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Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a
German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
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 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)
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 The Federal Republic of Germany, as a federal state, consists of sixteen partly sovereign federated states (german: Land (state), plural (sta ...
, 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 The Feast of the Ascension of Jesus Christ, also called Ascension Day, Ascension Thursday, or sometimes Holy Thursday, commemorates the Christian belief of the bodily Ascension of Jesus into heaven. It is one of the ecumenical (i.e., universally cel ...
(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 A woodcut depicting the confirmation of Lutheran youth In Christian denominations that practice infant baptism, confirmation is seen as the sealing of the covenant created in baptism. It is an affirmation of commitment and belief. Those bei ...

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 to the level of an adult expert. The term ''wunderkind'' (from German ''Wunderkind''; literally "wonder ...
. In his memorial on Gauss,
Wolfgang Sartorius von Waltershausen Wolfgang Sartorius Freiherr (; male, abbreviated as ), (; his wife, abbreviated as , literally "free lord" or "free lady") and (, his unmarried daughters and maiden aunts) are designations used as title of nobility, titles of nobility in the ...
says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an
arithmetic series An Arithmetic progression (AP) 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 ...

arithmetic series
problem (commonly said to be ) faster than anyone else in his class of 100 students. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. 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 's ''The Creation of Adam ''The Creation of Adam'' () is a fresco Fresco (plural ''frescos'' or ''frescoes'') is a technique of Mural, mural painting executed upon freshly laid ("wet") lime plaster. Water is used as the vehicle for the dry-po ...

magnum opus
, ''
Disquisitiones Arithmeticae Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'', in 1798, at the age of 21—though it was not published until 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 A duke (male) can either be a monarch ranked below the emperor An emperor (from la, imperator, via fro, empereor) is a monarch, and usually the sovereignty, sovereign ruler of an empire or another type of imperial realm. Empress, the f ...
, who sent him to the Collegium Carolinum (now
Braunschweig University of Technology The Technische Universität Braunschweig (unofficially University of Braunschweig – Institute of Technology), commonly referred to as TU Braunschweig, is the oldest ' (comparable to an institute of technology in the American system) in Germany ...
), which he attended from 1792 to 1795, and to the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...
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 (mathematics), plane Shape, 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 (mathematic ...

polygon
can be constructed by
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr ...
if the number of its sides is the product of distinct Fermat primes and a
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
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 civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
, and the discovery ultimately led Gauss to choose mathematics instead of
philology Philology is the study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing system composed o ...
as a career. Gauss was so pleased with this result that he requested that a regular
heptadecagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

heptadecagon
be inscribed on his tombstone. The
stonemason Stonemasonry or stonecraft is the creation of buildings, structures, and sculpture using rock (geology), stone as the primary material. It is one of the oldest activities and professions in human history. Many of the long-lasting, ancient Shelte ...
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 #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, 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 (mathematics), 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 ...
s are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and Cube (algebra)#In integers, cube numbers. The th triangular number ...
s 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, which 150 years later led to the
Weil conjectures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.


Later years and death

Gauss remained mentally active into his old age, even while suffering from
gout Gout is a form of characterized by recurrent attacks of a red, tender, hot, and . Pain typically comes on rapidly, reaching maximal intensity in less than 12 hours. The is affected in about half of cases. It may also result in , s, or . Go ...

gout
and 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 In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and Ray tracing (physics), ray tracing of light through an optical system (such as a lens (optics), lens). A paraxial ray is a Ray (optics), r ...
(
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 directions north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to East and West. ''North'' is a noun, adjective, o ...
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 i ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
'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 , is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and community outreach. ...
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 Hemodynamics American and British English spelling differences#ae and oe, or haemodynamics are the Fluid dynamics, dynamics of blood flow. The circulatory sys ...

heart attack
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 The Federal Republic of Germany, as a federal state, consists of sixteen partly sovereign federated states (german: Land (state), plural (sta ...
); 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 oriental studies, orientalist, Protestant theology, theologian, and Biblical exegete. He studied at the University of Göttingen. In 1827 he became extraordinary professor there, ...

Heinrich Ewald
, and
Wolfgang Sartorius von Waltershausen Wolfgang Sartorius Freiherr (; male, abbreviated as ), (; his wife, abbreviated as , literally "free lord" or "free lady") and (, his unmarried daughters and maiden aunts) are designations used as title of nobility, titles of nobility in the ...
, who was Gauss's close friend and biographer. Gauss's brain was preserved and was studied by
Rudolf Wagner Rudolf Friedrich Johann Heinrich Wagner (30 July 1805 – 13 May 1864) was a German anatomist and physiologist and the co-discoverer of the germinal vesicle. He made important investigations on Ganglion, ganglia, nerve-endings, and the Sympathetic ...

Rudolf Wagner
, who found its mass to be slightly above average, at 1,492 grams, and the cerebral area equal to 219,588 square millimeters (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius.


Religious views

Gauss was a
Lutheran Lutheranism is one of the largest branches of Protestantism that identifies with the teachings of Jesus Christ and was founded by Martin Luther, a 16th-century German monk and Protestant Reformers, reformer whose efforts to reform the theology ...
Protestant Protestantism is a form of that originated with the 16th-century , a movement against what its followers perceived to be in the . Protestants originating in the Reformation reject the Roman Catholic doctrine of , but disagree among themselves ...
, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had previously defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by the grace of the Lord." One of his biographers, G. Waldo Dunnington, described Gauss's religious views as follows:
For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength, it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life, it brought him confidence. Gauss's God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report the truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.
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 Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the p ...
with very unorthodox views, while Dunnington (though 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 that identifies with the teachings of Jesus Christ and was founded by Martin Luther, a 16th-century German monk and Protestant Reformers, reformer whose efforts to reform the theology ...
. In connection to this, there is a record of a conversation between
Rudolf Wagner Rudolf Friedrich Johann Heinrich Wagner (30 July 1805 – 13 May 1864) was a German anatomist and physiologist and the co-discoverer of the germinal vesicle. He made important investigations on Ganglion, ganglia, nerve-endings, and the Sympathetic ...

Rudolf Wagner
and Gauss, in which they discussed
William Whewell William Whewell ( ; 24 May 17946 March 1866) was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval Eng ...

William Whewell
'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 ''fides'' and ''feid'', is confidence or trust in a , thing, or In the context of , one can define faith as " in or in the doctrines or teachings of religion". Religious people often think of faith as confidence based on ...

faith
, 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 Lutheranism is one of the largest branches of Protestantism that identifies with the teachings of Martin Luther, a 16th-century German Protestant Reformers, reform ...

Paul Gerhardt
than by
Moses Moses he, מֹשֶׁה, ''Mōše''; also known as Moshe Rabbenu ( he, מֹשֶׁה רַבֵּנוּ "Moshe our Teacher"); syr, ܡܘܫܐ, ''Mūše''; ar, موسى '; el, Mωϋσῆς, ' () is considered the most important prophet in Judais ...

Moses
. Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the
New Testament The New Testament grc, Ἡ Καινὴ Διαθήκη, Transliteration, transl. ; la, Novum Testamentum. (NT) is the second division of the Christian biblical canon. It discusses the teachings and person of Jesus in Christianity, Jesus, as ...

New Testament
. Two religious works which Gauss read frequently were Braubach's ''Seelenlehre'' (Giessen, 1843) and Süssmilch's ''Gottliche'' (Ordnung gerettet A756); he also devoted considerable time to the New Testament in the original Greek. Dunnington further elaborates on Gauss's religious views by writing:
Gauss's religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.
Gauss declared he firmly believed in the
afterlife The afterlife (also referred to as life after death or the world to come) is an existence in which the essential part of an individual's identity Identity may refer to: Social sciences * Identity (social science), personhood or group a ...

afterlife
, and saw spirituality as something essentially important for human beings. He was quoted stating: ''"The world would be nonsense, the whole creation an absurdity without immortality,"'' and for this statement he was severely criticized by the atheist
Eugen DühringEugen Karl Dühring (12 January 1833, Berlin Berlin (; ) is the Capital city, capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it t ...
who judged him as a narrow superstitious man. Though he was not a church-goer, Gauss strongly upheld
religious tolerance Toleration is the allowing, permitting, or acceptance of an action, idea, object, or person which one dislikes or disagrees with. Political scientist Andrew R. Murphy explains that "We can improve our understanding by defining "toleration" as a ...
, 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 he, 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 in the Midwest, he learned the Sioux language. Later, he moved to
Missouri Missouri is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in ...

Missouri
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 Missouri is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State ( ...

St. Louis
. 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 Eric Temple Bell (7 February 1883 – 21 December 1960) was a Scottish-born mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
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 city, capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it the List of cities in the European Union by ...

Berlin
in 1828. However, 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 abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
. On Gauss's recommendation,
Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astro ...
was awarded an honorary doctor 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qu ...

Sophie Germain
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 Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'', where he states that all analysis (i.e., 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 Napoléon Bonaparte (15 August 1769 – 5 May 1821) was a French military and political leader. He rose to prominence during the French Revolution The French Revolution ( ) refers to the period that began with the Estates General o ...
, whom he saw as an outgrowth of revolution. Gauss summarized his views on the pursuit of knowledge in a letter to
Farkas Bolyai Farkas Bolyai (; 9 February 1775 – 20 November 1856; also known as Wolfgang Bolyai in Germany) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 19 ...

Farkas Bolyai
dated 2 September 1808 as follows:
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.


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 The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
which states that every non-constant single-variable
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
with complex coefficients has at least one complex
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large group ...
. Mathematicians including
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Enc ...
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 In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting loop (topology), continuous loop in the plane. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" reg ...

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 devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
with his 1801 book ''
Disquisitiones Arithmeticae Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'' (
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

Latin
, 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 featuring the equal sign The equals sign (British English, Unicode Consortium) or equal sign (American English), ...
symbol for
congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
and used it in a clean presentation of
modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
, developed the theories of binary and ternary
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, stated the
class number problem In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having cl ...
for them, and showed that a regular
heptadecagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

heptadecagon
(17-sided polygon) can be constructed with straightedge and compass. It appears that Gauss already knew the
class number formulaIn number theory, the class number formula relates many important invariants of a number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, stru ...
in 1801. In addition, he proved the following conjectured theorems: *
Fermat polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the su ...
for ''n'' = 3 *
Fermat's last theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
for ''n'' = 5 * Descartes's rule of signs *
Kepler conjecture#REDIRECT Kepler conjecture The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician A mathematician i ...
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 arc (geometry), arcs, all of whose internal and external angles, internal angles are right angles. This shape was described by ...
(se
University of Bielefeld website
* developed an algorithm for determining the
date of Easter As a moveable feast A moveable feast or movable feast is an observance in a Christian liturgical calendar, borrowed from the Hebrew Lunisolar calendar, which therefore occurs on a different date (relative to the Roman Civil calendar, civil or ...
* 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 number, composite size N = N_1N_2 in ...
for calculating the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
s 160 years before Cooley and Tukey


Astronomy

On 1 January 1801, Italian astronomer
Giuseppe Piazzi Giuseppe Piazzi ( , ; 16 July 1746 – 22 July 1826) was an Italian Catholic The Catholic Church, often referred to as the Roman Catholic Church, is the List of Christian denominations by number of members, largest Christian church, wi ...

Giuseppe Piazzi
discovered the
dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun – something 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 d ...
Ceres Ceres most commonly refers to: * Ceres (dwarf planet) Ceres (; minor-planet designation: 1 Ceres) is the smallest recognized dwarf planet, the closest dwarf planet to the Sun, and the List of notable asteroids, largest object in the main astero ...
. 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 (; male, abbreviated as ), (; his wife, abbreviated as , literally "free lord" or "free lady") and (, his unmarried daughters and maiden aunts) are designations used as titles of nobility ...
on 31 December at
Gotha Gotha () is the fifth-largest city in Thuringia Thuringia (; german: Thüringen ), officially the Free State of Thuringia ( ), is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazi ...
, and one day later by
Heinrich Olbers Heinrich Wilhelm Matthias Olbers (; ; 11 October 1758 – 2 March 1840) was a German physician A physician (American English), medical practitioner (English in the Commonwealth of Nations, Commonwealth English), medical doctor, or simply d ...
in
Bremen Bremen (, also ; Low German : : : : : , minority = (70,000) (30,000) (8,000) , familycolor = Indo-European , fam2 = Germanic Germanic may refer to: * Germanic peoples, an ethno-linguistic group identified by t ...
. This confirmation eventually led to the classification of Ceres as
minor-planet designation A formal minor-planet designation is, in its final form, a number–name combination given to a minor planet (asteroid, centaur (minor planet), centaur, trans-Neptunian object and dwarf planet but not comet). Such designation always features a le ...
1 Ceres: the first
asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Historically, these terms have been applied to any astronomical object orbiting the Sun that did not resolve into a disc in a telescope and was not observ ...

asteroid
(now dwarf planet) ever discovered.
Gauss's method In orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculat ...
involved determining a
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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). 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 t ...
. While this method is attributed to a 1965 paper by
James Cooley James William Cooley (1926 – June 29, 2016) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (USA), co ...
and
John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the Cooley–Tukey FFT algorithm, Fast Fourier Transform (FFT) algorithm and box plot. The Tukey's range test, ...
, 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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, R ...

Joseph Fourier
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 The Gaussian gravitational constant (symbol ) is a parameter used in the orbital mechanics of the solar system The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astrono ...
, and contained an influential treatment of the
method of 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 resid ...
, 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 Statistics is the discipline that co ...
. Gauss proved the method under the assumption of
normally distributed In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

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 (set theory), class of linear regression model, linear ...
; see also
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 eponymous ...
). 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 named a ...
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 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.


Non-Euclidean geometries

Gauss also claimed to have discovered the possibility of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
but never published it. This discovery was a major
paradigm shift A paradigm shift, a concept identified by the American physicist and philosopher Thomas Kuhn Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American whose 1962 book ' was influential in both academic and popular circles, ...
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 physicists of all time. Einstein is known for developing the theory of relativity The theo ...

Einstein
's theory of general relativity, which describes the universe as non-Euclidean. His friend 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 János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the ...
, 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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, fields of mathematics dealing with
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s and
surfaces Water droplet lying on a damask. Surface tension">damask.html" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most gener ...
. Among other things, he came up with the notion of
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

Gaussian curvature
. This led in 1828 to an important theorem, the
Theorema Egregium without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area. Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry Differential geometry is a Mathe ...
(''remarkable theorem''), establishing an important property of the notion of
curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

curvature
. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s and
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
s 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 ( Swedish: ''Kungliga Vetenskapsakademien'') is one of the royal academies of Sweden Sweden ( sv, Sverige ), officially the Kingdom of Sweden ( sv, links=no, Konungariket Sverige ), is a Nordic co ...
. Gauss was elected a Foreign Honorary Member of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences, founded 1780, (abbreviation: AAAS) is one of the oldest learned societies A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization ...

American Academy of Arts and Sciences
in 1822.


Magnetism

In 1831, Gauss developed a fruitful collaboration with the physics professor , leading to new knowledge in
magnetism Magnetism is a class of physical attributes that are mediated by magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For in ...

magnetism
(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 circuit An electrical network is an interconnection of electronic compon ...
in electricity. It was during this time that he formulated his namesake
law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its bounda ...
. 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, geophysics, geophysical, oceanography and volcanology are examples of disciplines for which observatories have been cons ...

observatory
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 In astronomy and planetary science, a magnetosphere is a region of space surrounding an astronomical object in which charged particles are affected by that object's magnetic field. It is created by a star or planet with an active interior Dynamo t ...

magnetospheric
) sources of Earth's magnetic field.


Appraisal

The British mathematician
Henry John Stephen Smith Prof Henry John Stephen Smith FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be " ...

Henry John Stephen Smith
(1826–1883) gave the following appraisal of Gauss:


Anecdotes

There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances. Another 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
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s in
arithmetic progression An Arithmetic progression (AP) 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 ...

arithmetic progression
; 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 . 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 Wolfgang Sartorius Freiherr (; male, abbreviated as ), (; his wife, abbreviated as , literally "free lord" or "free lady") and (, his unmarried daughters and maiden aunts) are designations used as title of nobility, titles of nobility in the ...
source and the changes in other versions), and some authors, such as Joseph J. Rotman in his book ''A First Course in Abstract Algebra''(2000), 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 In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality :e^ + 1 = 0 where : is E (mathematical constant), Euler's number, the base of natural logarithms, : is the imaginary unit, which by defini ...
as a benchmark pursuant to becoming a first-class mathematician.


Commemorations

From 1989 through 2001, Gauss's portrait, a
normal distribution curve In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density, probability de ...

normal distribution curve
and some prominent Göttingen buildings were featured on the German ten-mark banknote. The reverse featured the approach for Kingdom of Hanover, Hanover. 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. A film version directed by Detlev Buck was released in 2012. In 2007 a Bust (sculpture), bust of Gauss was placed in the Walhalla temple. The List of things named after Carl Friedrich Gauss, 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 * Gauss (unit), gauss, the Centimetre gram second system of units, CGS unit for magnetic field In 1929 the Polish mathematician Marian Rejewski, who helped to solve the German Enigma machine, Enigma cipher machine in December 1932, began studying actuarial statistics at Göttingen. 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 honoured Gauss in his would-be 241st birthday with a Google Doodle 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 tables used up into the early 1980s.


Writings

* 1799: Doctoral dissertation on the
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
, 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 Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'' (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 [Introduces 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 [Determination 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: [Elementary 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 2005, mit Anmerkungen von Neumamn, (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)


See also

* Gaussian elimination * German inventors and discoverers * List of things named after Carl Friedrich Gauss * Romanticism in science * Seconds pendulum * Gauss (unit)


References


Notes


Citations


Sources

* *


Further reading

* * * * * * * *


External links

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


*

– Biography at Fermat's Last Theorem Blog

by Jürgen Schmidhuber
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 Carl Friedrich Gauss, 1777 births 1855 deaths 18th-century German mathematicians 19th-century German mathematicians Technical University of Braunschweig alumni Corresponding Members of the St Petersburg Academy of Sciences German deists Differential geometers Fellows of the American Academy of Arts and Sciences Fellows of the Royal Society German astronomers German Lutherans German physicists Honorary Members of the St 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 Linear algebraists Optical physicists People from Braunschweig People from the Duchy of Brunswick People on banknotes 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