# Carl Friedrich Gauss

Carl Friedrich Gauß, portrait by Gottlieb Biermann (1887). Gauss's signature:
Carl Friedrich Gauß, bust, bronze (around 1900), German Research Center for Geosciences

Johann Carl Friedrich Gauß ( Latinized Carolus Fridericus Gauss; born April 30, 1777 in Braunschweig , † February 23, 1855 in Göttingen ) was a German mathematician , statistician , astronomer , geodesist and physicist . Because of his outstanding scientific achievements, he was already considered Princeps Mathematicorum ("Prince of Mathematicians"; "First among mathematicians") during his lifetime .

At the age of 18, Gauss developed the basics of modern adjustment calculations and mathematical statistics ( method of least squares ), with which he made the rediscovery of the first asteroid Ceres possible in 1801 . Walking on Gauss non-Euclidean geometry , numerous mathematical functions, integral equations , the normal distribution , the first solutions for elliptic integrals and Gaussian curvature back. In 1807 he was appointed university professor and director of the observatory in Göttingen and was later entrusted with surveying the Kingdom of Hanover . In addition to number and potential theory , he researched a. a. the earth's magnetic field .

As early as 1856, the King of Hanover had commemorative medals struck with the image of Gauss and the inscription “ Mathematicorum Principi ” (German: “The Prince of Mathematicians”). Since Gauss published only a fraction of his discoveries, posterity did not fully understand the depth and scope of his work until his diary was discovered and evaluated in 1898 , and when the estate became known.

Many mathematical and physical phenomena and solutions are named after Gauß, several surveying and observation towers, numerous schools, research centers and scientific honors such as the Carl Friedrich Gauß Medal of the Braunschweigische Akademie and the festive Gauß lecture , which takes place every semester German university takes place.

## Life

### Parents, childhood and youth

The house where Carl Friedrich Gauß was born at Wilhelmstrasse  30 in Braunschweig ; it was completely destroyed in World War II.
Memorial plaque on the former site of the birthplace

Carl Friedrich was the only child of the married couple Gebhard Dietrich Gauß (1744–1808) and Dorothea Gauß née Bentze (1743–1839) and was born in the house at Wilhelmstrasse 30 in Braunschweig. The mother Dorothea was the daughter of a stonemason from Velpke , who died early, and was portrayed as clever, with a cheerful disposition and a firm character. Throughout his life, Gauss had a close relationship with his mother, who last lived with him at the Göttingen observatory. She initially worked as a maid before becoming Gebhard Dietrich Gauß's second wife. He had many jobs, including a gardener, butcher, bricklayer, assistant businessman and treasurer of a small insurance company. Some anecdotes say that the three-year-old Carl Friedrich was already correcting his father's payroll. Gauss later said of himself that he learned arithmetic before speaking. Throughout his life he retained the gift of performing even the most complicated calculations in his head.

An anecdote, the origin of which goes back to the stories of Wolfgang Sartorius von Waltershausen , describes the early mathematical talent of little Carl Friedrich:

At the age of seven, Gauss started school . When he was nine years old, his teacher Büttner had given the students the task of adding up the numbers from 1 to 100 for longer employment. However, Gauss solved it after a very short time by  forming 50 pairs with the sum 101 (1 + 100, 2 + 99, ..., 50 + 51) and got 5050 as the result. He put the answer with the words in Braunschweig Low German "Ligget se" (svw: "Here she is") on the teacher's table.

The resulting formula is sometimes referred to as “ the little Gauss ”. Gauß 'teacher Büttner recognized and promoted his extraordinary mathematical talent by procuring (among other things) a special arithmetic book for him from Hamburg and, supported by his assistant Martin Bartels , ensured that Gauß could visit the Katharineum in 1788 .

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

Ornate name of the 17-year-old Gauss

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

At the age of 18, Gauss was the first to prove the constructability of the regular seventeenth-octagon on the basis of purely algebraic considerations - a sensational discovery, because there had been little progress in this area since ancient times. He then concentrated on studying mathematics, which he completed in 1799 with his doctoral thesis at the Academia Julia , the University of Helmstedt . Mathematics was represented here by Johann Friedrich Pfaff - who became his doctoral supervisor - and, last but not least, Gauß 'patron, the Duke of Braunschweig, insisted that Gauß should not do his doctorate at a university abroad.

### Marriages, families and children

Daughter Therese

In November 1804 became engaged he was with Johanna Elisabeth Rosina Osthoff (* May 8, 1780; † 11 October 1809), the daughter of a white Gerber from Brunswick, and married her on October 9, 1805. On August 21, 1806 her in Braunschweig first child Joseph († July 4, 1873) born, named after Giuseppe Piazzi , the discoverer of the dwarf planet Ceres , whose recovery Gauss's orbit calculation made possible in 1801.

After the family moved to Göttingen, their daughter Wilhelmine , called Minna, was born on February 29, 1808 , and their son Louis was born on September 10, 1809. On October 11, 1809 Johanna Gauß died of the consequences of childbirth, Louis himself died on March 1, 1810. As a result of these events Gauß fell into a depression for a time , in which he received a 1927 from his grandson Carl August Gauß (1849–1927 ) wrote "Lament for the Dead" written on tear-covered paper for his deceased wife. Carl August Gauß was his only grandson born in Germany, landowner (Gut Lohnde near Hanover) and son of Joseph. Wilhelmine married the orientalist Heinrich Ewald , who later left the Kingdom of Hanover as one of the Göttingen Seven and became a professor in Tübingen .

On August 4, 1810, the widower, who had two small children to look after, married Friederica Wilhelmine Waldeck (called Minna; * April 15, 1788; † September 12, 1831), daughter of the Göttingen legal scholar Johann Peter Waldeck , who was his best friend deceased woman. He had three children with her. As a law student, Eugen fell out with his father and emigrated to America in 1830, where he lived as a merchant and founded the “First National Bank” of St. Charles . Wilhelm followed Eugen to America in 1837 and also achieved prosperity. His youngest daughter Therese ran the household for her father after her mother's death until his death. Minna Gauß had died of tuberculosis after suffering for 13 years .

### Later years

Göttingen observatory (around 1835)

After his doctorate , Gauß lived in Braunschweig on the small salary that the Duke paid him and worked on his work Disquisitiones Arithmeticae .

Gauss turned down a call to the Petersburg Academy of Sciences out of gratitude to his patron, the Duke of Braunschweig, and probably in the hope that he would build him an observatory in Braunschweig. After the sudden death of the duke after the battle of Jena and Auerstedt , Gauß became professor at the Georg-August University in Göttingen and director of the observatory there in November 1807 . There he had to give lectures, but he developed an aversion to them. The practical astronomy was represented there by Karl Ludwig Harding , the mathematics chair was held by Bernhard Friedrich Thibaut . Several of his students became influential mathematicians, including Richard Dedekind and Bernhard Riemann .

As he grew older, he became increasingly interested in literature and was an avid newspaper reader. His favorite writers were Jean Paul and Walter Scott . He spoke fluent 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), in which he finally learned Russian and tried Sanskrit , which but 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 the Royal Society of Edinburgh . In 1808 he was elected a corresponding member and in 1820 a foreign member of the Bavarian Academy of Sciences and in 1822 he was elected to the American Academy of Arts and Sciences . In 1838 he received the Copley Medal from the Royal Society. In 1842 he was accepted into the peace class of the order Pour le Mérite . In the same year he turned down an offer to Vienna . In 1845 he became Privy Councilor and in 1846 dean of the Philosophical Faculty for the third time . In 1849 he celebrated his golden doctoral jubilee and became an honorary citizen of Braunschweig and Göttingen. In 1852 he undertook his last scientific work, repeating Foucault's pendulum experiment to prove the earth's rotation.

He collected all kinds of numerical and statistical data and, for example, kept lists of the life expectancy of famous men (calculated in days). So he wrote on December 7, 1853 to his friend and Chancellor of his order Alexander von Humboldt a . a .: "The day after tomorrow when you, my dear friend, pass into an area into which none of the luminaries of the exact sciences has penetrated, the day when you will reach the same age as Newton's through 30,766 days measured earthly career has closed. And Newton's powers were utterly exhausted at this stage: to the greatest delight of the whole scientific world, you are still enjoying your admirable power to the full. May you continue to enjoy 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 dealt with stock speculation and so on his death left behind a considerable fortune of 170,000 thalers (with a basic professor's salary of 1000 thalers annually) mostly in securities, many of them from railways. There is one of the few places in the correspondence in which he expresses himself critical of politics and banks that cooperate with them, because the railway shares he had acquired in Hessen-Darmstadt lost drastically in value when it became known that the railway could be nationalized at any time.

He was still scientifically active towards the end of his life and gave lectures on the least squares method in 1850/51. Two of his most important students, Bernhard Riemann (who received his doctorate from Gauß in 1851 and made a big impression on Gauß in 1854 with his habilitation lecture on the fundamentals of Riemannian geometry) and Richard Dedekind , he only had towards the end of his career.

Gauss was very conservative and monarchist; he did not approve of the revolution of 1848 .

### death

Grave of Carl Friedrich Gauß in the historic Albani cemetery , adjacent to the Cheltenhampark in Göttingen

Gauss suffered from heart failure (diagnosed as dropsy ) and insomnia in his final years . In June 1854 he traveled with his daughter Therese to the construction site of the railway line from Hanover to Göttingen , where the passing railway made the horses shy and overturned the carriage, the coachman was seriously injured, Gauss and his daughter were uninjured. Gauss took part in the inauguration of the railway line on July 31, 1854, after which he was increasingly restricted to his house due to illness. He died on February 23, 1855 at 1:05 a.m. in his armchair in Göttingen. He was buried there in the Albani cemetery .

## Services

### Justification and contributions to non-Euclidean geometry

Lithograph by Gauss in the Astronomische Nachrichten, 1828 by Bendixen

Gauss already distrusted twelve years of evidence in elementary geometry and guessed at sixteen, that in addition to the Euclidean yet another, non-Euclidean geometry had to be.

He deepened this work in the 1820s: independently of János Bolyai and Nikolai Iwanowitsch Lobatschewski , he noticed that the Euclidean axiom of parallels is not necessary. However , he did not publish his thoughts on non-Euclidean geometry , according to the reports of his confidants, presumably out of fear that his contemporaries would not understand. When his college friend Wolfgang Bolyai , with whom he corresponded, told him about the work of his son János Bolyai, he praised him, but could not fail to mention that he himself had thought of it much earlier ("[the work To praise your son is to praise myself ”). He hadn't published anything about it because he “shy away from the screams of the Boiotians”. Gauss found Lobachevski's work so interesting that he learned Russian at an advanced age in order to study it.

### Prime number distribution and least squares method

When he was 18, he discovered some properties of the prime distribution and found the least squares method , which is about minimizing the sum of the squares of deviations. For the time being he refrained from publication. After Adrien-Marie Legendre had published his "Méthode des moindres carrés" in a treatise in 1805 and Gauss did not publish his results until 1809, a dispute over priority arose.

Using this method, the most likely 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 areas under curves (numerical integration), which led him to arrive at the Gaussian bell curve . The associated function is known as the density of the normal distribution and is used in many probability calculations , where it is the (asymptotic, i.e. valid for sufficiently large amounts of data) distribution function of data that randomly scatter around a mean value. Gauss himself made use of this, among other things, in his successful administration of the widows and orphans' fund at Göttingen University. He made a thorough analysis over several years, in which he concluded that pensions could be increased slightly. In doing so, Gauss also laid the foundations for actuarial mathematics .

### Introduction of the elliptical functions

As a 19-year-old, he introduced the lemniscate sine functions, the historically first so-called elliptic functions , in 1796 when considering the arc length on a lemniscate depending on the distance from the curve point to the origin . However, he never published his notes about it. This work is related to his investigation of the arithmetic-geometric mean. The actual development of the theory of elliptical functions, the inverse functions of the elliptical integrals, which have long been known, was carried out by Niels Henrik Abel (1827) and Carl Gustav Jacobi .

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

Gauss recognized the usefulness of complex numbers early , for example in his doctoral thesis from 1799, 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 inadequate, but even his own proof did not yet meet the later requirements 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 Jean-Robert Argand had found in 1806 and Caspar Wessel in 1797. In the letter to Bessel , in which he communicated this, it also became clear that he was familiar with other important concepts in function theory such as the curve integral in the complex and Cauchy's integral theorem and first approaches to periods of integrals. But he did not publish anything about it 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 anticipated him in the publication of the Justification of Complex Analysis . In 1849 he published an improved version of his dissertation on the fundamental theorem of algebra for his golden jubilee, in which, in contrast to the first version, he explicitly used complex numbers.

### Contributions to number theory

Announcement of the constructibility in the intelligence sheet of the Allgemeine Literatur-Zeitung (1796)
17-corner star on the Braunschweig Gaußdenkmal

On March 30, 1796, a month before his nineteenth birthday, he demonstrated the constructability of the regular seventeenth-corner and thus provided the first noteworthy addition to Euclidean constructions in 2000 years. But this was only a side result of the work for his work Disquisitiones Arithmeticae, which was much more far-reaching in number theory .

The first announcement of this work was found on June 1, 1796 in the intelligence paper of the Allgemeine Literatur-Zeitung in Jena. 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 in the course of his life . In addition to building elementary number theory on modular arithmetic, there is a discussion of continued fractions and the division of circles, with a famous suggestion of similar theorems in the lemniscate and other elliptical functions that Niels Henrik Abel and others later suggested. The theory of quadratic forms, whose gender theory he developed, takes up a large part of the work.

There are, however, many other deep-seated results, often only briefly indicated, in this book, which stimulated the work of later generations of number theorists in many ways. The number theorist Peter Gustav Lejeune Dirichlet reported that he had always had the Disquisitiones to hand at work throughout his life. The same applies to the two works on biquadratic reciprocity laws of 1825 and 1831, in which he introduces the Gaussian numbers (integer lattice in a complex number plane). The works are probably part of a planned sequel to the Disquisitiones , which never appeared. Gotthold Eisenstein then gave proof of these laws in 1844.

André Weil suggested reading this work (and some passages in the diary, where it is hidden in the solution of equations for finite bodies) according to his own information about his work on the Weil conjectures. Gauss knew the prime number theorem , but did not publish it.

Gauss promoted one of the first modern mathematicians in this area, Sophie Germain . Gauss corresponded with her from 1804 on number theory, initially using a male pseudonym. It was not until 1806 that she revealed her female identity when she used Braunschweig's French commander for his security after the occupation. 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 money he received with the Lalande Prize .

### Contributions to astronomy

After completing the Disquisitiones , Gauss turned to astronomy . The reason 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 determining the orbit and his compensation calculations based on the method of least squares in such a way that Franz Xaver von Zach confirmed it on December 7, 1801 and - confirmed - on December 31. December 1801 could find again. Heinrich Wilhelm Olbers confirmed this independently of Zach through observation on January 1 and 2, 1802.

The problem of finding the Ceres as such lay in the fact that neither the location, part of the path, nor the distance are known from the observations , only the directions of the observation. This leads to the search for an ellipse and not for a circle as suggested by Gauss's competitors. One of the focal points of the ellipse is known (the sun itself), and the arcs of the orbit of the Ceres between the directions of observation are traversed according to Kepler's second law , that is, the times behave like the surfaces swept by the guide beam. In addition, it is known for the computational solution that the observations themselves assume a conic section in space, the earth's orbit itself.

In principle, the problem leads to an equation of the eighth degree, the trivial solution of which is the earth's orbit itself. Thanks to extensive constraints and the method of least squares developed by Gauß, the 24-year-old managed to specify the location he had calculated for the orbit of the Ceres for November 25 to December 31, 1801. This enabled Zach to find Ceres on the last day of the forecast. The place was no less than 7 degrees (i.e. 13.5  full moon widths ) east of where the other astronomers had suspected Ceres, which not only Zach but also Olbers duly paid tribute to.

This work, which Gauß undertook before his appointment as director of the observatory in Göttingen, made him known in Europe even more than his number theory at one stroke and, among other things, earned him an invitation to the Academy in Saint Petersburg , of which he became a member in 1802.

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

Gauss then dealt with the orbit of the asteroid Pallas , on the calculation of which the Paris Academy had offered prize money, but could not find the solution. However, his experiences with determining the orbit of celestial bodies culminated in his work Theoria motus corporum coelestium in sectionibus conicis solem ambientium in 1809 .

### Contributions to potential theory

In potential theory and physics, the Gaussian integral theorem (1835, published only 1867) is fundamental. In a vector field, it identifies 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.

### Land surveying and invention of the heliotrope

Back of the 10 DM note - at the bottom right a sketch of the triangulation of northern Germany by Gauss

In the field of geodesy , Gauss gained his first experience between 1797 and 1801 when he assisted the French Quartermaster General Lecoq as an advisor for his 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 measurement in Danish territory. Subsequently, from 1820 to 1826 Gauss was in charge of the land surveying of the Kingdom of Hanover (" Gaußsche Landesaufnahme "), where he was temporarily assisted by his son Joseph , who was an artillery officer in the Hanoverian army . This survey continued the Danish one on the Hanoverian area to the south, whereby Gauss also used the Braaker base measured by Schumacher . With the method of least squares 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 the practical implementation: He invented the heliotrope illuminated by sun mirrors as a measuring instrument .

### Gaussian curvature and geodesy

The Gaussian point in Bremen

During these years he dealt - inspired by geodesy and map theory - with the theory of the differential geometry of surfaces, introduced, among other things, Gaussian curvature and proved his theorema egregium . This means that the Gaussian curvature, which is defined by the main curvatures of a surface in space, is solely determined by the dimensions of the internal geometry , i.e. H. can be determined by measurements within the area. Therefore, the Gaussian curvature is independent of the embedding of the surface in the three-dimensional space, so it does not change when surfaces are mapped onto one another with true- length .

Memorial plaque on the Brocken

Wolfgang Sartorius von Waltershausen reports that on the occasion of the Hanover land survey, Gauss empirically looked for a deviation of the sum of the angles of particularly large triangles from the Euclidean value 180 ° - such as the flat triangle measured by Gauss, that of the Brocken in the Harz , the Inselsberg in the Thuringian Forest and the Hohen Hagen near Dransfeld . Max Jammer wrote about this Gaussian measurement and its result:

“He measured [...] a triangle formed by three mountains, the Brocken, the Hohen Hagen and the Inselberg, the sides of which measured 69, 85 and 107 km. It hardly needs to be specifically stated that he did not discover a deviation of 180 ° within the error limit and drew the conclusion from this that the structure of real space is, as far as experience allows a statement, Euclidean. "

The excess angle in this triangle is only 0.25 arc minutes due to the size of the earth. The above-mentioned guess on motivation is the subject of speculation.

### Magnetism, electricity and telegraphy

From 1831 he worked together with Wilhelm Eduard Weber in the field of magnetism . Gauss invented the magnetometer with Weber and thus connected his observatory with the physical institute in 1833 . He exchanged messages with Weber via electromagnetically influenced compass needles: the first telegraph connection in the world. With him he developed the CGS system of units , which was determined in 1881 at an international congress in Paris as the basis for electrotechnical units of measurement. He organized a worldwide network of observation stations ( Magnetischer Verein ) to measure the terrestrial magnetic field.

In his experiments on electricity in 1833, Gauss found Kirchhoff's rules for electrical circuits before Gustav Robert Kirchhoff (1845) .

### Others

He developed the Gaussian Easter formula for calculating the Easter date , and he also developed a Passover formula .

## Working method of Gauss

Gaussian seal Pauca sed Matura ("Little, but mature")

Gauss worked in many fields, but did not publish his results until he thought a theory was complete. This led to the fact that he occasionally pointed out to colleagues that he had proven this or that result for a long time, just not having presented it because of the incompleteness of the underlying theory or the lack of carelessness necessary to work quickly.

Significantly, Gauss had a seal showing a tree hung with a few fruits and bearing the motto Pauca sed Matura (“ Little but ripe”). According to an anecdote, he refused to replace this motto with acquaintances who knew about Gauss' extensive work, e.g. B. by Multa nec immatura (“Much, but not immature”), since, according to his testimony, he preferred to leave a discovery to someone else than to publish it under his name in a fully elaborated manner. This saved him time in areas that Gauss viewed as marginal issues, so that he could use this time on his original work.

Gauß 'scientific papers are kept in the special collections of the Lower Saxony State and University Library in Göttingen.

## Others

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

In autumn 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 specimens were kept in jars with formaldehyde in the anatomical collection of the Göttingen University Clinic. The original Gauss brain was in the jar labeled “C. H. Fuchs ”, and the Fuchs brain was labeled“ C. F. Gauss ”. This means that the previous research results on Gauss's brain are obsolete. The scientist Renate Schweizer looked again at the specimens because of the MRI images made of the supposed brain of Gauss , which showed a rare division of the central groove in two , and discovered that this abnormality was missing in drawings made shortly after Gauss's death .

## Gauss as the namesake

Portrait on a surveying stone on Wilseder Berg

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

Carl Friedrich Gauß, the Gaussian normal distribution and the Göttingen observatory on the 10 DM note. Portrait image reversed
• the Gaussian normal equations , a quadratic system of equations, the solution of which is the least squares solution
• the Gaussian normal distribution , also known as the Gaussian bell curve, or Gaussian distribution (the bell curve, placed next to the portrait of Carl Friedrich Gauß, was the last 10 DM banknote of the Federal Republic of Germany from 1989 to 2001 )
• the Gaussian numbers , an extension of the whole numbers to the complex numbers
• the Gaussian plane of numbers as a geometric interpretation of the set of complex numbers
• the Gaussian bracket , a function that rounds numbers down to the nearest whole number
• the Gaussian process , a stochastic process whose finite-dimensional distributions are normal distributions
• the lemma of Gauss , a step in one of its proofs of quadratic reciprocity law
• the Gaussian empirical formula , also known as the "little Gaussian", a formula for the sum of the first natural numbers
• the Gaussian sum , a certain type of finite sum of roots of unity
• the Gauss-Markov theorem on the existence of a best linear unbiased estimator within the class of linear unbiased estimators

Methods and ideas that are partly based on his work are:

Survey ship Gauss

Named in his honor are:

## Fonts

### Complete edition

Volumes 10 and 11 contain detailed comments 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 .

### Translations

• Research generales on the surfaces courbes. Bachelier, Paris 1852 (French translation of Disquisitiones generales circa superficies curvas. 1828; in Gallica ).
• Méthode des moindres carrés. Mallet-Bachelier, Paris 1855 (French translation of Theoria combinationis observationum erroribus minimis obnoxiae. 1823/1828, and others by Joseph Bertrand ; at Google Books, ditto ).
• Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. Little, Brown and Company, Boston 1857 (English translation of Theoria motus corporum coelestium in sectionibus conicis solem ambientium. 1809, by Charles Henry Davis; on Google Books, ditto; in the Internet archive , ditto, ditto ).
• Carl Haase (Ed.): Theory of the movement of the heavenly bodies which revolve around the sun in conic sections. Carl Meyer, Hannover 1865 (German translation of Theoria motus corporum coelestium in sectionibus conicis solem ambientium. 1809, by Carl Haase; on Google Books ); Facsimile reprint Verlag Kessel, 2009, ISBN 978-3-941300-13-2 .
• Anton Börsch, Paul Simon (Ed.): Treatises on the least squares method by Carl Friedrich Gauss. P. Stankiewicz, Berlin 1887 (German translation of Theoria combinationis observationum erroribus minimis obnoxiae. 1823/1828, and others; in the internet archive ).
• Heinrich Simon (ed.): General investigations into the infinite series, etc.${\ displaystyle \ textstyle {1 + {\ frac {\ alpha \ beta} {1. \ gamma}} x + {\ frac {\ alpha (\ alpha +1) \ beta (\ beta +1)} {1 \. \ 2 \. \ \ Gamma (\ gamma +1)}} xx + {\ frac {\ alpha (\ alpha +1) (\ alpha +2) \ beta (\ beta +1) (\ beta +2)} { 1 \. \ 2 \. \ 3 \. \ \ Gamma (\ gamma +1) (\ gamma +2)}} x ^ {3} +}}$ Julius Springer, Berlin 1888 (German translation of Disquisitiones generales circa seriem infinitam 1 +… 1813, by Heinrich Simon; in the Internet archive ).
• Hermann Maser (Ed.): Carl Friedrich Gauss' investigations into higher arithmetic . Julius Springer, Berlin 1889 (German translation of Disquisitiones Arithmeticae. 1801, and others; in the Internet archive ); Facsimile reprint Verlag Kessel, 2009, ISBN 978-3-941300-09-5 .
• Albert Wangerin (ed.): General area theory (Disquisitiones generales circa superficies curvas). Wilhelm Engelmann, Leipzig 1889 (German translation; at the University of Michigan; in the Internet archive, ditto ).
• Eugen Netto (Ed.): The four Gaussian proofs for the decomposition of whole algebraic functions into real factors of the first or second degree (1799–1849). Wilhelm Engelmann, Leipzig 1890 (German translation of the doctoral thesis, 1799, and other works; at the University of Michigan; in the Internet archive, ditto, ditto ).
• Eugen Netto (Ed.): Six proofs of the fundamental theorem about quadratic residues by Carl Friedrich Gauss. Wilhelm Engelmann, Leipzig 1901 (German translation from Disquisitiones Arithmeticae. 1801, and others with annotations; at the University of Michigan; in the Internet archive, ditto, ditto, ditto ).
• General investigations of curved surfaces of 1827 and 1825. The Princeton University Library, 1902 (English translation of Disquisitiones generales circa superficies curvas. 1828, and New general investigations of curved surfaces. 1900, by James Caddall Morehead and Adam Miller Hiltebeitel; at the University of Michigan; in the Internet archive, ditto ).
• Heinrich Weber (Ed.): General principles of a theory of the shape of liquids in the state of equilibrium. Wilhelm Engelmann, Leipzig 1903 (German translation of Principia generalia theoriae figurae fluidorum in statu aequilibrii. 1830, by Rudolf Heinrich Weber ; in the Internet archive, ditto ).

## Monuments

Memorial in Braunschweig
Göttingen memorial plaque (Kurz-Geismar-Straße 25)
Postage stamp (1955) for the 100th anniversary of death
Postage stamp (1977) for the 200th birthday

### Written culture of remembrance

• On the front of the 10 DM banknote of the fourth series of the Deutsche Mark there is an image of Gauss, together with a representation of the bell curve and important buildings of Göttingen. Two commemorative coins also commemorate him, which were issued in 1977 on the occasion of his 200th birthday in the Federal Republic of Germany (5 DM) and in the GDR (20 M).
• In Germany three postage stamps commemorate Gauss: In 1955 the Deutsche Bundespost issued a 10 pfennig stamp on the occasion of the 100th anniversary of his death; In 1977 the GDR commemorated the 200th birthday with a 20 pfennig postage stamp, as did the Deutsche Bundespost with a 40 pfennig postage stamp.
• Memorial plaque at the location of the birthplace at Wilhelmstrasse 30 in Braunschweig.
• Three Göttingen memorial plaques .
• Two memorial plaques on the former home of Gauß 'doctoral supervisor Johann Friedrich Pfaff in Helmstedt .
• 1977 on a 5 DM commemorative coin for the 200th birthday of Carl Friedrich Gauss, edition 8,000,000 pieces, 250,000 pieces in mirror finish April 1977 issue

### Gauss stones

The numerous surveying stones set up on the instructions of Gauss include:

### Portraits

There are relatively many portraits of Gauss, including:

## literature

Commons : Carl Friedrich Gauß  - Album with pictures, videos and audio files
Wikisource: Carl Friedrich Gauß  - Sources and full texts
Wikisource: Johann Carl Friedrich Gauß  - Sources and full texts (Latin)

## Individual evidence

1. Sartorius von Waltershausen : Gauss to the memory.
2. ^ Sartorius von Waltershausen: Gauss to the memory. 1856, p. 12.
3. ^ Brian Hayes: Gauss's Day of Reckoning. In: American Scientist. 94, 2006, p. 200, doi: 10.1511 / 2006.3.200 .
4. Horst Michling: Carl Friedrich Gauß. 2nd ed. Göttingen, 1982, pp. 67-68.
5. Printed for example in W. K. Bühler, Gauß, Springer, p. 186, Kurt-R. Biermann: Gauß, 1990, p. 79
6. Gausschildren.org (accessed July 22, 2011)
7. Wyneken Family Tree (accessed July 22, 2011)
8. ^ List of members since 1666: Letter G. Académie des sciences, accessed on November 17, 2019 (French).
9. ^ Entry on Gauss, Karl Friedrich (1777 - 1855) in the archive of the Royal Society , London
10. ^ Fellows Directory. Biographical Index: Former RSE Fellows 1783–2002. Royal Society of Edinburgh, accessed December 7, 2019 .
11. Member entry by Prof. Dr. Carl Friedrich Gauß at the Bavarian Academy of Sciences , accessed on February 7, 2016.
12. ^ Letter No. 45 to Alexander von Humboldt dated December 7, 1853
13. Wussing, Gauß, 1989, p. 81
14. ^ WK Bühler, Gauß, p. 151
15. ^ Letter to Wolfgang von Bolyai dated March 6, 1832, excerpt from Gauß: Werke. Volume 8. P. 220–224, completely in Schmidt, Stäckel (Ed.): Correspondence between Carl Friedrich Gauss and Wolfgang Bolyai. 1899, pp. 108-113 ( at the University of Michigan ; on the Internet archive ).
16. ^ Letter to Friedrich Wilhelm Bessel dated January 27, 1829, excerpt from Gauß: Werke. Volume 8. P. 200, completely in Auwers (Ed.): Correspondence between Gauss and Bessel. 1880, pp. 487-490 ( in the Internet archive ). " Boeotier " is proverbial for "rural, rough, uneducated people".
17. ^ Letter to Bessel dated December 18, 1811, Gauß, Werke, Volume 8, pp. 155–160 (online).
18. ^ Jean-Luc Verley: Analytical functions. In: History of Mathematics 1700–1900. Vieweg, 1985, p. 145.
19. Magnus Georg Paucker : Geometric distortion of the regular seventeen-corner and two hundred and fifty-seven corner in the circle. Annual negotiations of the Courland Society for Literature and Art, Volume 2, 1822, pp. 160–219, specifically p. 219 (online). Retrieved October 7, 2014.
20. ^ W. Sartorius von Waltershausen : Gauss to the memory. Verlag von S. Hirzel, Leipzig, 1856. p. 16 (digitized online). Retrieved October 8, 2014.
21. It can be found in a letter to Johann Franz Encke dated December 24, 1849, printed in: Gauß: Werke. Volume 2. pp. 444–447 ( online in the Google book search).
22. ^ Moritz CantorGauß: Karl Friedrich G. In: General German Biography (ADB). Volume 8, Duncker & Humblot, Leipzig 1878, pp. 430–445., Here p. 436.
23. Paul Karlson: Magic of Numbers. Ullstein-Verlag, Berlin-West. Ninth, revised and expanded edition, 1967, p. 390 f.
24. ^ Foreign members of the Russian Academy of Sciences since 1724. Carl Friedrich Gauss. Russian Academy of Sciences, accessed August 15, 2015 (Russian).
25. Dieter Lelgemann : Gauß and the art of measuring. PRIMUS Verlag GmbH, Darmstadt, 2011, pp. 72-73.
26. ^
27. Max Jammer: The problem of space. Darmstadt 1960, p. 164.
28. Erhard Scholz thinks it is quite possible that Gauß thought of it (see arxiv : math.HO / 0409578 ), although Gauß himself stated in a letter to Olbers of March 1, 1827, quoted in Bühler p. 97, that the measurement errors are too large for such a determination of deviations.
29. ^ Dunnington: Gauss - Titan of Science. American Mathematical Society, p. 161.
30. ^ Wolfgang Hänicke, Jens Frahm and Axel D. Wittmann: Magnetic resonance tomography of the brain by Carl Friedrich Gauß. In: MPI News 5. Issue 12 (1999). Online version, Internet archive. ( Memento from July 19, 2011 in the Internet Archive ).
31. From HNA.de of October 28, 2013: Unexpected discovery: Wrong brain in a glass
32. ^ Hannoversche Allgemeine Zeitung, October 29, 2013.
33. Lotte Burkhardt: Directory of eponymous plant names . Extended Edition. Botanic Garden and Botanical Museum Berlin, Free University Berlin Berlin 2018. bgbm.org
34. ^ Archives of the Gauss lectures at the German Mathematicians Association.
35. kalliope-verbund.info Available in digital form
36. ^ Gauß-Weber monument on the part of the city of Göttingen
37. Gauss bust erected in the Walhalla. (PDF; 297 kB) Press release from the City of Göttingen from September 12, 2007.
38. ^ Hermann Müller-Bohn : The monuments of Berlin in words and pictures. Publishing house by IM Spaeth, Berlin.
39. A. Wietzke: The re-found image of youth by Carl Friedrich Gauß . In: Annual report of the German Mathematicians Association. 41 (1932), pp. 1-2.
40. In addition to Gauß, whose findings on the earth's magnetic field are presented, four other scientists who made discoveries in geoscience : Pierre Simon de Laplace , who deciphered the formation of the earth, Léon-Philippe Teisserenc de Bort and Auguste Piccard , explorer of the stratosphere and Emil Wiechert , Inventor of the seismograph.
 This version was added to the list of articles worth reading on July 21, 2005 .