- Carl Friedrich Gauss
box_width = 300px
name = Carl Friedrich Gauss
caption = Johann Carl Friedrich Gauss (1777-1855), painted by
Christian Albrecht Jensen
birth_date = birth date|1777|4|30|df=y
Braunschweig, Electorate of Brunswick-Lüneburg, Holy Roman Empire
death_date = death date and age|1855|2|23|1777|4|30|df=y
Göttingen, Kingdom of Hanover
Kingdom of Hanover
nationality = German
fields = Mathematician and physicist
University of Göttingen
University of Helmstedt
Johann Friedrich Pfaff
Johann Christian Martin Bartels
Friedrich Bessel Christoph Gudermann Christian Ludwig Gerling Richard DedekindJohann EnckeJohann Listing Bernhard RiemannChristian Peters Moritz Cantor
August Ferdinand Möbius Julius Weisbach L. C. Schnürlein
Johann Carl Friedrich Gauss (IPAEng|ˈɡaʊs, audio|De-carlfriedrichgauss.ogg|Audio, _de. Gauß, _la. Carolus Fridericus Gauss) (
30 April 1777– 23 February 1855) was a German mathematicianand scientistwho contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. Sometimes known as the "princeps mathematicorum" [cite book
title=Oxford User's Guide to Mathematics
publisher=Oxford University Press
pages=1188 ] (
Latin, usually translated as "the Prince of Mathematicians", although Latin "princeps" also can simply mean "the foremost") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.Dunnington, G. Waldo. (May, 1927). " [http://www.mathsong.com/cfgauss/Dunnington/1927/ The Sesquicentennial of the Birth of Gauss] ". "Scientific Monthly" XXIV: 402–414. Retrieved on 29 June 2005. Comprehensive biographical article.]
Gauss was a
child prodigy. There are many anecdotes pertaining to his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed " Disquisitiones Arithmeticae", his magnum opus, in 1798 at the age of 21, though it would not be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Early years (1777–1798)
Carl Friedrich Gauss was born in
Braunschweig, in the Electorate of Brunswick-Lüneburg, now part of Lower Saxony, Germany, as the only son of poor working-class parents. [cite web |url=http://www.math.wichita.edu/history/men/gauss.html|title=Carl Friedrich Gauss|accessmonthday= |accessyear= |last= |first= |date= |work= |publisher=Wichita State University ] There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, in his head, an error his father had made on paper while calculating finances.
Another famous story, and one that has evolved in the telling, has it that in
primary schoolhis teacher, J.G. Büttner, tried to occupy pupils by making them add a list of integers. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels. Gauss's presumed method, which supposes the list of numbers was from 1 to 100, was to realise that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050 ("see arithmetic series and summation"). [http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning/2 for discussion of original Wolfgang Sartorius von Waltershausensource.] However whilst the method works, the incident itself is probably apocryphal; some, such as Joseph Rotman in his book "A first course in Abstract Algebra", question whether it ever happened.
As his father wanted him to follow in his footsteps and become a mason, he was not supportive of Gauss's schooling in mathematics and science. Gauss was primarily supported by his mother in this effort and by the Duke of Braunschweig, who awarded Gauss a fellowship to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795, and subsequently he moved to the University of Göttingen from 1795 to 1798. While in university, Gauss independently rediscovered several important theorems;Fact|date=July 2007 his breakthrough occurred in 1796 when he was able to show that any regular
polygonwith a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philologyas a career. Gauss was so pleased by this result that he requested that a regular heptadecagonbe inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.Fact|date=September 2008
The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the
heptadecagonon March 30. [Carl Friedrich Gauss §§365-366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965.] He invented modular arithmetic, greatly simplifying manipulations in number theory. He became the first to prove the quadratic reciprocitylaw on April 8. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on May 31, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on July 10and then jotted down in his diary the famous words, "Heureka! num = ." On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which ultimately led to the Weil conjectures150 years later.
Middle years (1799–1830)
In his 1799 doctorate "in absentia", "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", Gauss proved the
fundamental theorem of algebrawhich states that every non-constant single-variable polynomialover the complex numbers has at least one root. Mathematicians including Jean le Rond d'Alemberthad produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally considered rigorous. His attempts clarified the concept of complex numbers considerably along the way.
Gauss also made important contributions to
number theorywith his 1801 book " Disquisitiones Arithmeticae" ( Latin, Arithmetical Investigations), which contained a clean presentation of modular arithmeticand the first proof of the law of quadratic reciprocity., a post he held for the remainder of his life.
The discovery of Ceres by Piazzi on
January 1, 1801led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 under the name "Theoria motus corporum coelestium in sectionibus conicis solem ambientum" (theory of motion of the celestial bodies moving in conic sections around the sun). Piazzi had only been able to track Ceres for a couple of months, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit.
Gauss, who was 23 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—- just about a year after its first sighting—and this turned out to be accurate within a half-degree. In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—- published a few years later as "Theory of Celestial Movement"—- remains a cornerstone of astronomical computation.Fact|date=July 2007 It introduced the
Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss was able to prove the method in 1809 under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendrein 1805, but Gauss claimed that he had been using it since 1795.Fact|date=July 2007
Gauss was a prodigious
mental calculator. Reputedly, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used logarithms." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!"Fact|date=September 2008
In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the state of Hanover, linking up with previous Danish surveys. To aid in the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, , a life-long student of Gauss, successfully proves in "Gauss, Titan of Science" that Gauss was in fact in full possession of non-Euclidian geometry long before it was published by János, but that he refused to publish any of it because of his fear of controversy.
The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with
curves and surfaces. This led in 1828 to an important theorem, the Theorema Egregium("remarkable theorem" in Latin), establishing an important property of the notion of curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space.
Later years and death (1831–1855)
In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in
magnetism(including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of Kirchhoff's circuit lawsin electricity. They constructed the first electromagnetic telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic observatoryto be built in the garden of the observatory, and with Weber founded the magnetischer Verein("magnetic club" in German), 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 has been in use well into the second half of the 20th century and worked out the mathematical theory for separating the inner (core and crust) and outer (magnetospheric) sources of Earth's magnetic field.
Gauss died in Göttingen, Hannover (now part of
Lower Saxony, Germany) in 1855 and is interred in the cemetery Albanifriedhofthere. Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewaldand Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. His brain was preserved and was studied by Rudolf Wagnerwho found its weight to be 1,492 grams and the cerebral area equal to 219,588 square millimeters [This reference from 1891 (cite journal
title=Anatomical Observations on the Brain and Several Sense-Organs of the Blind Deaf-Mute, Laura Dewey Bridgman
journal=The American Journal of Psychology
publisher=E. C. Sanford
accessdate=) says: "Gauss, 1492 grm. 957 grm. 219588. sq. mm. ", i.e the unit is "square mm". In the later reference: Dunnington (1927), the unit is erroneously reported as square cm, which gives an unreasonably large area, the 1891 reference is more reliable.] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius. [Dunnington, 1927]
Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a depression from which he never fully recovered. He married again, to Johanna's best friend named Friederica Wilhelmine Waldeck but commonly known as Minna. When his second wife died in 1831 after a long illness, [cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gauss.html |title=Gauss biography|publisher=Groups.dcs.st-and.ac.uk |date= |accessdate=2008-09-01] one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1817 until her death in 1839.
Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). Of all of Gauss's children, Wilhelmina was said to have come closest to his talent, but she died young. With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene emigrated to the
United Statesabout 1832 after a falling out with his father.Fact|date=July 2007 Wilhelm also settled in Missouri, starting as a farmerand later becoming wealthy in the shoe business in St. Louis. Therese kept house for Gauss until his death, after which she married.
Gauss eventually had conflicts with his sons, two of whom migrated to the United States. He did not want any of his sons to enter mathematics or science for "fear of sullying the family name".Fact|date=July 2007 Gauss wanted Eugene to become a
lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and emigrated to the United States, where he was quite successful. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues. See also on September 3, 1912.
Gauss was an ardent perfectionist and a hard worker. According to
Isaac Asimov, Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done." [cite book
title=Biographical Encyclopedia of Science and Technology; the Lives and Achievements of 1195 Great Scientists from Ancient Times to the Present, Chronologically Arranged.
year=1972 ] This anecdote is briefly discussed in Waldo Dunnington's "Gauss, Titan of Science" where it is suggested that it is an apocryphal story.
He was never a prolific writer, refusing to publish works which he did not consider complete and above criticism. This was in keeping with his personal motto "pauca sed matura" ("few, but ripe"). His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Mathematical historian
Eric Temple Bellestimated that had Gauss timely published all of his discoveries, Gauss would have advanced mathematics by fifty years. [cite book
last=Bell |first=E. T.
chapter=Ch. 14: The Prince of Mathematicians: Gauss
title=Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré
publisher=Simon and Schuster
Though he did take in a few students, Gauss was known to dislike teaching. It is said that he attended only a single scientific conference, which was in
Berlinin 1828. However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel. Before she died, Sophie Germainwas recommended by Gauss to receive her honorary degree.
Gauss usually declined to present the intuition behind his often very elegant proofs—-he preferred them to appear "out of thin air" and erased all traces of how he discovered them.Fact|date=July 2007 This is fully, however briefly, explained by Gauss himself in his "Disquisitiones Arithmeticae", where he states that all analysis (i.e. the paths one travelled to reach the solution of a problem) must be suppressed for sake of brevity.
Gauss supported monarchy and opposed Napoleon, whom he saw as an outgrowth of
From 1989 until the end of 2001, his portrait and a normal distribution curve as well as some prominent buildings of
Göttingenwere featured on the German ten-mark banknote. The other side of the note features the heliotrope and a triangulationapproach for Hannover. Germany has issued three stamps honouring Gauss, as well. A righteous stamp (no. 725), was issued in 1955 on the hundredth anniversary of his death; two other stamps, no. 1246 and 1811, were issued in 1977, the 200th anniversary of his birth.
In 2007, his bust was introduced to the
Walhalla temple. [http://www.stmwfk.bayern.de/downloads/aviso/2004_1_aviso_48-49.pdf]
Things named in honour of Gauss:
* The CGS unit for magnetic induction was named gauss in his honour.
* Gauss crater on the
Moon[Andersson, L. E.; Whitaker, E. A., (1982). NASA Catalogue of Lunar Nomenclature. NASA RP-1097.]
Asteroid 1001 Gaussia.
* The ship "Gauss", used in the
Gauss expeditionto the Antarctic.
Gaussberg, an extinct volcano discovered by the above mentioned expedition
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 Computingis named in honour of Gauss.
* In University of California, Santa Cruz, in Crown College, a dormitory building is named after Gauss.
* The Gauss Haus, an NMR center at the
University of Utah.
* The Carl-Friedrich-Gauß School for Mathematics, Computer Science, Business Administration, Economics, and Social Sciences of University of Braunschweig
List of topics named after Carl Friedrich Gauss
Doctoral dissertationon the Fundamental theorem of algebra, with the title: "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse" ("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")
*1801: [http://resolver.sub.uni-goettingen.de/purl?PPN235993352 "Disquisitiones Arithmeticae"]
*1809: [http://books.google.com/books?id=ORUOAAAAQAAJ&dq=Theoria+Motus+Corporum+Coelestium+in+sectionibus+conicis+solem+ambientium&source=gbs_summary_s&cad=0 "Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium"] (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), English translation by C. H. Davis, reprinted 1963, Dover, New York.
*1821, 1823 und 1826: "Theoria combinationis observationum erroribus minimis obnoxiae". Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. English translation by G. W. Stewart, 1987, Society for Industrial Mathematics.
*1827: [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN35283028X_0006_2NS "Disquisitiones generales circa superficies curvas"] , Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99-146. " [http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABR1255 General Investigations of Curved Surfaces] " (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.
*1843/44: " [http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D39018 Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung] ", [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN250442582_0002 Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band] , pp. 3-46
*1846/47: " [http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D39036 Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung] ", [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN250442582_0003 Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band] , pp. 3-44
*"Mathematisches Tagebuch 1796–1814", Ostwaldts Klassiker, Harri Deutsch Verlag 2005, mit Anmerkungen von Neumamn, ISBN 978-3-8171-3402-1 (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)
* [http://dz-srv1.sub.uni-goettingen.de/cache/toc/D38910.html Gauss' collective works are online here] This includes German translations of Latin texts and commentaries by various authorities
*citebook |last=Dunnington|first=G. Waldo.|title=Carl Friedrich Gauss: Titan of Science
publisher=The Mathematical Association of America|year=2003|isbn=088385547X |oclc=53933110
*citebook |last=Gauss|first=Carl Friedrich|others=tr. Arthur A. Clarke|title=
Disquisitiones Arithmeticae|publisher=Yale University Press|year=1965|isbn=0300094736
title=Carl Friedrich Gauss: A Biography
title=Die Vermessung der Welt
title=The Giant Book of Scientists: The 100 Greatest Minds of All Time
publisher=The Book Company
*planetmath reference |id=5594 |title=Carl Friedrich Gauss
* [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235957348 Complete works]
* [http://www.gausschildren.org Gauss and his children]
* [http://www.corrosion-doctors.org/Biographies/GaussBio.htm Gauss biography]
* [http://fermatslasttheorem.blogspot.com/2005/06/carl-friedrich-gauss.html Carl Friedrich Gauss] , Biography at Fermat's Last Theorem Blog.
* [http://www.idsia.ch/~juergen/gauss.html Gauss: mathematician of the millennium] , by
* [http://books.google.com/books?id=yh0PAAAAIAAJ English translation of Waltershausen's 1862 biography]
* [http://www.gauss.info Gauss] general website on Gauss
* [http://adsabs.harvard.edu//full/seri/MNRAS/0016//0000080.000.html MNRAS 16 (1856) 80] Obituary
* [http://www.americanscientist.org/template/AssetDetail/assetid/50686?&print=yes A discussion of childhood problem and the sources]
* [http://www-personal.umich.edu/~jbourj/money1.htm Carl Friedrich Gauss on the 10 Deutsche Mark banknote]
NAME= Gauss, Johann Carl Friedrich
SHORT DESCRIPTION= Mathematician and physicist
DATE OF BIRTH= birth date|1777|4|30|df=y
PLACE OF BIRTH=
DATE OF DEATH= death date|1855|2|23|df=y
PLACE OF DEATH=
Göttingen, Hannover, Germany
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