Carl Friedrich Gauss
Overview
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and scientist
Scientist
A scientist in a broad sense is one engaging in a systematic activity to acquire knowledge. In a more restricted sense, a scientist is an individual who uses the scientific method. The person may be an expert in one or more areas of science. This article focuses on the more restricted use of the word...
who contributed significantly to many fields, including number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, differential geometry
Differential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques 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...
, geodesy
Geodesy
Geodesy , also named 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...
, geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
, electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
and optics
Optics
Optics is the branch of physics which involves 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 referred to as the Princeps mathematicorum (Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
, "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.
Discussions
Quotations
We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
Letter to Friedrich Bessel|Friedrich Wilhelm Bessel (1830)
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 .
Letter to Farkas Bolyai|Farkas Bolyai, on his son János Bolyai|János Bolyai's 1832 publishings on non-Euclidean geometry.
Ask her to wait a moment— I am almost done.
When told, while working, that his wife was dying. As quoted in Men of Mathematics (1937) by E. T. Bell
I have had my results for a long time: but I do not yet know how I am to arrive at them.
The Mind and the Eye (1954) by A. Arber
Encyclopedia
Johann Carl Friedrich Gauss (icon; ) (30 April 1777 23 February 1855) was a German
mathematician
and scientist
who contributed significantly to many fields, including number theory
, statistics
, analysis
, differential geometry
, geodesy
, geophysics
, electrostatics
, astronomy
and optics
.
Sometimes referred to as the Princeps mathematicorum (Latin
, "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".
, in the duchy of Braunschweig-Wolfenbüttel, now part of Lower Saxony
, Germany
, as the son of poor working-class parents. Indeed, 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 itself occurs 40 days after Easter
. Gauss would later solve this puzzle for his birthdate in the context of finding the date of Easter
, deriving methods to compute the date in both past and future years. He was christened and confirmed in a church near the school he attended as a child.
Gauss was a child prodigy
. 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 Arithmeticae
, his magnum opus
, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig
, who sent him to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795, and to the University of Göttingen
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 polygon
with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power
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 Greeks
, and the discovery ultimately led Gauss to choose mathematics instead of philology
as a career.
Gauss was so pleased by this result that he requested that a regular heptadecagon
be inscribed on his tombstone
. The stonemason declined, stating that the difficult construction would essentially look like a circle.
The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon
on March 30. He invented modular arithmetic
, greatly simplifying manipulations in number theory. He became the first to prove the quadratic reciprocity
law on 8 April. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem
, conjectured on 31 May, gives a good understanding of how the 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 number
s on 10 July and then jotted down in his diary
the famous words, "ΕΥΡΗΚΑ
! num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with coefficients in finite field
s, which ultimately led to the Weil conjectures
150 years later.
which states that every non-constant single-variable polynomial
over the complex number
s has at least one root. 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 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
with his 1801 book Disquisitiones Arithmeticae
(Latin
, Arithmetical Investigations), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic
, had the first two proofs of the law of quadratic reciprocity
, developed the theories of binary and ternary quadratic form
s, stated the class number problem for them, and showed that a regular heptadecagon
(17-sided polygon) can be constructed with straightedge and compass.
In that same year, Italian
astronomer Giuseppe Piazzi
discovered the dwarf planet
Ceres. Piazzi had only been able to track Ceres for a few 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 when it was rediscovered by Franz Xaver von Zach on 31 December in Gotha
, and one day later by Heinrich Olbers
in Bremen
.
Gauss's method involved determining a conic section
in space, given one focus (the sun) and the conic's intersection with three given lines (lines of sight from the earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work Gauss used comprehensive approximation methods which he created for that purpose.
One such method was the fast Fourier transform
. While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier
on the subject in 1807.
Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had 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 Göttingen
, a post he held for the remainder of his life.
The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (theory of motion of the celestial bodies moving in conic sections around the sun). In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant
, and contained an influential treatment of the method of least squares
, a procedure used in all sciences to this day to minimize the impact of measurement error
. Gauss was able to prove the method under the assumption of normally distributed errors (see Gauss–Markov theorem
; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre
in 1805, but Gauss claimed that he had been using it since 1795.
In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey
of the state of Hanover
, linking up with previous Danish
surveys. To aid in the survey, Gauss invented the heliotrope
, 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 geometries
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, Einstein
's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai
with whom Gauss had sworn "brotherhood and the banner of truth" as a student had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai
, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." This unproved statement put a strain on his relationship with 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 Dunnington
, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidian geometry long before it was published by János Bolyai
, 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 geometry
, a field of mathematics dealing with curve
s and surface
s. Among other things he came up with the notion of Gaussian curvature
. This led in 1828 to an important theorem, the Theorema Egregium
(remarkable theorem in Latin
), establishing an important property of the notion of curvature
. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angle
s and distance
s on the surface. That is, curvature does not depend on how the surface might be embedded
in 3-dimensional space or 2-dimensional space.
In 1821, he was made a foreign member of the Royal Swedish Academy of Sciences
.
, leading to new knowledge in magnetism
(including finding a representation for the unit of magnetism in terms of mass, length 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
to be built in the garden of the observatory, and with Weber founded the "Magnetischer Verein" (magnetic club in German
), 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 (core
and crust
) and outer (magnetospheric
) sources of Earth's magnetic field.
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
). Among his results, Gauss showed that under a paraxial approximation that an optical system can be characterized by its cardinal points
and he derived the Gaussian lens formula.
In 1854, Gauss notably selected the topic for Bernhard Riemann
's now famous Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.
Gauss died in Göttingen, Hannover (now part of Lower Saxony
, Germany) in 1855 and is interred in the cemetery Albanifriedhof
there. Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewald
and Wolfgang Sartorius von Waltershausen
, who was Gauss's close friend and biographer. His brain was preserved and was studied by Rudolf Wagner
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.
According to Dunnington, Gauss's religion was based upon the search for truth. He believed in "the immortality of the spiritual individuality, in a personal permanence after death, in a last order of things, in an eternal, righteous, omniscient and omnipotent God". Gauss also upheld religious tolerance, believing it wrong to disturb others who were at peace with their own beliefs.
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 shared a good measure of Gauss' talent in languages and computation. Therese kept house for Gauss until his death, after which she married.
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". Gauss wanted Eugene to become a lawyer
, 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, in about 1832, emigrated to the United States, where he was quite successful. Wilhelm also 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.
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. Mathematical historian Eric Temple Bell
estimated that, had Gauss 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
in 1828. However, several of his students became influential mathematicians, among them Richard Dedekind
, Bernhard Riemann
, and Friedrich Bessel
. Before she died, Sophie Germain
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 Arithmeticae
", 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 monarchy
and opposed Napoleon
, whom he saw as an outgrowth of revolution
.
Another famous 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
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
.
Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050.
However, the details of the story are at best uncertain (see for discussion of the original Wolfgang Sartorius von Waltershausen
source and the changes in other versions); some authors, such as Joseph Rotman
in his book A first course in Abstract Algebra, question whether it ever happened.
According to Isaac Asimov
, 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 Dunnington
's Gauss, Titan of Science where it is suggested that it is an apocryphal story.
buildings were featured on the German ten-mark banknote. The reverse featured the heliotrope
and a triangulation
approach for Hannover. 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
.
In 2007 a bust
of Gauss was placed in the Walhalla temple
.
Things named in honor of Gauss include:
In 1929 the Polish
mathematician Marian Rejewski
, who would solve the German Enigma cipher machine
in December 1932, began studying actuarial statistics at Göttingen
. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss's grave.
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and scientist
Scientist
A scientist in a broad sense is one engaging in a systematic activity to acquire knowledge. In a more restricted sense, a scientist is an individual who uses the scientific method. The person may be an expert in one or more areas of science. This article focuses on the more restricted use of the word...
who contributed significantly to many fields, including number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, differential geometry
Differential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques 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...
, geodesy
Geodesy
Geodesy , also named 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...
, geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
, electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
and optics
Optics
Optics is the branch of physics which involves 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 referred to as the Princeps mathematicorum (Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
, "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".
Early years (1777–1798)
Carl Friedrich Gauss was born on April 30, 1777 in BraunschweigBraunschweig
Braunschweig , is a city of 247,400 people, located in the federal-state of 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 duchy of Braunschweig-Wolfenbüttel, now part of Lower Saxony
Lower Saxony
Lower Saxony is a German state situated in north-western Germany and is second in area and fourth in population among the sixteen states of Germany...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, as the son of poor working-class parents. Indeed, 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 itself occurs 40 days after Easter
Easter
Easter is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...
. Gauss would later solve this puzzle for his birthdate in the context of finding the date of Easter
Computus
Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....
, deriving methods to compute the date in both past and future years. He was christened and confirmed in a church near the school he attended as a child.
Gauss was a child prodigy
Child prodigy
A child prodigy is someone who, at an early age, masters one or more skills far beyond his or her level of maturity. One criterion for classifying prodigies is: a prodigy is a child, typically younger than 18 years old, who is performing at the level of a highly trained adult in a very demanding...
. 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 Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
, his magnum opus
Magnum opus
Magnum opus , from the Latin meaning "great work", refers to the largest, and perhaps the best, greatest, most popular, or most renowned achievement of a writer, artist, or composer.-Related terms:Sometimes the term magnum opus is used to refer to simply "a great work" rather than "the...
, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig
Charles William Ferdinand, Duke of Brunswick
Charles William Ferdinand , Duke of Brunswick-Wolfenbüttel, was a sovereign prince of the Holy Roman Empire, and a professional soldier who served as a Generalfeldmarschall of the Kingdom of Prussia...
, who sent him to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795, and to the University of Göttingen
Georg-August University of Göttingen
The University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.Founded in 1734 by King George II of Great Britain and the Elector of Hanover, it opened for classes 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 polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power
Exponentiation
Exponentiation is a mathematical operation, written as 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 Greeks
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
, and the discovery ultimately led Gauss to choose mathematics instead of philology
Philology
Philology is the study of language in written historical sources; it is a combination of literary studies, history and linguistics.Classical philology is the philology of Greek and Classical Latin...
as a career.
Gauss was so pleased by this result that he requested that a regular heptadecagon
Heptadecagon
In 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 at the age of 19....
be inscribed on his tombstone
Headstone
A headstone, tombstone, or gravestone is a marker, usually stone, that is placed over a grave. In most cases they have the deceased's name, date of birth, and date of death inscribed on them, along with a personal message, or prayer.- 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 heptadecagon
Heptadecagon
In 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 at the age of 19....
on March 30. He invented modular arithmetic
Modular arithmetic
In 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 reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
law on 8 April. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
, conjectured on 31 May, gives a good understanding of how the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s are distributed among the integers.
Gauss also discovered that every positive integer is representable as a sum of at most three triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
s on 10 July and then jotted down in his diary
Gauss's diary
Gauss's diary was a record of the mathematical discoveries of C. F. Gauss from 1796 to 1814. It was rediscovered in 1897 and published by , and reprinted in volume X1 of his collected works.-Sample entries:...
the famous words, "ΕΥΡΗΚΑ
Eureka (word)
"Eureka" is an interjection used to celebrate a discovery, a transliteration of a word attributed to Archimedes.-Etymology:The word comes from ancient Greek εὕρηκα heúrēka "I have found ", which is the 1st person singular perfect indicative active of the verb heuriskō "I find"...
! num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with coefficients in finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, which ultimately led to the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
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 algebraFundamental theorem of algebra
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 polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s has at least one root. Mathematicians including Jean le Rond d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. 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 theorem
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
. 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
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
with his 1801 book Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
(Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
, Arithmetical Investigations), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, had the first two proofs of the law of quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
, developed the theories of binary and ternary quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s, stated the class number problem for them, and showed that a regular heptadecagon
Heptadecagon
In 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 at the age of 19....
(17-sided polygon) can be constructed with straightedge and compass.
In that same year, Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
astronomer Giuseppe Piazzi
Giuseppe Piazzi
Giuseppe Piazzi was an Italian Catholic priest of the Theatine order, 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 planet
Dwarf planet
A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...
Ceres. Piazzi had only been able to track Ceres for a few 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 when it was rediscovered by Franz Xaver von Zach on 31 December in Gotha
Gotha Observatory
Gotha Observatory was a German astronomical observatory located on Seeberg hill near Gotha, Thuringia, Germany...
, and one day later by Heinrich Olbers
Heinrich Wilhelm Matthäus Olbers
Heinrich 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 Bremen
Bremen
The City Municipality of Bremen is a Hanseatic city in northwestern Germany. A commercial and industrial city with a major port on the river Weser, Bremen is part of the Bremen-Oldenburg metropolitan area . Bremen is the second most populous city in North Germany and tenth in Germany.Bremen is...
.
Gauss's method involved determining a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
in space, given one focus (the sun) and the conic's intersection with three given lines (lines of sight from the earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work Gauss used comprehensive approximation methods which he created for that purpose.
One such method was the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
. While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
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 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 Göttingen
Göttingen Observatory
Göttingen Observatory is a German astronomical observatory located in Göttingen, Lower Saxony, Germany.-History:...
, a post he held for the remainder of his life.
The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (theory of motion of the celestial bodies moving in conic sections around the sun). In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant
Gaussian gravitational constant
The 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 squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
, a procedure used in all sciences to this day to minimize the impact of measurement error
Observational error
Observational 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 under the assumption of normally distributed errors (see Gauss–Markov theorem
Gauss–Markov theorem
In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator of the coefficients is given by the...
; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
in 1805, but Gauss claimed that he had been using it since 1795.
In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...
of the state of Hanover
Kingdom of Hanover
The 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. It succeeded the former Electorate of Brunswick-Lüneburg , and joined with 38 other sovereign states in the German...
, linking up with previous Danish
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
surveys. To aid in the survey, Gauss invented the heliotrope
Heliotrope (instrument)
The 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 geometries
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
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, Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai
Farkas Bolyai
Farkas Bolyai was a Hungarian mathematician, mainly known for his work in geometry, and of his son János Bolyai.-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 Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
, 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 Dunnington
G. Waldo Dunnington
Guy Waldo Dunnington was a writer, historian and professor of German known for his writings on the famous German mathematician Carl Friedrich Gauss. Dunnington wrote several articles about Gauss and later a biography entitled Gauss: Titan of Science...
, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidian geometry long before it was published by János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
, 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 geometry
Differential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques 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 curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s and surface
Surface
In 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 R3 — for example, the surface of a ball...
s. Among other things he came up with the notion of Gaussian curvature
Gaussian curvature
In 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 surface, not on the way...
. This led in 1828 to an important theorem, the Theorema Egregium
Theorema Egregium
Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
(remarkable theorem in Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
), establishing an important property of the notion of curvature
Curvature
In 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 angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s and distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
s on the surface. That is, curvature does not depend on how the surface might be embedded
Embedding
In 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 Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien 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...
.
Later years and death (1831–1855)
In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm WeberWilhelm Eduard Weber
Wilhelm 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 magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
(including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of Kirchhoff's circuit laws
Kirchhoff's circuit laws
Kirchhoff'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. It was during this time that he formulated his namesake law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
. They constructed the first electromechanical telegraph
Electrical telegraph
An electrical telegraph is a telegraph that uses electrical signals, usually conveyed via telecommunication lines or radio. The electromagnetic telegraph is a device for human-to-human transmission of coded text messages....
in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic observatory
Observatory
An observatory is a location used for observing terrestrial 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 German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
), 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 (core
Planetary core
The planetary core consists of the innermost layer of a planet.The core may be composed of solid and liquid layers, while the cores of Mars and Venus are thought to be completely solid as they lack an internally generated magnetic field. In our solar system, core size can range from about 20% to...
and crust
Crust (geology)
In geology, the crust is the outermost solid shell of a rocky planet or natural satellite, which is chemically distinct from the underlying mantle...
) and outer (magnetospheric
Magnetosphere
A magnetosphere is formed when a stream of charged particles, such as the solar wind, interacts with and is deflected by the intrinsic magnetic field of a planet or similar body. Earth is surrounded by a magnetosphere, as are the other planets with intrinsic magnetic fields: Mercury, Jupiter,...
) sources of Earth's magnetic field.
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
Paraxial approximation
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system ....
(Gaussian optics
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. In this approximation, trigonometric functions can...
). Among his results, Gauss showed that under a paraxial approximation that an optical system can be characterized by its cardinal points
Cardinal point (optics)
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of an ideal, rotationally symmetric, focal, optical system...
and he derived the Gaussian lens formula.
In 1854, Gauss notably selected the topic for Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
's now famous Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.
Gauss died in Göttingen, Hannover (now part of Lower Saxony
Lower Saxony
Lower Saxony is a German state situated in north-western Germany and is second in area and fourth in population among the sixteen states of Germany...
, Germany) in 1855 and is interred in the cemetery Albanifriedhof
Albanifriedhof
Albanifriedhof 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.The cemetery is named after St...
there. Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewald
Heinrich Ewald
Georg 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 Waltershausen
Wolfgang Sartorius von Waltershausen
Wolfgang 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 Wagner
Rudolf Wagner
Rudolf 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.
Religion
Bühler writes that, according to correspondence with Rudolf Wagner, Gauss did not appear to believe in a personal god. He further asserts that although Gauss firmly believed in the immortality of the soul and in some sort of life after death, it was not in a fashion that could be interpreted as Christian.According to Dunnington, Gauss's religion was based upon the search for truth. He believed in "the immortality of the spiritual individuality, in a personal permanence after death, in a last order of things, in an eternal, righteous, omniscient and omnipotent God". Gauss also upheld religious tolerance, believing it wrong to disturb others who were at peace with their own beliefs.
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 depressionClinical depression
Major depressive disorder is a mental disorder characterized by an all-encompassing low mood accompanied by low self-esteem, and by 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 shared a good measure of Gauss' talent in languages and computation. Therese kept house for Gauss until his death, after which she married.
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". Gauss wanted Eugene to become a lawyer
Lawyer
A lawyer, according to Black's Law Dictionary, is "a person learned in the law; as an attorney, counsel or solicitor; a person who is practicing law." Law is the system of rules of conduct established by the sovereign government of a society to correct wrongs, maintain the stability of political...
, 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, in about 1832, emigrated to the United States, where he was quite successful. Wilhelm also settled in Missouri
Missouri
Missouri is a US state located in the Midwestern United States, bordered by Iowa, Illinois, Kentucky, Tennessee, Arkansas, Oklahoma, Kansas and Nebraska. With a 2010 population of 5,988,927, Missouri is the 18th most populous state in the nation and the fifth most populous in the Midwest. It...
, starting as a farmer
Farmer
A farmer is a person engaged in agriculture, who raises living organisms for food or raw materials, generally including livestock husbandry and growing crops, such as produce and grain...
and later becoming wealthy in the shoe business in St. Louis
St. Louis, Missouri
St. Louis is an independent city on the eastern border of Missouri, United States. With a population of 319,294, it was the 58th-largest U.S. city at the 2010 U.S. Census. The Greater St...
. 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 (psychology)
Perfectionism, in psychology, is a belief that a state of completeness and flawlessness 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. 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. Mathematical historian Eric Temple Bell
Eric Temple Bell
Eric 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 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
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
in 1828. However, several of his students became influential mathematicians, among them Richard Dedekind
Richard Dedekind
Julius 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 Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, and Friedrich Bessel
Friedrich Bessel
-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...
. Before she died, Sophie Germain
Sophie Germain
Marie-Sophie Germain was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a gender-biased society, she gained education from books in her father's library and from correspondence with famous mathematicians such as...
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 Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by 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 monarchy
Monarchy
A monarchy is a form of government in which the office of head of state is usually held until death or abdication and is often hereditary and includes a royal house. In some cases, the monarch is elected...
and opposed Napoleon
Napoleon I of France
Napoleon Bonaparte was a French military and political leader during the latter stages of the French Revolution.As Napoleon I, he was Emperor of the French from 1804 to 1815...
, whom he saw as an outgrowth of revolution
Revolution
A 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:...
.
Mythology
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 after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task : add a list of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s in arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is 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 Bartels
Johann Christian Martin Bartels
Johann 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 Waltershausen
Wolfgang Sartorius von Waltershausen
Wolfgang 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
Joseph Rotman
Joseph Louis Rotman, O.C., LL.D. , is a noted Canadian businessman and philanthropist. Rotman has been the founder, benefactor and member of many successful organizations, such as the Clairvest Group Inc., the Rotman Research Institute, and the Rotman School of Management...
in his book A first course in Abstract Algebra, question whether it ever happened.
According to Isaac Asimov
Isaac Asimov
Isaac Asimov was an American author and professor of biochemistry at Boston University, best known for his works of science fiction and for his popular science books. Asimov was one of the most prolific writers of all time, having written or edited more than 500 books and an estimated 90,000...
, 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 Dunnington
G. Waldo Dunnington
Guy Waldo Dunnington was a writer, historian and professor of German known for his writings on the famous German mathematician Carl Friedrich Gauss. Dunnington wrote several articles about Gauss and later a biography entitled Gauss: Titan of Science...
's Gauss, Titan of Science where it is suggested that it is an apocryphal story.
Commemorations
From 1989 through 2001, Gauss's portrait, a normal distribution curve and some prominent GöttingenGöttingen
Göttingen is a university 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:...
buildings were featured on the German ten-mark banknote. The reverse featured the heliotrope
Heliotrope (instrument)
The 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 triangulation
Triangulation
In 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 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
Daniel Kehlmann
Daniel Kehlmann is a German language author of both Austrian and German nationality. His work Die Vermessung der Welt is the best 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 World
Measuring the World
Measuring the World is a 2005 novel by German author Daniel Kehlmann. The novel re-imagines the lives of German mathematician Carl Friedrich Gauss and German geographer Alexander von Humboldt – who was accompanied on his journeys by Aimé Bonpland – and their many groundbreaking ways of...
(2006), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer Alexander von Humboldt
Alexander von Humboldt
Friedrich Wilhelm Heinrich Alexander Freiherr von Humboldt was a German naturalist and explorer, and the younger brother of the Prussian minister, philosopher and linguist Wilhelm von Humboldt...
.
In 2007 a bust
Bust (sculpture)
A 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...
of Gauss was placed in the Walhalla temple
Walhalla temple
The Walhalla temple is a hall of fame that honors laudable and distinguished Germans, famous personalities in German history — politicians, sovereigns, scientists and artists of the German tongue". The hall is housed in a neo-classical building above the Danube River, east of Regensburg, in...
.
Things named in honor of Gauss include:
- The CGSCentimetre gram second system of unitsThe 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...
unitUnits of measurementA unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
for magnetic fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
was named gaussGauss (unit)The 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; it equals 1 tesla...
in his honour, - The crater GaussGauss (crater)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 MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
, - AsteroidAsteroidAsteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
1001 Gaussia1001 Gaussia-External links:*...
, - The ship GaussGauss (ship)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, - GaussbergGaussbergGaussberg 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 TowerGauss TowerThe 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 DransfeldDransfeldDransfeld 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....
, GermanyGermanyGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, - In Canadian junior high schools, an annual national mathematics competition (Gauss Mathematics Competition) administered by the Centre for Education in Mathematics and ComputingCentre for Education in Mathematics and ComputingThe Centre for Education in Mathematics and Computing , has become Canada's largest and most recognized outreach organization for promoting and creating activities and materials in mathematics and computer science. The CEMC is housed within the Faculty of Mathematics at the University of Waterloo...
is named in honour of Gauss, - In University of California, Santa Cruz, in Crown CollegeCrown College, University of California, Santa CruzCrown 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 him, - The Gauss Haus, an NMRNuclear magnetic resonanceNuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
center at the University of UtahUniversity 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,
- The Gauss Building - University of IdahoUniversity of IdahoThe University of Idaho is the State of Idaho's flagship and oldest public university, located in the rural city of Moscow in Latah County in the northern portion of the state...
(College of Engineering).
In 1929 the Polish
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
mathematician Marian Rejewski
Marian Rejewski
Marian Adam Rejewski was a Polish mathematician and cryptologist who in 1932 solved the plugboard-equipped Enigma machine, the main cipher device used by Germany...
, who would solve the German Enigma cipher machine
Enigma machine
An Enigma machine is any of a family of related electro-mechanical rotor cipher machines used for the encryption and decryption of secret messages. Enigma was invented by German engineer Arthur Scherbius at the end of World War I...
in December 1932, began studying actuarial statistics at Göttingen
Göttingen
Göttingen is a university 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:...
. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss's grave.
Writings
- 1799: Doctoral dissertation on the Fundamental theorem of algebraFundamental theorem of algebraThe 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. German translation by H. Maser , pp. 1–453. English translation by Arthur A. Clarke .
- 1808: . German translation by H. Maser , pp. 457–462 [Introduces Gauss's lemmaGauss's lemma (number theory)Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity....
, 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), English translation by C. H. Davis, reprinted 1963, Dover, New York.
- 1811: . German translation by H. Maser , pp. 463–495 [Determination of the sign of the quadratic Gauss sumQuadratic Gauss sumIn number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general...
, 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 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.
- 1828: . German translation by H. Maser , pp. 511–533 [Elementary facts about biquadratic residues, proves one of the supplements of the law of biquadratic reciprocity (the biquadratic character of 2)]
- 1832: . German translation by H. Maser , pp. 534–586 [Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i]
- 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
- Romanticism in scienceRomanticism in scienceRomanticism, also known as the “Age of Reflection,” describes the intellectual movement from 1800-1840 that originated in Western Europe as a counter-movement to the Enlightenment of the late 18th century...
- German inventors and discoverersGerman inventors and discoverersThis is a list of German inventors and discoverers. The following list comprises people from Germany or German-speaking Europe, also of people of predominantly German heritage, in alphabetical order of the surname. The main section includes existing articles, indicated by blue links, and possibly...
- List of topics named after Carl Friedrich Gauss
- Carl Friedrich Gauss PrizeCarl Friedrich Gauss PrizeThe 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
- Complete works
- Gauss and his children
- Gauss biography
- Carl Friedrich Gauss, Biography at Fermat's Last Theorem Blog.
- Gauss: mathematician of the millennium, by Jürgen SchmidhuberJürgen SchmidhuberJürgen Schmidhuber is a computer scientist and artist known for his work on machine learning, universal Artificial Intelligence , artificial neural networks, digital physics, and low-complexity art. His contributions also include generalizations of Kolmogorov complexity and the Speed Prior...
- English translation of Waltershausen's 1862 biography
- Gauss general website on Gauss
- MNRAS 16 (1856) 80 Obituary
- Carl Friedrich Gauss on the 10 Deutsche Mark banknote
- Carl Friedrich Gauss at WikiquoteWikiquoteWikiquote is one of a family of wiki-based projects run by the Wikimedia Foundation, running on MediaWiki software. Based on an idea by Daniel Alston and implemented by Brion Vibber, the goal of the project is to produce collaboratively a vast reference of quotations from prominent people, books,...
- 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]
- 1808: . German translation by H. Maser , pp. 457–462 [Introduces Gauss's lemma