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Sunday, 3 July 2016

337) Johann Carl Friedrich Gauss: A 10 Mark German Banknote issued commemorating his contributions in 1990 which was in circulation till 2001:

337) 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 18th 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 onethereby 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 20th 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 HabilitationvortragUber 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 “1st 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 100th 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 2005Die 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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