Carl Friedrich Gauss

"Gauss" redirects here. For things named after Carl Friedrich Gauss, see List of things named after Carl Friedrich Gauss. For other persons or things named Gauss, see Gauss (disambiguation).
Johann Carl Friedrich Gauss

Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Born Johann Carl Friedrich Gauss
(1777-04-30)30 April 1777
Brunswick, Duchy of Brunswick-Wolfenbüttel, Holy Roman Empire
Died 23 February 1855(1855-02-23) (aged 77)
Göttingen, Kingdom of Hanover
Residence Kingdom of Hanover
Nationality German
Fields Mathematics and physics
Institutions University of Göttingen
Alma mater University of Helmstedt
Thesis Demonstratio nova... (1799)
Doctoral advisor Johann Friedrich Pfaff
Other academic advisors Johann Christian Martin Bartels
Doctoral students Johann Listing
Christian Ludwig Gerling
Richard Dedekind
Bernhard Riemann
Christian Peters
Moritz Cantor
Other notable students Johann Encke
Christoph Gudermann
Peter Gustav Lejeune Dirichlet
Gotthold Eisenstein
Carl Wolfgang Benjamin Goldschmidt
Gustav Kirchhoff
Ernst Kummer
August Ferdinand Möbius
L. C. Schnürlein
Julius Weisbach
Known for See full list
Influenced Friedrich Bessel
Sophie Germain
Ferdinand Minding
Notable awards Lalande Prize (1810)
Copley Medal (1838)

Signature

Johann Carl Friedrich Gauss (/ɡaʊs/; German: Gauß, pronounced [ɡaʊs]; Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics.

Sometimes referred to as the Princeps mathematicorum[1] (Latin, "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.[2]

Early years

Statue of Gauss at his birthplace, Brunswick

Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), as the son of poor working-class parents.[3] His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which itself occurs 39 days after Easter. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.[4] He was christened and confirmed in a church near the school he attended as a child.[5]

Gauss was a child prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100.[6][7] There are many other anecdotes about 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.

Gauss's intellectual abilities attracted the attention of the Duke of Brunswick,[2] who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems.[8] His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if and only 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; 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.[9]

The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.[10] He further advanced modular arithmetic, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law. 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 note: "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

Middle years

Algebra

Title page of Gauss's Disquisitiones Arithmeticae

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 with complex coefficients has at least one complex 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, among other things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary 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.

Astronomy

Gauss's portrait published in Astronomische Nachrichten 1828

In the same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could only track Ceres for somewhat more than a month, 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 24 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 when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen.

Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work Gauss used comprehensive approximation methods which he created for that purpose.[11]

One such method was the fast Fourier transform. While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata,[12] was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807.[13]

Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had up to that point been financially supported by his 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 Göttingen, a post he held for the remainder of his life.

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work remains a cornerstone of astronomical computation. 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.

Four Gaussian distributions in statistics

Gauss proved the method under the assumption of normally distributed errors (see Gauss–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 1794 or 1795.[14] In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares."[15]

Geodesic survey

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the Kingdom of Hanover, linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.

Non-Euclidean geometries

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, János 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 Bolyai who thought that Gauss was "stealing" his idea.[16]

Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.[17][18]

Theorema Egregium

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade,[19] fueled Gauss's interest in differential geometry and topology, fields 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), 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

Daguerreotype of Gauss on his deathbed, 1855.
Grave of Gauss at Albanifriedhof in Göttingen, 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, charge, and time) and the discovery of Kirchhoff's circuit laws in electricity. It was during this time that he formulated his namesake law. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. 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 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 1840, Gauss published his influential Dioptrische Untersuchungen,[20] in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics).[21] Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points[22] and he derived the Gaussian lens formula.[23]

In 1845, he became associated member of the Royal Institute of the Netherlands; when that became the Royal Netherlands Academy of Arts and Sciences in 1851, he joined as a foreign member.[24]

In 1854, Gauss selected the topic for Bernhard Riemann's Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen.[25] On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.[26]

Gauss died in Göttingen, in the Kingdom of Hanover (now part of Lower Saxony, Germany) on 23 February 1855[3] and is interred in the Albanifriedhof cemetery 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 (slightly above average) and the cerebral area equal to 219,588 square millimeters[27] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.[28]

Religious views

Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen.[29] Potential evidence that Gauss believed in God comes from his response after solving a problem that had previously stumped him: "Finally, two days ago, I succeeded— not on account of my hard efforts, but by the grace of the Lord."[30] One of his biographers G. Waldo Dunnington describes Gauss's religious views in these terms:

For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life it brought him confidence. Gauss' God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.[31]

Apart from his correspondence, there are not many known details about Gauss' personal creed. Many biographers of Gauss disagree with his religious stance, with Bühler and others considering him a deist with very unorthodox views,[32][33][34] while Dunnington (though admitting that Gauss did not believe literally in all Christian dogmas and that it is unknown what he believed on most doctrinal and confessional questions) points out that he was, at least, a nominal Lutheran.[35]

In connection to this, there's a record of a conversation between Rudolf Wagner and Gauss, in which they discussed William Whewell's book Of the Plurality of Worlds. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe.[32][36] This later led them to discuss the topic of faith, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Lutheran minister Paul Gerhardt than by Moses.[37] Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament.[38]

Dunnington further elaborates on Gauss's religious views by writing:

Gauss' religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.[39]

Gauss declared he firmly believed in the afterlife, and saw spirituality as something essentially important for human beings.[40] He was quoted stating: "The world would be nonsense, the whole creation an absurdity without immortality,"[41] and for this statement he was severely criticized by the atheist Eugen Dühring who judged him as a narrow superstitious man.[42]

Though he was not a church-goer,[43] Gauss strongly upheld religious tolerance, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble."[2] When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved of this, saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.[44]

Family

Gauss's daughter Therese (1816—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, Friederica Wilhelmine Waldeck, commonly known as Minna. When his second wife died in 1831 after a long illness,[45] one of his daughters, Therese, took over the household and cared for Gauss for the rest of his life. His mother lived in his house from 1817 until her death in 1839.[2]

Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene shared a good measure of Gauss's talent in languages and computation.[46] Therese kept house for Gauss until his death, after which she married.

Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.[46] 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, in about 1832, emigrated to the United States, where he was quite successful. While working for the American Fur Company in the Midwest, he learned the Sioux language. Later, he moved to Missouri and became a successful businessman. Wilhelm also moved to America in 1837 and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. 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

Carl Gauss was an ardent perfectionist and a hard worker. 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 said that had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.[47]

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 and Bernhard Riemann.

On Gauss's recommendation, Friedrich Bessel was awarded an honorary doctor degree from Göttingen in March 1811.[48] Around that time, the two men engaged in an epistolary correspondence.[49] However, when they met in person in 1825, they quarrelled; the details are not known.[50]

Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree; she never received it.[51]

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. 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 the monarchy and opposed Napoleon, whom he saw as an outgrowth of revolution.

Anecdotes

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 story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: 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 × 101 = 5050. However, the details of the story are at best uncertain (see[7] 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.

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."[52] 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 referred to mathematics as "the queen of sciences"[53] and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.[54]

Commemorations

German 10-Deutsche Mark Banknote (1993; discontinued) featuring Gauss
Gauss (aged about 26) on East German stamp produced in 1977. Next to him: heptadecagon, compass and straightedge.

From 1989 through 2001, Gauss's portrait, a normal distribution curve and some prominent Göttingen buildings were featured on the German ten-mark banknote. The reverse featured the approach for Hanover. Germany has also issued three postage stamps honoring Gauss. One (no. 725) appeared in 1955 on the hundredth anniversary of his death; two others, nos. 1246 and 1811, in 1977, the 200th anniversary of his birth.

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World (2006), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer Alexander von Humboldt. A film version directed by Detlev Buck was released in 2012.[55]

In 2007 a bust of Gauss was placed in the Walhalla temple.[56]

Things named in honor of Gauss include:

In 1929 the Polish mathematician Marian Rejewski, who helped to solve the German Enigma cipher machine in December 1932, began studying actuarial statistics at Göttingen. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss's grave.[58]

Writings

Gauss's collective works are online at dz-srv1.sub.uni-goettingen.de Uni-goettingen.de includes German translations of Latin texts and commentaries by various authorities.

See also

Notes

  1. ↑ Zeidler, Eberhard (2004). Oxford Users' Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0-19-850763-1.
  2. 1 2 3 4 Dunnington, G. Waldo. (May 1927). The Sesquicentennial of the Birth of Gauss at the Wayback Machine (archived February 26, 2008) Scientific Monthly XXIV: 402–414. Retrieved on 29 June 2005. Now available at "The Sesquicentennial of the Birth of Gauss". Retrieved 23 February 2014. Comprehensive biographical article.
  3. 1 2 "Carl Friedrich Gauss". Wichita State University.
  4. ↑ "Gauss Birthday Problem".
  5. ↑ Susan Chamberless (2000-03-11). "Letter:WORTHINGTON, Helen to Carl F. Gauss – 1911-07-26". Susan D. Chambless. Retrieved 2011-09-14.
  6. ↑ "Gauss, Carl Friedrich (1777-1855)." (2014). In The Hutchinson Dictionary of scientific biography. Abington, United Kingdom: Helicon.
  7. 1 2 Brian Hayes (14 November 2009). "Gauss's Day of Reckoning". American Scientist. doi:10.1511/2006.3.200. Retrieved 30 October 2012.
  8. ↑ O'Connor, John J.; Robertson, Edmund F., "Carl Friedrich Gauss", MacTutor History of Mathematics archive, University of St Andrews.
  9. ↑ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
  10. ↑ Carl Friedrich Gauss §§365–366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965.
  11. ↑ Klein, Felix; Hermann, Robert (1979). Development of mathematics in the 19th century. Math Sci Press. ISBN 978-0-915692-28-6.
  12. ↑ http://lseet.univ-tln.fr/~iaroslav/Gauss_Theoria_interpolationis_methodo_nova_tractata.php
  13. ↑ Heideman, M.; Johnson, D.; Burrus, C. (1984). "Gauss and the history of the fast fourier transform". IEEE ASSP Magazine 1 (4): 14–21. doi:10.1109/MASSP.1984.1162257.
  14. ↑ Oscar Sheynin, History of Statistics, Berlin: NG Verlag Berlin, 2012, p. 81.
  15. ↑ Stephen M. Stigler, "Gauss and the Invention of Least Squares," Ann. Statist., 9(3), 1981, pp. 465–474.
  16. ↑ Steven G. Krantz (1 April 2010). An Episodic History of Mathematics: Mathematical Culture through Problem Solving. MAA. pp. 171–. ISBN 978-0-88385-766-3. Retrieved 9 February 2013.
  17. ↑ Halsted, G. B. (1912). "Duncan M. Y. Sommerville". American Mathematical Monthly 19: 1–4. doi:10.2307/2973871. jstor.org
  18. ↑ Sondow, J. (2014). "From the Monthly Over 100 Years Ago…". American Mathematical Monthly 121: 963. doi:10.4169/amer.math.monthly.121.10.963.jstor.org arXiv "Gauss and the eccentric Halsted".
  19. ↑ The Prince of Mathematics. The Door to Science by keplersdiscovery.com.
  20. ↑ Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag. pp. 144–145. ISBN 0-387-10662-6.
  21. ↑ Hecht, Eugene (1987). Optics. Addison Wesley. p. 134. ISBN 0-201-11609-X.
  22. ↑ Bass, Michael; DeCusatis, Casimer; Enoch, Jay; Lakshminarayanan, Vasudevan (2009). Handbook of Optics. McGraw Hill Professional. p. 17.7. ISBN 0-07-149889-3.
  23. ↑ Ostdiek, Vern J.; Bord, Donald J. (2007). Inquiry Into Physics. Cengage Learning. p. 381. ISBN 0-495-11943-1.
  24. ↑ "C.F. Gauss (1797 - 1855)". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
  25. ↑ Monastyrsky, Michael (1987). Riemann, Topology, and Physics. Birkhäuser. pp. 21–22. ISBN 0-8176-3262-X.
  26. ↑ Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag. p. 154. ISBN 0-387-10662-6.
  27. ↑ 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): 248–294. doi:10.2307/1411270. JSTOR 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.
  28. ↑ Bardi, Jason (2008). The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons, Inc. p. 189. ISBN 978-0-470-46736-7.
  29. ↑ Guy Waldo Dunnington (1955). Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work. Exposition Press, pp. 300
  30. ↑ "WikiQuotes". WikiQuotes.
  31. ↑ Guy Waldo Dunnington (1955). Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work. Exposition Press, pp. 298-301
  32. 1 2 Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag. p. 153. ISBN 0-387-10662-6.
  33. ↑ Gerhard Falk (1995). American Judaism in Transition: The Secularization of a Religious Community. University Press of America. p. 121. ISBN 978-0-7618-0016-3. Gauss told his friend Rudolf Wagner, a professor of biology at Gottingen University, that he did not fully believe in the Bible but that he had meditated a great deal on the future of the human soul and speculated on the possibility of the soul being reincarnated on another planet. Evidently, Gauss was a Deist with a good deal of skepticism concerning religion but incorporating a great deal of philosophical interests in the Big Questions, that is. the immortality of the soul, the afterlife and the meaning of man's existence.
  34. ↑ Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag. p. 152. ISBN 0-387-10662-6. Closely related to Gauss's political and social views were his religious beliefs. Despite his religious beliefs. Despite his strong roots in the Enlightenment, Gauss was not an atheist, rather a deist with very unorthodox convictions, unorthodox even if measured against the very liberal persuasions of the contemporary Protestant church.
  35. ↑ Guy Waldo Dunnington (2004). Carl Friedrich Gauss: Titan of Science. MAA. p. 305. ISBN 9780883855478. It is not known just what Gauss believed on most doctrinal and confessional questions. He did not believe literally in all Christian dogmas. Officially he was a member of St. Albans Church (Evangelical Lutheran) in Gottingen. All baptisms, burials, and weddings in his family occurred there. It is also not known whether he attended church regularly or contributed financially. A faculty colleague called Gauss a deist, but there is good reason to believe that this label did not fit well. Gauss possessed strong religious tolerance which he carried over to every belief originating in the depths of the human heart. This tolerance is not to be confused with religious indifference. He took special interest in the religious development of the human race, especially in his own century. With reference to the manifold denominations, which frequently did not agree with his views, he always emphasized that one is not justified in disturbing the faith of others in which they find consolation for earthly sufferings and a safe refuge in days of misfortune
  36. ↑ Guy Waldo Dunnington (2004). Carl Friedrich Gauss: Titan of Science. MAA. p. 305. ISBN 9780883855478. league, I believe you are more believing in the Bible than I. I am not, and, he added, with the expression of great inner emotion, you are much happier than I. I must say that so often in earlier times when I saw people of the lower classes, simple manual laborers who could believe so rightly with their hearts, I always envied them, and now, he continued, with soft voice and that naive childlike manner peculiar to him, while a tear came into his eye, tell me how does one begin this?...
  37. ↑ Guy Waldo Dunnington (2004). Carl Friedrich Gauss: Titan of Science. MAA. p. 356. ISBN 9780883855478. I must confess that such old theologians and song writers as Paul Gerhard have always made a great impression on me; a song by Paul Gerhard always exerted a wonderful power on me, much more than, for example, Moses, against whom as a man of God I have all sorts of qualms.
  38. ↑ Guy Waldo Dunnington (2004). Carl Friedrich Gauss: Titan of Science. MAA. p. 305. ISBN 9780883855478. " Two religious works which Gauss read frequently were Braubach's Seelenlehre (Giessen, 1843) and Siissmilch's Gottliche (Ordnung gerettet A756); he also devoted considerable time to the New Testament in the original Greek.
  39. ↑ Guy Waldo Dunnington; Jeremy Gray; Fritz-Egbert Dohse (2004). Carl Friedrich Gauss: Titan of Science. MAA. p. 300. ISBN 978-0-88385-547-8. Gauss' religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.
  40. ↑ Morris Kline (1982). Mathematics: The Loss of Certainty. Oxford University Press. p. 73. ISBN 978-0-19-503085-3.
  41. ↑ Dunnington. 2004:357
  42. ↑ Dunnington. 2004:359
  43. ↑ "Gauss, Carl Friedrich". Complete Dictionary of Scientific Biography. 2008. Retrieved 29 July 2012. In seeming contradiction, his religious and philosophical views leaned toward those of his political opponents. He was an uncompromising believer in the priority of empiricism in science. He did not adhere to the views of Kant, Hegel and other idealist philosophers of the day. He was not a churchman and kept his religious views to himself. Moral rectitude and the advancement of scientific knowledge were his avowed principles.
  44. ↑ Guy Waldo Dunnington (1955). Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work. Exposition Press, pp. 311
  45. ↑ "Gauss biography". Groups.dcs.st-and.ac.uk. Retrieved 2008-09-01.
  46. 1 2 "Letter:GAUSS, Charles Henry to Florian Cajori – 1898-12-21". Susan D. Chambless. 2000-03-11. Retrieved 2011-09-14.
  47. ↑ 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 Poincaré. New York: Simon and Schuster. pp. 218–269. ISBN 0-671-46400-0.
  48. ↑ Bessel never had a university education.
  49. ↑ Helmut Koch, Introduction to Classical Mathematics I: From the Quadratic Reciprocity Law to the Uniformization Theorem, Springer, p. 90.
  50. ↑ Oscar Sheynin, History of Statistics, Berlin: NG Verlag Berlin, 2012, p. 88.
  51. ↑ Mackinnon, Nick (1990). "Sophie Germain, or, Was Gauss a feminist?". The Mathematical Gazette 74 (470): 346–351, esp. p. 347.
  52. ↑ 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.
  53. ↑ Quoted in Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8. ISSN B0000BN5SQ ASIN: B0000BN5SQ.
  54. ↑ Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p. 202. ISBN 0-309-08549-7.
  55. ↑ baharuka (25 October 2012). "Die Vermessung der Welt (2012) - IMDb". IMDb.
  56. ↑ "Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst: Startseite" (PDF). Stmwfk.bayern.de. Retrieved 2009-07-19.
  57. ↑ Andersson, L. E.; Whitaker, E. A., (1982). NASA Catalogue of Lunar Nomenclature. NASA RP-1097.
  58. ↑ Władysław Kozaczuk, Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two, Frederick, Maryland, University Publications of America, 1984, p. 7, note 6.

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