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                          JOHANN CARL FRIEDRICH GAUSS
                                       
   
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  Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany)
  Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)
  
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     Carl Friedrich Gauss worked in a wide variety of fields in both
     mathematics and physics incuding number theory, analysis,
     differential geometry, geodesy, magnetism, astronomy and optics. His
     work has had an immense influence in many areas.
     
   At the age of seven, Carl Friedrich started elementary school, and his
   potential was noticed almost immediately. His teacher, Büttner, and
   his assistant, Martin Bartels, were amazed when Gauss summed the
   integers from 1 to 100 instantly by spotting that the sum was 50 pairs
   of numbers each pair summing to 101.
   
   In 1788 Gauss began his education at the Gymnasium with the help of
   Büttner and Bartels, where he learnt High German and Latin. After
   receiving a stipend from the Duke of Brunswick- Wolfenbüttel, Gauss
   entered Brunswick Collegium Carolinum in 1792. At the academy Gauss
   independently discovered Bode's law, the binomial theorem and the
   arithmetic- geometric mean, as well as the law of quadratic
   reciprocity and the prime number theorem.
   
   In 1795 Gauss left Brunswick to study at Göttingen University. Gauss's
   teacher there was Kaestner, whom Gauss often ridiculed. His only known
   friend amongst the students was Farkas Bolyai. They met in 1799 and
   corresponded with each other for many years.
   
   Gauss left Göttingen in 1798 without a diploma, but by this time he
   had made one of his most important discoveries - the construction of a
   regular 17-gon by ruler and compasses This was the most major advance
   in this field since the time of Greek mathematics and was published as
   Section VII of Gauss's famous work, Disquisitiones Arithmeticae .
   
   Gauss returned to Brunswick where he received a degree in 1799. After
   the Duke of Brunswick had agreed to continue Gauss's stipend, he
   requested that Gauss submit a doctoral dissertation to the University
   of Helmstedt. He already knew Pfaff, who was chosen to be his advisor.
   Gauss's dissertation was a discussion of the fundamental theorem of
   algebra.
   
   With his stipend to support him, Gauss did not need to find a job so
   devoted himself to research. He published the book Disquisitiones
   Arithmeticae in the summer of 1801. There were seven sections, all but
   the last section, referred to above, being devoted to number theory.
   
   In June 1801, Zach, an astronomer whom Gauss had come to know two or
   three years previously, published the orbital positions of Ceres, a
   new "small planet" which was discovered by G Piazzi, an Italian
   astronomer on 1 January, 1801. Unfortunately, Piazzi had only been
   able to observe 9 degrees of its orbit before it disappeared behind
   the Sun. Zach published several predictions of its position, including
   one by Gauss which differed greatly from the others. When Ceres was
   rediscovered by Zach on 7 December 1801 it was almost exactly where
   Gauss had predicted. Although he did not disclose his methods at the
   time, Gauss had used his least squares approximation method.
   
   In June 1802 Gauss visited Olbers who had discovered Pallas in March
   of that year and Gauss investigated its orbit. Olbers requested that
   Gauss be made director of the proposed new observatory in Göttingen,
   but no action was taken. Gauss began corresponding with Bessel, whom
   he did not meet until 1825, and with Sophie Germain.
   
   Gauss married Johanna Ostoff on 9 October, 1805. Despite having a
   happy personal life for the first time, his benefactor, the Duke of
   Brunswick, was killed fighting for the Prussian army. In 1807 Gauss
   left Brunswick to take up the position of director of the Göttingen
   observatory.
   
   Gauss arrived in Göttingen in late 1807. In 1808 his father died, and
   a year later Gauss's wife Johanna died after giving birth to their
   second son, who was to die soon after her. Gauss was shattered and
   wrote to Olbers asking him give him a home for a few weeks,
   
     to gather new strength in the arms of your friendship - strength for
     a life which is only valuable because it belongs to my three small
     children.
     
   Gauss was married for a second time the next year, to Minna the best
   friend of Johanna, and although they had three children, this marriage
   seemed to be one of convenience for Gauss.
   
   Gauss's work never seemed to suffer from his personal tragedy. He
   published his second book, Theoria motus corporum coelestium in
   sectionibus conicis Solem ambientium, in 1809, a major two volume
   treatise on the motion of celestial bodies. In the first volume he
   discussed differential equations, conic sections and elliptic orbits,
   while in the second volume, the main part of the work, he showed how
   to estimate and then to refine the estimation of a planet's orbit.
   Gauss's contributions to theoretical astronomy stopped after 1817,
   although he went on making observations until the age of 70.
   
   Much of Gauss's time was spent on a new observatory, completed in
   1816, but he still found the time to work on other subjects. His
   publications during this time include Disquisitiones generales circa
   seriem infinitam , a rigorous treatment of series and an introduction
   of the hypergeometric function, Methodus nova integralium valores per
   approximationem inveniendi , a practical essay on approximate
   integration, Bestimmung der Genauigkeit der Beobachtungen , a
   discussion of statistical estimators, and Theoria attractionis
   corporum sphaeroidicorum ellipticorum homogeneorum methodus nova
   tractata . The latter work was inspired by geodesic problems and was
   principally concerned with potential theory. In fact, Gauss found
   himself more and more interested in geodesy in the 1820's.
   
   Gauss had been asked in 1818 to carry out a geodesic 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.
   
   Because of the survey, Gauss invented the heliotrope which worked by
   reflecting the Sun's rays using a design of mirrors and a small
   telescope. However, inaccurate base lines were used for the survey and
   an unsatisfactory network of triangles. Gauss often wondered if he
   would have been better advised to have pursued some other occupation
   but he published over 70 papers between 1820 and 1830.
   
   In 1822 Gauss won the Copenhagen University Prize with Theoria
   attractionis... together with the idea of mapping one surface onto
   another so that the two are similar in their smallest parts . This
   paper was published in 1825 and led to the much later publication of
   Untersuchungen über Gegenstände der Höheren Geodäsie (1843 and 1846).
   The paper Theoria combinationis observationum erroribus minimis
   obnoxiae (1823), with its supplement (1828), was devoted to
   mathematical statistics, in particular to the least squares method.
   
   From the early 1800's Gauss had an interest in the question of the
   possible existence of a non-Euclidean geometry. He discussed this
   topic at length with Farkas Bolyai and in his correspondence with
   Gerling and Schumacher. In a book review in 1816 he discussed proofs
   which deduced the axiom of parallels from the other Euclidean axioms,
   suggesting that he believed in the existence of non-Euclidean
   geometry, although he was rather vague. Gauss confided in Schumacher,
   telling him that he believed his reputation would suffer if he
   admitted in public that he believed in the existence of such a
   geometry.
   
   In 1831 Farkas Bolyai sent to Gauss his son János Bolyai's work on the
   subject. Gauss replied
   
     to praise it would mean to praise myself .
     
   Again, a decade later, when he was informed of Lobachevsky's work on
   the subject, he praised its "genuinely geometric" character, while in
   a letter to Schumacher in 1846, states that he
   
     had the same convictions for 54 years
     
   indicating that he had known of the existence of a non-Euclidean
   geometry since he was 15 years of age (this seems unlikely).
   
   Gauss had a major interest in differential geometry, and published
   many papers on the subject. Disquisitiones generales circa superficies
   curva (1828) was his most renowned work in this field. In fact, this
   paper rose from his geodesic interests, but it contained such
   geometrical ideas as Gaussian curvature. The paper also includes
   Gauss's famous theorema egregrium:
   
     If an area in E ^3 can be developed (i.e. mapped isometrically) into
     another area of E ^3 , the values of the Gaussian curvatures are
     identical in corresponding points.
     
   The period 1817-1832 was a particularly distressing time for Gauss. He
   took in his sick mother in 1817, who stayed until her death in 1839,
   while he was arguing with his wife and her family about whether they
   should go to Berlin. He had been offered a position at Berlin
   University and Minna and her family were keen to move there. Gauss,
   however, never liked change and decided to stay in Göttingen. In 1831
   Gauss's second wife died after a long illness.
   
   In 1831, Wilhelm Weber arrived in Göttingen as physics professor
   filling Tobias Mayer's chair. Gauss had known Weber since 1828 and
   supported his appointment. Gauss had worked on physics before 1831,
   publishing Uber ein neues allgemeines Grundgesetz der Mechanik , which
   contained the principle of least constraint, and Principia generalia
   theoriae figurae fluidorum in statu aequilibrii which discussed forces
   of attraction. These papers were based on Gauss's potential theory,
   which proved of great importance in his work on physics. He later came
   to believe his potential theory and his method of least squares
   provided vital links between science and nature.
   
   In 1832, Gauss and Weber began investigating the theory of terrestrial
   magnetism after Alexander von Humboldt attempted to obtain Gauss's
   assistance in making a grid of magnetic observation points around the
   Earth. Gauss was excited by this prospect and by 1840 he had written
   three important papers on the subject: Intensitas vis magneticae
   terrestris ad mensuram absolutam revocata (1832), Allgemeine Theorie
   des Erdmagnetismus (1839) and Allgemeine Lehrsätze in Beziehung auf
   die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden
   Anziehungs- und Abstossungskräfte (1840). These papers all dealt with
   the current theories on terrestrial magnetism, including Poisson's
   ideas, absolute measure for magnetic force and an empirical definition
   of terrestrial magnetism. Dirichlet's principal was mentioned without
   proof.
   
   Allgemeine Theorie... showed that there can only be two poles in the
   globe and went on to prove an important theorem, which concerned the
   determination of the intensity of the horizontal component of the
   magnetic force along with the angle of inclination. Gauss used the
   Laplace equation to aid him with his calculations, and ended up
   specifying a location for the magnetic South pole.
   
   Humboldt had devised a calendar for observations of magnetic
   declination. However, once Gauss's new magnetic observatory (completed
   in 1833 - free of all magnetic metals) had been built, he proceeded to
   alter many of Humboldt's procedures, not pleasing Humboldt greatly.
   However, Gauss's changes obtained more accurate results with less
   effort.
   
   Gauss and Weber achieved much in their six years together. They
   discovered Kirchhoff's laws, as well as building a primitive telegraph
   device which could send messages over a distance of 5000 ft. However,
   this was just an enjoyable pastime for Gauss. He was more interested
   in the task of establishing a world-wide net of magnetic observation
   points. This occupation produced many concrete results. The
   Magnetischer Verein and its journal were founded, and the atlas of
   geomagnetism was published, while Gauss and Weber's own journal in
   which their results were published ran from 1836 to 1841.
   
   In 1837, Weber was forced to leave Göttingen when he became involved
   in a political dispute and, from this time, Gauss's activity gradually
   decreased. He still produced letters in response to fellow scientists'
   discoveries usually remarking that he had known the methods for years
   but had never felt the need to publish. Sometimes he seemed extremely
   pleased with advances made by other mathematicians, particularly that
   of Eisenstein and of Lobachevsky.
   
   Gauss spent the years from 1845 to 1851 updating the Göttingen
   University widow's fund. This work gave him practical experience in
   financial matters, and he went on to make his fortune through shrewd
   investments in bonds issued by private companies.
   
   Two of Gauss's last doctoral students were Moritz Cantor and Dedekind.
   Dedekind wrote a fine description of his supervisor
   
     ... usually he sat in a comfortable attitude, looking down, slightly
     stooped, with hands folded above his lap. He spoke quite freely,
     very clearly, simply and plainly: but when he wanted to emphasise a
     new viewpoint ... then he lifted his head, turned to one of those
     sitting next to him, and gazed at him with his beautiful,
     penetrating blue eyes during the emphatic speech. ... If he
     proceeded from an explanation of principles to the development of
     mathematical formulas, then he got up, and in a stately very upright
     posture he wrote on a blackboard beside him in his peculiarly
     beautiful handwriting: he always succeeded through economy and
     deliberate arrangement in making do with a rather small space. For
     numerical examples, on whose careful completion he placed special
     value, he brought along the requisite data on little slips of paper.
     
   Gauss presented his golden jubilee lecture in 1849, fifty years after
   his diploma had been granted by Hemstedt University. It was
   appropriately a variation on his dissertation of 1799. From the
   mathematical community only Jacobi and Dirichlet were present, but
   Gauss received many messages and honours.
   
   From 1850 onwards Gauss's work was again of nearly all of a practical
   nature although he did approve Riemann's doctoral thesis and heard his
   probationary lecture. His last known scientific exchange was with
   Gerling. He discussed a modified Foucalt pendulum in 1854. He was also
   able to attend the opening of the new railway link between Hanover and
   Göttingen, but this proved to be his last outing. His health
   deteriorated slowly, and Gauss died in his sleep early in the morning
   of 23 February, 1855.
   
   References (67 books/articles)
   
   Some pages from works by Gauss:
   
   A letter from Gauss to Taurinus discussing the possibility of
   non-Euclidean geometry.
   An extract from Theoria residuorum biquadraticorum 
   (1828-32)
   
   References elsewhere in this archive:
   
   You can see another picture of Gauss in 1803.
   
   Tell me about the Prime Number Theorem
   
   Show me Gauss's estimate for the density of primes and compare it with
   Legendre's
   
   Tell me about Gauss's part in investigating prime numbers
   
   Tell me about Gauss's part in the development of group theory and
   matrices and determinants
   
   Tell me about his work on non-Euclidean geometry and topology
   
   Tell me about Gauss's work on the fundamental theorem of algebra
   
   Tell me about his work on orbits and gravitation
   
   Other Web sites:
   
   You can find out about the Prime Number Theorem at University of
   Tennessee, USA
   
   
   
   
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   JOC/EFR December 1996