Johann Carl Friedrich Gauss
Article
by: J J O'Connor and E F Robertson (University of St. Andrews, Scotland)
Johann
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 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 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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