Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
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
![Statue-of-Gauss-in-Braunschweig](https://upload.wikimedia.org/wikipedia/commons/9/9b/Statue-of-Gauss-in-Braunschweig.jpg)
Johann Carl Friedrich Gauss was born on 30 April 1777 in
Brunswick (Braunschweig), in the
Duchy of Brunswick-Wolfenbüttel
A duchy, also called a dukedom, is a medieval country, territory, fief, or domain ruled by a duke or duchess, a ruler hierarchically second to the king or queen in Western European tradition.
There once existed an important difference between " ...
(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 Christian denominations that practice infant baptism, confirmation is seen as the sealing of the covenant created in baptism. Those being confirmed are known as confirmands. For adults, it is an affirmation of belief. It involves laying on ...
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
Wolfgang Sartorius Freiherr von Waltershausen (17 December 180916 March 1876) was a German geologist.
Life and work
Waltershausen was born at Göttingen and educated at this city's university. There he devoted his attention to physical and natur ...
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, ''
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 ranke ...
,
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
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 ...
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 toge ...
can be constructed by
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
if the number of its sides is the product of distinct
Fermat primes
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 4294967 ...
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 cultu ...
, and the discovery ultimately led Gauss to choose mathematics instead of
philology
Philology () is the study of language in oral and writing, written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defin ...
as a career.
Gauss was so pleased with this result that he requested that a regular
heptadecagon
In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.
Regular heptadecagon
A '' regular heptadecagon'' is represented by the Schläfli symbol .
Construction
As 17 is a Fermat prime, the regular heptadecagon is a ...
be inscribed on his tombstone. The
stonemason
Stonemasonry or stonecraft is the creation of buildings, structures, and sculpture using stone as the primary material. It is one of the oldest activities and professions in human history. Many of the long-lasting, ancient shelters, temples, mo ...
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 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 quantifying ...
, 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.
![Eureka Gauss](https://upload.wikimedia.org/wikipedia/commons/c/cd/Eureka_Gauss.png)
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 numbers. The th triangular number is the number of dots in ...
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, 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, subtr ...
s, which 150 years later led to the
Weil conjectures.
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 intensit ...
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 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 rig ...
'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.
![Carl Friedrich Gauss on his Deathbed, 1855](https://upload.wikimedia.org/wikipedia/commons/5/5c/Carl_Friedrich_Gauss_on_his_Deathbed%2C_1855.jpg)
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 tr ...
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, and
Wolfgang Sartorius von Waltershausen
Wolfgang Sartorius Freiherr von Waltershausen (17 December 180916 March 1876) was a German geologist.
Life and work
Waltershausen was born at Göttingen and educated at this city's university. There he devoted his attention to physical and natur ...
, 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 ganglia, nerve-endings, and the sympathetic nerves ...
, 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
Guy Waldo Dunnington (January 15, 1906, Bowling Green, Missouri – April 10, 1974, Natchitoches, Louisiana) was a writer, historian and professor of German known for his writings on the famous German mathematician Carl Friedrich Gauss. Dunnin ...
, 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 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 reformer whose efforts to reform the theology and practice of the Catholic Church launched th ...
.
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 ganglia, nerve-endings, and the sympathetic nerves ...
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 achieved dist ...
'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 often ...
, 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 pro ...
. Other religious influences included Wilhelm Braubach,
Johann Peter Süssmilch Johann Peter Süßmilch or Süssmilch (September 3, 1707 in Zehlendorf – March 22, 1767 in Berlin) was a German Protestant pastor, statistician and demographer.
Education and career
Süßmilch studied medicine and theology at Jena and Hal ...
, 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
![Therese Gauss](https://upload.wikimedia.org/wikipedia/commons/a/a3/Therese_Gauss.jpg)
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 U.S. state, state in the Midwestern United States, Midwestern region of the United States. Ranking List of U.S. states and territories by area, 21st in land area, it is bordered by eight states (tied for the most with Tennessee ...
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. 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-16](https://upload.wikimedia.org/wikipedia/commons/8/87/Gauss-16.jpg)
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. His ...
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 rig ...
.
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 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 who ...
, 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 Hungarian mathematician, mainly known for his work in geometry.
Biography
Bolyai was born in Bolya, a village near Hermannstadt, Grand Pr ...
dated 2 September 1808 as follows:
Career and achievements
Algebra
![Disqvisitiones-800](https://upload.wikimedia.org/wikipedia/commons/e/e3/Disqvisitiones-800.jpg)
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 exa ...
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 sur ...
. Mathematicians including
Jean le Rond d'Alembert 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
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 the ...
, Arithmetical Investigations), which, among other things, introduced the
triple bar 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 mod ...
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, 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 a ...
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 c ...
for them, and showed that a regular
heptadecagon
In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.
Regular heptadecagon
A '' regular heptadecagon'' is represented by the Schläfli symbol .
Construction
As 17 is a Fermat prime, the regular heptadecagon is a ...
(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
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 sum ...
for ''n'' = 3
*
Fermat's Last Theorem 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 Log ...
(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 approxi ...
* invented the
Cooley–Tukey FFT algorithm 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
![Bendixen - Carl Friedrich Gauß, 1828](https://upload.wikimedia.org/wikipedia/commons/3/33/Bendixen_-_Carl_Friedrich_Gau%C3%9F%2C_1828.jpg)
On 1 January 1801, Italian astronomer
Giuseppe Piazzi discovered the
dwarf planet Ceres
Ceres most commonly refers to:
* Ceres (dwarf planet), the largest asteroid
* Ceres (mythology), the Roman goddess of agriculture
Ceres may also refer to:
Places
Brazil
* Ceres, Goiás, Brazil
* Ceres Microregion, in north-central Goiás st ...
. 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 on 31 December at
Gotha, 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 consis ...
. 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 specia ...
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 orbits ...
). 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 th ...
. While this method is attributed to a 1965 paper by
James Cooley
James William Cooley (1926 – June 29, 2016) was an American mathematician. Cooley received a B.A. degree in 1949 from Manhattan College, Bronx, NY, an M.A. degree in 1951 from Columbia University, New York, NY, and a Ph.D. degree in 1961 in app ...
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 fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...
, 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 people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
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.
![Normal Distribution PDF](https://upload.wikimedia.org/wikipedia/commons/7/74/Normal_Distribution_PDF.svg)
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.
It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.
The value of histori ...
, 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.
Gauss proved the method under the assumption of
normally distributed errors (see
Gauss–Markov theorem; 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 eponymo ...
). The method had been described earlier by
Adrien-Marie Legendre 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
![Gauß-Stein Garlste](https://upload.wikimedia.org/wikipedia/commons/5/53/Gau%C3%9F-Stein_Garlste.jpg)
In 1818 Gauss, putting his calculation skills to practical use, carried out a
geodetic survey
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiva ...
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.
![10 DM Serie4 Rueckseite](https://upload.wikimedia.org/wikipedia/commons/f/fe/10_DM_Serie4_Rueckseite.jpg)
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 pol ...
, Gauss first defined a physical approximation for the
figure of the Earth
Figure of the Earth is a Jargon, 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 Spherical Earth, sphere is a well-k ...
as the surface everywhere perpendicular to the direction of gravity (of which
mean sea level
There are several kinds of mean in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...
makes up a part), later called the ''
geoid''.
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 geo ...
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
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consisten ...
, 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 multili ...
and
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 (ge ...
s and
surfaces. 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 .
F ...
.
This led in 1828 to an important theorem, the
Theorema Egregium (''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 canonic ...
. 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 Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s and
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
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 ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
. 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, and ...
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 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 Royal Society of London, FRS FRSE Royal Astronomical Society, FRAS LLD (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadrati ...
(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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s in
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
Martin Bartels
Johann Christian Martin Bartels (12 August 1769 – ) was a German mathematician. He was the tutor of Carl Friedrich Gauss in Braunschweig, Brunswick and the educator of Nikolai Ivanovich Lobachevsky, Lobachevsky at the Kazan State University, ...
. 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 von Waltershausen (17 December 180916 March 1876) was a German geologist.
Life and work
Waltershausen was born at Göttingen and educated at this city's university. There he devoted his attention to physical and natur ...
source and the changes in other versions), and some authors, such as
Joseph J. Rotman Joseph J. Rotman (May 26, 1934 – October 16, 2016) was a Professor of Mathematics at the University of Illinois at Urbana–Champaign and also a published author of 10 textbooks.
Rotman was born in Chicago. He did his undergraduate and graduate ...
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
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circum ...
as a benchmark pursuant to becoming a first-class mathematician.
Commemorations
![10 DM Serie4 Vorderseite](https://upload.wikimedia.org/wikipedia/commons/0/0d/10_DM_Serie4_Vorderseite.jpg)
From 1989 through 2001, Gauss's portrait, a
normal distribution curve and some prominent
Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
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, p ...
. A film version directed by
Detlev Buck was released in 2012.
In 2007 a
bust of Gauss was placed in the
Walhalla temple.
The
numerous things named in honor of Gauss include:
* the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, 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 unit ...
, the most common of the several electromagnetic unit systems based on
CGS units.
*
gauss, 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
Marian Adam Rejewski (; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine, aided by limited documents obtained by French mili ...
, who helped to solve the German
Enigma cipher machine in December 1932, began studying
actuarial statistics
In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of death ...
at
Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
. At the request of his
Poznań University
Poznań () is a city on the River Warta in west-central Poland, within the Greater Poland region. The city is an important cultural and business centre, and one of Poland's most populous regions with many regional customs such as Saint John ...
professor,
Zdzisław Krygowski
Zdzisław Jan Ewangelii Antoni Krygowski (Lwów, 1872 — 1955, Poznań) was a Polish mathematician, rector of the Lwów Polytechnic (1917–18), and professor at Poznań University (1919–38, 1946–55).
Enigma
Krygowski has become famous in ...
, 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. ...
honored Gauss on 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 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 i ...
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 Gauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss:
*
*
*
* A generalization of Euclid's lemma is sometimes called Gauss's lemma
See also
* List of topics named after Carl Friedrich Gauss
Carl Friedrich Gauss ( ...
, 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''
* 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, wa ...
2005, mit Anmerkungen von Neumamn, (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)
See also
*
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
*
List of things named after Carl Friedrich Gauss
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 ...
**
Gauss (unit)
**
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
**
Gaussian integral
**
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 observatio ...
**
Gaussian quadrature
**
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 .
F ...
References
Notes
Citations
Sources
*
*
*
*
*
*
*
Further reading
*
*
*
*
*
*
External links
*
*
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 Artif ...
English translation of Waltershausen's 1862 biographyGaussgeneral 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
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German deists
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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
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Recipients of the Copley Medal
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Ceres (dwarf planet)
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