Johann Carl Friedrich Gauss (30 April 1777 23 February 1855) was a
GermanThe German people are an ethnic group, in the sense of sharing a common German culture, descent, and speaking the German language as a mother tongue. Within Germany, Germans are defined by citizenship , distinguished from people of German ancestry...
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
and
scientistA scientist, in the broadest sense, is any person who engages in a systematic activity to acquire knowledge or an individual that engages in such practices and traditions that are linked to schools of thought or philosophy. In a more restricted sense, a scientist is an individual who uses the...
who contributed significantly to many fields, including
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
,
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
,
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
,
differential geometryDifferential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis...
,
geodesyGeodesy , also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...
,
geophysicsGeophysics, a major discipline of the Earth sciences and a subdiscipline of physics, is the study of the whole Earth by the quantitative observation of its physical properties. Geophysical data are used in academics to observe tectonic plate motions, study the internal structure of the Earth,...
,
electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
,
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
and
opticsOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
. Sometimes known as the
Princeps mathematicorum (
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe...
, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences."
Gauss was a
child prodigyA child prodigy is someone who at an early age masters one or more skills at an adult level. One heuristic for classifying prodigies is: a prodigy is a child, typically younger than 15 years old, who is performing at the level of a highly trained adult in a very demanding field of endeavor...
. There are many anecdotes pertaining to his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed
Disquisitiones ArithmeticaeThe Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
, his
magnum opusMagnum opus , from the Latin meaning great work, refers to the largest, and perhaps the best, greatest, most popular, or most renowned achievement of an author, artist, or composer.The term Great Work is also used in several...
, in 1798 at the age of 21, though it would not be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Early years (1777–1798)
Johann Carl Friedrich Gauss was born on April 30, 1777 in
BraunschweigBraunschweig , known as Brunswiek in Low German, is a city of 245,810 people , located in Lower Saxony, Germany. It is located north of the Harz mountains at the farthest navigable point of the Oker river, which connects to the North Sea via the rivers Aller and Weser...
, in the Electorate of Brunswick-Lüneburg, now part of
Lower SaxonyLower Saxony lies in north-western Germany and is second in area and fourth in population among the sixteen Bundesländer of Germany...
,
GermanyGermany , officially the Federal Republic of Germany , is a country in Central Europe. It is bordered to the north by the North Sea, Denmark, and the Baltic Sea; to the east by Poland and the Czech Republic; to the south by Austria and Switzerland; and to the west by France, Luxembourg, Belgium,...
, as the second son of poor working-class parents. He was christened and confirmed in a church near the school he had attended as a child. 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 famous story has it that in primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s in
arithmetic progressionIn mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant...
; 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 BartelsJohann Christian Martin Bartels was a German Mathematician. He was the tutor of Carl Friedrich Gauss in Brunswick and the educator of Lobachevsky at the University of Kazan.- Biography :...
. 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 WaltershausenWolfgang Sartorius Freiherr von Waltershausen was a German geologist.-Life and Work:Waltershausen was born at Göttingen and educated at the university in that city. There he devoted his attention to physical and natural science, and in particular to mineralogy...
source and the changes in other versions); some authors, such as Joseph Rotman in his book
A first course in Abstract Algebra, question whether it ever happened.
As his father wanted him to follow in his footsteps and become a pastor, he was not supportive of Gauss's schooling in mathematics and science. Gauss was primarily supported by his mother in this effort and by the
Duke of BraunschweigCharles II William Ferdinand, Duke of Brunswick-Wolfenbuettel was a sovereign prince of the Holy Roman Empire, and a professional soldier who served as a Generalfeldmarschall of the Kingdom of Prussia...
, who awarded Gauss a fellowship to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795, and subsequently he moved to the
University of GöttingenThe University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.It was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and was then opened in 1737. The University of Göttingen soon grew in size and popularity...
from 1795 to 1798.
While in university, Gauss independently rediscovered several important theorems; his breakthrough occurred in 1796 when he was able to show that any regular
polygonIn geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments . These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners...
with a number of sides which is a
Fermat primeIn mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the formwhere n is a nonnegative integer. The first ten Fermat numbers are :...
(and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a
powerExponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n...
of 2) can be constructed by compass and straightedge. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the
Ancient GreeksAncient Greece is the civilisation belonging to the period of Greek history lasting from the Greek Dark Ages ca. 1100 BC and the Dorian invasion, to 146 BC and the Roman conquest of Greece after the Battle of Corinth. It is generally considered to be the seminal culture which provided the...
, and the discovery ultimately led Gauss to choose mathematics instead of
philologyPhilology considers both form and meaning in linguistic expression, combining linguistics and literary studies.Classical philology is the philology of the Greek, Latin and Sanskrit languages...
as a career.
Gauss was so pleased by this result that he requested that a regular
heptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796....
be inscribed on his
tombstoneA headstone, tombstone, or gravestone is a marker, normally carved from stone, placed over or next to the site of a burial in a cemetery or elsewhere.- Use :...
. The stonemason declined, stating that the difficult construction would essentially look like a circle.
The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the
heptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796....
on March 30. He invented
modular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...
, greatly simplifying manipulations in number theory. He became the first to prove the
quadratic reciprocityThe law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the...
law on 8 April. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The
prime number theoremIn number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a rough description of how the primes are distributed....
, conjectured on 31 May, gives a good understanding of how the
prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s are distributed among the integers.
Gauss also discovered that every positive integer is representable as a sum of at most three
triangular numberA triangular number is the number of dots in an equilateral triangle evenly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number...
s on 10 July and then jotted down in his diary the famous words, "
HeurekaEureka is an exclamation used as an interjection to celebrate a discovery.-Archimedes:...
! num = Δ + Δ + Δ." On October 1 he published a result on the number of solutions of polynomials with coefficients in
finite fieldIn abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, which ultimately led to the
Weil conjecturesIn mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions derived from counting the number of points on algebraic varieties over finite fields.A variety V over a finite field with q...
150 years later.
Middle years (1799–1830)
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 algebraIn mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
which states that every non-constant single-variable
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
over the
complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s has at least one root. Mathematicians including
Jean le Rond d'AlembertJean le Rond d'Alembert was a French mathematician, mechanician, physicist and philosopher. He was also co-editor with Denis Diderot of the Encyclopédie...
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 implicit use of the
Jordan curve theoremIn topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside" region, and any path connecting a point of one region to a point of the other intersects that loop somewhere.The first to give a proof was...
. 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 theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
with his 1801 book
Disquisitiones ArithmeticaeThe Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
(
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe...
, Arithmetical Investigations), which contained a clean presentation of
modular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...
and the first proof of the law of
quadratic reciprocityThe law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the...
.
In that same year,
ItalianItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
astronomer
Giuseppe PiazziGiuseppe Piazzi was an Italian Theatine monk, mathematician, and astronomer. He was born in Ponte in Valtellina, and died in Naples. He established an observatory at Palermo, now the Osservatorio Astronomico di Palermo – Giuseppe S...
discovered the
dwarf planetA dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be rounded by its own gravity but has not cleared its neighbouring region of planetesimals and is not a satellite. More explicitly, it has to have sufficient mass to...
Ceres, but could only watch it for a few days. Gauss predicted correctly the position at which it could be found again, and it was rediscovered by
Franz Xaver von ZachBaron Franz Xaver von Zach was a German astronomer born at Pest in Hungary....
on 31 December 1801 in
GothaGotha Observatory was a German astronomical observatory located on Seeberg hill near Gotha, Thuringia, Germany...
, and one day later by
Heinrich OlbersHeinrich Wilhelm Matthäus Olbers was a German physician and astronomer.-Life and career:Olbers was born in Arbergen, near Bremen, and studied to be a physician at Göttingen. After his graduation in 1780, he began practicing medicine in Bremen, Germany...
in
BremenThe City Municipality of Bremen is a Hanseatic city in northwestern Germany. A port city along the river Weser, about south from its mouth on the North Sea, Bremen is part of the Bremen-Oldenburg metropolitan area...
. Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again." Though Gauss had been up to that point supported by the 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
GöttingenGöttingen is a college town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
, a post he held for the remainder of his life.
The discovery of Ceres by Piazzi on 1 January 1801 led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 under the name
Theoria motus corporum coelestium in sectionibus conicis solem ambientum (theory of motion of the celestial bodies moving in conic sections around the sun). Piazzi had only been able to track Ceres for a couple of months, 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, who was 23 at the time, 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. In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—published a few years later as
Theory of Celestial Movement—remains a cornerstone of astronomical computation. It introduced the
Gaussian gravitational constantThe Gaussian gravitational constant is an astronomical constant first proposed by German polymath Carl Friedrich Gauss in his 1809 work Theoria motus corporum coelestium in sectionibus conicis solem ambientum , although he had already used the concept to great success in predicting the...
, and contained an influential treatment of the
method of least squaresThe method of least squares is applied to approximate solutions of overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis....
, a procedure used in all sciences to this day to minimize the impact of
measurement errorObservational error is the difference between a measured value of quantity and its true value. In statistics, an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process.-Science and experiments:...
. Gauss was able to prove the method in 1809 under the assumption of
normally distributedIn probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
errors (see
Gauss–Markov theoremIn statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear model in which the errors have expectation zero and are uncorrelated and have equal variances, a best linear unbiased estimator of the coefficients is given by the ordinary least...
; see also
Gaussian). The method had been described earlier by
Adrien-Marie LegendreAdrien-Marie Legendre was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis....
in 1805, but Gauss claimed that he had been using it since 1795.
Gauss was a prodigious
mental calculatorMental calculators are people with a prodigious ability in some area of mental calculation, such as multiplying large numbers or factoring large numbers. Some rare mental calculators are autistic savants, with a narrow area of great skill and poor mental development in other directions, but many...
. Reputedly, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used
logarithmIn mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number....
s." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!"
In 1818 Gauss, putting his calculation skills to practical use, carried out a
geodesic surveySurveying or land surveying is the technique and science of accurately determining the terrestrial or three-dimensional space position of points and the distances and angles between them...
of the state of
HanoverThe Kingdom of Hanover was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era...
, linking up with previous
DanishDenmark is a Scandinavian country in Northern Europe and the senior member of the Kingdom of Denmark. It is the southernmost of the Nordic countries; southwest of Sweden and south of Norway, and it is bordered to the south by Germany. Denmark borders both the Baltic and the North Sea...
surveys. To aid in the survey, Gauss invented the
heliotropeThe heliotrope is an instrument that uses a mirror to reflect sunlight over great distances to mark the positions of participants in a land survey. The heliotrope was invented by the German mathematician Carl Friedrich Gauss....
, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.
Gauss also claimed to have discovered the possibility of
non-Euclidean geometriesA non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the...
but never published it. This discovery was a major paradigm shift 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,
EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
's theory of general relativity, which describes the universe as non-Euclidean. His friend
Farkas Wolfgang BolyaiFarkas Bolyai was a Hungarian mathematician, mainly known for his work in geometry.-Biography:...
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 BolyaiJános Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in Kolozsvár, Transylvania, Kingdom of Hungary, Habsburg Empire , the son of a well-known mathematician, Farkas Bolyai.- Life :By the age of 13, he had mastered calculus and other forms of...
, 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 János Bolyai (who thought that Gauss was "stealing" his idea), but it is now generally taken at face value. Letters by Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines.
Waldo DunningtonGuy Waldo Dunnington was a life-long student of Carl Friedrich Gauss, a famous German mathematician. Dunnington wrote several articles about Gauss and later a biography entitled Carl Frederick Gauss: Titan of Science .Dunnington originally became interested in Gauss through one of his elementary...
, a life-long student of Gauss, successfully proves in
Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidian geometry long before it was published by János, but that he refused to publish any of it because of his fear of controversy.
The survey of Hanover fueled Gauss's interest in
differential geometryDifferential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis...
, a field of mathematics dealing with
curveIn mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave...
s and
surfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R
3 — for example, the surface of a ball...
s. Among other things he came up with the notion of
Gaussian curvatureIn differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the...
. This led in 1828 to an important theorem, the
Theorema EgregiumGauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
(
remarkable theorem in
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe...
), establishing an important property of the notion of
curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring
angleIn geometry and trigonometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle...
s and
distanceDistance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical...
s on the surface. That is, curvature does not depend on how the surface might be
embeddedIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in 3-dimensional space or 2-dimensional space.
In 1821, he was made a foreign member of the
Royal Swedish Academy of SciencesThe Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademin is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2 June...
.
Later years and death (1831–1855)
In 1831 Gauss developed a fruitful collaboration with the physics professor
Wilhelm WeberWilhelm Eduard Weber was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph.-Early years:...
, leading to new knowledge in
magnetismIn physics, the term magnetism is used to describe how materials respond on the microscopic level to an applied magnetic field; to categorize the magnetic phase of a material. For example, the most well known form of magnetism is ferromagnetism such that some ferromagnetic materials produce their...
(including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of
Kirchhoff's circuit lawsKirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...
in electricity. They constructed the first
electromagnetic telegraphThe electrical telegraph is a telegraph that uses electric signals. The electromagnetic telegraph is a device for human-to-human transmission of coded text messages over wire...
in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic
observatoryAn observatory is a location used for observing terrestrial and/or celestial events. Astronomy, climatology/meteorology, geology, oceanography and volcanology are examples of disciplines for which observatories have been constructed...
to be built in the garden of the observatory, and with Weber founded the magnetischer Verein (
magnetic club in
GermanGerman is a West Germanic language, thus related to and classified alongside English and Dutch. It is one of the world's major languages and the most widely spoken first language in the European Union. Around the world, German is spoken by approximately 105 million native speakers and also by...
), 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 has been in use well into the second half of the 20th century and worked out the mathematical theory for separating the inner (
coreThe planetary core consists of the innermost layer of a planet.The core may be a solid or a liquid layer, as is Mercury's, while the cores of Mars and Venus are thought to be completely solid as they lack an internally generated magnetic field...
and
crustIn geology, a crust is the outermost solid shell of a rocky planet or moon, which is chemically distinct from the underlying mantle. The crusts of Earth, our Moon, Mercury, Venus, Mars, Io, and other planetary bodies have been generated largely by igneous processes, and these crusts are richer in...
) and outer (
magnetosphericA magnetosphere is a highly magnetized region around and possessed by an astronomical object. Earth is surrounded by a magnetosphere, as are the magnetized planets Mercury, Jupiter, Saturn, Uranus, and Neptune. Jupiter's moon Ganymede is magnetized, but too weak to trap solar wind plasma. Mars has...
) sources of Earth's magnetic field.
Gauss died in Göttingen, Hannover (now part of
Lower SaxonyLower Saxony lies in north-western Germany and is second in area and fourth in population among the sixteen Bundesländer of Germany...
, Germany) in 1855 and is interred in the cemetery
AlbanifriedhofAlbanifriedhof is a cemetery in Göttingen, Germany just outside the city wall to the southeast. It is most famous as the final resting place of Carl Friedrich Gauss.Well known final resting place of:...
there. Two individuals gave eulogies at his funeral, Gauss's son-in-law
Heinrich EwaldGeorg Heinrich August Ewald was a German orientalist and theologian.-Life:Ewald was born at Göttingen where his father was a linen weaver. In 1815 he was sent to the gymnasium, and in 1820 he entered the University of Göttingen, where he studied with J.G. Eichhorn and T. C. Tychsen, specialising...
and
Wolfgang Sartorius von WaltershausenWolfgang Sartorius Freiherr von Waltershausen was a German geologist.-Life and Work:Waltershausen was born at Göttingen and educated at the university in that city. There he devoted his attention to physical and natural science, and in particular to mineralogy...
, who was Gauss's close friend and biographer. His brain was preserved and was studied by
Rudolf WagnerRudolf Wagner 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.-Life:...
who found its mass to be 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 was suggested as the explanation of his genius.
Family
Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a
depressionMajor depressive disorder is a mental disorder characterized by an all-encompassing low mood accompanied by low self-esteem, and loss of interest or pleasure in normally enjoyable activities...
from which he never fully recovered. He married again, to Johanna's best friend named Friederica Wilhelmine Waldeck but commonly known as Minna. When his second wife died in 1831 after a long illness, one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1817 until her death in 1839.
Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). Of all of Gauss's children, Wilhelmina was said to have come closest to his talent, but she died young. With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene emigrated to the
United StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
about 1832 after a falling out with his father. Wilhelm also settled in
MissouriMissouri is a state in the Midwest region of the United States bordered by Iowa, Illinois, Kentucky, Tennessee, Arkansas, Oklahoma, Kansas and Nebraska. Missouri is the 18th most populous state with a 2008 estimated population of 5,911,605. It comprises 114 counties and one independent city....
, starting as a
farmerA farmer is a person who raises living organisms for food or raw materials.- Definition :The term farmer usually applies to a person who grows field crops, and/or manages orchards or vineyards, or raises livestock or poultry such as chicken and cows...
and later becoming wealthy in the shoe business in
St. LouisSt. Louis is an independent city in the U.S. state of Missouri. With an estimated population of 354,361 in 2008, it is the principal municipality of Greater St. Louis, population 2,866,517, the largest urban area in Missouri and sixteenth largest in the United States...
. Therese kept house for Gauss until his death, after which she married.
Gauss eventually had conflicts with his sons, two of whom migrated to the United States. He did not want any of his sons to enter mathematics or science for "fear of sullying the family name". Gauss wanted Eugene to become a
lawyerA lawyer, according to Black's Law Dictionary, is "a person learned in the law; as an attorney, counsel or solicitor; a person licensed to practice law." Law is the system of rules of conduct established by the sovereign government of a society to correct wrongs, maintain stability, and deliver...
, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and emigrated to the United States, where he was quite successful. 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
perfectionistPerfectionism, in psychology, is a belief that perfection can and should be attained. In its pathological form, perfectionism is a belief that work or output that is anything less than perfect is unacceptable...
and a hard worker. According to
Isaac AsimovIsaac Asimov , was an American author and professor of biochemistry, best known for his works of science fiction and for his popular science books...
, Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done." This anecdote is briefly discussed in
G. Waldo DunningtonGuy Waldo Dunnington was a life-long student of Carl Friedrich Gauss, a famous German mathematician. Dunnington wrote several articles about Gauss and later a biography entitled Carl Frederick Gauss: Titan of Science .Dunnington originally became interested in Gauss through one of his elementary...
's
Gauss, Titan of Science where it is suggested that it is an apocryphal story.
He was never a prolific writer, refusing to publish works 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. Mathematical historian
Eric Temple BellEric Temple Bell was a mathematician and science fiction author born in Scotland who lived in the U.S. for most of his life...
estimated that had Gauss timely published all of his discoveries, Gauss 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
BerlinBerlin is the capital city and one of sixteen states of Germany. With a population of 3.4 million within its city limits, Berlin is Germany's largest city. It is the second most populous city and the eighth most populous urban area in the European Union...
in 1828. However, several of his students became influential mathematicians, among them
Richard DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
,
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
, and
Friedrich BesselFriedrich Wilhelm Bessel was a German mathematician, astronomer, and systematizer of the Bessel functions . He was a contemporary of Carl Gauss, also a mathematician and astronomer...
. Before she died,
Sophie GermainMarie-Sophie Germain was a French mathematician who made important contributions to the fields of differential geometry and number theory, and to the study of Fermat's Last Theorem.-Biography:...
was recommended by Gauss to receive her honorary degree.
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 ArithmeticaeThe Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
", where he states that all analysis (i.e. the paths one travelled to reach the solution of a problem) must be suppressed for sake of brevity.
Gauss supported
monarchyThe person who heads a monarchy is called a monarch. It was a common form of government in the world during the ancient and medieval times. A Monarchy is a form of government in which supreme power is absolutely or nominally lodged with an individual, who is the head of state, often for life or...
and opposed
NapoleonNapoleon Bonaparte later known as Napoleon I, and previously Napoleone di Buonaparte, was a military and political leader of France whose actions shaped European politics in the early 19th century.Born in Corsica and trained as an artillery officer in mainland France, Bonaparte rose to prominence...
, whom he saw as an outgrowth of
revolutionA revolution is a fundamental change in power or organizational structures that takes place in a relatively short period of time.Aristotle described two types of political revolution:...
.
Commemorations
From 1989 until the end of 2001, his portrait and a normal distribution curve as well as some prominent buildings of
GöttingenGöttingen is a college town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
were featured on the German ten-mark banknote. The other side of the note features the
heliotropeThe heliotrope is an instrument that uses a mirror to reflect sunlight over great distances to mark the positions of participants in a land survey. The heliotrope was invented by the German mathematician Carl Friedrich Gauss....
and a
triangulationIn trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
approach for Hannover. Germany has issued three stamps honouring Gauss, as well. A righteous stamp (no. 725), was issued in 1955 on the hundredth anniversary of his death; two other stamps, no. 1246 and 1811, were issued in 1977, the 200th anniversary of his birth.
Daniel KehlmannDaniel Kehlmann is a German language author of both Austrian and German nationality. His work Die Vermessung der Welt is the biggest selling novel in the German language since Patrick Süskind's Perfume was released in 1985...
's 2005 novel
Die Vermessung der Welt, translated into English as
Measuring the WorldMeasuring the World is a 2005 novel by German author Daniel Kehlmann. This novel deals with Carl Friedrich Gauss and Alexander von Humboldt - who was accompanied on his journeys by Aimé Bonpland - and their two different ways of taking the world's measure, as well as their travels in South America...
: a Novel in 2006, explores Gauss's life and work through a lens of historical fiction, contrasting it with the German explorer
Alexander von Humboldtwas a German naturalist and explorer, and the younger brother of the Prussian minister, philosopher, and linguist, Wilhelm von Humboldt...
.
In 2007, his
bustA bust is a sculpted or cast representation of the upper part of the human figure, depicting a person's head and neck, as well as a variable portion of the chest and shoulders. The piece is normally supported by a plinth. These forms recreate the likeness of an individual...
was introduced to the
Walhalla templeThe Walhalla is a hall of fame for "laudable and distinguished Germans" resp. "famous personalities in German history – politicians, sovereigns, scientists and artists" "of the German tongue", housed in a neo-classical building above the Danube River east of Regensburg, in Bavaria, Germany...
.
Things named in honour of Gauss include:
- The CGS
The centimetre-gram-second system is a metric system of physical units based on centimetre as the unit of length, gram as a unit of mass, and second as a unit of time...
unitA measurement unit is a scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number....
for magnetic inductionFaraday's law of induction is a basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. The law states:...
was named gaussThe gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter.-Unit name and convention:This unit is named after Carl Friedrich Gauss...
in his honour.
- The crater Gauss
Gauss is a large lunar crater, named after Carl Friedrich Gauss, that is located near the northeastern limb of the Moon's near side. It belongs to a category of lunar formations called a walled plain, meaning that it has a diameter of at least 110 kilometers, with a somewhat sunken floor and little...
on the MoonThe Moon is Earth's only natural satellite and the fifth largest satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is , about thirty times the diameter of the Earth. The common centre of mass of the system is located at about —a quarter the Earth's...
- Asteroid
thumb|260px|right|[[253 Mathilde]], a [[C-type asteroid]] measuring about across. Photograph taken in 1997 by the [[NEAR Shoemaker]] probe.Asteroids, sometimes called minor planets or planetoids, are small Solar System bodies in orbit around the Sun, especially in the inner Solar System; they are...
1001 Gaussia1001 Gaussia is a main-belt asteroid orbiting the Sun. Initially it received the designation 1923 OA. Later it was named after the mathematician Carl F. Gauss. It has a mean visual magnitude of 9.77.-External links:*...
.
- The ship Gauss
Gauss was a ship used for the Gauss expedition to Antarctica. led by Arctic veteran and geology professor Erich von Drygalski....
, used in the Gauss expedition to the Antarctic.
- Gaussberg
Gaussberg is an extinct volcanic cone, 370 metres high, fronting on Davis Sea immediately west of the Posadowsky Glacier in Kaiser Wilhelm II Land in Antarctica....
, an extinct volcano discovered by the above mentioned expedition
- Gauss Tower
The Gauss Tower is a reinforced concrete observation tower on the summit of the High Hagens in Dransfeld, Germany. The tower can be reached directly by car...
, an observation tower in DransfeldDransfeld is a town in the district of Göttingen, in Lower Saxony, Germany. It is situated approx. 12 km west of Göttingen.Dransfeld is also the seat of the Samtgemeinde Dransfeld....
, GermanyGermany , officially the Federal Republic of Germany , is a country in Central Europe. It is bordered to the north by the North Sea, Denmark, and the Baltic Sea; to the east by Poland and the Czech Republic; to the south by Austria and Switzerland; and to the west by France, Luxembourg, Belgium,...
.
- In Canadian junior high schools, an annual national mathematics competition administered by the Centre for Education in Mathematics and Computing
The Centre for Education in Mathematics and Computing , founded in 1995 and hosted at the University of Waterloo, administers mathematics and computing contests for Canadian high school students....
is named in honour of Gauss.
- In University of California, Santa Cruz, in Crown College
Crown College is one of the residential colleges that makes up the University of California, Santa Cruz, USA.Despite its thematic grounding in natural science and technology, like at all UCSC colleges, Crown students major in subjects across all disciplines...
, a dormitory building is named after Gauss.
- The Gauss Haus, an NMR
Nuclear magnetic resonance is a property that magnetic nuclei have in a magnetic field and applied electromagnetic pulse, which cause the nuclei to absorb energy from the EM pulse and radiate this energy back out...
center at the University of UtahThe University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
.
- The Carl-Friedrich-Gauß School for Mathematics, Computer Science, Business Administration, Economics, and Social Sciences of University of Braunschweig.
Writings
- 1799: Doctoral dissertation on the Fundamental theorem of algebra
In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, 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
- 1809: Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), English translation by C. H. Davis, reprinted 1963, Dover, New York.
- 1812: Disquisitiones Generales Circa Seriem Infinitam
- 1821, 1823 und 1826: Theoria combinationis observationum erroribus minimis obnoxiae. Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. 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 A.M.Hiltebeitel and J.C.Morehead.
- 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, Harri Deutsch Verlag 2005, mit Anmerkungen von Neumamn, ISBN 978-3-8171-3402-1 (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)
- Gauss' collective works are online here This includes German translations of Latin texts and commentaries by various authorities
See also
- German inventors and discoverers
This is a list of German Inventions and Discoveries of German people or inventors/discoverers of German heritage in alphabetical order of the surname. The main section includes existing articles, indicated by blue links and possibly non-existing, indicated by red links...
- List of topics named after Carl Friedrich Gauss
- Carl Friedrich Gauss Prize
The Carl Friedrich Gauss Prize for Applications of Mathematics is a mathematics award, granted jointly by the International Mathematical Union and the German Mathematical Society for "outstanding mathematical contributions that have found significant applications outside of mathematics". The award...
External links