Search Results for Gauss

Biographies

1. Gauss biography
   
   • Johann Carl Friedrich Gauss .
     
   • At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately.
     
   • His teacher, Buttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.
     
   • In 1788 Gauss began his education at the Gymnasium with the help of Buttner and Bartels, where he learnt High German and Latin.
     
   • After receiving a stipend from the Duke of Brunswick- Wolfenbuttel, Gauss entered Brunswick Collegium Carolinum in 1792.
     
   • At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.
     
   • In 1795 Gauss left Brunswick to study at Gottingen University.
     
   • Gauss's teacher there was Kastner, whom Gauss often ridiculed.
     
   • Gauss left Gottingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae.
     
   • Gauss returned to Brunswick where he received a degree in 1799.
     
   • After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt.
     
   • Gauss's dissertation was a discussion of the fundamental theorem of algebra.
     
   • With his stipend to support him, Gauss did not need to find a job so devoted himself to research.
     
   • In June 1801, Zach, an astronomer whom Gauss had come to know two or three years previously, published the orbital positions of Ceres, a new "small planet" which was discovered by G Piazzi, an Italian astronomer on 1 January, 1801.
     
   • Zach published several predictions of its position, including one by Gauss which differed greatly from the others.
     
   • When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted.
     
   • Although he did not disclose his methods at the time, Gauss had used his least squares approximation method.
     
   • In June 1802 Gauss visited Olbers who had discovered Pallas in March of that year and Gauss investigated its orbit.
     
   • Olbers requested that Gauss be made director of the proposed new observatory in Gottingen, but no action was taken.
     
   • Gauss began corresponding with Bessel, whom he did not meet until 1825, and with Sophie Germain.
     
   • Gauss married Johanna Ostoff on 9 October, 1805.
     
   • In 1807 Gauss left Brunswick to take up the position of director of the Gottingen observatory.
     
   • Gauss arrived in Gottingen in late 1807.
     
   • In 1808 his father died, and a year later Gauss's wife Johanna died after giving birth to their second son, who was to die soon after her.
     
   • Gauss was shattered and wrote to Olbers asking him to give him a home for a few weeks, .
     
   • Gauss was married for a second time the next year, to Minna the best friend of Johanna, and although they had three children, this marriage seemed to be one of convenience for Gauss.
     
   • Gauss's work never seemed to suffer from his personal tragedy.
     
   • Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70.
     
   • Much of Gauss's time was spent on a new observatory, completed in 1816, but he still found the time to work on other subjects.
     
   • In fact, Gauss found himself more and more interested in geodesy in the 1820s.
     
   • Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid.
     
   • Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations.
     
   • Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope.
     
   • Gauss often wondered if he would have been better advised to have pursued some other occupation but he published over 70 papers between 1820 and 1830.
     
   • In 1822 Gauss won the Copenhagen University Prize with Theoria attractionis..
     
   • From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry.
     
   • Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry.
     
   • In 1831 Farkas Bolyai sent to Gauss his son Janos Bolyai's work on the subject.
     
   • Gauss replied .
     
   • Gauss had a major interest in differential geometry, and published many papers on the subject.
     
   • The paper also includes Gauss's famous theorema egregrium: .
     
   • The period 1817-1832 was a particularly distressing time for Gauss.
     
   • Gauss, however, never liked change and decided to stay in Gottingen.
     
   • In 1831 Gauss's second wife died after a long illness.
     
   • Gauss had known Weber since 1828 and supported his appointment.
     
   • Gauss had worked on physics before 1831, publishing Uber ein neues allgemeines Grundgesetz der Mechanik, which contained the principle of least constraint, and Principia generalia theoriae figurae fluidorum in statu aequilibrii which discussed forces of attraction.
     
   • These papers were based on Gauss's potential theory, which proved of great importance in his work on physics.
     
   • In 1832, Gauss and Weber began investigating the theory of terrestrial magnetism after Alexander von Humboldt attempted to obtain Gauss's assistance in making a grid of magnetic observation points around the Earth.
     
   • Gauss was excited by this prospect and by 1840 he had written three important papers on the subject: Intensitas vis magneticae terrestris ad mensuram absolutam revocata (1832), Allgemeine Theorie des Erdmagnetismus (1839) and Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskrafte (1840).
     
   • Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.
     
   • However, once Gauss's new magnetic observatory (completed in 1833 - free of all magnetic metals) had been built, he proceeded to alter many of Humboldt's procedures, not pleasing Humboldt greatly.
     
   • However, Gauss's changes obtained more accurate results with less effort.
     
   • Gauss and Weber achieved much in their six years together.
     
   • However, this was just an enjoyable pastime for Gauss.
     
   • The Magnetischer Verein and its journal were founded, and the atlas of geomagnetism was published, while Gauss and Weber's own journal in which their results were published ran from 1836 to 1841.
     
   • In 1837, Weber was forced to leave Gottingen when he became involved in a political dispute and, from this time, Gauss's activity gradually decreased.
     
   • Gauss spent the years from 1845 to 1851 updating the Gottingen University widow's fund.
     
   • Two of Gauss's last doctoral students were Moritz Cantor and Dedekind.
     
   • Gauss presented his golden jubilee lecture in 1849, fifty years after his diploma had been granted by Helmstedt University.
     
   • From the mathematical community only Jacobi and Dirichlet were present, but Gauss received many messages and honours.
     
   • From 1850 onwards Gauss's work was again nearly all of a practical nature although he did approve Riemann's doctoral thesis and heard his probationary lecture.
     
   • His health deteriorated slowly, and Gauss died in his sleep early in the morning of 23 February, 1855.
     
   • Gauss's Disquisitiones Arithmeticae .
     
   • Gauss's estimate for the density of primes .
     
   • A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry.
     
   • Honours awarded to Carl Friedrich Gauss .
     
   • Lunar featuresCrater Gauss .
     
   • Gauss Werke (1863 in Latin and German) .
     
   • S D Chambless (An obituary of Gauss's son) and an account of his life in the USA .
     
   • http://www-history.mcs.st-andrews.ac.uk/Biographies/Gauss.html .
     

2. Lobachevsky biography
   
   • Bartels was a school teacher and friend of Gauss, and the two corresponded.
     
   • We shall return later to discuss ideas of some historians, for example M Kline, that Gauss may have given Lobachevsky hints regarding directions that he might take in his mathematical work through the letters exchanged between Bartels and Gauss.
     
   • This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non-euclidean geometry which is just simply false.
     
   • There is a coincidence which arises from the fact that we know that Gauss himself discovered non-euclidean geometry but told very few people, only his closest friends.
     
   • This coincidence has prompted speculation that both Lobachevsky and Bolyai were led to their discoveries by Gauss.
     
   • Also Laptev in [Dedicated to the memory of Lobachevskii 1 (Kazan, 1992), 35-40.',29)">29] has examined the correspondence between Bartels and Gauss and shown that Bartels did not know about Gauss's results in non-euclidean geometry.
     
   • For example in [Priroda (7) (1993), 4-11.',25)">25] the claims that Lobachevsky was in correspondence with Gauss ( Gauss appreciated Lobachevsky's works very highly but had no personal correspondence with him), that Gauss studied Russian to read Lobachevsky's Russian papers as claimed for example in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1] (actually, Gauss had studied Russian before he had even heard of Lobachevsky), and that Gauss was a "good propagandist" of Lobachevsky's works in Germany (Gauss never commented publicly on Lobachevsky's work) are shown to be false.
     
   • In 1866, ten years after Lobachevsky's death, Houel published a French translation of Lobachevsky's Geometrische Untersuchungen together with some of Gauss's correspondence on non-euclidean geometry.
     

3. Listing biography
   
   • Soon Listing was attending mathematics courses given by Gauss and he was quickly spotted by Gauss as being both a very able and a very hard working student.
     
   • Gauss invited him to join his circle of friends who included Weber.
     
   • Listing was not the only student invited into close friendship with Gauss.
     
   • The influence of Gauss on Listing was, however, very marked.
     
   • It was from Gauss that Listing began to learn topological concepts.
     
   • He also was a talented experimenter and collaborated with Gauss in physics experiments, particularly those relating to terrestrial magnetism.
     
   • Gauss became the supervisor of Listing's dissertation De superficiebus secundi ordinis which was on surfaces of the second degree and ternary forms.
     
   • He had also agreed to collect data on terrestrial magnetism for Gauss on this trip.
     
   • He worked in his spare time on mathematics, in particular working on the topological ideas which had first been suggested by Gauss.
     
   • Weber, the professor of physics at Gottingen and a close collaborator of Gauss, was one of seven professors at the university to sign a protest and all seven were dismissed.
     
   • However his professorship of physics had to be filled and after a while Gauss was asked to suggest possible candidates.
     

4. Riemann biography
   
   • This was granted, however, and Riemann then took courses in mathematics from Moritz Stern and Gauss.
     
   • Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius.
     
   • thesis, supervised by Gauss, was submitted in 1851.
     
   • However it was not only Gauss who strongly influenced Riemann at this time.
     
   • The Dirichlet Principle did not originate with Dirichlet, however, as Gauss, Green and Thomson had all made use if it.
     
   • In his report on the thesis Gauss described Riemann as having:- .
     
   • On Gauss's recommendation Riemann was appointed to a post in Gottingen and he worked for his Habilitation, the degree which would allow him to become a lecturer.
     
   • Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry.
     
   • Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts.
     
   • Gauss's chair at Gottingen was filled by Dirichlet in 1855.
     

5. Eisenstein biography
   
   • In 1842 he bought a French translation of Gauss's Disquisitiones arithmeticae and, like Dirichlet, he became fascinated by the number theory which he read there.
     
   • In June 1844 Eisenstein went to Gottingen for two weeks to visit Gauss.
     
   • Gauss had a reputation for being extremely hard to impress, but Eisenstein had sent some of his papers to Gauss before the visit and Gauss was full of praise.
     
   • the whole trouble is that, when I learned of [Jacobi's] work on cyclotomy, I did not immediately and publicly acknowledge him as the originator, while I frequently have done this in the case of Gauss.
     
   • He was receiving many honours, for example Gauss proposed Eisenstein for election to the Gottingen Academy and he was elected in 1851.
     
   • He worked on the theory of forms with the aim of generalising the results obtained by Gauss in Disquisitiones arithmeticae for the theory of quadratic forms.
     
   • He examined the higher reciprocity laws, with the aim of generalising Gauss's results on quadratic reciprocity, again contained in Disquisitiones arithmeticae.
     
   • These two topics on which Eisenstein worked were both strongly motivated by Gauss's Disquisitiones arithmeticae and the paper [Algebraic number theory (Boston, MA, 1989), 463-469.',13)">13] discusses the copy of this work which Eisenstein owned from his days at school which is now in the mathematical library in Giessen.
     

6. Legendre biography
   
   • Of course today we attribute the law of quadratic reciprocity to Gauss and the theorem concerning primes in an arithmetic progression to Dirichlet.
     
   • However, Gauss published his version of the least squares method in 1809 and, while acknowledging that it appeared in Legendre's book, Gauss still claimed priority for himself.
     
   • For example Gauss had proved the law of quadratic reciprocity in 1801 after making critical remarks about Legendre's proof of 1785 and Legendre's much improved proof of 1798 in the first edition of Theorie des nombres.
     
   • Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man.
     
   • Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof.
     
   • To his credit Legendre used Gauss's proof of quadratic reciprocity in the 1808 edition of Theorie des nombres giving proper credit to Gauss.
     
   • Again Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but certainly it was Legendre who first brought these ideas to the attention of mathematicians.
     

7. Bolyai biography
   
   • In 1816 Farkas wrote to his friend Gauss asking him if he would let Janos live with him and take him on as a pupil so that he might receive the best possible mathematical education.
     
   • Gauss, however, rejected the idea.
     
   • By 20 June 1831 the Appendix had been published for on that day Farkas Bolyai sent a reprint to Gauss who, on reading the Appendix, wrote to a friend saying:- .
     
   • To Farkas Bolyai, however, Gauss wrote:- .
     
   • There is no doubt that Gauss was simply stating facts here.
     
   • The clearest reference in Gauss's letters to his work on non-euclidean geometry, which shows the depth of his understanding, occurs in a letter he wrote to Taurinus on 8 November 1824 when he wrote:- .
     
   • The discovery that Gauss had anticipated much of his work, however, greatly upset Bolyai who took it as a severe blow.
     
   • They include his complaint that he was wronged, his suspicion that Lobachevsky did not exist at all, and that everything was the spiteful machinations of Gauss: it is the tragic lament of an ingenious geometrician who was aware of the significance of his discovery but failed to get support from the only person who could have appreciated his merits.
     

8. Abel biography
   
   • Abel sent this pamphlet to several mathematicians including Gauss, who he intended to visit in Gottingen while on his travels.
     
   • It had been Abel's intention to travel with Crelle to Paris and to visit Gauss in Gottingen on the way.
     
   • However, news got back to Abel that Gauss was not pleased to receive his work on the insolubility of the quintic, so Abel decided that he would be better not to go to Gottingen.
     
   • It is uncertain why Gauss took this attitude towards Abel's work since he certainly never read it - the paper was found unopened after Gauss's death.
     
   • the first possibility is that Gauss had proved the result himself and was willing to let Abel take the credit.
     
   • The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a symbol for the root of the equation and then saying that the equation had a root equal to the symbol.
     

9. Dedekind biography
   
   • Gauss also taught courses in mathematics, but mostly at an elementary level.
     
   • In the autumn term of 1850, Dedekind attended his first course given by Gauss.
     
   • fifty years later Dedekind remembered the lectures as the most beautiful he had ever heard, writing that he had followed Gauss with constantly increasing interest and that he could not forget the experience.
     
   • Dedekind did his doctoral work in four semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals.
     
   • He received his doctorate from Gottingen in 1852 and he was to be the last pupil of Gauss.
     
   • Gauss died in 1855 and Dirichlet was appointed to fill the vacant chair at Gottingen.
     
   • Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Peter Dirichlet, Carl Gauss, and Georg Riemann.
     

10. Pfaff biography
    
    • One student who studied at Helmstedt was Gauss.
      
    • After studying at Gottingen, Gauss came to Helmstedt in 1798.
      
    • Pfaff recommended Gauss's doctoral dissertation and, when necessary, greatly assisted him; Gauss always retained a friendly memory of Pfaff both as a teacher and as a man.
      
    • By the time Gauss studied with Pfaff at Helmstedt, the university was under threat of closure.
      
    • In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral.
      
    • This failure to recognise the importance of the work is strange, particularly given the very positive review which Gauss wrote of the work shortly after it was published.
      

11. Dirichlet biography
    
    • Dirichlet set off for France carrying with him Gauss's Disquisitiones arithmeticae a work he treasured and kept constantly with him as others might do with the Bible.
      
    • It was therefore something of a relief when, on Gauss's death in 1855, he was offered his chair at Gottingen.
      
    • However he received no quick reply to his modest request so he wrote to Gottingen accepting the offer of Gauss's chair.
      
    • Around this time he also published a paper inspired by Gauss's work on the law of biquadratic reciprocity.
      
    • Details are given in [The Mathematical Intelligencer 10 (1988), 13-26.',13)">13] where Rowe discusses the importance of the intellectual and personal relationship between Gauss and Dirichlet.
      
    • This had been conjectured by Gauss.
      

12. Germain biography
    
    • However, Germain's most famous correspondence was with Gauss.
      
    • During their correspondence, Gauss gave her number theory proofs high praise, an evaluation he repeated in letters to his colleagues.
      
    • Germain's true identity was revealed to Gauss only after the 1806 French occupation of his hometown of Braunschweig.
      
    • Recalling Archimedes' fate and fearing for Gauss's safety, she contacted a French commander who was a friend of her family.
      
    • When Gauss learnt that the intervention was due to Germain, who was also "M.
      

13. Borchardt biography
    
    • He did important research on the arithmetic geometric mean continuing work in this area which had been begun by Gauss and Lagrange.
      
    • In 1881 Borchardt published an algorithm for the arithmetic-geometric mean of two elements from (two) sequences, although it was actually first proposed by Gauss in a letter to Pfaff written in 1800.
      
    • Although Gauss's letter is lost we know its contents through Pfaff's reply which was published in Gauss's Complete Works and indicates that Gauss had discovered the result.
      

14. Smith biography
    
    • He had been most influenced by the work of Gauss.
      
    • If we except the great name of Newton (and the exception is one that the great Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute vigorousness in demonstration..
      
    • Influenced by Gauss, Smith's most important contributions are in number theory where he worked on elementary divisors.
      
    • Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms.
      

15. Schlesinger biography
    
    • He was an admirer of Gauss and he wrote related essays, like Uber Gauss' Arbeiten zur Funktionentheorie 222 S Berlin, J Springer (C F Gauss.
      
    • published in 1933 or C F Gauss: Fragmente zur Theorie des arithmetisch- geometrischen Mittels aus den Jahren 1797-1799 in Gottinger Nachrichten published in 1912.
      

16. Bolyai Farkas biography
    
    • There he was taught by Kastner and became a life long friend of Gauss, a fellow student at Gottingen.
      
    • He discussed these issues with Gauss and his later writing show how important he considered their friendship to be for his mathematical development.
      
    • The only mathematical pleasures in his difficult life were the letters he exchanged with Gauss and, in later years, the mathematical achievements of his son.
      
    • In 1804 he believed that he had a proof that it could be deduced from the other axims, but he sent his proof to Gauss who discovered the error.
      

17. Jacobi biography
    
    • He now wrote to Gauss to tell him of the results on cubic residues which he had obtained, having been inspired by Gauss's results on quadratic and biquadratic residues.
      
    • Gauss was impressed, so much so that he wrote to Bessel to obtain more information about the young Jacobi.
      
    • On the journey to Paris he had visited Gauss in Gottingen.
      

18. Taurinus biography
    
    • Taurinus not only corresponded on mathematical topics with his uncle but he also corresponded with Gauss about his ideas on geometry.
      
    • like those of his uncle, Schweikart, represent a middle stage in the historical development of this problem between the efforts of Saccheri and Lambert, on the one hand, and those of Gauss, Lobachevsky, and Bolyai, on the other.
      
    • A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry.
      

19. Fuss biography
    
    • It was in this role that Fuss wrote to Gauss offering him a post at the Academy.
      
    • 4 (1977), 37-41.',3)">3] contains Gauss's reply to Fuss written on 4 April 1803, in which he declines employment in St Petersburg and discusses his observations of the asteroid Pallas.
      
    • The reason for refusing the offer of a position was Gauss's commitment to Karl Wilhelm Ferdinand, the Duke of Braunschweig.
      

20. Mobius biography
    
    • In 1813 Mobius travelled to Gottingen where he studied astronomy under Gauss.
      
    • Gauss was the director of the Observatory in Gottingen but of course the greatest mathematician of his day, so again Mobius studied under an astronomer whose interests were mathematical.
      
    • From Gottingen Mobius went to Halle where he studied under Johann Pfaff, Gauss's teacher.
      

21. Beltrami biography
    
    • Influenced by Cremona, Lobachevsky, Gauss and Riemann, Beltrami contributed to work in differential geometry on curves and surfaces.
      
    • He translated Gauss's work on conformal representation into Italian.
      

22. Sierpinski biography
    
    • Gauss proved in 1837 that d ≤ 1.
      
    • Let us digress for a moment to discuss some further work which flowed from this result of Sierpinski on what is often called the 'Gauss circle problem'.
      

23. Kummer biography
    
    • In 1855 Dirichlet left Berlin to succeed Gauss at Gottingen.
      
    • He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations.
      

24. Minding biography
    
    • His work, which continued Gauss's study of 1828 on the differential geometry of surfaces, greatly influenced Peterson.
      
    • In fact Gauss had proved these results, before either Minding of Bonnet, in 1825 but he had not published them.
      

25. Jarnik biography
    
    • One of the problems which he worked on extensively was related to the Gauss circle problem.
      
    • Gauss proved in 1837 that d <= 1.
      

26. Dahlquist biography
    
    • As an example let us note the publication of On summation formulas due to Plana, Lindelof and Abel, and related Gauss-Christoffel rules in BIT in three parts (1997, 1997, 1999).
      
    • Gauss quadrature rules are designed for each of them.
      

27. Ricci-Curbastro biography
    
    • The initial contributions had been made by Gauss, then the ideas had been developed in Riemann's 1854 Probevorlesung and in an 1861 paper which he wrote for a prize contest of the Paris Academie des Sciences.
      
    • But the methods themselves and the advantages they offer have their raison d'etre and their source in the intimate relationships that join them to the notion of an n-dimensional variety, which we owe to the brilliant minds of Gauss and Riemann.
      

28. Clausen biography
    
    • Clausen's work was recognised by many of the top scientists of the day including Olbers, Gauss, Bessel, Hansen, Crelle, von Humboldt and Arago.
      
    • Gauss was impressed with Clausen, describing him as .
      

29. Cantor Moritz biography
    
    • Moritz Cantor entered Heidelberg University in 1848, then worked under Gauss, Weber and Stern at Gottingen.
      
    • This fourth volume again stopped just before a highly significant development since 1799 is the year of Gauss's doctoral thesis.
      

30. Delsarte biography
    
    • One of the most surprising of Delsarte's results was a generalisation of a result due to Gauss.
      
    • Gauss had shown that if a continuous function f on Rn has at each point x a value equal to its mean value on every sphere of centre x, then f is harmonic.
      

31. Chern biography
    
    • He worked on characteristic classes during his 1943-45 visit to Princeton and, also at this time, he gave a now famous proof of the Gauss-Bonnet formula.
      
    • He made Gauss-Bonnet a household word, .
      

32. Adrain biography
    
    • After publishing further work on Diophantine algebra, he published a paper on the normal law of errors in 1808, one year before Gauss.
      
    • It is unfortunate that despite Adrain's priority over Gauss, it is the latter who has received the credit for this important statistical contribution.
      

33. Galois biography
    
    • Galois' brother and his friend Chevalier copied his mathematical papers and sent them to Gauss, Jacobi and others.
      
    • It had been Galois' wish that Jacobi and Gauss should give their opinions on his work.
      

34. Kahler biography
    
    • He gave these to Kahler who, through studying them, learnt about the work of Gauss and about elliptic and abelian functions.
      
    • He also read works by Gauss, Abel, Weierstrass, Riemann, Lagrange and got to know Artin personally.
      

35. Qin Jiushao biography
    
    • We should not underestimate [Qin's] revolutionary advance, because from [Sun Zi's] single remainder problem, we come at once to the general procedure for solving the remainder problem, even more advanced than Gauss's method, and there is not the slightest indication of gradual evolution.
      
    • How impressive is this work? Well suffice to say that Euler failed to provide a satisfactory solution to these problems and it was left to Gauss, Lebesgue and Stieltjes to rediscovered this method of solving systems of congruences.
      

36. Christoffel biography
    
    • He generalised Gauss's method of quadrature and expressed the polynomials which are involved as a determinant.
      
    • Nevertheless, it is widely recognised, at least in the German speaking countries of Europe, that Riemann was the best mathematician of the 19th century, behind Gauss and ahead of Weierstrass.
      

37. Chowla biography
    
    • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
      
    • Among the theorems to which Chowla's name have been attached are the Bruck-Chowla-Ryser theorem on designs (1950); the Ankeny-Artin-Chowla theorem on the class number of real quadratic number fields (1952); the Chowla-Mordell theorem on Gauss sums (1962); and the Chowla-Selberg formula for the product of certain values of the Dedekind eta function.
      

38. Bachmann biography
    
    • In 1856 he went from Berlin to Gottingen so that he could continue to study courses by Dirichlet who had just left Berlin to succeed to Gauss's chair in Gottingen.
      
    • In Gottingen Bachmann became close friends with Dedekind who had only a few years earlier been awarded his doctorate under Gauss's supervision.
      

39. Meders biography
    
    • Meders was also interested in the history of mathematics and he wrote an important paper Direkte und indirekte Beziehungen zwischen Gauss und der Dorpater Universitat (Direct and indirect connections between Gauss and the University of Dorpat) in 1928.
      

40. Knapowski biography
    
    • On experimental evidence, after extensive calculation, Legendre in 1798 and Gauss in 1793 (according to a letter he wrote 50 years later) suggested that for large n the density of primes behaves like the function 1/log(n).
      
    • Gauss's estimate, the logarithmic integral .
      

41. Wessel biography
    
    • It was rediscovered by Argand in 1806 and again by Gauss in 1831.
      
    • (It is worth noting that Gauss redid another part of Wessel's work, for he retriangulated Oldenburg in around 1824.) .
      

42. Artin biography
    
    • Also in 1927 he gave a general law of reciprocity which included all previously known laws of reciprocity which had been discovered from the time that Gauss produced his first law.
      
    • Given any integer g not 1 or -1, and g not a power of some other integer, then Artin conjectured that there are infinitely many prime numbers p such that g is a primitive root modulo p in the sense of Gauss.
      

43. Kaestner biography
    
    • He was an excellent expositor of mathematics although it is reported that Gauss did not bother to go to his lectures as he found them too elementary.
      
    • However he did influence Gauss, in particular with his interest in Euclid's parallel postulate.
      

44. Euler biography
    
    • Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively.
      
    • Many unpublished results by Euler in this area were rediscovered by Gauss.
      

45. Dase biography
    
    • He requested a grant to allow him to undertake this work and Gauss was asked by the Academy if this was a worthwhile task.
      
    • Gauss replied (see for example [he American Journal of Psychology 4 (1) (1891), 1-59.',4)">4]):- .
      

46. Jordan biography
    
    • Although given Jordan's work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not.
      
    • The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying.
      

47. Libri biography
    
    • It was an impressive start for the remarkable young mathematician, and his first contribution received considerable praise from many of the leading mathematicians of the day such as Babbage, Cauchy, and Gauss.
      
    • To neglect the path by which human nature ought to have passed to arrive at such and such a discovery - for example, not to stop at a mathematical theorem, until at the hands of a Lagrange or a Gauss it has received a definitive form - would be to act as a naturalist who attempted only to study insects under the shape of beautiful butterflies, without giving the slightest attention to the caterpillars, to those less perfect larvae which at a later period are to be transformed into those self-same lepidoptera ..
      

48. Bonnet biography
    
    • A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem.
      
    • Gauss published a special case.
      

49. Mathews biography
    
    • In his two volume work Theory of numbers (1892) topics covered included Gauss's theory of quadratic forms and their development by mathematicians such as Dirichlet, Eisenstein and Smith.
      

50. Richard Louis biography
    
    • Under Richard's guidance, Hermite read papers by Euler, Gauss and Lagrange rather than work for his formal examinations, and he published two mathematics papers while a student at Louis-le-Grand.
      

51. Von Dyck biography
    
    • He made significant contributions to the Gauss-Bonnet theorem.
      

52. Morera biography
    
    • He was interested also in the Cauchy integral for the representation of functions of a complex variable, in the discontinuity of the differentials of the potential function and in the Gauss representation formula.
      

53. Bauer biography
    
    • the theory of multidimensional Lebesgue integration as a tool for handling integrals involved in problems of analysis and mathematical statistics (the gamma function, the Gauss distribution function, potential theory, the volume of the n-dimensional sphere, etc.).
      

54. Hirst biography
    
    • He met Weber and Gauss at this time.
      

55. Eisenhart biography
    
    • Riemann proposed the generalisation of the theory of surfaces as developed by Gauss, to spaces of any order, and introduced certain fundamental ideas in this general theory.
      

56. Darboux biography
    
    • Relying on the classical results of Monge, Gauss, and Dupin, Darboux fully used, in his own creative way, the results of his colleagues Bertrand, Bonnet, Ribaucour, and others.
      

57. Weingarten biography
    
    • In fact Darboux said that Weingarten's work was worthy of Gauss, a compliment indeed.
      

58. Ramanujan biography
    
    • Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series.
      

59. Krylov Aleksei biography
    
    • Krylov's practical interests were combined with a deep understanding of the ideas and methods of classical mathematics and mechanics of the seventeenth, eighteenth, and nineteenth centuries; and in the world of Newton, Euler, and Gauss, he found forgotten methods that were applicable to the solution of contemporary problems.
      

60. Wolf biography
    
    • In 1838 he visited Gauss, then in the following year he became a lecturer in mathematics and physics at the University of Bern.
      

61. Redei biography
    
    • Gauss had proved that the number of even invariants of the class group of a quadratic number field is one less than the number of prime factors occurring in the discriminant of the number field.
      

62. Codazzi biography
    
    • The formulas give two relations between the first and second quadratic forms over a surface together with an equation, already found by Gauss, which gives necessary and sufficient conditions for the existence of a surface which admits two given quadratic forms.
      

63. Ito biography
    
    • DMV/IMU Gauss Prize2006 .
      

64. Wantzel biography
    
    • Gauss had stated that the problems of duplicating a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs.
      

65. Vandermonde biography
    
    • Vandermonde considers the intertwining of the curves generated by the moving knight and his work in this area marks the beginning of ideas which would be extended first by Gauss and then by Maxwell in the context of electrical circuits.
      

66. Wangerin biography
    
    • As examples of Wangerin's historical writing, in addition to the articles on his teacher Franz Neumann which we mentioned above, we should point in particular to the article he wrote on Eduard Heine in 1928 as well as to his input to editing the works of Gauss, Euler, Lambert, and Lagrange.
      

67. Gibbs biography
    
    • The method was applied to find the orbit of Swift's comet of 1880 and involved less computation than Gauss's method.
      

68. Killing biography
    
    • He also read works by Hesse and he read Gauss's Disquisitiones Arithmeticae.
      

69. Calugareanu biography
    
    • In differential topology, starting from an invariant introduced by Gauss, Calugareanu discovered a system of invariants which found applications in knot theory and molecular biology.
      

70. Zhu Shijie biography
    
    • In dealing with simultaneous equations, Zhu certainly presented improvements, giving a method essentially equivalent to Gauss's pivotal condensation.
      

71. Cajori biography
    
    • S D Chambless (A letter from Cajori to Gauss's grandson) and the reply .
      

72. Lerch biography
    
    • His favourite topics in number theory included binary quadratic forms, quadratic residues, Gauss sums and Fermat quotients.
      

73. Genocchi biography
    
    • He did not adopt the methods of Riemann and Weierstrass, but rather worked in the tradition of Euler, Lagrange, Gauss and Cauchy.
      

74. Alling biography
    
    • The first two parts offer a detailed and scrupulous presentation of the historical development of the theory of elliptic integrals and functions in the 18th and 19th centuries, from Giulio Fagnano and Euler through Legendre, Gauss, Abel and Jacobi to Riemann and Weierstrass.
      

75. Heilbronn biography
    
    • The first of the two papers proved a conjecture of Gauss on imaginary quadratic number fields using ideas of Hecke, Deuring and Mordell.
      

76. Stieltjes biography
    
    • Stieltjes started his studies at the Polytechnical School of Delft in 1873 but spent his student years reading Gauss and Jacobi in the library rather than attending lectures.
      

77. Aleksandrov Aleksandr biography
    
    • Because of the depth of this theory, the importance of its applications and the breadth of its generality, Aleksandrov comes second only to Gauss in the history of the development of the theory of surfaces.
      

78. Whiston biography
    
    • This is of interest because his study precedes the publication of 'Meyer's method' by thirty-one years and the Legendre-Gauss 'method of least squares' by eighty-six years.
      

79. Bessel-Hagen biography
    
    • Bessel-Hagen was also involved in editing the works of Gauss.
      

80. Fatou biography
    
    • Using existance theorems for the solutions to differential equations, Fatou was able to prove rigorously certian results on planetary orbits which Gauss had suggested by only verified with an intuitive argument.
      

81. Cunha biography
    
    • In the same year, Gauss wrote a letter to Bessel in which he commented positively on da Cunha's definitions of the exponential and logarithmic functions.
      

82. Korkin biography
    
    • He had read, and with his wonderful memory could then recall, most works by Abel, Dirichlet, Euler, Fourier, Gauss, Jacobi, Lagrange, Laplace, Legendre, Monge, and Poisson.
      

83. Nugel biography
    
    • He became the first mathematician since Gauss to turn down a chair at Berlin, considered up to that time as the ultimate goal a mathematician could reach.
      

84. Blum biography
    
    • The point of view of this book is that the Turing model (we call it "classical") with its dependence on 0's and 1's, is fundamentally inadequate for giving such a foundation for modern scientific computation, where most of the algorithms - with origins in Newton, Euler, Gauss, et al.
      

85. Lemaitre biography
    
    • In 1942 he published L'iteration rationnelle in which he discussed Gauss's method of successive approximations applied to a system of two equations in two unknowns to determine the orbit of a planet from three observations.
      

86. Smithies biography
    
    • Since the first generally accepted proof of this result was given by Gauss in 1799, Wood's paper deserves careful examination.
      

87. Helmholtz biography
    
    • The following year, fully sharing the mathematical itinerary that, through Gauss, Riemann, Lobachevsky and Beltrami, led to the creation of the new geometry, he proposed to spread this knowledge among philosophers while at the same time criticizing the Kantian system.
      

88. Fraenkel biography
    
    • He wrote on Gauss's work in algebra in 1920, then in 1930, he published an important biography of Cantor.
      

89. Weber biography
    
    • Equally important was the fact that Carl Friedrich Gauss also attended Weber's lecture and immediately saw the tremendous potential displayed by the young physicist.
      
    • At this time Gauss was interested in geomagnetism and he realised that Weber would make an outstanding co-worker.
      
    • There followed six years of close friendship and collaboration between Weber and Gauss.
      
    • In 1832 Weber and Gauss published a joint paper which introduced absolute units of measurement of magnetism for the first time.
      
    • Another joint venture by Weber and Gauss of fundamental importance was their founding of the Gottingen Magnetische Verein in 1833.
      
    • In [Gauss-Ges.
      
    • 20-21 (1983/84), 44-53.',7)">7] the Gauss-Weber telegraph design is discussed.
      
    • Gauss and Weber jointly published Atlas Des Erdmagnetismus: Nach Den Elementen Der Theorie Entworfen in 1840 which contains magnetic maps constructed using a network of magnetic observatories which they had organized from 1836 onwards to correlate measurements of terrestrial magnetism around the world.
      
    • Not all Weber's work during this time was with Gauss, for he also collaborated with his younger brother Eduard, an anatomist and physiologist, who was interested in the physics of human locomotion, particularly the mechanism of walking.
      
    • Gauss and von Humboldt appealed to the King to reinstate Weber and the King agreed to do so provided Weber make a public retraction of the views expressed in the letter of protest.
      
    • Gauss and Weber's systematic assignment of absolute units, overshadowed by our four-unit systems, allowed the nineteenth-century analytic formulation of physical laws (a fundamental requisite for theoretical predictions) ..
      
    • By this time Gauss was over seventy years of age and rather too old for the two scientists to restart the remarkably fruitful collaboration which had begun nearly twenty years earlier.
      
    • Gauss died in 1855, and shortly before this Weber began a collaboration with Rudolph Hermann Arndt Kohlrausch who was then at Marburg.
      

90. Herschel Caroline biography
    
    • She was now a celebrity in the world of science and she was visited by many scientists including Gauss.
      

91. Blumenthal biography
    
    • Blumenthal was given the works of Gauss, nicely bound:- .
      

92. Fuchs biography
    
    • Fuchs was a gifted analyst whose works form a bridge between the fundamental researches od Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincare, Painleve, and Emile Picard.
      

93. Freitag biography
    
    • A colleague, with echoes of Gauss's description of Mathematics as the Queen of the Sciences, and Number Theory as the Queen of Mathematics, named Herta Freitag as the Queen of the Fibonacci Association.
      

94. Hsu biography
    
    • One concerned what is now known as the Behrens-Fisher problem, while the second Hsu examined the problem of optimal estimators of the variance in the Gauss-Markov model.
      

95. Frattini biography
    
    • On this latter topic he simplified the classical work by Euler, Lagrange and Gauss (anyone would be proud to improve on the work of these three mathematicians!).
      

96. Hopf biography
    
    • He was awarded many prizes including the Gauss-Weber medal and the Lobachevsky award.
      

97. Mertens biography
    
    • Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name.
      

98. Kruskal William biography
    
    • Some of his later publications include When are Gauss-Markov and least squares estimators identical? A coordinate-free approach (1968), The geometry of generalized inverses (1975), and Miracles and statistics: the casual assumption of independence (1988).
      

99. Kellogg biography
    
    • After his death Converses of Gauss' theorem on the arithmetic mean was published in the Transactions of the American Mathematical Society.
      

100. Duarte biography
     
     • He also observed that the interpolation formula of Everett is a consequence of the interpolation formula of Gauss.
       

101. Vinogradov biography
     
     • He often returned to the topic of his first research paper on the error term in an asymptotic formula discovered by Gauss.
       

102. Lame biography
     
     • Lame was considered the leading French mathematician of his time by many, in particular Gauss who was never one to give praise easily held this opinion.
       

103. Waring biography
     
     • This is, in essence, the first result in the theory of symmetric functions (beyond the basic building blocks which appeared in Chapter 1), a theory whose systematic development was not to appear until the 19th century (Lagrange, Gauss, and others) and was ultimately followed by the theory of permutation groups (Galois, Jordan, ..
       

104. Siegel biography
     
     • The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau.
       

105. Seidel biography
     
     • This was valuable coaching for Seidel, particularly since Schnurlein was a good mathematician who had studied under Gauss.
       

106. Weierstrass biography
     
     • Gauss had promised a proof of this in 1831 but had failed to give one.
       

107. Graffe biography
     
     • In 1824 Graffe went to Gottingen where he attended lectures by Gauss and Thibaut.
       

108. Heine biography
     
     • After one semester he went to the University of Gottingen where he attended lectures by Gauss and by M A Stern on number theory.
       

109. Minkowski biography
     
     • Already at this stage in his education he was reading the work of Dedekind, Dirichlet and Gauss.
       

110. Konigsberger biography
     
     • On leaving Greifswald, the students presented him with a beautifully bound edition of the first four volumes of Gauss's works to show their appreciation.
       

111. Hilbert biography
     
     • Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann.
       

112. Kramp biography
     
     • As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers.
       

113. Peirce Benjamin biography
     
     • The course he set up was impressive, including the study of works of Lacroix, Cauchy, Monge, Biot, Hamilton, Laplace, Poisson, Gauss, Le Verrier, Bessel, Adams, Airy, MacCullagh and Franz Neumann.
       

114. De Bruijn biography
     
     • During the summer of 1959 he was Visiting Gauss Professor at the University of Gottingen.
       

115. Fox Leslie biography
     
     • It contains chapters on: Matrix algebra; Elimination methods of Gauss, Jordan, and Aitken; Compact elimination methods of Doolittle, Crout, Banachiewicz and Cholesky; Orthogonalization methods; Condition, accuracy and precision; Comparison of methods, measure of work; Iterative and gradient methods; Iterative methods for latent roots and vectors; and Notes on error analysis for latent roots and vectors.
       

116. Meissel biography
     
     • Meissel must be judged as a classical mathematician, continuing a tradition from an earlier epoch associated with names like Euler, Laplace, Legendre, Gauss, Jacobi, and Dirichlet.
       

117. Bianchi biography
     
     • His methods were based on the theory of the two fundamental differential quadratic forms of Gauss.
       

118. Henrici Peter biography
     
     • He keeps reminding us to ask what Gauss would have done with a parallel computer - or with a pocket calculator.
       

119. Carleson biography
     
     • Carl Friedrich Gauss once described mathematics at the queen of science, and for a servant of this queen like me to stand here in these beautiful surroundings and receive the grand Abel Prize from a real queen is really an overwhelming event in my life.
       

120. Slutsky biography
     
     • Leontovich was a physiologist who had been studying the statistical ideas of Gauss and Pearson and he gave Slutsky material on statistical techniques.
       

121. Hermite biography
     
     • In some ways Hermite was similar to Galois for he preferred to read papers by Euler, Gauss and Lagrange rather than work for his formal examinations.
       

122. Bessel biography
     
     • A doctorate was awarded by the University of Gottingen on the recommendation of Gauss, who had met Bessel in Bremen in 1807 and recognised his talents.
       

123. Frohlich biography
     
     • This was not the only book which he published in 1983, for Central extensions, Galois groups, and ideal class groups of number fields appeared in the same year, as did Gauss sums and p-adic division algebras with Classgroups and Hermitian modules being published in the following year.
       

124. Hsiung biography
     
     • The central themes are the Gauss-Bonnet formula and uniqueness theorems for the Minkowski and Christoffel problems.
       

125. Bertrand biography
     
     • In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French.
       

126. Schwarz biography
     
     • In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results.
       

127. Bunyakovsky biography
     
     • His work in number theory was important and he gave a new proof of Gauss's law of quadratic reciprocity.
       

History Topics

1. Fund theorem of algebra
   
   • Gauss is usually credited with the first proof of the FTA.
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   • Of Euler's proof Gauss says .
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   • Gauss himself does not claim to give the first proper proof.
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   • Gauss's proof of 1799 is topological in nature and has some rather serious gaps.
     
   • Two years after Argand's proof appeared Gauss published in 1816 a second proof of the FTA.
     
   • Gauss uses Euler's approach but instead of operating with roots which may not exist, Gauss operates with indeterminates.
     
   • A third proof by Gauss also in 1816 is, like the first, topological in nature.
     
   • Gauss introduced in 1831 the term 'complex number'.
     
   • Gauss's criticisms of the Lagrange-Laplace proofs did not seem to find immediate favour in France.
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   • Lagrange's 1808 2nd Edition of his treatise on equations makes no mention of Gauss's new proof or criticisms.
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   • Even the 1828 Edition, edited by Poinsot, still expresses complete satisfaction with the Lagrange-Laplace proofs and no mention of the Gauss criticisms.
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   • In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots.
     
   • The proof is similar to the first proof given by Gauss.
     
   • It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest.
     
   • Gauss called them a shadow of shadows.
     
   • I regard it as unjust to ascribe this proof exclusively to Gauss, who merely added the finishing touches.
     

2. Non-Euclidean geometry
   
   • The first person to really come to understand the problem of the parallels was Gauss.
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   • However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates.
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   • Perhaps most surprisingly of all Gauss never published this work but kept it a secret.
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   • At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.
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   • Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate.
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   • Gauss, after reading the 24 pages, described Janos Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order .
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   • Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published.
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   • Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication.
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   • Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
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3. Group theory
   
   • Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and Janos Bolyai among others.
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   • Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups.
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   • Gauss also examined other abelian groups.
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   • Gauss examined the behaviour of forms under transformations and substitutions.
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   • Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds.
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   • In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.
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4. Orbits
   
   • However Gauss in a brilliant piece of work was able to compute an orbit from the small number of observations.
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   • In fact Gauss' s method requires only 3 observations and is still essentially that used today in calculating orbits.
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   • Ceres, so named by Piazzi, was found to be where Gauss predicted by Olbers.
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   • Johann Encke, a student of Gauss, computed (using Gauss's method) an elliptical orbit for the comet of 1818.
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   • Encke, Gauss's student referred to above, suggested Neptune as a name.
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5. Prime numbers
   
   • Legendre and Gauss both did extensive calculations of the density of primes.
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   • Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers).
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   • Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n).
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   • while Gauss's estimate is in terms of the logarithmic integral .
     
   • You can see the Legendre estimate and the Gauss estimate and can compare them.
     

6. Matrices and determinants
   
   • The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms.
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   • In the same work Gauss lays out the coefficients of his quadratic forms in rectangular arrays.
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   • Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas.
     
   • Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns.
     
   • Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix.
     

7. Non-Euclidean geometry references
   
   • V F Kagan, The construction of non-Euclidean geometry by Lobachevskii, Gauss and Bolyai (Russian), Akad.
     
   • H Karzel, Development of non-Euclidean geometries since Gauss, Proceedings of the 2nd Gauss Symposium (Berlin, 1995).
     
   • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
     

8. Non-Euclidean geometry references
   
   • V F Kagan, The construction of non-Euclidean geometry by Lobachevskii, Gauss and Bolyai (Russian), Akad.
     
   • H Karzel, Development of non-Euclidean geometries since Gauss, Proceedings of the 2nd Gauss Symposium (Berlin, 1995).
     
   • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
     

9. Fund theorem of algebra references
   
   • J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss, Publ.
     
   • A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math.
     
   • R C F Kooistra, Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr.
     
   • I Schneider, Herausragende Einzelleistungen im Zusammenhang mit der Kreisteilungsgleichung, dem Fundamentalsatz der Algebra und der Reihenkonvergenz, in Carl Friedrich Gauss (1777-1855) (Munich, 1981), 37-63.
     

10. Fund theorem of algebra references
    
    • J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss, Publ.
      
    • A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math.
      
    • R C F Kooistra, Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr.
      
    • I Schneider, Herausragende Einzelleistungen im Zusammenhang mit der Kreisteilungsgleichung, dem Fundamentalsatz der Algebra und der Reihenkonvergenz, in Carl Friedrich Gauss (1777-1855) (Munich, 1981), 37-63.
      

11. Mental arithmetic
    
    • Other mathematicians who have exhibited great powers in mental arithmetic include Ampere, Hamilton and Gauss.
      
    • He is of particular interest since his talents were investigated by Gauss, Encke and other mathematicians.
      
    • Gauss commented that he thought that someone skilled in calculation could have done the 100 digit example in about half that time with pencil and paper.
      

12. Cartography references
    
    • H Kautzleben, Carl Friedrich Gauss und die Astronomie, Geodasie und Geophysik seiner Zeit, in Festakt und Tagung aus Anlass des 200 Geburtstages von Carl Friedrich Gauss, Berlin, 1977 (Berlin, 1978), 123-136.
      

13. Topology history references
    
    • E Breitenberger, Gauss und Listing : Topologie und Freundschaft, Gauss-Ges.
      

14. Cartography references
    
    • H Kautzleben, Carl Friedrich Gauss und die Astronomie, Geodasie und Geophysik seiner Zeit, in Festakt und Tagung aus Anlass des 200 Geburtstages von Carl Friedrich Gauss, Berlin, 1977 (Berlin, 1978), 123-136.
      

15. Ring Theory
    
    • For example Legendre and Gauss investigated integer congruences in 1801.
      
    • Gauss had proved around 1801 that numbers of the form a + b√-1, where a, b are integers, could be written uniquely as a product of prime numbers of the form a + b√-1 in an analogous manner to the unique decomposition of an integer as a product of prime integers.
      

16. Topology history
    
    • Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology.
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17. Trisecting an angle
    
    • The construction of regular polygons using ruler and compass was certainly one of the major aims of Greek mathematics and it was not until the discoveries of Gauss that further polygons were constructed with ruler and compass which the ancient Greeks had failed to find.
      
    • Gauss had stated that the problems of doubling a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs.
      

18. Hirst's diary
    
    • Gauss .
      
    • Carl Friedrich Gauss: .
      

19. Topology history references
    
    • E Breitenberger, Gauss und Listing : Topologie und Freundschaft, Gauss-Ges.
      

20. General relativity
    
    • Einstein then remembered that he had studied Gauss's theory of surfaces as a student and suddenly realised that the foundations of geometry have physical significance.
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21. Abstract linear spaces references
    
    • E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
      

22. Infinity references
    
    • W C Waterhouse, Gauss on infinity, Historia Math.
      

23. Abstract linear spaces references
    
    • E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
      

24. Longitude2 references
    
    • R Eck, Tobias Mayer, Johann David Michaelis, Carsten Niebuhr und die Gottinger Methode der Langenbestimmung, Gauss-Ges.
      

25. Nine chapters references
    
    • V I Ilyushchenko, Gauss elimination method (1849 AD) in the ancient Chinese script Mathematics in nine chapters (152 BC), Communications of the Joint Institute for Nuclear Research (Dubna, 1992).
      

26. Quadratic etc equations references
    
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
      

27. Debating topics
    
    • How many days did Gauss live? .
      

28. Bolzano publications.html
    
    • Evidently Bolzano had no inkling of the work of his great contemporaries: Gauss, Bolyai and Lobachevsky.
      

29. Infinity references
    
    • W C Waterhouse, Gauss on infinity, Historia Math.
      

30. Matrices and determinants references
    
    • E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
      

31. History overview
    
    • Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences.
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32. Knots and physics
    
    • He was interested in knots because of electromagnetic considerations and in a letter to Tait written on the 4 December 1867 he rediscovered an integral formula counting the linking number of two closed curves which Gauss had discovered, but had not published, in 1833.
      

33. Longitude1 references
    
    • R Eck, Tobias Mayer, Johann David Michaelis, Carsten Niebuhr und die Gottinger Methode der Langenbestimmung, Gauss-Ges.
      

34. Infinity
    
    • Gauss, in a letter to Schumacher in 1831, argued against the actual infinite:- .
      

35. Indian mathematics
    
    • These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral.
      

36. Bolzano's manuscripts references
    
    • P Bussotti, The problem of the foundations of mathematics at the beginning of the nineteenth century : Two lines of thought: Bolzano and Gauss (Italian), Teoria (N.S.) 20 (1) (2000), 83-95.
      

37. Doubling the cube
    
    • Gauss had stated that the problems of doubling a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs.
      

38. Orbits references
    
    • D Gerdes, Wilhelm Olbers' erste Berechnung einer Kometenbahn aus dem Jahre 1779, Gauss-Ges.
      

39. Fermat's last theorem
    
    • [Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).] .
      

40. Quadratic etc equations references
    
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
      

41. Bolzano's manuscripts references
    
    • P Bussotti, The problem of the foundations of mathematics at the beginning of the nineteenth century : Two lines of thought: Bolzano and Gauss (Italian), Teoria (N.S.) 20 (1) (2000), 83-95.
      

42. Longitude2 references
    
    • R Eck, Tobias Mayer, Johann David Michaelis, Carsten Niebuhr und die Gottinger Methode der Langenbestimmung, Gauss-Ges.
      

43. Nine chapters references
    
    • V I Ilyushchenko, Gauss elimination method (1849 AD) in the ancient Chinese script Mathematics in nine chapters (152 BC), Communications of the Joint Institute for Nuclear Research (Dubna, 1992).
      

44. Orbits references
    
    • D Gerdes, Wilhelm Olbers' erste Berechnung einer Kometenbahn aus dem Jahre 1779, Gauss-Ges.
      

45. Longitude1 references
    
    • R Eck, Tobias Mayer, Johann David Michaelis, Carsten Niebuhr und die Gottinger Methode der Langenbestimmung, Gauss-Ges.
      

46. Matrices and determinants references
    
    • E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
      

Famous Curves

1. Frequency
   
   • It was also studied with Laplace and Gauss.
     

Societies etc

1. DMV/IMU Gauss Prize
   
   • IMU Carl Friedrich Gauss Prize for Applications of Mathematics .
     
   • The International Mathematical Union and the the Deutsche Mathematiker-Vereinigung awards the Carl Friedrich Gauss Prize for Applications of Mathematics for:- .
     
   • DMV/IMU Gauss Prize .
     

2. Gauss
   
   • Karl Friedrich Gauss .
     

3. Lunar features
   
   • (W) (L) Gauss .
     

4. Lunar features
   
   • Gauss .
     

5. IMU Nevanlinna Prize
   
   • DMV/IMU Gauss Prize .
     

6. Hungarian Academy of Sciences
   
   • The first such foreign member was Babbage, elected in 1833, followed by Gauss and Poncelet in 1847 and John Herschel and Quetelet in 1858.
     

7. BMC 2004
   
   • Patterson, S JSquaring circles and circling squares: Gauss's Circle Problem .
     

8. Lunar features
   

9. Fellows of the RSE
   
   • Karl Friedrich Gauss1820More infoMacTutor biography .
     

10. Fellows of the RSE
    
    • Karl Friedrich Gauss1820More infoMacTutor biography .
      

11. Fellows of the RSE
    
    • Karl Friedrich Gauss1820More infoMacTutor biography .
      

12. Fellow of the Royal Society
    
    • Carl F Gauss 1804 .
      

13. DVR Honorary Members
    
    • DMV/IMU Gauss Prize .
      

14. Copley Medal
    
    • 1838 Karl Gauss .
      

15. Catalan Mathematical Society
    
    • It publishes books including translations into Catalan of classic texts such as Gauss' Disquisitiones Arithmeticae and Descartes' La Geometrie .
      

16. Royal Astronomical Society (London)
    
    • They include Bouvard from Paris, Delambre from Paris, Gauss from Gottingen, and Olbers from Bremen.
      

17. AMS Cole Prize in Number Theory
    
    • for his paper "Gauss's class number problem for imaginary quadratic fields".
      

References

1. References for Gauss
   
   • References for Carl Friedrich Gauss .
     
   • W K Buhler, Gauss: A Biographical Study (Berlin, 1981).
     
   • G W Dunnington, Carl Friedrich Gauss : Titan of Science (New York, 1955).
     
   • T Hall, Carl Friedrich Gauss : A Biography (1970).
     
   • G M Rassias (ed.), The mathematical heritage of C F Gauss (Singapore, 1991).
     
   • H Reichardt, Gauss, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
     
   • H Reichardt (ed.), C F Gauss Gedenkband anlasslich des 100.
     
   • W S von Waltershausen, Gauss, a Memorial (Colorado Springs, Colo., 1966).
     
   • C Agostinelli, Some aspects of the life and work of Carl Friedrich Gauss and that of other illustrious members of the Academy (Italian), Atti Accad.
     
   • G V Bagratuni, Carl Friedrich Gauss, his works on geodesy and his geodetic research (Russian), Izv.
     
   • W Benham, The Gauss anagram : an alternative solution, Ann.
     
   • H J M Bos, Carl Friedrich Gauss : a biographical note (Dutch), Nieuw Tijdschr.
     
   • E Breitenberger, Gauss und Listing: Topologie und Freundschaft, Gauss-Ges.
     
   • E Breitenberger, Gauss's geodesy and the axiom of parallels, Arch.
     
   • E Buissant des Amorie, Gauss' formula for π (Dutch), Nieuw Tijdschr.
     
   • D A Cox,Gauss and the arithmetic - geometric mean, Notices Amer.
     
   • D A Cox, The arithmetic-geometric mean of Gauss, Enseign.
     
   • H S M Coxeter, Gauss as a geometer, Historia Math.
     
   • J Dieudonne, Carl Friedrich Gauss : a bicentenary, Southeast Asian Bull.
     
   • P J de Doelder, Gauss and function theory (especially with regard to the lemniscate functions) (Dutch), Nieuw Tijdschr.
     
   • J Dutka, On Gauss' priority in the discovery of the method of least squares, Arch.
     
   • M Folkerts, C F Gauss' Beitrag zur Besetzung von Professuren an der Universitat Gottingen, Gauss-Ges.
     
   • E G Forbes, The astronomical work of Carl Friedrich Gauss (1777-1855), Historia Math.
     
   • E G Forbes, Gauss and the discovery of Ceres, J.
     
   • A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math.
     
   • G D Garland, The contributions of Carl Friedrich Gauss to geomagnetism, Historia Math.
     
   • S Gindikin, Carl Friedrich Gauss (on the 200th anniversary of his birth) (Russian), Kvant 8 (1977), 2-14.
     
   • H Grauert, Wie Gauss die alte Gottinger Mathematik schuf, Proceedings of the 2nd Gauss Symposium.
     
   • H-J Felber, Die beiden Ausnahmebestimmungen in der von C F Gauss aufgestellten Osterformel, Sterne 53 (1) (1977), 22-34.
     
   • H-J Treder, Gauss und die Gravitationstheorie, Sterne 53 (1) (1977), 9-14.
     
   • F Henneman, Gauss' law of errors and the method of least squares : a historical sketch (Dutch), Nieuw Tijdschr.
     
   • S H Hollingdale, C F Gauss (1777-1855) : a bicentennial tribute, Bull.
     
   • K-R Biermann, Aus der Gauss-Forschung, Gauss-Ges.
     
   • K-R Biermann, Zu den Beziehungen von C F Gauss und A v Humboldt zu A F Mobius, NTM Schr.
     
   • K-R Biermann, Die Gauss-Briefe in Goethes Besitz, NTM Schr.
     
   • K-R Biermann, C F Gauss in seinem Verhaltnis zur britischen Wissenschaft und Literatur, NTM Schr.
     
   • K-R Biermann, Zu Dirichlets geplantem Nachruf auf Gauss, NTM Schr.
     
   • R Kooistra, C F Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr.
     
   • R Lehti, Gauss's 'Disquisitiones arithmeticae' (Finnish), Arkhimedes 29 (2) (1977), 49-66.
     
   • A F Monna, Gauss and the physical sciences (Dutch), Nieuw Tijdschr.
     
   • P Muursepp, Gauss and Tartu University, Historia Math.
     
   • P Muursepp, Gauss' letter to Fuss of 4 April 1803, Historia Math.
     
   • W Narkiewicz, The work of C F Gauss in algebra and number theory, Festakt und Tagung aus Anlass des 200.
     
   • Geburtstages von Carl Friedrich Gauss (Berlin, 1978), 75-82.
     
   • J G O'Hara, Gauss and the Royal Society : the reception of his ideas on magnetism in Britain (1832-1842), Notes and Records Roy.
     
   • R L Plackett, The influence of Laplace and Gauss in Britain, Bull.
     
   • K Reich, Gauss und seine Zeit, Sterne und Weltraum 16 (5) (1977), 148-157.
     
   • N Ritsema, Gauss and the cyclotomic equation (Dutch), Nieuw Tijdschr.
     
   • D E Rowe, Gauss, Dirichlet and the Law of Biquadratic Reciprocity, The Mathematical Intelligencer 10 (1988), 13-26.
     
   • H Schimank, Carl Friedrich Gauss (German), Gauss-Gesellschaft Gottingen, Mitteilungen 8 (1971), 6-31.
     
   • V G Selihanovic, Carl Friedrich Gauss (on the occasion of the 200th anniversary of his birth) (Russian), Izv.
     
   • O Sheynin, C F Gauss and geodetic observations, Arch.
     
   • O B Sheynin, C F Gauss and the theory of errors, Arch.
     
   • D A Sprott, Gauss's contributions to statistics, Historia Math.
     
   • H B Stauffer, Carl Friedrich Gauss, Bull.
     
   • G W Stewart, Gauss, statistics, and Gaussian elimination, J.
     
   • S M Stigler, Gauss and the invention of least squares, Ann.
     
   • S M Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math.
     
   • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
     
   • W A van der Spek, The Easter formulae of C F Gauss (Dutch), Nieuw Tijdschr.
     
   • W A van der Spek, Gauss' logarithms (Dutch), Nieuw Tijdschr.
     
   • F van der Blij, Gauss and analytic number theory (Dutch), Nieuw Tijdschr.
     
   • W Waterhouse, Gauss's first argument for least squares, Arch.
     
   • W Waterhouse, Gauss on infinity, Historia Math.
     
   • H Wussing, Carl Friedrich Gauss - Leben und Wirken, Festakt und Tagung aus Anlass des 200.
     
   • Geburtstages von Carl Friedrich Gauss (Berlin, 1978), 151-160.
     
   • K Zormbala, Gauss and the definition of the plane concept in Euclidean elementary geometry, Historia Math.
     
   • http://www-history.mcs.st-andrews.ac.uk/References/Gauss.html .
     

2. References for Weber
   
   • G Beuermann and R Gorke, Der elektromagnetische Telegraph von Gauss und Weber aus dem Jahre 1833, Gauss-Ges.
     
   • W Gresky, Die Gauss-Webersche Telegraphenleitung, erneutes Verlegen zum Universitatsjubilaum 1887, Gauss-Ges.
     
   • K H Wiederkehr, Wilhelm Weber und die Entwicklung in der Geomagnetik und Elektrodynamik, Gauss-Ges.
     

3. References for Legendre
   
   • A Aubry, Sur les travaux arithmetiques de Lagrange, de Legendre et de Gauss, Enseignement mathematique 11 (1909), 430-450.
     
   • O B Sheinin, Previous publication of Legendre's attack on Gauss: 'An attack on Gauss, published by Legendre in 1820' by S M Stigler, Historia Math.
     
   • S M Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math.
     

4. References for Laplace
   
   • L Brandt, Uber das Bahnbestimmungsproblem bei Gauss und Laplace.
     
   • Eine Gegenuberstellung ihrer Methoden, Gauss-Ges.
     
   • R L Plackett, The influence of Laplace and Gauss in Britain, Bull.
     

5. References for Taurinus
   
   • F Engel and P Stackel, Die Theorie der Parallellinien von Euklid bis Gauss (Leipzig,1895).
     
   • S Xambo, Non-Euclidean geometry : from Euclid to Gauss (Catalan), Butl.
     

6. References for Lalande
   
   • Geburtstag von Carl Friedrich Gauss, Gauss-Ges.
     

7. References for Listing
   
   • E Breitenberger, Gauss und Listing: Topologie und Freundschaft, Gauss-Ges.
     

8. References for Cunha
   
   • A P Yushkevich, C F Gauss et J A da Cunha (French), Rev.
     
   • A P Yushkevich, C F Gauss and J A da Cunha (Russian), Istor.-Mat.
     

9. References for Bessel
   
   • M Galuzzi, Trigonometric interpolation in Gauss, Bessel and Cauchy (Italian), Writings in honor of Giovanni Melzi, Sci.
     
   • M Kussner, Friedrich Wilhelm Bessels Beziehungen zu Gottingen und Erinnerungen an ihn, Gauss-Ges.
     

10. References for Khayyam
    
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
      

11. References for Eisenstein
    
    • R C Laubenbacher, Gauss, Eisenstein, and the 'third' proof of the quadratic reciprocity theorem, The Mathematical intelligencer 16 (1994), 67-72.
      

12. References for Mobius
    
    • K-R Biermann, Zu den Beziehungen von C F Gauss und A v.
      

13. References for Landen
    
    • G Almkvist, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π and the Ladies Diary, Amer.
      

14. References for Fuss
    
    • P Muursepp, Gauss' letter to Fuss of 4 April 1803, Historia Math.
      

15. References for Bolyai
    
    • V F Kagan, The construction of non-Euclidean geometry by Lobachevsky, Gauss and Bolyai (Russian), Proc.
      

16. References for Maskelyne
    
    • E G Forbes, The correspondence between Carl Friedrich Gauss and the Rev Nevil Maskelyne (1802-5), Annals of Science 27 (1971), 213-237.
      

17. References for Dirichlet
    
    • D E Rowe, Gauss, Dirichlet and the Law of Biquadratic Reciprocity, The Mathematical Intelligencer 10 (1988), 13-26.
      

18. References for Al-Biruni
    
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
      

19. References for Descartes
    
    • M Bartolozzi and R Franci, The rule of signs, from its statement by R Descartes (1637) to its proof by C F Gauss (1828) (Italian), Arch.
      

20. References for Mayer Tobias
    
    • R Eck, Tobias Mayer, Johann David Michaelis, Carsten Niebuhr und die Gottinger Methode der Langenbestimmung, Gauss-Ges.
      

21. References for Archimedes
    
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
      

22. References for Lobachevsky
    
    • J M Montesinos Amilibia, Non-Euclidean geometries : Gauss, Lobachevskii and Bolyai (Spanish), in History of mathematics in the XIXth century (Part 1) (Madrid, 1992), 65-114.
      

23. References for Bolzano
    
    • P Bussotti, The problem of the foundations of mathematics at the beginning of the nineteenth century : Two lines of thought: Bolzano and Gauss (Italian), Teoria (N.S.) 20 (1) (2000), 83-95.
      

24. References for Kaestner
    
    • G Goe, Kaestner, Forerunner of Gauss, Pasch, Hilbert, Proceedings 10th International Congress of the History of Science II (Paris, 1964), 659-661.
      

25. References for Dedekind
    
    • W Fuchs, Zum 150-jahrigen Geburtstag von Richard Dedekind, Gauss-Ges.
      

26. References for Stackel
    
    • N Ya Tsyganova, Gauss's principle of least force in the research of P Stackel (Russian), Differential equations and applied problems (Tula, 1985), 68-74.
      

Additional material

1. Gauss: 'Disquisitiones Arithmeticae
   
   • Gauss: Disquisitiones Arithmeticae .
     
   • In 1801 Carl Friedrich Gauss published his classic work Disquisitiones Arithmeticae.
     
   • A second edition of Gauss' masterpiece appeared in 1870, fifteen years after his death.
     
   • The first reprint was published under my direction in 1863 as the first volume of Gauss' Works.
     
   • The eighth section, to which Gauss makes frequent reference and which he had intended to publish with the others, was found among his manuscripts.
     
   • I believe that this was justified because Gauss had made such a point of economising on space.
     
   • A version of Gauss' Dedication and Preface are given below essentially following the translation by Clark: .
     
   • C F GAUSS .
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Gauss_Disquisitiones.html .
     

2. IMU Carl Friedrich Gauss Prize for Applications of Mathematics
   
   • IMU Carl Friedrich Gauss Prize for Applications of Mathematics .
     
   • The International Mathematical Union and the the Deutsche Mathematiker-Vereinigung awards the Carl Friedrich Gauss Prize for Applications of Mathematics for:- .
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Gauss_prize.html .
     

3. Leonard J Savage: 'Foundations of Statistics
   
   • It was pushed forward in the nineteenth century by Laplace, Gauss, and others, and it has been subject to a fervour of activity since the early twenties of this century, when it received great impetus from the work of R A Fisher.
     
   • It goes back at least as far as Gauss [Abhandlungen zur Methode der kleinsten Quadrate von Carl Friedrich Gauss (Berlin, 1887).
     
   • These same reasons are, to say the least, suggestive even for problems of pure science-a precedent for this idea can be seen in Gauss [Abhandlungen zur Methode der kleinsten Quadrate von Carl Friedrich Gauss (Berlin, 1887).
     
   • Carl Friedrich Gauss, Abhandlungen zur Methode der kleinsten Quadrate von Carl Friedrich Gauss (Berlin, 1887).
     

4. Publications of Albert Wangerin
   
   • A Wangerin (ed.), C F Gauss: Allgemeine Flachentheorie (W Engelmann, Leipzig, 1889).
     
   • A Wangerin (ed.) , C F Gauss: Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des Quadrates der Entfernung wirkenden Anziehungs- und Abstossungs-Krafte (W Engelmann, Leipzig, 1889).
     
   • Laplace, Ivory, Gauss, Chasles und Dirichlet: Uber die Anziehung homogener Ellipsoide (W Engelmann, Leipzig, 1890).
     
   • A Wangerin (ed.), Lagrange (1779) und Gauss (1822): Abhandlungen uber Kartenprojektion (W Engelmann, Leipzig, 1894).
     
   • A Wangerin, Uber die auf die Theorie der conformen Abbildung bezuglichen Arbeiten von Lambert, Lagrange und Gauss.
     
   • A Wangerin (ed.), C F Gauss: Allgemeine Flachentheorie (Disquisitiones generales circa superficies curvas 1827), 2 Aufl.
     

5. Heinrich Tietze on Numbers, Part 2
   
   • Dedekind's work (the theory of ideals) was in a field in which Gauss had been a pioneer; it was algebraic in origin and dealt among other things with problems of factorization analogous to the familiar problem of factoring a number into its prime factors, e.g.: 84 = 2.2.3.7.
     
   • Richard Dedekind was born in Brunswick (the birthplace of the great Gauss), 6 October 1831, where he lived until he was 19.
     
   • For many years Dedekind was the only surviving student of Gauss.
     
   • At 19 he had heard Gauss lecture in the small auditorium of the Gottingen Observatory.
     

6. P G Tait's obituary of Listing
   
   • One of the few remaining links that still continued to connect our time with that in which Gauss had made Gottingen one of the chief intellectual centres Of the civilised world has just been broken by the death of Listing.
     
   • It seems not unlikely, though we can find no proof of it, that Listing was led to such researches by the advice or example of Gauss himself; for Gauss, so long ago as 1833, pointed out their connection with his favourite electromagnetic inquiries.
     
   • The numerals on the scale of Gauss' Magnetometer must, in order to appear to the observer in their natural position, be both perverted and inverted, in consequence of the perversion by reflection and the inversion by the telescope.
     

7. Dubreil-Jacotin on Sophie Germain
   
   • Nevertheless, later on, when she wanted to write Gauss, after the publication of his Disquisitiones arithmeticae in 1801, to discuss with him results she had obtained in the theory of numbers, she once more concealed herself under the pseudonym of Le Blanc, Polytechnic student.
     
   • But Gauss, too, learned the true identity of Le Blanc.
     
   • It was during the time of the German campaign and French - troops were entering Brunswick, Gauss's city; Sophie Germain, haunted by the memory of Archimedes' death, began to fear for the scholar and wrote to a friend of her father, General Pernety, who was at the very moment in Brunswick, to commend her master to him and to entreat him to watch out for his safety.
     

8. R A Fisher: 'History of Statistics
   
   • A direct result of Laplace's study of the distribution of the resultant of numerous independent causes was the recognition of the normal law of error, a law more usually ascribed, with some reason, to his great contemporary, Gauss.
     
   • Gauss, moreover, approached the problem of statistical estimation in an empirical spirit, raising the question of the estimation not only of probabilities but of other quantitative parameters.
     
   • Gauss, further, perfected the systematic fitting of regression formulae, simple and multiple, by the method of least squares, which, in the cases to which it is appropriate, is a particular example of the method of maximum likelihood.
     

9. Mathematicians and Music
   
   • He referred to the cultures of mathematics and music "not merely as having arithmetic for their common parent but as similar in their habits and affections." "May not Music be described," he wrote, "as the Mathematic of Sense, Mathematic as the Music of reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, - Music the dream, Mathematic the working life, - each to receive its consummation from the other when the human intelligence, elevated to the perfect type, shall shine forth glorified in some future Mozart-Dirichlet, or Beethoven-Gauss - a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz"! .
     
   • The late G B Mathews knew music as thoroughly as most professional musicians; his copies of Gauss and Bach were placed together on the same shelf.
     

10. Von Neumann: 'The Mathematician
    
    • The idea that in at least one significant sense-and in spite of all mathematico-logical analyses-the decision for or against Euclid may have to be empirical, was certainly present in the mind of the greatest mathematician, Gauss.
      
    • An inexact, semi-physical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period, against this inexact, mathematically inadequate background! Some of the leading mathematical spirits of the period were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi.
      

11. G H Hardy addresses the British Association in 1922
    
    • There are similar questions, of course, for squares, but the answers to these, were found long ago by Euler and by Gauss, and belong to the classical mathematics.
      
    • Mersenne, he observes, was a good mathematician, but not an Euler or a Gauss, and he inclines to attribute the discovery to the exceptional genius of Fermat, the only mathematician of the age whom anyone could suspect of being hundreds of years ahead of his time.
      

12. Tietze: 'Famous Problems of Mathematics
    
    • Or when, in composing the section on Gauss, the war-torn era was described in which the first years of Gauss' activity at Gottingen fell, we were again in the midst of an epoch in which the slogan of creating a "new Europe" by military means was being advanced.
      

13. Henry Baker addresses the British Association in 1913, Part 2
    
    • We know how the great Gauss, whose lynx eye was laboriously turned upon all the physical science of his time, has left it on record that in order to settle the law of a plus or minus sign in one of the formulae of his theory of numbers he took up the pen every week for four years.
      
    • But in the land of Kummer and Gauss and Dirichlet the subject to-day claims the allegiance of many eager minds.
      

14. Charles Tweedie on James Stirling
    
    • Binet, in a celebrated memoir on Definite Integrals, has shown Stirling's place as a pioneer of Gauss.
      
    • Gauss himself had most unwillingly to make use of Stirling's Series, though its lack of convergence was anathema to him.
      

15. Percy MacMahon addresses the British Association in 1901, Part 2
    
    • A combination of great intellects Legendre, Gauss, Eisenstein, Stephen Smith, etc.
      
    • Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincare, and Klein are great names that will be for ever associated with it.
      

16. Horace Lamb addresses the British Association in 1904
    
    • The classical style of memoir, after the manner of Lagrange, or Poisson, or Gauss, complete in itself and deliberately composed like a work of art, is continually becoming rarer.
      
    • It is upon these men that we must look as the immediate intellectual ancestors of Stokes, for, although Gauss and Franz Neumann were in their full vigour, the interaction of German and English science was at that time not very great.
      

17. Carathéodory: 'Conformal representation
    
    • In 1822 Gauss (1777-1855) stated and completely solved the general problem of finding all conformal transformations which transform a sufficiently small neighbourhood of a point on an arbitrary analytic surface into a plane area.
      
    • This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane.
      

18. Andrew Forsyth addresses the British Association in 1905, Part 2
    
    • When the wonderful school of French physicists, composed of Monge, Sadi Carnot, Fourier, Poisson, Poinsot, Ampere, and Fresnel (to mention only some names), together with Gauss, Kirchhoff, and von Helmholtz in Germany, and Ivory, Green, Stokes, Maxwell, and others in England, applied their mathematics to various branches of physics, for the most part its development was that of an ancillary subject.
      
    • The bead-roll of names in that science - Gauss; Abel, Jacobi; Cauchy, Riemann, Weierstrass, Hermite; Cayley, Sylvester; Lobachevsky, Lie - will on only the merest recollection of the work with which their names are associated show that an age has been reached where the development of human thought is deemed as worthy a scientific occupation of the human mind as the most profound study of the phenomena of the material universe.
      

19. Carl B Boyer
    
    • Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Geometrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
      

20. Cajori: 'A history of mathematics' Introduction
    
    • The easy credulity with which a young student supposes that of course every algebraic equation must have a root gives place finally to a delight in the slow conquest of the realm of imaginary numbers, and in the youthful genius of a Gauss who could demonstrate this once obscure fundamental proposition." The history of mathematics is important also as a valuable contribution to the history of civilisation.
      

21. Henry Baker addresses the British Association in 1913
    
    • And, although the principle of Thomson and Dirichlet, which relates to the potential problem referred to, was expounded by Gauss, and accepted by Riemann, and remains to-day in our standard treatise on Natural Philosophy, there can be no doubt that, in the form in which it was originally stated, it proves just nothing.
      

22. Bolzano's publications
    
    • Evidently Bolzano had no inkling of the work of his great contemporaries: Gauss, Bolyai and Lobachevsky.
      

23. G H Hardy addresses the British Association in 1922, Part 2
    
    • Mersenne, he observes, was a good mathematician, but not an Euler or a Gauss, and he inclines to attribute the discovery to the exceptional genius of Fermat, the only mathematician of the age whom anyone could suspect of being hundreds of years ahead of his time.
      

24. Finlay Freundlich's Inaugural Address, Part 2
    
    • About one century ago, however, three mathematicians (Gauss, Bolyai and Lobachevsky) nearly simultaneously came to the conviction that one of the axioms in Euclid's theory could be dropped, i.e.
      

25. E C Titchmarsh: 'Aftermath
    
    • Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, Cauchy or Riemann; few careers can have been more satisfying than theirs.
      

26. Bronowski and retrodigitisation
    
    • Already Johann Carl Friedrich Gauss (1777--1855), in Disquisitiones Arithmeticae, raised the question of determining the primes p for which 10 is a primitive root modulo p.
      

27. David Hilbert: 'Mathematical Problems
    
    • The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky.
      

28. Levi-Civita: 'Lezioni di calcolo differenziale assoluto
    
    • Einstein's discovery of the gravitational equations was announced by him in the famous note "Zur allgemeinen Relativitatstheorie" in the following words: "Sie bedeutet einen wahren Triumph der durch Gauss, Riemann, Christoffel, Ricci ..
      

29. G H Hardy addresses the British Association in 1922, Part 1
    
    • There are similar questions, of course, for squares, but the answers to these, were found long ago by Euler and by Gauss, and belong to the classical mathematics.
      

30. Sommerville obituary.html
    
    • Note on Gauss's Proof of the Reciprocity of Parallelism, 32, pp.
      

31. Carl B Boyer: 'Foremost Modern Textbook
    
    • Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Geometrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
      

32. Percy MacMahon addresses the British Association in 1901
    
    • Whereas in 1801 on the Continent there were the leaders Lagrange, Laplace and Legendre, and of rising men, Fourier, Ampere, Poisson and Gauss, we could only claim Thomas Young and Ivory as men who were doing notable work in research.
      

33. Born Inaugural
    
    • Gauss has frankly expressed his opinion that the axioms of geometry have no superior position as compared with the laws of physics, both being formulations of experience, the former stating the general rules of the mobility of rigid bodies and giving the conditions for measurements in space.
      

34. A de Lapparent: 'Wantzel
    
    • He proved, for the first time the impossibility (already affirmed, but not demonstrated, by Gauss) of obtaining, with rule and compass, the duplication of a cube or the trisection of an angle.
      

35. Bertrand's work on probability' Introduction
    
    • In 1855 he translated into French Gauss's writings on the theory of errors and method of least squares.
      

Quotations

1. Quotations by Gauss
   
   • Quotations by Carl Friedrich Gauss .
     
   • http://www-history.mcs.st-andrews.ac.uk/Quotations/Gauss.html .
     

2. Quotations by Jacobi
   
   • Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is.
     
   • When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain ..
     

3. Quotations by Bell
   
   • Archimedes, Newton, and Gauss, these three, are in a class by .
     
   • three, Archimedes, Newton, and Gauss.
     

4. A quotation by Germain
   
   • Letter to Gauss (1807) .
     

5. Quotations by Boltzmann
   
   • A mathematician will recognise Cauchy, Gauss, Jacobi or Helmholtz after reading a few pages, just as musicians recognise, from the first few bars, Mozart, Beethoven or Schubert.
     

6. Quotations by Abel
   
   • [About Gauss' mathematical writing style] .
     

7. A quotation by Tietze
   
   • The story was told that the young Dirichlet had as a constant companion all his travels, like a devout man with his prayer book, an old, worn copy of the Disquisitiones Arithmeticae of Gauss.
     

8. Quotations by Klein
   
   • The greatest mathematicians, as Archimedes, Newton, and Gauss, .
     

9. Quotations by Titchmarsh
   
   • Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs.
     

Chronology

1. Mathematical Chronology
   
   • Gauss, in 1820, also investigated the normal distribution.
     
   • Gauss gives the first correct proof of the law of quadratic reciprocity.
     
   • Gauss proves the fundamental theorem of algebra and notes that earlier proofs, such as by d'Alembert in 1746, could easily be corrected.
     
   • Gauss publishes Disquisitiones Arithmeticae (Discourses on Arithmetic).
     
   • Gauss computes its orbit from the few observations that had been made leading to Ceres being rediscovered in almost exactly the position predicted by Gauss.
     
   • Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.
     
   • Gauss describes the least-squares method which he uses to find orbits of celestial bodies in Theoria motus corporum coelestium in sectionibus conicis Solem ambientium (Theory of the Movement of Heavenly Bodies).
     
   • When Bolyai discovers that Gauss had anticipated much of his work, but not published anything, he delays publication.
     
   • Gauss introduces differential geometry and publishes Disquisitiones generales circa superficies.
     
   • The paper also includes Gauss's famous theorema egregrium.
     
   • Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.
     

2. Chronology for 1800 to 1810
   
   • Gauss publishes Disquisitiones Arithmeticae (Discourses on Arithmetic).
     
   • Gauss computes its orbit from the few observations that had been made leading to Ceres being rediscovered in almost exactly the position predicted by Gauss.
     
   • Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.
     
   • Gauss describes the least-squares method which he uses to find orbits of celestial bodies in Theoria motus corporum coelestium in sectionibus conicis Solem ambientium (Theory of the Movement of Heavenly Bodies).
     

3. Chronology for 1820 to 1830
   
   • When Bolyai discovers that Gauss had anticipated much of his work, but not published anything, he delays publication.
     
   • Gauss introduces differential geometry and publishes Disquisitiones generales circa superficies.
     
   • The paper also includes Gauss's famous theorema egregrium.
     

4. Chronology for 1780 to 1800
   
   • Gauss gives the first correct proof of the law of quadratic reciprocity.
     
   • Gauss proves the fundamental theorem of algebra and notes that earlier proofs, such as by d'Alembert in 1746, could easily be corrected.
     

5. Chronology for 1840 to 1850
   
   • Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.
     

6. Chronology for 1720 to 1740
   
   • Gauss, in 1820, also investigated the normal distribution.