magnetism in terms of mass, charge, and time. They founded the

2904:

cannot be transformed to a plane without distortion, which causes a fundamental problem in designing

2764: 2550: 2506: 2011: 1977: 630: 88: 38: 12327: 12187: 10885: 10237:"Alexander von Humboldt und Carl Friedrich Gauss als Wegbereiter der neuen Disziplin Erdmagnetismus" 6247:

Wittmann, Axel (2005). "Carl Friedrich Gauß und sein Wirken als Astronom". In Mittler, Elmar (ed.).

4740:[On achromatic double lenses with special regard to a more complete dispersion of colours]. 4700:

Experiments on the Law of falling Bodies, on the Resistance of Air, and of the Rotation of the Earth

1327: 12694: 12181: 11891: 11785: 6530: 6411: 5124: 4100: 3831: 3782:

of both towns of Brunswick and Göttingen. Soon after his death a medal was issued by order of King

2941: 2913: 2530: 2491: 2389: 2377: 2190:

on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an

2168: 1860: 1104: 810: 798: 718: 468: 342: 13238: 12626: 11885: 10500: 10375: 8947: 8718: 8561:(2 August 2019). "Iterative Methods for Linear Systems of Equations: A Brief Historical Journey". 7871: 7831: 7565: 7523: 7522:(2005). "Carl Friedrich Gauß' Beschäftigung mit der russischen Sprache". In Mittler, Elmar (ed.). 7164: 6405: 6248: 5791: 3610: 3300: 2445:

While Gauss's contributions to theoretical astronomy came to an end, more practical activities in

1920: 1108: 1096:

in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000

1005: 856: 505: 13308: 13208: 13073: 12922: 12562: 10881: 10371: 8158: 8001: 4842: 4532: 3958: 3764: 3760: 3476: 3440:

of the used glass types. In a short article from 1817 Gauss dealt with the problem of removal of

2929: 2814: 2795: 2743: 2715: 2590: 2446: 2401: 2191: 2110: 2005: 1859:

When Olbers encouraged Gauss in 1816 to compete for a prize from the French Academy on proof for

1763: 1431: 1030: 957: 706: 568: 352: 164: 12738: 12714: 11739: 8507: 7915:(2007). "The Unpublished Section Eight: On the Way for Function Fields over a Finite Field". In 7306: 5622: 5280:

Once Gauss drew a lecture scene with professor Kästner producing errors in a simple calculation.

5170:. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. 5096: 4737: 4696:

Versuche über das Gesetz des Falls, über den Widerstand der Luft und über die Umdrehung der Erde

3208: 3199:

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the

2990:

The first publications on non-Euclidean geometry in the history of mathematics were authored by

2813:

as the surface everywhere perpendicular to the direction of gravity; later his doctoral student

778:. Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect 13439: 12514: 10435: 10341: 10313: 10290: 10236: 8055: 7827: 5829: 4713:"Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata" 3752: 3468: 3381: 3268: 3158: 2984: 2953: 2277: 2187: 2015: 1195:, who found its mass to be slightly above average, at 1,492 grams (3.29 lb). Wagner's son 953: 941: 790:

in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped.

757: 606: 591: 587: 405: 317: 63: 11342: 10499:

Timm, Arnulf (2005). "Der elektrische Telegraph von Gauß und Weber". In Mittler, Elmar (ed.).

10336: 9780: 3736:

from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".

2767:

near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.

13339: 12754: 12586: 12376: 11998: 8647: 7073: 5085: 5051: 4627: 4482:. Translated by J. C. Morehead and A. M. Hiltebeitel. The Princeton University Library. 1902. 4427:. Translated by G. W. Stewart. Philadelphia: Society for Industrial and Applied Mathematics. 4388: 4006: 3526: 3196:

in many regions of the world with equal methods at arranged dates in the years 1836 to 1841.

3141: 3110: 3091: 2838: 2676: 2609: 2502: 2479: 1848:

One of Gauss's first results was the empirically found conjecture of 1792 – the later called

1813: 1786: 1771: 1767: 1273: 1204: 1056: 909: 852: 848: 709:, whose lectures in classics Gauss attended with pleasure. Fellow students of this time were 556: 13158: 4988:. Acta Historica Astronomica (in German). Vol. 53. Leipzig: Akademische Verlaganstalt. 3192:), an international working group of several observatories, which supported measurements of 2958: 2061: 1569: 1263:; he then moved to Missouri, started as a farmer and became wealthy in the shoe business in 1034:

The new Göttingen observatory of 1816; Gauss's living rooms were in the western wing (right)

13598: 13593: 13113: 13023: 12879: 12866: 12546: 12152: 12128: 11487: 11457: 11364: 10815: 9729: 8943: 8915: 8821: 8777: 8717:(2005). "Der praktische Gauß – Präzisionsmessung für den Alltag". In Mittler, Elmar (ed.). 8381: 8095: 7928: 7344: 6843: 5981: 5838: 5544: 5453: 5419: 5074:

und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext

4593: 4408: 3923: 3779: 3756: 3546: 3441: 3425: 3328: 3261: 3170: 3071: 2937: 2799: 2756: 2561:. Gauss's solution is the first-ever result in the metrical theory of continued fractions. 2514: 2455: 2340: 2336: 2269: 2160: 2034: 1899: 1895: 1849: 1806: 1502: 1252: 1070: 1042: 997: 993: 897: 864: 753: 576: 544: 420: 10502:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 10377:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 8949:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 8720:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 8181: 7873:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 7833:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 7567:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 7525:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 7193:

Althoen, Steven C.; McLaughlin, Renate (1987), "Gauss–Jordan reduction: a brief history",

7166:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 6749: 6250:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 5793:"Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen 4422: 4106: 3371:, comparable to Newton's law of gravitation, but his attempt ended in a "tragic failure". 2719: 1894:

of spheres in the three-dimensional space is given when the centers of the spheres form a

1613:

he was also acquainted with modern languages. At the age of 62, he began to teach himself

1558:

charming, open-minded host. Gauss abominated polemic natures; together with his colleague

1523:

Gauss refused to publish work that he did not consider complete and above criticism. This

8: 13228: 12810: 12730: 12702: 12092: 12087: 12057: 11934: 11765: 11604: 10595:

Roche, John J. (1990). "A critical study of the vector potential". In Roche, John (ed.).

9877: 9230:"Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth" 8631:(in Latin). Göttingen: K. Gesellschaft der Wissenschaften zu Göttingen. pp. 265–327. 8373: 8369: 7920: 7916: 5908: 5358: 5120: 4654: 4570: 4553: 4165: 4112: 3783: 3717: 3682: 3492: 3437: 3349: 3341: 3291: 3253: 3177: 3095: 2991: 2810: 2534: 2487: 2475: 2393: 2332: 2218: 2211: 2203: 2094: 2055: 1985: 1907: 1880: 1751: 1659: 1643: 1618: 1537: 1264: 1123: 1074: 1066: 921: 806: 390: 347: 11491: 11141: 10819: 9733: 9708:"János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword" 8946:(2005). "Gauß, die Konzertsaalakustik und der Asteroid Palls". In Mittler, Elmar (ed.). 8919: 8781: 6847: 6773: 5985: 5842: 5496:

Brendel (1929) thought this cipher to be insoluble, but actually decoding was very easy.

5097:"The Correspondence between Carl Friedrich Gauss and the Rev. Nevil Maskelyne (1802–05)" 4717: