Carl Friedrich Gauss

gigatos | June 24, 2022

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Summary

Johann Carl Friedrich Gauss (* April 30, 1777 in Brunswick, Principality of Brunswick-Wolfenbüttel; † February 23, 1855 in Göttingen, Kingdom of Hanover) was a German mathematician, statistician, astronomer, geodesist, electrical engineer and physicist.Because of his outstanding scientific achievements, he was already considered Princeps mathematicorum (Prince of Mathematicians) during his lifetime. His activities extended not only to pure mathematics but also to applied fields, for example he was in charge of the land survey of the Kingdom of Hanover, together with Wilhelm Eduard Weber he was one of the first to invent electromagnetic telegraphy and both were the first to use it over longer distances, he developed magnetometers and he initiated a worldwide network of stations for the study of geomagnetism.

At the age of 18, Gauss developed the foundations of the modern calculus of equations and mathematical statistics (method of least squares), with which he enabled the rediscovery of the first asteroid Ceres in 1801. Non-Euclidean geometry, numerous mathematical functions, integral theorems, the normal distribution, first solutions for elliptic integrals and Gaussian curvature can be traced back to Gauss. In 1807 he was appointed university professor and observatory director in Göttingen and later entrusted with the land surveying of the Kingdom of Hanover. In addition to number theory and potential theory, he researched, among other things, the earth”s magnetic field.

As early as 1856, the King of Hanover had medals minted with the image of Gauss and the inscription Mathematicorum Principi (the Prince of Mathematicians). Since Gauss published only a fraction of his discoveries, the profundity and scope of his work only became fully known to posterity when his diary was discovered in 1898 and the estate became known.

Many mathematical-physical phenomena and solutions are named after Gauss, several surveying and observation towers, numerous schools, furthermore research centers and scientific honors such as the Carl Friedrich Gauss Medal of the Braunschweig Academy and the festive Gauss Lecture, which takes place every semester at a German university.

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Parents, childhood and youth

Carl Friedrich was born in Braunschweig on April 30, 1777, the son of Mr. and Mrs. Gauss. His birthplace at Wendengraben in Wilhelmstraße 30 – in the first floor of which the Gauss Museum was later established – did not survive the Second World War. There he grew up as the only common child of his parents; from an earlier marriage of his father there was still an older stepbrother. His father Gebhard Dietrich Gauss (1744-1808) practiced various professions, including gardener, butcher, bricklayer, assistant merchant, and treasurer of a small insurance company. Dorothea Bentze (1743-1839), one year older, worked as a maid before marriage and became his second wife. She was the daughter of a stonemason from Velpke who died early, and is described as bright, of cheerful mind and firm character. Gauss”s relationship with his mother remained close throughout his life; the 96-year-old last lived with him in Göttingen.

Anecdotes say that even the three-year-old Carl Friedrich corrected his father in the payroll. Later, Gauss jokingly said of himself that he had learned to calculate before he learned to speak. He still had the gift of performing even the most complicated calculations in his head at an advanced age. According to a story by Wolfgang Sartorius von Waltershausen, little Carl Friedrich”s mathematical talent was noticed when he entered the arithmetic class at the Catherinen Volksschule after two years of elementary school:

There the teacher Büttner used to occupy his pupils with longer arithmetic problems while he walked up and down with a carbat in his hand. One task was the summation of an arithmetic series; whoever had finished put his blackboard on the desk with the calculations for the solution. With the words “Ligget se.” in Braunschweig Low German, the nine-year-old Gauss amazingly quickly placed his on the table, which bore only a single number. After Gauss”s extraordinary talent was recognized, another arithmetic book was first procured from Hamburg before the assistant Martin Bartels procured usable mathematical books for joint study – and ensured that Gauss was able to attend the Martino-Katharineum Braunschweig in 1788.

The elegant procedure with which “little Gauss” calculated the solution so quickly in his head is today called the Gaussian summation formula. In order to calculate the sum of an arithmetic series, for example of the natural numbers from 1 to 100, pairs of equal partial sums are formed, for example 50 pairs with the sum 101 (1 + 100, 2 + 99, …, 50 + 51), with which 5050 can be quickly obtained as the result.

When the “boy wonder” Gauss was fourteen years old, he was introduced to Duke Karl Wilhelm Ferdinand of Brunswick. He then supported him financially. Thus, Gauss was able to study at the Collegium Carolinum (Brunswick) from 1792 to 1795, which is located between a secondary school and a university and is the predecessor of today”s Technical University in Brunswick. There it was Professor Eberhard August Wilhelm von Zimmermann who recognized his mathematical talent, encouraged him and became a fatherly friend.

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Academic years

In October 1795, Gauss transferred to the Georg August University in Göttingen. There he listened to lectures on classical philology by Christian Gottlob Heyne, which at that time interested him as much as mathematics. The latter was represented by Abraham Gotthelf Kästner, who was also a poet. With Georg Christoph Lichtenberg he heard experimental physics in the summer semester of 1796 and very probably astronomy in the following winter semester. In Göttingen he became friends with Wolfgang Bolyai.

At the age of 18, Gauss was the first to succeed in proving the possibility of constructing the regular heptagon with compass and ruler, on the basis of purely algebraic reasoning – a sensational discovery; for there had been little progress in this field since antiquity. He then concentrated on the study of mathematics, which he completed in 1799 with his doctoral thesis at the University of Helmstedt. Mathematics was represented by Johann Friedrich Pfaff, who became his doctoral advisor. And the Duke of Brunswick attached importance to the fact that Gauss should not receive his doctorate at a “foreign” university.

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Marriages, family and children

In November 1804, he became engaged to Johanna Elisabeth Rosina Osthoff († October 11, 1809), the daughter of a white tanner from Braunschweig, whom he had courted for some time, and married her on October 9, 1805. On August 21, 1806, their first child was born in Braunschweig, Joseph Gauss († July 4, 1873). The son was given his first name after Giuseppe Piazzi, the discoverer of Ceres, a minor planet whose rediscovery in 1801 had made Gauss”s orbital calculation possible.

Soon after the family moved to Göttingen, their daughter Wilhelmine, called Minna, was born on February 29, 1808, and their son Louis was born the following year on September 10, 1809. A month later, on October 11, 1809, Johanna Gauss died in childbirth, Louis a few months later on March 1, 1810. Due to Johanna”s death, Gauss fell into a depression for a while; from October 1809 comes a moving lament written by Gauss, which was found in his estate. The finder Carl August Gauss (1849-1927) was his only German-born grandson, son of Joseph and owner of the Lohne estate near Hanover. Wilhelmine married the orientalist Heinrich Ewald, who later left the Kingdom of Hanover as one of the Göttingen Seven and became a professor at the University of Tübingen.

On August 4, 1810, the widower, who had two small children to support, married Friederica Wilhelmine Waldeck († September 12, 1831), daughter of the Göttingen jurist Johann Peter Waldeck, who had been the best friend of his late wife. With her he had three children. Eugen Gauss fell out with his father as a law student and emigrated to America in 1830, where he lived as a merchant and founded the “First National Bank” of St. Charles. Wilhelm Gauss followed Eugen to the United States in 1837 and also became prosperous. His youngest daughter Therese Staufenau managed her father”s household after her mother”s death until his death. Minna Gauss had died of tuberculosis after 13 years of suffering.

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Later years

After his doctorate, Gauss lived in Brunswick on the small salary paid to him by the Duke and worked on his Disquisitiones Arithmeticae.

Gauss declined a call to the Petersburg Academy of Sciences out of gratitude to the Duke of Brunswick, probably also in the hope that the latter would build him an observatory in Brunswick. After the sudden death of the Duke after the Battle of Jena and Auerstedt, Gauss became a professor at the Georg August University of Göttingen and director of the Göttingen Observatory in November 1807. There he had to give lectures, against which he developed an aversion. Practical astronomy was represented there by Karl Ludwig Harding, and the mathematical chair was held by Bernhard Friedrich Thibaut. Several of his students became influential mathematicians, including Richard Dedekind and Bernhard Riemann, as well as the mathematics historian Moritz Cantor.

At an advanced age, he became increasingly involved with literature and was an avid reader of newspapers. His favorite writers were Jean Paul and Walter Scott. He was fluent in English and French and, in addition to his familiarity with the classical languages of antiquity from his youth, read several modern European languages (Spanish, Italian, Danish, Swedish), most recently learning Russian and tentatively studying Sanskrit, which, however, did not appeal to him.

From 1804 he was a corresponding member of the Académie des sciences and from 1820 associé étranger of the Academy. Also in 1804 he became a fellow of the Royal Society and in 1820 of the Royal Society of Edinburgh. In 1808 he was elected a corresponding and in 1820 a foreign member of the Bavarian Academy of Sciences and in 1822 to the American Academy of Arts and Sciences.

In 1838 he received the Copley Medal of the Royal Society. In 1842 he was admitted to the Peace Class of the Order Pour le Mérite. In the same year he turned down a call to the University of Vienna. In 1845 he became Privy Councillor and in 1846 Dean of the Faculty of Philosophy for the third time. In 1849 he celebrated his Golden Doctoral Jubilee and became an honorary citizen of Brunswick and Göttingen. His last scientific exchange was about an improvement of the Foucault pendulum in a letter to Alexander von Humboldt in 1853.

He collected numerical and statistical data of all kinds and, for example, kept lists of the life expectancy of famous men (calculated in days). Thus, on December 7, 1853, he wrote to his friend and chancellor of his order Alexander von Humboldt, among other things: “It is the day after tomorrow when you, my esteemed friend, will pass into a region into which none of the luminaries of the exact sciences has yet penetrated, the day when you will reach the same age at which Newton closed his earthly career measured by 30,766 days. And Newton”s forces were completely exhausted at this stage: you still stand in the full enjoyment of your admirable power to the highest joy of the whole scientific world. May you remain in this enjoyment for many years to come.” Gauss was interested in music, attended concerts and sang a lot. Whether he played an instrument is not known. He was involved in stock speculation and at his death left a considerable fortune of 170,000 talers (on a professor”s basic salary of 1000 talers a year) mainly in securities, including many from railroads. This is one of the few passages in his correspondence in which he is critical of politics and the banks cooperating with it, because the Hesse-Darmstadt railroad shares he had purchased lost value drastically when it became known that the railroad could be nationalized at any time.

He was still scientifically active towards the end of his life and in 1850 held

Gauss was very conservative and monarchist, the German Revolution 1848

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Death

In his last years, Gauss suffered from heart failure (diagnosed as dropsy) and insomnia. In June 1854, he traveled with his daughter Therese Staufenau to the construction site of the railroad from Hanover to Göttingen, where the passing railroad caused the horses to spook and overturn the carriage, the coachman was seriously injured, Gauss and his daughter remained unharmed. Gauss still participated in the inauguration of the railroad on July 31, 1854, after which he was increasingly confined to his home by illness. He died on February 23, 1855 at 1:05 in the morning in Göttingen in his armchair.

The tomb in the Albani Cemetery was not erected until 1859 and was designed by the Hanoverian architect Heinrich Köhler. It was soon considered a Göttingen landmark.

Justification and contributions to non-Euclidean geometry

Gauss distrusted the proofs of elementary geometry already at the age of twelve and suspected at the age of sixteen that there had to be a non-Euclidean geometry besides Euclidean geometry.

He deepened this work in the 1820s: Independently of János Bolyai and Nikolai Ivanovich Lobachevsky, he noticed that Euclid”s parallel axiom was not dense. However, he did not publish his thoughts on non-Euclidean geometry, according to the reports of his confidants, presumably for fear of not being understood by his contemporaries. However, when his student friend Wolfgang Bolyai, with whom he corresponded, told him about the work of his son János Bolyai, he praised him, but could not refrain from mentioning that he himself had come up with it much earlier (” to praise would be to praise myself”). He had not published anything about it, because he “shied away from the shouting of the Boeotians”. Gauss found Lobachevsky”s work so interesting that he learned the Russian language at an advanced age in order to study it.

Prime number distribution and method of least squares

At the age of 18, he discovered some properties of the prime number distribution and found the method of least squares, which involves minimizing the sum of squares of divergences. He refrained from publishing for the time being. After Adrien-Marie Legendre had published his “Méthode des moindres carrés” in a treatise in 1805 and Gauss did not make his results known until 1809, a priority dispute arose.

According to this method, for example, the most probable result for a new measurement can be determined from a sufficiently large number of previous measurements. On this basis, he later investigated theories for calculating the area under curves (numerical integration), which led him to the Gaussian bell curve. The associated function is known as the density of the normal distribution and is applied in many probability calculation tasks, where it is the (asymptotic, i.e. valid for sufficiently large data sets) distribution function of the sum of data randomly scattering about a mean value. Gauss himself made use of it, among other things, in his successful administration of the widows” and orphans” fund of Göttingen University. He made a thorough analysis over several years, concluding that pensions could be increased slightly. In this way, Gauss also laid foundations in actuarial mathematics.

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Introduction of the elliptic functions

In 1796, at the age of 19, while considering the arc length on a lemniscate as a function of the distance of the curve point from the origin, he introduced what are historically called the first elliptic functions, the lemniscated sine functions. However, he never published his notes on them. This work is related to his study of the arithmetic-geometric mean. The actual development of the theory of elliptic functions, the inverse functions of the elliptic integrals already known for a long time, was done by Niels Henrik Abel (1827) and Carl Gustav Jacobi.

Fundamental theorem of algebra, contributions to the use of complex numbers

Gauss grasped the usefulness of complex numbers early on, as in his 1799 doctoral thesis, which contains a proof of the Fundamental Theorem of Algebra. This theorem states that every algebraic equation with degree greater than zero has at least one real or complex solution. Gauss criticized the older proof by Jean-Baptiste le Rond d”Alembert as insufficient, but even his own proof did not yet meet later demands for topological rigor. Gauss came back to the proof of the Fundamental Theorem several times and gave new proofs in 1815 and 1816.

By 1811 at the latest, Gauss knew the geometric representation of complex numbers in a number plane (Gaussian number plane), which had already been found by Jean-Robert Argand in 1806 and Caspar Wessel in 1797. In the letter to Bessel in which he communicates this, it also became clear that he knew other important concepts of function theory such as the curve integral in the complex and Cauchy”s integral theorem, and first approaches to periods of integrals. However, he did not publish anything about this until 1831, when he introduced the name complex number in his essay on number theory Theoria biquadratorum. In the meantime Augustin-Louis Cauchy (1821, 1825) had preceded him in the publication of the foundation of complex analysis. In 1849, on the occasion of his Golden Jubilee, he published an improved version of his dissertation on the Fundamental Theorem of Algebra, in which, in contrast to the first version, he explicitly used complex numbers.

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Contributions to number theory

On March 30, 1796, a month before his nineteenth birthday, he proved the constructibility of the regular seventeenth, providing the first significant addition to Euclidean constructions in 2000 years. But this was only a side result in the work for his number-theoretically much more far-reaching work Disquisitiones Arithmeticae.

A first announcement of this work was found in the Intelligenzblatt of the Allgemeine Literatur-Zeitung in Jena on June 1, 1796. The Disquisitiones, published in 1801, became fundamental for the further development of number theory, to which one of his main contributions was the proof of the quadratic reciprocity law, which describes the solvability of quadratic equations “mod p” and for which he found almost a dozen different proofs during his lifetime. In addition to building elementary number theory on modular arithmetic, there is a discussion of continued fractions and circular division, with a famous hint about similar theorems in Lemniskate and other elliptic functions, which later inspired Niels Henrik Abel and others. A large part of the work is taken up by the theory of quadratic forms, whose gender theory he develops.

However, there are many more profound results, often only briefly hinted at, in this book, which fertilized the work of later generations of number theorists in many ways. The number theorist Peter Gustav Lejeune Dirichlet reported that he always had the Disquisitiones handy at work throughout his life. The same is true for the two works on biquadratic reciprocity laws of 1825 and 1831, in which he introduces the Gaussian numbers (integer lattice in complex number plane). The works are probably part of a planned sequel to the Disquisitiones, which never appeared. Proofs of these laws were then given by Gotthold Eisenstein in 1844.

According to André Weil, the reading of these works (and of some passages in the diary, where it is about solving equations over finite bodies in a hidden form) inspired his work on the Weil conjectures. Gauss knew the prime number theorem, but did not publish it.

Gauss promoted one of the first female mathematicians of the modern era in this field, Sophie Germain. Gauss corresponded with her about number theory from 1804 on, although she first used a male pseudonym. She did not reveal her female identity until 1806, when she pleaded for his safety with the French commander after the occupation of Brunswick. Gauss praised her work and her deep understanding of number theory and asked her to get him an accurate pendulum clock in Paris in 1810 for the prize money he received with the Lalande Prize.

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Contributions to astronomy

After the completion of the Disquisitiones, Gauss turned to astronomy. The occasion for this was the discovery of the dwarf planet Ceres by Giuseppe Piazzi on January 1, 1801, whose position in the sky the astronomer had lost again shortly after its discovery. The 24-year-old Gauss managed to calculate the orbit with the help of a new indirect method of orbit determination and his balancing calculations based on the method of least squares in such a way that Franz Xaver von Zach was able to find it again on December 7, 1801 and – confirmed – on December 31, 1801. Heinrich Wilhelm Olbers confirmed this independently of Zach by observation on January 1 and 2, 1802.

The problem of the recovery of Ceres as such lay in the fact that by the observations neither the place, a piece of the orbit, nor the distance are known, but only the directions of the observation. This leads to the search of an ellipse and not of a circle, as Gauss” competitors assumed. One of the foci of the ellipse is known (the Sun itself), and the arcs of Ceres” orbit between the directions of observation are traversed according to Kepler”s second law, that is, the times behave like the areas swept by the guiding ray. Moreover, for the computational solution, it is known that the observations themselves start from a conic section in space, the Earth”s orbit itself.

In principle, the problem leads to an eighth-degree equation whose trivial solution is the Earth”s orbit itself. Using extensive constraints and the least squares method developed by Gauss, the 24-year-old succeeded in giving the location he calculated for Ceres” orbit for November 25 to December 31, 1801. This enabled Zach to find Ceres on the last day of the prediction. The location was no less than 7° (i.e. 13.5 full moon latitudes) east of where the other astronomers had suspected Ceres to be, which not only Zach but also Olbers duly acknowledged.

This work, which Gauss undertook even before his appointment as director of the observatory in Göttingen, made him even more famous than his number theory in Europe at a stroke and provided him, among other things, with an invitation to the Academy in St. Petersburg, of which he became a corresponding member in 1802.

The iterative method found by Gauss in this context is still used today because, on the one hand, it makes it possible to incorporate all known forces into the physical-mathematical model without considerable additional effort and, on the other hand, it is easy to handle in terms of computer technology.

Gauss then worked on the orbit of the asteroid Pallas, for whose calculation the Paris Academy had offered prize money, but was unable to find the solution. His experiences with the determination of the orbits of celestial bodies, however, resulted in his 1809 work Theoria motus corporum coelestium in sectionibus conicis solem ambientium.

Contributions to potential theory

In potential theory and physics, the Gaussian integral theorem (1835, published only in 1867) is fundamental. It identifies in a vector field the integral of the divergence (derivative vector applied to the vector field) over a volume with the integral of the vector field over the surface of this volume.

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Land survey and invention of the heliotrope

Gauss gained his first experience in the field of geodesy between 1797 and 1801, when he served as an advisor to the French Quartermaster General Lecoq during his national survey of the Duchy of Westphalia. In 1816, his former student Heinrich Christian Schumacher was commissioned by the King of Denmark to carry out a latitude and longitude survey in Danish territory. Subsequently, from 1820 to 1826, Gauss was put in charge of the national survey of the Kingdom of Hanover (“gaußsche Landesaufnahme”), at times assisted by his son Joseph, who was an artillery officer in the Hanoverian army. This survey continued the Danish one on Hanoverian territory to the south, with Gauss using the Braaker base measured by Schumacher. Through the method of least squares, which he invented, and the systematic solution of extensive systems of linear equations (Gaussian elimination method), he achieved a considerable increase in accuracy. He was also interested in practical implementation: He invented the heliotrope illuminated via solar mirrors as a measuring instrument.

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Gaussian curvature and geodesy

During these years, inspired by geodesy and map theory, he dealt with the theory of differential geometry of surfaces, introduced among other things the Gaussian curvature and proved his Theorema egregium. This states that Gaussian curvature, defined by the principal curvatures of a surface in space, can be determined solely by measures of internal geometry, that is, by measurements within the surface. Therefore, the Gaussian curvature is independent of the embedding of the surface in the three-dimensional space, i.e. it does not change for length-frequency mappings of surfaces to each other.

Wolfgang Sartorius von Waltershausen reports that Gauss, on the occasion of the Hanoverian national survey, empirically searched for a deviation of the angular sum of especially large triangles from the Euclidean value of 180° – as, for example, in the case of the plane triangle measured by Gauss, which is formed by the Brocken in the Harz Mountains, the Inselsberg in the Thuringian Forest and the Hoher Hagen near Dransfeld. Max Jammer wrote about this Gaussian measurement and its result:

The angular excess in this triangle is only 0.25 angular minutes due to the size of the Earth. The above mentioned conjecture about the motivation is subject to speculation.

Magnetism, electricity and telegraphy

Together with Wilhelm Eduard Weber, he worked in the field of magnetism from 1831. Weber, together with Gauss, invented an electromagnetic telegraph system with a relay-like principle in 1833, which connected his observatory with the physical institute over a distance of 1100 meters. They used galvanometers and magnetometers adapted to telegraphy and developed several versions. The conductor consisted of two copper wires (later iron wires), each connecting two coils: one in Weber”s cabinet and one in Gauss” observatory. Both coils were loosely wound around a magnetic rod and could be moved along the rod. The principle of electromagnetic induction, discovered two years earlier, triggered a surge of current when the transmitter coil, which was wound around a bar magnet, moved; the current was conducted over the wire to the other coil, where it was translated back into motion. The deflection of the bar magnet with coil fixed in a wooden frame at the receiver (which was a relay or magnetometer or mirror galvanometer like principle) was thereby magnified and made visible by a system of mirrors and telescopes. Letters were represented by a binary code corresponding to the direction of the current (the mirror in the receiver was turned to the left or right in each case). The first message was probably knowledge before mine, being before seeming – this message was found in Gauss”s records in binary code. According to other sources, they announced the arrival of a servant who otherwise delivered the messages (Michelmann forthcoming). Already two years before Gauss and Weber, Joseph Henry and one year before Gauss and Weber, Paul Ludwig Schilling of Cannstatt developed an electromagnetic telegraphy apparatus, but it did not come to any application over longer distances for both of them, and it did not attract more attention. In 1845, Gauss and Weber”s equipment was destroyed by a lightning strike, which also set fire to a lady”s hat. A stable, which the line passed, was spared, however, which might otherwise have caused a possible town fire. Commercial application, however, was by others, notably Samuel Morse in the U.S. a few years after Gauss and Weber”s invention. Gauss, however, saw the possibilities of application, for example, in the large-scale Russian Empire and for railroads, and they wrote a memorandum to that effect, which, however, did not materialize in Germany at that time because of the cost of the lines. Although they also published about it, the telegraph invention of Gauss and Weber was almost forgotten in the following years and others claimed the invention for themselves.

Together with Weber, he developed the CGS system of units, which was designated as the basis for electrotechnical units of measurement at an international congress in Paris in 1881. He organized a worldwide network of observation stations (Magnetic Association) to measure the geomagnetic field.

Gauss found Kirchhoff”s rules for electric circuits in his experiments on electricity in 1833 before Gustav Robert Kirchhoff (1845).

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Other

From him came the Gaussian Easter formula for calculating the date of Easter, and he also developed a Passover formula.

Gauss worked in many fields, but published his results only when a theory was, in his opinion, complete. This led to him occasionally pointing out to colleagues that he had long since proved this or that result, but had not yet presented it because of the incompleteness of the underlying theory or because he lacked the recklessness necessary to work quickly.

Significantly, Gauss possessed a petschaft showing a tree hung with few fruits with the motto Pauca sed Matura (“Few, but Ripe”). According to an anecdote, he refused to replace this motto with, for example, Multa nec immatura (“Much, but not immature”) to acquaintances who knew about Gauss”s extensive work, because, according to him, he preferred to leave a discovery to someone else than not to publish it fully elaborated under his name. This saved him time in areas that Gauss considered more marginal, so that he could spend that time on his original work.

Gauss”s scientific estate is preserved in the Special Collections of the Göttingen State and University Library.

After his death, the brain was removed. It was examined several times, most recently in 1998, using various methods, but without any particular finding that would explain his mathematical abilities. It is now kept separately, preserved in formalin, in the Department of Ethics and History of Medicine of the Medical Faculty of the University of Göttingen.

In the fall of 2013, a mix-up was uncovered at the University of Göttingen: The brain preparations of the mathematician Gauss and the Göttingen physician Conrad Heinrich Fuchs, which were more than 150 years old at the time, were mixed up – probably soon after they were taken. Both preparations were stored in the Anatomical Collection of the Göttingen University Hospital in jars containing formaldehyde. The original Gauss brain was in the jar labeled “C. H. Fuchs,” and the Fuchs brain was labeled “C. F. Gauss.” This makes the previous research results on Gauss” brain obsolete. Because of the MRI images made of Gauss” supposed brain, which showed a rare bisection of the central furrow, the scientist Renate Schweizer looked again at the specimens and discovered that this conspicuous feature was missing in drawings made shortly after Gauss” death.

Methods or ideas developed by Gauss that bear his name are:

Methods and ideas based in part on his work include:

Named in his honor are:

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Complete edition

Volumes 10 and 11 contain detailed commentaries by Paul Bachmann (number theory), Ludwig Schlesinger (function theory), Alexander Ostrowski (algebra), Paul Stäckel (geometry), Oskar Bolza (calculus of variations), Philipp Maennchen (Gauss as a calculator), Harald Geppert (mechanics, potential theory), Andreas Galle (geodesy), Clemens Schaefer (physics) and Martin Brendel (astronomy). The editor was first Ernst Schering, then Felix Klein.

Gauss stones

Among the numerous survey stones erected on the instructions of Gauss:

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Portraits

There are relatively many portraits of Gauss, among others:

Sources