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Carl Friedrich Gauss

2007 Schools Wikipedia Selection. Related subjects: Mathematicians

   CAPTION: Johann Carl Friedrich Gauss

   Carl Friedrich Gauss
   Carl Friedrich Gauss
         Born        30 April 1777
                     Brunswick, Germany
         Died        23 February 1855
                     Göttingen, Hanover, Germany
       Residence     Germany
      Nationality    German
         Field       Mathematician and physicist
      Institution    Georg-August University
      Alma Mater     Helmstedt University
   Doctoral Advisor  Johann Friedrich Pfaff
   Doctoral Students Friedrich Bessel
                     Christoph Gudermann
       Known for     Number theory
                     The Gaussian
                     Magnetism

   Carl Friedrich Gauss (Gauß)  ( 30 April 1777 – 23 February 1855) was a
   German mathematician and scientist of profound genius who contributed
   significantly to many fields, including number theory, analysis,
   differential geometry, geodesy, magnetism, astronomy and optics.
   Sometimes known as "the prince 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.

   Gauss was a child prodigy, of whom there are many anecdotes pertaining
   to his astounding precocity while a mere toddler, and made his first
   ground-breaking mathematical discoveries while still a teenager. He
   completed Disquisitiones Arithmeticae, his magnum opus, at the age of
   twenty-one (1798), though it would not be published until 1801.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.

Biography

Early years

   Statue of Gauss in Brunswick
   Enlarge
   Statue of Gauss in Brunswick

   Gauss was born in Brunswick, in the Duchy of Brunswick-Lüneburg (now
   part of Lower Saxony, Germany), as the only son of uneducated
   lower-class parents. According to legend, his gifts became very
   apparent at the age of three when he corrected, in his head, an error
   his father had made on paper while calculating finances.

   Another famous story, and one that has evolved in the telling, has it
   that in primary school his teacher, J.G. Büttner tried to occupy pupils
   by making them add up the integers from 1 to 100. The young Gauss
   produced the correct answer within seconds by a flash of mathematical
   insight, to the astonishment of all. Gauss had realized 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 (see arithmetic series and
   summation). (For more information, see for discussion of original
   Wolfgang Sartorius von Waltershausen source.)

   The Duke of Brunswick awarded Gauss a fellowship to the Collegium
   Carolinum (now Technische Universität Braunschweig), which he attended
   from 1792 to 1795, and from there went on to the University of
   Göttingen from 1795 to 1798. While in college, 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. 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.

   1796 was probably the most productive year for both Gauss and number
   theory. The construction of the heptadecagon was discovered on March
   30. He invented modular arithmetic, greatly simplifying manipulations
   in number theory. He became the first to prove the quadratic
   reciprocity law on April 8. This remarkably general law (the previous
   discovery of which by Legendre in 1788 was unknown to Gauss) allows
   mathematicians to determine the solvability of any quadratic equation
   in modular arithmetic. The prime number theorem, conjectured on May 31,
   gives a good understanding of how the prime numbers are distributed
   among the integers. Gauss also discovered that every positive integer
   is representable as a sum of at most three triangular numbers on July
   10 and then jotted down in his diary the famous words, " Heureka! num=
   Δ + Δ + Δ." On October 1 he published a result on the number of
   solutions of polynomials with coefficients in finite fields (this
   ultimately led to the Weil conjectures 150 years later).

Middle years

   Title page of Gauss's Disquisitiones Arithmeticae
   Enlarge
   Title page of Gauss's Disquisitiones Arithmeticae

   In his 1799 dissertation, A New Proof That Every Rational Integer
   Function of One Variable Can Be Resolved into Real Factors of the First
   or Second Degree, Gauss gave a proof of the fundamental theorem of
   algebra. This important theorem states that every polynomial over the
   complex numbers must have at least one root. Other mathematicians had
   tried to prove this before him, e.g. Jean le Rond d'Alembert. Gauss's
   dissertation contained a critique of d'Alembert's proof, but his own
   attempt would not be accepted owing to implicit use of the Jordan curve
   theorem. Gauss over his lifetime produced three more proofs, probably
   due in part to this rejection of his dissertation; his last proof in
   1849 is generally considered rigorous by today's standard. 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, which contained a clean presentation
   of modular arithmetic and the first proof of the law of quadratic
   reciprocity. In that same year, Italian astronomer Giuseppe Piazzi
   discovered the planetoid 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 Zach on December 31,
   1801 in Gotha, and one day later by Heinrich Olbers in Bremen. Zach
   noted that "without the intelligent work and calculations of Doctor
   Gauss we might not have found Ceres again." Though Gauss had up to this
   point been 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 by Piazzi on January 1, 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 couldn't locate it:
   the mathematical tools of the time weren't 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
   head-on. 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 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 in 1809 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.

   Gauss had been asked in the late 1810s to carry out a geodetic survey
   of the state of Hanover to link up with the existing Danish grid. Gauss
   was pleased to accept and took personal charge of the survey, making
   measurements during the day and reducing them at night, using his
   extraordinary mental capacity for calculations. He regularly wrote to
   Schumacher, Olbers and Bessel, reporting on his progress and discussing
   problems. As part of the survey, Gauss invented the heliotrope which
   worked by reflecting the Sun's rays using a set of mirrors and a small
   telescope.

   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 nowadays
   generally taken at face value.
   Gaussian distribution in statistics.
   Enlarge
   Gaussian distribution in statistics.

   The survey of Hanover later led to the development of the Gaussian
   distribution, also known as the normal distribution, for describing
   measurement errors. Moreover, it fuelled Gauss's interest in
   differential geometry, a field of mathematics dealing with curves and
   surfaces. In this field, he came up with an important theorem, the
   theorema egregrium (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 angles and distances on the surface; that is, curvature does
   not depend on how the surface might be embedded in (3-dimensional)
   space.

Later years, death, and afterwards

   In 1831 Gauss developed a fruitful collaboration with the physics
   professor Wilhelm Weber; it led to new knowledge in the field of
   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. Gauss and Weber constructed the first
   electromagnetic 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"), 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.

   Gauss died in Göttingen, Hanover (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
   Robert Wagner who found its weight to be 1,492 grams and the cerebral
   area equal to 219,588 square centimetres. There were also found highly
   developed convolutions, which in the early 20th century was suggested
   as the explanation of his genius (Dunnington, 1927).

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 depression from which he never fully
   recovered. He married again, to a friend of his first wife named
   Friederica Wilhelmine Waldeck (Minna), but this second marriage does
   not seem to have been very happy. 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, three by each wife. 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 immigrated to the United States about 1832
   after a falling out with his father, eventually settling in St.
   Charles, Missouri, where he became a well respected member of the
   community. Wilhelm came to settle in Missouri somewhat later, starting
   as a farmer and later becoming wealthy in the shoe business in St.
   Louis. Therese kept house for Gauss until his death, after which she
   married.

Personality

   Gauss was an ardent perfectionist and a hard worker. There is a famous
   anecdote of Gauss being 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 'til I'm through". 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). A study of his personal diaries reveal that he had in
   fact discovered several important mathematical concepts years or
   decades before they were published by his contemporaries. Prominent
   mathematical historian Eric Temple Bell estimated that had Gauss made
   known all of his discoveries, mathematics would have been advanced by
   fifty years. (Bell, 1937.)

   Another criticism of Gauss is that he did not support the younger
   mathematicians who followed him. He rarely, if ever, collaborated with
   other mathematicians and was considered aloof and austere by many.
   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 turned out to be 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 generally did not get along with his male relatives. His father
   had wanted him to follow in his footsteps, i.e., to become a mason. He
   was not supportive of Gauss's schooling in mathematics and science.
   Gauss was primarily supported by his mother in this effort. Likewise,
   he 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". His conflict with Eugene was
   particularly bitter. 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
   immigrated 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 September 3, 1912.

   Unlike modern mathematicians, 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.

   Gauss was deeply religious and conservative. He supported monarchy and
   opposed Napoleon whom he saw as an outgrowth of revolution.

Commemorations

   The cgs unit for magnetic induction was named gauss in his honour.

   From 1989 until the end of 2001, his portrait and a normal distribution
   curve were featured on the German ten-mark banknote. Germany has issued
   three stamps honoring Gauss, as well. A 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.

   G. Waldo Dunnington was a lifelong student of Gauss. He wrote many
   articles, and a biography: Carl Frederick Gauss: Titan of Science. This
   book was reissued in 2003, after having been out of print for almost 50
   years.

   In 2007, his bust is going to be introduced to the Walhalla.

   Places, vessels and events named in honour of Gauss:
     * Gauss crater on the Moon
     * Asteroid 1001 Gaussia.
     * The First German Antarctica Expedition's ship Gauss
     * Gaussberg, an extinct volcano discovered by the above mentioned
       expedition
     * Gauss Tower, an observation tower
     * In Canadian junior high schools, an annual national mathematics
       competition administered by the Centre for Education in Mathematics
       and Computing is named in honour of Gauss.

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