# Carl Friedrich Gauss: Wikis

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# Encyclopedia

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (1777â€“1855), painted by Christian Albrecht Jensen
Born 30 April 1777
Braunschweig, Electorate of Brunswick-LÃ¼neburg, Holy Roman Empire
Died 23 February 1855 (aged 77)
GÃ¶ttingen, Kingdom of Hanover
Residence Kingdom of Hanover
Nationality German
Fields Mathematician and physicist
Institutions University of GÃ¶ttingen
Alma mater University of Helmstedt
Doctoral students Friedrich Bessel
Christoph Gudermann
Christian Ludwig Gerling
Richard Dedekind
Johann Encke
Johann Listing
Bernhard Riemann
Christian Peters
Moritz Cantor
Other notable students August Ferdinand MÃ¶bius
Julius Weisbach
L. C. SchnÃ¼rlein
Known for See full list
Influenced Sophie Germain
Notable awards Copley Medal (1838)
Signature

Johann Carl Friedrich Gauss (pronounced /ËˆÉ¡aÊŠs/; German: GauÃŸ listen , Latin: Carolus Fridericus Gauss) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Sometimes referred to as the Princeps mathematicorum[1] (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") 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.[2] He referred to mathematics as "the queen of sciences."[3]

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 was not 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 (1777â€“1798)

Statue of Gauss at his birthplace, Braunschweig

Carl Friedrich Gauss was born on April 30, 1777 in Braunschweig, in the Electorate of Brunswick-LÃ¼neburg, now part of Lower Saxony, Germany, as the son of poor working-class parents.[4] He was christened and confirmed in a church near the school he attended as a child.[5] There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances.

Another famous story has it that in primary school his teacher, J.G. BÃ¼ttner, tried to occupy pupils by making them add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. 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 was to realize 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 Ã— 101 = 5050. However, the details of the story are at best uncertain (see [6] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions); some authors, such as Joseph Rotman in his book A first course in Abstract Algebra, question whether it ever happened.

Carl's intellectual abilities attracted the attention of the Duke of Braunschweig,[2] who sent him to the Collegium Carolinum (now Technische UniversitÃ¤t Braunschweig), which he attended from 1792 to 1795, and to the University of GÃ¶ttingen from 1795 to 1798. While in university, Gauss independently rediscovered several important theorems;[citation needed] his breakthrough occurred in 1796 when he was able to show that any regular polygon with 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 philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.[7]

The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on March 30.[8] He invented modular arithmetic, greatly simplifying manipulations in number theory.[citation needed] He became the first to prove the quadratic reciprocity law on 8 April. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, 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 10 July and then jotted down in his diary the famous words, "Heureka! num = Î” + Î” + Î”." 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 conjectures 150 years later.

## Middle years (1799â€“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 algebra which states that every non-constant single-variable polynomial over the complex numbers has at least one root. Mathematicians including Jean le Rond d'Alembert had 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 rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which contained a clean presentation of modular arithmetic and the first proof of the law of quadratic reciprocity.

Title page of Gauss's Disquisitiones Arithmeticae

In that same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres, but could only watch it for a few days. Gauss predicted correctly the position at which it could be found again, and it was rediscovered by Franz Xaver von Zach on 31 December 1801 in Gotha, and one day later by Heinrich Olbers in Bremen. Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again." Though Gauss had been up to that point supported by the stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in GÃ¶ttingen, a post he held for the remainder of his life.

The discovery of Ceres by Piazzi on 1 January 1801 led 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 dataâ€”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 1801â€”just about a year after its first sightingâ€”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 workâ€”published a few years later as Theory of Celestial Movementâ€”remains a cornerstone of astronomical computation.[citation needed] 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 Gaussâ€“Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.[citation needed]

Gauss' portrait published in Astronomische Nachrichten 1828

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!"[citation needed]

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, JÃ¡nos Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

This unproved statement put a strain on his relationship with JÃ¡nos Bolyai (who thought that Gauss was "stealing" his idea), but it is now generally taken at face value.[citation needed] Letters by Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, 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 JÃ¡nos, 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. Among other things he came up with the notion of Gaussian curvature. 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 or 2-dimensional space.

In 1821, he was made a foreign member of the Royal Swedish Academy of Sciences.

## Later years and death (1831â€“1855)

Grave of Gauss at Albanifriedhof in GÃ¶ttingen, Germany.

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 laws in electricity. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in GÃ¶ttingen. Gauss ordered a magnetic observatory to 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 GÃ¶ttingen, Hannover (now part of Lower Saxony, Germany) in 1855 and is interred in the cemetery Albanifriedhof there. Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. His brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters[9] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.[10]

## Family

Gauss' daughter Therese (1816â€”1864)

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,[11] 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.[2]

Gauss had six children. With Johanna (1780â€“1809), his children were Joseph (1806â€“1873), Wilhelmina (1808â€“1846) and Louis (1809â€“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 (1811â€“1896), Wilhelm (1813â€“1879) and Therese (1816â€“1864). Eugene emigrated to the United States about 1832 after a falling out with his father.[citation needed] Wilhelm also settled in Missouri, starting as a farmer and 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".[citation needed] 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 the letter from Robert Gauss to Felix Klein on 3 September 1912.

## Personality

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."[12] This anecdote is briefly discussed in G. 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 work 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 Bell estimated that had Gauss timely published all of his discoveries, Gauss would have advanced mathematics by fifty years.[13]

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 Berlin in 1828. However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel. Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree.

Gauss usually declined to present the intuition behind his often very elegant proofsâ€”he preferred them to appear "out of thin air" and erased all traces of how he discovered them.[citation needed] This is justified, if unsatisfactorily, by Gauss 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 revolution.

## Commemorations

A 10 Deutsche Mark banknote from Germany 1993 (discontinued) showing Gauss
Gauss (about 26) on an East-German stamp produced in 1977. Heptadecagon, compass and straightedge are shown next to him.

From 1989 until the end of 2001, his portrait and a normal distribution curve as well as some prominent buildings of GÃ¶ttingen were featured on the German ten-mark banknote. The other side of the note features the heliotrope and a triangulation approach 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.

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World: a Novel in 2006, explores Gauss's life and work through a lens of historical fiction, contrasting it with the German explorer Alexander von Humboldt.

In 2007, his bust was introduced to the Walhalla temple.[14]

Things named in honour of Gauss include:

## Notes

1. ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631.
2. ^ a b c Dunnington, G. Waldo. (May, 1927). "The Sesquicentennial of the Birth of Gauss". Scientific Monthly XXIV: 402â€“414. Retrieved on 29 June 2005. Comprehensive biographical article.
3. ^ Smith, S. A., et al. 2001. Algebra 1: California Edition. Prentice Hall, New Jersey. ISBN 0130442631
4. ^ "Carl Friedrich Gauss". Wichita State University.
5. ^ Susan Chambless. "Author â€” Date". Homepages.rootsweb.ancestry.com. Retrieved 2009-07-19.
6. ^ http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning/2
7. ^ Pappas, Theoni: Mathematical Snippets, Page 42. Pgw 2008
8. ^ Carl Friedrich Gauss Â§Â§365â€“366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965.
9. ^ This reference from 1891 (Donaldson, Henry H. (1891). "Anatomical Observations on the Brain and Several Sense-Organs of the Blind Deaf-Mute, Laura Dewey Bridgman". The American Journal of Psychology (E. C. Sanford) 4 (2): 248â€“294. doi:10.2307/1411270. ) 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.
10. ^ Dunnington, 1927
11. ^ "Gauss biography". Groups.dcs.st-and.ac.uk. Retrieved 2008-09-01.
12. ^ Asimov, I. (1972). Biographical Encyclopedia of Science and Technology; the Lives and Achievements of 1195 Great Scientists from Ancient Times to the Present, Chronologically Arranged.. New York: Doubleday.
13. ^ Bell, E. T. (2009). "Ch. 14: The Prince of Mathematicians: Gauss". Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to PoincarÃ©. New York: Simon and Schuster. pp. 218â€“269. ISBN 0-671-46400-0.
14. ^ "Bayerisches Staatsministerium fÃ¼r Wissenschaft, Forschung und Kunst: Startseite". Stmwfk.bayern.de. Retrieved 2009-07-19.
15. ^ Andersson, L. E.; Whitaker, E. A., (1982). NASA Catalogue of Lunar Nomenclature. NASA RP-1097.

Awards and achievements
Preceded by
Antoine CÃ©sar Becquerel and John Frederic Daniell
Copley Medal
1838
Succeeded by
Robert Brown

# Quotes

Up to date as of January 14, 2010

### From Wikiquote

Carl Friedrich Gauss (or GauÃŸ)

Johann Carl Friedrich Gauss ( â€“ ) was a German mathematician, astronomer and physicist.

## Sourced

• The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
• Disquisitiones Arithmeticae (1801) Article 329
• The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it. But when a person of that sex, that, because of our mores and our prejudices, has to encounter infinitely more obstacles and difficulties than men in familiarizing herself with these thorny research problems, nevertheless succeeds in surmounting these obstacles and penetrating their most obscure parts, she must without doubt have the noblest courage, quite extraordinary talents and superior genius.
• It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.
• We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
• To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years .
• I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer J. Bolyai a genius of the first rank.
• Mathematics is the queen of the sciences.
• As quoted in Gauss zum GedÃ¤chtniss (1856) by Wolfgang Sartorius von Waltershausen; Variants: Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
Mathematics is the queen of the sciences and number theory is the queen of mathematics. ( Die Mathematik ist die KÃ¶nigin der Wissenschaften und die Zahlentheorie ist die KÃ¶nigin der Mathematik. )
• It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.
• Gauss-Schumacher Briefwechsel (1862)
• Ask her to wait a momentâ€” I am almost done.
• When told, while working, that his wife was dying. As quoted in Men of Mathematics (1937) by E. T. Bell
• I have had my results for a long time: but I do not yet know how I am to arrive at them.
• The Mind and the Eye (1954) by A. Arber
• If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
• The World of Mathematics (1956) Edited by J. R. Newman
• I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
• A reply to Olbers' 1816 attempt to entice him to work on Fermat's Theorem. As quoted in The World of Mathematics (1956) Edited by J. R. Newman
• There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
• As quoted in The World of Mathematics (1956) Edited by J. R. Newman
• Finally, two days ago, I succeededâ€” not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
• Mathematical Circles Squared (1972) by Howard W. Eves
• A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
• On higher arithmetic. Mathematical Circles Adieu (1977) by Howard W. Eves
• I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect. . . Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
• As quoted in Solid Shape' ' (1990) by Jan J. Koenderink
• You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
• As quoted in Calculus Gems (1992) by George F. Simmons
• I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half a proof is zero, and it is demanded for proof that every doubt becomes impossible.
• As quoted in Calculus Gems (1992) by George F. Simmons
• In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
• As quoted in Gauss, Werke, Bd. 8, page 298
• As quoted in Memorabilia Mathematica (or The Philomath's Quotation-Book) (1914) by Robert Edouard Moritz, quotation #1215
• As quoted in The First Systems of Weighted Differential and Integral Calculus (1980) by Jane Grossman, Michael Grossman, and Robert Katz, page ii

## Quotes of others about Gauss

• According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name BÅ“otians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically. ~ Wolfgang Sartorius von Waltershausen in Gauss zum GedÃ¤chtniss (1856)

### The Music of the Primes (2003)

Quotations about some of the work of Gauss from a book about Prime Numbers by Marcus du Sautoy, professor of mathematics at Oxford University

• Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians.
• Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.
• The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'
• Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built.

# Simple English

File:Gauss
Statue of Gauss in Brunswick

Carl Friedrich Gauss (pronunciation:  Carl Friedrich Gauss (GauÃŸ) (info • help)) (30 April 30 1777 â€“ 23 February 1855) was a famous mathematician from GÃ¶ttingen, Germany. Gauss contributed to many areas of learning. Most of his work was about number theory and astronomy.

## Biography

### Childhood

He was born in Braunschweig. That city was then part of the duchy of Braunschweig-LÃ¼neburg. Today the city is part of Lower Saxony. As a child, he was a prodigy, meaning he was very clever. When he was 3 years old, he told his father that he had incorrectly measured something on his complicated payroll. Gauss was correct. Gauss also taught himself to read.

When he was in elementary school, his teacher once tried to keep the children busy, telling them to add up all the numbers from 1 to 100. Gauss did it quickly, like this: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on. There were a total of 50 pairs, so 50 Ã— 101 = 5,050. The formula is $\frac 1 2 * \left(n*\left(n+1\right)\right)$. According to this website, the problem given to Gauss was actually more difficult to do.

The Duke of Brunswick gave Gauss a fellowship to the Collegium Carolinum, where he attended from 1792 to 1795. This meant that the Duke paid for the education of Carl Friedrich Gauss at the Collegium. After this, Gauss went to the University of GÃ¶ttingen, from 1795 to 1798.

## Work

Gauss wrote Disquisitiones Arithmeticae which is a book about number theory. In that book he proved the law of Quadratic reciprocity. He also was the first mathematician to explain Modular arithmetic in a very detailed way. Before Gauss, mathematicians had used modular arithmetic in some cases but did not know much about using it broadly.