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Johann Carl Friedrich Gauss: A 10 Mark German Banknote issued commemorating his
contributions in 1990 which was in circulation till 2001:
About
Johann Carl Friedrich Gauss (30.04.1777-23.02.1855)
Johann Carl Friedrich
Gauss was a German mathematician who made significant contributions to various
fields including the Number theory, Algebra, Statistics, Differential Geometry,
Geodesy, Electrostatics, Astronomy, Matrix Theory and Optics.
He is often referred to as
the “princeps mathematicorum”
(meaning the “Prince of Mathematics”). He is also referred to as “the foremost
of Mathematicians” or “the greatest Mathematician since antiquity”.
These accolades have been
showered upon him because of his immense contributions in several fields of
Mathematics and Science.
Born to poor, illiterate
parents who could not even write down his date of birth, Gauss was a child
prodigy and was educated.
His
Early Studies and Path-breaking work:
At the age of 14, his
work in ground-breaking discoveries
in Mathematical Theory attracted the attention of the Duke of Brunswick,
Carl Wilhelm, who became his patron and supported his intellectual pursuits and
his higher education for the next 16 years.
In 1792, by the
time, Gauss joined the “Brunswick
Collegium Carolinum” on a scholarship, he possessed a scientific and
classical education far beyond that usual for his age at that time.
He
was already familiar with Elementary Geometry, Algebra and Analysis (often
having discovered important theorems before reaching them in his studies), but
in addition he possessed a wealth of Arithmetical information and many Number-theoretic
insights.
Till 1795, he
studied at the Collegium where he continued his Empirical Arithmetic and on
one occasion found the square root in two different ways to 50 Decimal
places by ingenuous expansions and interpolations. He also formulated the Principle of Least Squares.
In 1795, by the
time he joined the University of
Gottingen, where he studied till 1798, he had rediscovered the Law of Quadratic Reciprocity
(conjectured by Lagrange in 1785), independently rediscovered several important
theorems, related the Arithmetic-Geometric
Mean to Infinite Series Expansions and conjectured the Prime Number Theorem.
In 1796, he showed
that a regular polygon can be
constructed by compass and straightedge if the number of sides is the
product of distinct Fermat primes and a power of 2. This was a major discovery
in an important field of Mathematics. Some of his most important findings had
practical implications, as he proposed several theorems on shapes that have a direct impact on Architecture and
Construction as Construction problems had engaged Mathematicians since
the days of the ancient Greeks and this discovery helped immensely in finding
solutions.
Also, in 1796, he
discovered a construction of the
Heptadecagon (17-sided figure). He was the first Mathematician to
construct a 17-sided Heptadecagon using a compass and a straight edge and was
the first to prove the Laws of
Quadratic Reciprocity.
His Prime
Numbers Theorem gave an understanding of how Prime Numbers are
distributed among the integers. His theorem broadly applies to Mathematics even
today.
His most influential writing was drafted when he was only 21
years of age and still defines the understanding of Number theory even today.
He also discovered that every
positive integer is representable as a sum of at most three triangular numbers.
He published several solutions, which 150 years later, led to the “Weil Conjectures”.
In 1799, in his
doctorate “A new proof of the theorem
that every integral rational algebraic function of one variable can be resolved
into real factors of the first or second degree” he proved the Fundamental Theorem of Algebra. His
work was instrumental to the understanding of Algebra, as he proved its central
theorem which states that “every non-constant single-variable polynomial with
complex coefficients has at least one complex root”.
In 1801, he
completed “Disquisitiones Arithmeticae”, his Magnum Opus
which, inter alia, introduced a symbol for Congruence and used it in a clean
presentation of modular Arithmetic, contained the first two proofs of the law
of Quadratic forms, stated the class number problem for them and showed that a
regular heptadecagon (17-sided polygon) can be constructed with straightedge
and compass.
Before the age of 25, he
had already achieved fame as a Mathematician and an Astronomer.
One of his most important contributions to Astronomy
stemmed from using conic equations to track dwarf planet Ceres, whose own
discoverer Giuseppe Piazzi could not locate it months after its discovery due
to limitations of available tools.
In 1809, in “Theoria
motus corporum coelestium in sectionibus conicis solem ambientum”
(Theory of Motion of the celestial bodies moving in conic sections around the
Sun), he discussed the motion of planetoids disturbed by large planets. His
work on conic sections originating from the position of the Sun, replaced the
difficult Mathematical formulae that had been used in astronomy until then and
so streamlined the cumbersome Mathematics of the 18^{th} Century Orbital prediction, that it
remains a cornerstone of Astronomical computation, even today.
He introduced the “Gaussian Gravitational Constant”
and his work contained an influential treatment of the method of least squares,
a procedure still used in all Sciences to minimise the impact of measurement
error.
This brilliant work in
Astronomy led Gauss to be appointed on the position of Head of Astronomy at the
Observatory in Gottingen, which enabled him to take his work on planetary
motion forward.
In 1818, he carried
out a geodetic survey of the
Kingdom of Hanover, linking up with previous Danish surveys. For this purpose,
he invented the “Heliotrope”,
an instrument that uses a mirror to reflect sunlight over great distances to
measure positions. Among other researches, he came up with the concept of Gaussian curvature, which led to
a 1828 Theorem – the “Theorema Egregium” (meaning
“Remarkable Theorem”) – which established an important property of the notion
of curvature. Essentially, the Theorem puts forth that the curvature of a
surface can be determined entirely by measuring angles and distances on the
surface. In other words, curvature does not depend on how the surface might be
embedded in 3-Dimensional or 2-Dimensional space.
He claimed to have found some results that would hold
if “Euclidean Geometry were not the true one” thereby leading to the possibility
of non-Euclidean Geometries
but his research was never published, in deference to Euclid’s age-old works. Nevertheless,
this discovery marked a major paradigm shift in Mathematics, as it freed Mathematics from the
erroneous notion that Euclid’s axioms were the only way to make Geometry
consistent and non-contradictory.
In 1821, he was
made a foreign member of the Royal
Swedish Academy of Sciences.
In 1831, he
collaborated with a Physics Professor Wilhelm Weber, leading to new knowledge in Magnetism
(including finding a representation for the unit of magnetism in terms of mass,
charge and time) which culminated in the discovery of Kirchhoff’s Circuit Laws in Electricity.
In 1833, he and
Weber constructed the first
electromechanical telegraph which connected the Observatory with the
Institute for Physics in Gottingen.
In the garden of the
Observatory, Gauss had a Magnetic
Observatory built 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 was in use well
into the second half of the 20^{th} Century and worked out the
Mathematical theory for separating
the inner and outer (Magnetospheric) sources of Earth’s magnetic field.
In 1845, he became an
associated member of the “Royal
Institute of Netherlands”. Later, in 1851, when it was renamed
the “Royal Netherlands Academy of Arts
and Sciences”, he joined it as a foreign member.
In 1854, he
selected the topic for Bernhard Riemann’s Habilitationvortrag
“Uber die Hypothesen, welche der Geometrie
zu Grunde liegen”.
Passing
away and legacy:
On 23.02.1855,
Gauss passed away in Gottingen, in the Kingdom of Hanover (now part of Lower
Saxony, Germany).
He always referred to
Mathematics as the “Queen of Sciences”.
He was a perfectionist and
hard worker.
He never published any of
his works which he did not consider complete and above criticism.
A number of other major
discoveries in different related fields, including non-Euclid Geometry and
Gaussian Geometry which are important in Land Surveys and determining
Curvatures are attributed to him.
He was not very fond of
teaching and attended only a single scientific conference in Berlin in 1828. Nevertheless, several of his students
became influential and renowned mathematicians.
The
German Banknote Commemorating Gauss:
The following Banknote
honouring Gauss was part of the fourth Series of German Mark Banknotes, which were
introduced in 1990 by the Bundesbank. This Series contained advanced
security features for countering advances in counterfeiting technology. This Series
included Banknotes in the denominations of 5, 10, 20, 50, 100, 200, 500 and
1000 Deutsche Marks. This Series depicted prominent German Scientists and
Artists together with symbols, instruments and tools of their trade/craft.
Gauss’s portrait featured
on this Banknote from 1990 to 2001, after which, the Euro was the circulating
currency:
The Front of the German 10
Mark Banknote depicting Gauss, issued in 1993.
On the Front of the German 10 mark (Ten Mark) Banknote,
Carl Friedrich Gauss’s portrait with
his life years “1777-1855”, a Normal
Distribution Curve and some prominent
Gottingen Buildings were depicted. His name is mentioned in German as “Carl Fried Gaub”.
On the left periphery
rising upwards from the bottom is the name of the issuing Bank “DEUTSCHE
BUNDESBANK” followed by a stylised “BANKNOTE”. The denomination of
the Banknote is mentioned in numerals “10” and in German “ZEHN DEUTSCHE MARK”
(meaning “Ten Deutsche Marks”).
Gottingen:
is a University town in Lower Saxony, Germany. It is the capital of the
district of Gottingen. The river Leine passes through this town.
Gottingen is famous for
its old University (Georgia
Augusta or “Georg-August-Universitat”) which was founded in 1737 and
became the most visited University of Europe. In 1837, seven Professors
protested against the absolute Sovereignty of the Kings of Hanover, for which
they lost their offices, but became famous as the “Gottingen Seven”. Its alumni
included some well-known historical personalities – the Brothers Grimm,
Heinrich Ewald, Wilhelm Eduard Weber and Georg Gervinus. German Chancellors
Otto Von Bismarck and Gerhard Schroder studied at the Law school at the
Gottingen University. Karl Barth held his first professorship here. Some of the
most famous Mathematicians in history – Carl Friedrich Gauss, Bernhard Riemann
and David Hilbert were professors at Gottingen.
The Back of the German 10
Mark Banknote issued in 1993.
On the Back of the 10 Mark Banknote was depicted a Heliotrope designed by Gauss and a small map showing
the triangulation of the Kingdom of Hanover performed by Gauss.
This Banknote was issued
on “1 Oktober 1993” (meaning “1^{st} October 1993”). The logo of the
“DUETSCHE BUNDESBANK” the Eagle is pictured to the left of the Heliotrope.
Germany has also issued
three stamps on him one each in 1955, (on the 100^{th} Anniversary of
his passing away), 1811 and 1977 (Bicentenary of his birth).
The above is an image of
Gauss at the age of about 26 years on an East German stamp issued in 1977. Also
seen alongside his portrait are a heptadecagon, compass and straightedge
Some
other Commemoration:
- Daniel
Kehlmann in his novel published in 2005 “Die Vermessung der Weit” and translated into English as “Measuring
the World” (in 2006) has written about Gauss’s life and works through
historical fiction, contrasting them with those of the German explorer
Alexander von Humboldt. A movie directed by Detlev Buck was made on this story
in 2012.
- In
2007, a bust of Gauss was placed in the Walhalla temple (A Hall of Fame named after Valhalla of
Norse Mythology which honours laudable and distinguished people in German
history – politicians, sovereigns, scientists and artists etc. The Hall is a
neo-classical building above the Danube River, East of Regensburg in Bavaria.
Valhalla is a majestic, enormous Hall located in Asgard and ruled over by the
Norse God Odin).
- Prominent contributions
named after him include:
- The
Normal Distribution, Gaussian statistics (the bell curve)
- Gauss’s
Theorem, The Divergence Theorem
- The
Gauss Prize (one of the highest Honours in Mathematics)
- Gauss’s
Law and Gauss’s Law for Magnetism, two of Maxwell’s four equations
- Degaussing,
the process of eliminating a magnetic field
- The
CGS unit for magnetic field was named Gauss
- The
crater Gauss on the moon
- Asteroid
1011 Gaussia
- The
ship Gauss used in the Gauss Expedition to the Antarctic
- Gaussberg,
an extinct volcano discovered during the Gauss Expedition to the Antarctic
- Gauss
Tower, an observation tower in Dransfeld, Germany
- In
Canadian Junior High Schools, an annual National Mathematics Competition
administered by the centre for Education in Mathematics and computing is named
“Gauss Mathematics Competition” in his honour
- In
the University of California, Santa Cruz, in Crown College, a dormitory
building is named after him
- The
Gauss Haus, an NMR Centre (Nuclear Magnetic Resonance Centre) at the University
of Utah is named after him
- The
Carl-Friedrich-Gaub School for Mathematics, Computer Science, Business
Administration, Economics and Social Sciences of Braunschweig University of
Technology
- The
Gauss Building at the University of Idaho (College of Engineering)
- The
Carl-Friedrich-Gauss Gymnasium in Worms, Germany.
Interestingly, in 1929,
when the Polish Mathematician Marian Rejewski who helped solve the German
Enigma Cipher Machine in December 1932, began studying actuarial statistics at
Gottingen, on joining the University, he went to pay homage to Gauss at his
grave.
Gauss
has left behind several writings and works for mankind to benefit from and to
carry his works forward. Some of his works are:
In 1799, Doctoral
dissertation, on the Fundamental theorem of Algebra with the title “Demonstratio nova theorematis omnem
functionem algebraicam rationale integram unius variabillis in factores reales
primi vel secundi gradus resolvi posse” (meaning “New Proof of the
theorem that every integral algebraic function of one variable can be resolved
into real factors (i.e. polynomials) of the first or second degree”).
In 1801, “Disquisitiones Arithmeticae”
(meaning “Arithmetical Investigations”), which is a text-book of number theory
written in Latin.
In 1808, “Theorematis arithmetici demonstratio nova”.
In 1809, “Theoria Motus Corporum Coelestium in
sectionibus conicis solem ambientium” (meaning “Theory of the motion of
heavenly bodies Moving about the Sun in Conic Sections”).
In 1811, “Summatio serierun quarundam singularium”
In 1812, “Disquisitiones Generales Circa Seriem
Infinitam”
In 1818, “Theorematis fundamentallis in doctrina de
residuis quadraticis demonstrationes et amplicationes novae”
In 1821, 1823 and 1826,
“Theoria combinationis observationum
erroribus minimis obnoxiae”
In 1827, “Disquisitiones generales circa superficies
curvas” (meaning “General investigations of Curved Surfaces”)
In 1828, “Theoria residuorum biquadraticorum,
Commentatio prima” (meaning “Elementary facts about biquadratic
residues, proves one of the supplements of the law of biquadratic reciprocity”)
In 1832, “Theoria residuorum biquadraticorum, Commentatio
secunda” (“The law for biquadratic reciprocity proves the supplementary
law”)
In 1843-44, “Untersuchungen uber Gegenstande der Hoheren
Geodasie. Erste Abhandlung, Abhandlungen der Koniglichen Gesellschaft der
Wissenschaften in Gottingen Zweiter Band”
In 1846-47, “Untersuchungen uber Gegenstande der Hoheren
Geodasie, Zweite Abhandlung, Abhandlungen der Koniglichen Gesellschaft der
Wissenschaften in Gottingen Dritter Band.
(The above Banknote is
from the collection of Jayant Biswas. Banknote images scanned and post
researched and written by Rajeev Prasad)
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