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
