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Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article.

# Biography

## Early years

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of
Lower Saxony Lower Saxony (german: Niedersachsen ; nds, Neddersassen; stq, Läichsaksen) is a German state (') in northwestern Germany. It is the second-largest state by land area, with , and fourth-largest in population (8 million in 2021) among the 16 ...
, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years. He was christened and confirmed in a church near the school he attended as a child. Gauss was a
child prodigy A child prodigy is defined in psychology research literature as a person under the age of ten who produces meaningful output in some domain at the level of an adult expert. The term is also applied more broadly to young people who are extraor ...
. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen wrote that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, solved an arithmetic series problem faster than anyone else in his class of 100 pupils. There are many versions of this story, with various details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100. (See also under "Anecdotes" below.) There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his
magnum opus A masterpiece, ''magnum opus'' (), or ''chef-d’œuvre'' (; ; ) in modern use is a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or a work of outstanding creativity, ...
, '' Disquisitiones Arithmeticae'', in 1798, at the age of 21, and it was published in 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the
Duke of Brunswick Duke is a male title either of a monarch ruling over a duchy, or of a member of royalty, or nobility. As rulers, dukes are ranked below emperors, kings, grand princes, grand dukes, and sovereign princes. As royalty or nobility, they are rank ...
, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the
Ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cul ...
, and the discovery ultimately led Gauss to choose mathematics instead of
philology Philology () is the study of language in oral and written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defined as ...
as a career. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard ...
law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifyin ...
, conjectured on 31 May, gives a good understanding of how the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: " ΕΥΡΗΚΑ! ". On 1 October he published a result on the number of solutions of polynomials with coefficients in
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s, which 150 years later led to the
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. ...
.

## Later years and death

In 1807, Gauss became professor of mathematics at the university of Göttingen. Gauss remained mentally active into his old age, even while having
gout Gout ( ) is a form of inflammatory arthritis characterized by recurrent attacks of a red, tender, hot and swollen joint, caused by deposition of monosodium urate monohydrate crystals. Pain typically comes on rapidly, reaching maximal intensi ...
and suffering general unhappiness. For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential ''Dioptrische Untersuchungen'', in which he gave the first systematic analysis on the formation of images under a paraxial approximation (
Gaussian optics Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered ...
). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its
cardinal points The four cardinal directions, or cardinal points, are the four main compass directions: north, east, south, and west, commonly denoted by their initials N, E, S, and W respectively. Relative to north, the directions east, south, and west are a ...
and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands; when that became the
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ...
in 1851, he joined as a foreign member. In 1854, Gauss selected the topic for
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...
's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (''About the hypotheses that underlie Geometry''). On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement. He was elected as a member of the
American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
in 1853. On 23 February 1855, Gauss died of a
heart attack A myocardial infarction (MI), commonly known as a heart attack, occurs when blood flow decreases or stops to the coronary artery of the heart, causing damage to the heart muscle. The most common symptom is chest pain or discomfort which may ...
in Göttingen (then
Kingdom of Hanover The Kingdom of Hanover (german: Königreich Hannover) was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era. It succeeded the former Electorate of Han ...
and now
Lower Saxony Lower Saxony (german: Niedersachsen ; nds, Neddersassen; stq, Läichsaksen) is a German state (') in northwestern Germany. It is the second-largest state by land area, with , and fourth-largest in population (8 million in 2021) among the 16 ...
); he is interred in the Albani Cemetery there. Two people gave eulogies at his funeral: Gauss's son-in-law
Heinrich Ewald Georg Heinrich August Ewald (16 November 18034 May 1875) was a German orientalist, Protestant theologian, and Biblical exegete. He studied at the University of Göttingen. In 1827 he became extraordinary professor there, in 1831 ordinary profess ...
, and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be slightly above average, at , and the cerebral area equal to . Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius.

## Religious views

Gauss was nominally a member of the St. Albans Evangelical Lutheran church in Göttingen. One of his biographers, G. Waldo Dunnington, has described Gauss's religious views as follows: Apart from his correspondence, there are not many known details about Gauss's personal creed. Many biographers of Gauss disagree about his religious stance, with Bühler and others considering him a
deist Deism ( or ; derived from the Latin '' deus'', meaning "god") is the philosophical position and rationalistic theology that generally rejects revelation as a source of divine knowledge, and asserts that empirical reason and observation o ...
with very unorthodox views, while Dunnington (admitting that Gauss did not believe literally in all Christian dogmas and that it is unknown what he believed on most doctrinal and confessional questions) points out that he was, at least, a nominal
Lutheran Lutheranism is one of the largest branches of Protestantism, identifying primarily with the theology of Martin Luther, the 16th-century German monk and Protestant Reformers, reformer whose efforts to reform the theology and practice of the Cathol ...
. In connection to this, there is a record of a conversation between Rudolf Wagner and Gauss, in which they discussed
William Whewell William Whewell ( ; 24 May 17946 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge. In his time as a student there, he achieve ...
's book ''Of the Plurality of Worlds''. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe. This later led them to discuss the topic of
faith Faith, derived from Latin ''fides'' and Old French ''feid'', is confidence or trust in a person, thing, or In the context of religion, one can define faith as "belief in God or in the doctrines or teachings of religion". Religious people ofte ...
, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Lutheran minister
Paul Gerhardt Paul Gerhardt (12 March 1607 – 27 May 1676) was a German theologian, Lutheran minister and hymnodist. Biography Gerhardt was born into a middle-class family at Gräfenhainichen, a small town between Halle and Wittenberg. His father died in ...
than by
Moses Moses hbo, מֹשֶׁה, Mōše; also known as Moshe or Moshe Rabbeinu ( Mishnaic Hebrew: מֹשֶׁה רַבֵּינוּ, ); syr, ܡܘܫܐ, Mūše; ar, موسى, Mūsā; grc, Mωϋσῆς, Mōÿsēs () is considered the most important pr ...
. Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the
New Testament The New Testament grc, Ἡ Καινὴ Διαθήκη, transl. ; la, Novum Testamentum. (NT) is the second division of the Christian biblical canon. It discusses the teachings and person of Jesus, as well as events in first-century Christ ...
. Two religious works which Gauss read frequently were Braubach's ''Seelenlehre'' (Giessen, 1843) and Süssmilch's ''Gottliche'' (Ordnung gerettet, 1756); he also devoted considerable time to the New Testament in the original Greek. Dunnington further elaborates on Gauss's religious views by writing: Gauss believed in an omniscient source of creation however he claimed that belief or a lack of it did not affect his mathematics. Though he was not a church-goer, Gauss strongly upheld religious tolerance, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble." When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved of this, saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.

## Family

On 9 October 1805, Gauss married Johanna Osthoff (1780–1809), and had two sons and a daughter with her. Johanna died on 11 October 1809, and her youngest child, Louis, died the following year. Gauss plunged into a depression from which he never fully recovered. He then married Minna Waldeck (1788–1831) on 4 August 1810, and had three more children. Gauss was never quite the same without his first wife, and just like his father, grew to dominate his children. Minna Waldeck died on 12 September 1831. Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene shared a good measure of Gauss's talent in languages and computation. After his second wife's death in 1831 Therese took over the household and cared for Gauss for the rest of his life. His mother lived in his house from 1817 until her death in 1839. Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements. Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, for which Gauss refused to pay. The son left in anger and, in about 1832, emigrated to the United States. While working for the
American Fur Company The American Fur Company (AFC) was founded in 1808, by John Jacob Astor, a German immigrant to the United States. During the 18th century, furs had become a major commodity in Europe, and North America became a major supplier. Several British co ...
in the Midwest, he learned the Sioux language. Later, he moved to
Missouri Missouri is a state in the Midwestern region of the United States. Ranking 21st in land area, it is bordered by eight states (tied for the most with Tennessee): Iowa to the north, Illinois, Kentucky and Tennessee to the east, Arkansas to ...
and became a successful businessman. Wilhelm also moved to America in 1837 and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in
St. Louis St. Louis () is the second-largest city in Missouri, United States. It sits near the confluence of the Mississippi and the Missouri Rivers. In 2020, the city proper had a population of 301,578, while the bi-state metropolitan area, which ...
. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues. See also the letter from Robert Gauss to Felix Klein on 3 September 1912.

## Personality

Gauss was an ardent perfectionist and a hard worker. He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism. This was in keeping with his personal motto ''pauca sed matura'' ("few, but ripe"). His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Scottish-American mathematician and writer Eric Temple Bell said that if Gauss had published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years. Though he did take in a few students, Gauss was known to dislike teaching. It is said that he attended only a single scientific conference, which was in
Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
in 1828. Several of his students became influential mathematicians, among them
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. Hi ...
and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...
. On Gauss's recommendation, Friedrich Bessel was awarded an honorary doctoral degree from Göttingen in March 1811. Around that time, the two men engaged in a correspondence. However, when they met in person in 1825, they quarrelled; the details are unknown. Before she died,
Sophie Germain Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's lib ...
was recommended by Gauss to receive an honorary degree; she never received it. Gauss usually declined to present the intuition behind his often very elegant proofs—he preferred them to appear "out of thin air" and erased all traces of how he discovered them. This is justified, if unsatisfactorily, by Gauss in his '' Disquisitiones Arithmeticae'', where he states that all analysis (in other words, the paths one traveled to reach the solution of a problem) must be suppressed for sake of brevity. Gauss supported the monarchy and opposed
Napoleon Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader wh ...
, whom he saw as an outgrowth of revolution. Gauss summarized his views on the pursuit of knowledge in a letter to Farkas Bolyai dated 2 September 1808 as follows:

# Career and achievements

## Algebra

In his 1799 doctorate in absentia, ''A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree'', Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
with complex coefficients has at least one complex
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
. Mathematicians including
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopé ...
had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to the implicit use of the Jordan curve theorem. However, 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. Gauss also made important contributions to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
with his 1801 book '' Disquisitiones Arithmeticae'' (
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of t ...
, Arithmetical Investigations), which, among other things, introduced the
triple bar The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign  sign with a third line. The triple bar character in Unicode is code point .. The closely related code point ...
symbol for congruence and used it in a clean presentation of
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
, contained the first two proofs of the law of
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard ...
, developed the theories of binary and ternary
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. It appears that Gauss already knew the class number formula in 1801. In addition, he proved the following conjectured theorems: * Fermat polygonal number theorem for ''n'' = 3 *
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
for ''n'' = 5 * Descartes's rule of signs * Kepler conjecture for regular arrangements He also * explained the
pentagramma mirificum Pentagramma mirificum (Latin for ''miraculous pentagram'') is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book '' Mirifici ...
(se
University of Bielefeld website
* developed an algorithm for determining the
date of Easter As a moveable feast, the date of Easter is determined in each year through a calculation known as (). Easter is celebrated on the first Sunday after the Paschal full moon, which is the first full moon on or after 21 March (a fixed approx ...
* invented the
Cooley–Tukey FFT algorithm The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N_1N_2 in terms of ''N''1 ...
for calculating the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex ...
s 160 years before Cooley and Tukey

## Astronomy

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered the
dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to ...
Ceres. Piazzi could track Ceres for only somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit. Gauss heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by
Franz Xaver von Zach Baron Franz Xaver von Zach (''Franz Xaver Freiherr von Zach''; 4 June 1754 – 2 September 1832) was a Hungarian astronomer born at Pest, Hungary (now Budapest in Hungary). Biography Zach studied physics at the Royal University of Pest, and ...
on 31 December at
Gotha Gotha () is the fifth-largest city in Thuringia, Germany, west of Erfurt and east of Eisenach with a population of 44,000. The city is the capital of the district of Gotha and was also a residence of the Ernestine Wettins from 1640 until the ...
, and one day later by Heinrich Olbers in
Bremen Bremen (Low German also: ''Breem'' or ''Bräm''), officially the City Municipality of Bremen (german: Stadtgemeinde Bremen, ), is the capital of the German state Free Hanseatic City of Bremen (''Freie Hansestadt Bremen''), a two-city-state cons ...
. This confirmation eventually led to the classification of Ceres as minor-planet designation 1 Ceres: the first
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
(now dwarf planet) ever discovered. Gauss's method involved determining a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by
Kepler's Second Law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbit ...
). This problem 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. One such method was the
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the ...
. While this method is attributed to a 1965 paper by James Cooley and John Tukey, Gauss developed it as a trigonometric interpolation method. His paper, ''Theoria Interpolationis Methodo Nova Tractata'', was published only posthumously in Volume 3 of his collected works. This paper predates the first presentation by
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harm ...
on the subject in 1807. Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life. The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as ''Theoria motus corporum coelestium in sectionibus conicis solem ambientum'' (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of
measurement error Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics, an error is not necessarily a "mistak ...
. Gauss proved the method under the assumption of normally distributed errors (see
Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in th ...
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
). The method had been described earlier by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
in 1805, but Gauss claimed 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."

## Geodetic survey

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodetic survey of the
Kingdom of Hanover The Kingdom of Hanover (german: Königreich Hannover) was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era. It succeeded the former Electorate of Han ...
(), linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions. In 1828, when studying differences in
latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
, Gauss first defined a physical approximation for the
figure of the Earth Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxim ...
as the surface everywhere perpendicular to the direction of gravity (of which
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the ''arithm ...
makes up a part), later called the ''
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extend ...
''.

## Non-Euclidean geometries

Gauss also claimed to have discovered the possibility of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
but never published it. This discovery was a major
paradigm shift A paradigm shift, a concept brought into the common lexicon by the American physicist and philosopher Thomas Kuhn, is a fundamental change in the basic concepts and experimental practices of a scientific discipline. Even though Kuhn restricted t ...
in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things,
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." This unproved statement put a strain on his relationship with Bolyai who thought that Gauss was stealing his idea. Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in ''Gauss, Titan of Science'' (1955) that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.

## Theorema Egregium

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade, fueled Gauss's interest in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multil ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, fields of mathematics dealing with curves and
surfaces A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics * Surface (mathematics), a generalization of a plane which needs not be flat * Su ...
. Among other things, he came up with the notion of
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...
. This led in 1828 to an important theorem, the
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
(''remarkable theorem''), establishing an important property of the notion of
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canoni ...
. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space. In 1821, he was made a foreign member of the
Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for prom ...
. Gauss was elected a Foreign Honorary Member of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
in 1822.

## Magnetism

In 1831, Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in
magnetism Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
(including finding a representation for the unit of magnetism in terms of mass, charge, and time) and the discovery of
Kirchhoff's circuit laws Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
in electricity. It was during this time that he formulated his namesake law. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic
observatory An observatory is a location used for observing terrestrial, marine, or celestial events. Astronomy, climatology/meteorology, geophysical, oceanography and volcanology are examples of disciplines for which observatories have been constructed. His ...
to be built in the garden of the observatory, and with Weber founded the "Magnetischer Verein" (''magnetic association''), which supported measurements of Earth's magnetic field in many regions of the world. He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer ( magnetospheric) sources of Earth's magnetic field.

# Appraisal

The British mathematician
Henry John Stephen Smith Prof Henry John Stephen Smith FRS FRSE FRAS LLD (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass formu ...
(1826–1883) gave the following appraisal of Gauss:

# Anecdotes

There are several stories of his early genius. One story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: add a list of integers in
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common diffe ...
; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels. Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. However, the details of the story are at best uncertain (see for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions), and some authors, such as Joseph J. Rotman in his book ''A First Course in Abstract Algebra'' (2005), question whether it ever happened. He referred to mathematics as "the queen of sciences" and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.

# Commemorations

From 1989 through 2001, Gauss's portrait, a normal distribution curve and some prominent
Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The o ...
buildings were featured on the German ten-mark banknote. The reverse featured the approach for
Hanover Hanover (; german: Hannover ; nds, Hannober) is the capital and largest city of the German state of Lower Saxony. Its 535,932 (2021) inhabitants make it the 13th-largest city in Germany as well as the fourth-largest city in Northern Germany ...
. Germany has also issued three postage stamps honoring Gauss. One (no. 725) appeared in 1955 on the hundredth anniversary of his death; two others, nos. 1246 and 1811, in 1977, the 200th anniversary of his birth. Daniel Kehlmann's 2005 novel ''Die Vermessung der Welt'', translated into English as '' Measuring the World'' (2006), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer
Alexander von Humboldt Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 17696 May 1859) was a German polymath, geographer, naturalist, explorer, and proponent of Romantic philosophy and science. He was the younger brother of the Prussian minister, ...
. A film version directed by Detlev Buck was released in 2012. In 2007 a bust of Gauss was placed in the
Walhalla temple The Walhalla is a hall of fame that honours laudable and distinguished people in German history – "politicians, sovereigns, scientists and artists of the German tongue";Official Guide booklet, 2002, p. 3 Built decades before the foundation of ...
. The numerous things named in honor of Gauss include: * the normal distribution, also known as the Gaussian distribution, the most common bell curve in statistics; * the Gauss Prize, one of the highest honors in mathematics; *
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs un ...
, the most common of the several electromagnetic unit systems based on CGS units. *
gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, CGS unit for
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. In 1929 the Polish mathematician Marian Rejewski, who helped to solve the German Enigma cipher machine in December 1932, began studying actuarial statistics at
Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The o ...
. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss's grave. On 30 April 2018,
Google Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. I ...
honored Gauss on his would-be 241st birthday with a
Google Doodle A Google Doodle is a special, temporary alteration of the Google logo, logo on Google's home page, homepages intended to commemorate holidays, events, achievements, and notable historical figures. The first Google Doodle honored the 1998 edition ...
showcased in Europe, Russia, Israel, Japan, Taiwan, parts of Southern and Central America and the United States. Carl Friedrich Gauss, who also introduced the so-called Gaussian logarithms, sometimes gets confused with (1829–1915), a German geologist, who also published some well-known
logarithm table In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
s used up into the early 1980s.

# Writings

* 1799: Doctoral dissertation on the fundamental theorem of algebra, with the title: ''Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse'' ("New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors (i.e., polynomials) of the first or second degree") * 1801: '' Disquisitiones Arithmeticae'' (Latin). A German translation by H. Maser , pp. 1–453. English translation by Arthur A. Clarke * 1808: German translation by H. Maser , pp. 457–462 Gauss's_lemma,_uses_it_in_the_third_proof_of_quadratic_reciprocity.html" ;"title="Gauss's lemma (number theory)">Gauss's lemma, uses it in the third proof of quadratic reciprocity">Gauss's lemma (number theory)">Gauss's lemma, uses it in the third proof of quadratic reciprocity* 1809: ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium'' (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), ''Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections'' (English translation by C.H. Davis), reprinted 1963, Dover, New York. * * 1811: German translation by H. Maser , pp. 463–495 etermination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity] * 1812: ''Disquisitiones Generales Circa Seriem Infinitam'' $1+\frac+\mbox$ * 1818: . German translation by H. Maser , pp. 496–510 [Fifth and sixth proofs of quadratic reciprocity] * 1821, 1823 and 1826: ''Theoria combinationis observationum erroribus minimis obnoxiae''. Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation) English translation by G.W. Stewart, 1987, Society for Industrial Mathematics. * 1827: ''Disquisitiones generales circa superficies curvas'', Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146
"General Investigations of Curved Surfaces"
(published 1965), Raven Press, New York, translated by J. C. Morehead and A. M. Hiltebeitel. * 1828: . German translation by H. Maser * 1828: lementary facts about biquadratic residues, proves one of the supplements of the law of biquadratic reciprocity (the biquadratic character of 2)] * 1832: . German translation by H. Maser , pp. 534–586 [Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + ''i''] *
English translation
* * * * * * * * * 1843/44: ''Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung'', Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band, pp. 3–46 * 1846/47: ''Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung'', Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band, pp. 3–44 * * * * * ''Mathematisches Tagebuch 1796–1814'', Ostwaldts Klassiker,
Verlag Harri Deutsch The (VHD, HD) with headquarters in Frankfurt am Main, Germany, as well as in Zürich and Thun, Switzerland, was a German publishing house founded in 1961 and closed in 2013. Overview The ' with headquarters in Frankfurt am Main, Germany, ...
2005, mit Anmerkungen von Neumamn, (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)

*
Least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
*
Least-squares spectral analysis Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
* List of things named after Carl Friedrich Gauss **
Gauss (unit) The gauss, symbol (sometimes Gs), is a unit of measurement of magnetic induction, also known as '' magnetic flux density''. The unit is part of the Gaussian system of units, which inherited it from the older CGS-EMU system. It was named after t ...
** Gaussian distribution **
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
**
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /m ...
**
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
**
Gaussian mixture model In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation ...
**
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for mo ...
**
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...

# References

## Sources

* * * * * * *

* * * * * *

* *
Carl Friedrich Gauss Werke
– 12 vols., published from 1863 to 1933
Gauss and his children

*

– Biography at Fermat's Last Theorem Blog

by
Jürgen Schmidhuber Jürgen Schmidhuber (born 17 January 1963) is a German computer scientist most noted for his work in the field of artificial intelligence, deep learning and artificial neural networks. He is a co-director of the Dalle Molle Institute for Artific ...

English translation of Waltershausen's 1862 biography

Gauss
general website on Gauss

Obituary

*
"Carl Friedrich Gauss"
in the series ''A Brief History of Mathematics'' on BBC 4 *

at the Göttingen University {{DEFAULTSORT:Gauss, Carl Friedrich 1777 births 1855 deaths 18th-century German mathematicians 19th-century German mathematicians Technical University of Braunschweig alumni Corresponding members of the Saint Petersburg Academy of Sciences German deists Differential geometers Fellows of the American Academy of Arts and Sciences Fellows of the Royal Society 19th-century German astronomers German Lutherans 19th-century German physicists Honorary members of the Saint Petersburg Academy of Sciences Members of the Royal Netherlands Academy of Arts and Sciences Members of the Royal Swedish Academy of Sciences Mental calculators Number theorists Intuitionism Linear algebraists Optical physicists Scientists from Braunschweig People from the Duchy of Brunswick Recipients of the Copley Medal Recipients of the Pour le Mérite (civil class) University of Göttingen alumni University of Göttingen faculty University of Helmstedt alumni Ceres (dwarf planet) Recipients of the Lalande Prize Members of the Göttingen Academy of Sciences and Humanities