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Carl Friedrich Gauss
Born: 30
April 1777 in Brunswick, Duchy of Brunswick (now Germany)
Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)
At the age of seven, Carl
Friedrich Gauss 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 1820s.
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 1800s 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 E3
can be developed (i.e. mapped isometrically) into another area of
E3, 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 Über 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 principle 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.
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