高斯JohannCarlFriedrichGauss.docx

1、高斯JohannCarlFriedrichGaussJohann Carl Friedrich GaussBorn: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany)Died: 23 Feb 1855 in Gttingen, Hanover (now Germany)At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teache

2、r, Bttner, 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 Bttner and Bartels, where he learnt High Ger

3、man and Latin. After receiving a stipend from the Duke of Brunswick- Wolfenbttel, Gauss entered Brunswick Collegium Carolinum in 1792. At the academy Gauss independently discovered Bodes law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the

4、 prime number theorem. In 1795 Gauss left Brunswick to study at Gttingen University. Gausss teacher there was Kstner, whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai. They met in 1799 and corresponded with each other for many years. Gauss left Gttingen in 179

5、8 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 Gausss famous work, Disquisition

6、es Arithmeticae. Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gausss stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff, who was chosen to be his advisor. Gausss d

7、issertation 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. He published the book Disquisitiones Arithmeticae in the summer of 1801. There were seven sections, all but the last section, referred

8、 to above, being devoted to number theory. 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. Unfortunately, Piazzi had o

9、nly been able to observe 9 degrees of its orbit before it disappeared behind the Sun. 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 predic

10、ted. 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 obser

11、vatory in Gttingen, 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. Despite having a happy personal life for the first time, his benefactor, the Duke of Brunswick, was killed fi

12、ghting for the Prussian army. In 1807 Gauss left Brunswick to take up the position of director of the Gttingen observatory. Gauss arrived in Gttingen in late 1807. In 1808 his father died, and a year later Gausss wife Johanna died after giving birth to their second son, who was to die soon after her

13、. Gauss was shattered and wrote to Olbers asking him to give him a home for a few weeks, to gather new strength in the arms of your friendship - strength for a life which is only valuable because it belongs to my three small children. Gauss was married for a second time the next year, to Minna the b

14、est friend of Johanna, and although they had three children, this marriage seemed to be one of convenience for Gauss. Gausss work never seemed to suffer from his personal tragedy. He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a majo

15、r two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planets orbit. Gausss contribu

16、tions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70. Much of Gausss time was spent on a new observatory, completed in 1816, but he still found the time to work on other subjects. His publications during this time include Disquisitiones gener

17、ales circa seriem infinitam, a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi, a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen, a discussion of statistical es

18、timators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata. The latter work was inspired by geodesic problems and was principally concerned with potential theory. In fact, Gauss found himself more and more interested in geodesy in the 1820s. Gauss ha

19、d 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 calcula

20、tions. He regularly wrote to Schumacher, Olbers and Bessel, reporting on his progress and discussing problems. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Suns rays using a design of mirrors and a small telescope. However, inaccurate base lines were used for t

21、he survey and an unsatisfactory network of triangles. 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. together with the

22、idea of mapping one surface onto another so that the two are similar in their smallest parts. This paper was published in 1825 and led to the much later publication of Untersuchungen ber Gegenstnde der Hheren Geodsie (1843 and 1846). The paper Theoria combinationis observationum erroribus minimis ob

23、noxiae (1823), with its supplement (1828), was devoted to mathematical statistics, in particular to the least squares method. From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. He discussed this topic at length with Farkas Bolyai and in

24、his correspondence with Gerling and Schumacher. In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague. Gauss confided in Schumacher, telli

25、ng 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 Jnos Bolyais work on the subject. Gauss replied to praise it would mean to praise myself . Again, a decade later, when he was

26、 informed of Lobachevskys work on the subject, he praised its genuinely geometric character, while in a letter to Schumacher in 1846, states that he had the same convictions for 54 years indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age (this seem

27、s unlikely). Gauss had a major interest in differential geometry, and published many papers on the subject. Disquisitiones generales circa superficies curva (1828) was his most renowned work in this field. In fact, this paper rose from his geodesic interests, but it contained such geometrical ideas

28、as Gaussian curvature. The paper also includes Gausss famous theorema egregrium: If an area in E3 can be developed (i.e. mapped isometrically) into another area of E3, the values of the Gaussian curvatures are identical in corresponding points. The period 1817-1832 was a particularly distressing tim

29、e for Gauss. He took in his sick mother in 1817, who stayed until her death in 1839, while he was arguing with his wife and her family about whether they should go to Berlin. He had been offered a position at Berlin University and Minna and her family were keen to move there. Gauss, however, never l

30、iked change and decided to stay in Gttingen. In 1831 Gausss second wife died after a long illness. In 1831, Wilhelm Weber arrived in Gttingen as physics professor filling Tobias Mayers chair. Gauss had known Weber since 1828 and supported his appointment. Gauss had worked on physics before 1831, pub

31、lishing ber 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 Gausss potential theory, which proved of great impor

32、tance in his work on physics. He later came to believe his potential theory and his method of least squares provided vital links between science and nature. In 1832, Gauss and Weber began investigating the theory of terrestrial magnetism after Alexander von Humboldt attempted to obtain Gausss 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