# Carl Friedrich Gauss

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## Gauss, Carl Friedrich

**Gauss, Carl Friedrich** (kärl frēˈdrĭkh gous), born Johann Friederich Carl Gauss, 1777–1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ. of Göttingen, his education and early research being financed by the Duke of Brunswick. Following the death of the duke in 1806, Gauss became director (1807) of the astronomical observatory at Göttingen, a post he held until his death. Considered the greatest mathematician of his time and as the equal of Archimedes and Newton, Gauss showed his genius early and made many of his important discoveries before he was twenty. His greatest work was done in the area of higher arithmetic and number theory; his *Disquisitiones Arithmeticae* (completed in 1798 but not published until 1801) is one of the masterpieces of mathematical literature.

Gauss was extremely careful and rigorous in all his work, insisting on a complete proof of any result before he would publish it. As a consequence, he made many discoveries that were not credited to him and had to be remade by others later; for example, he anticipated Bolyai and Lobachevsky in non-Euclidean geometry, Jacobi in the double periodicity of elliptic functions, Cauchy in the theory of functions of a complex variable, and Hamilton in quaternions. However, his published works were enough to establish his reputation as one of the greatest mathematicians of all time. Gauss early discovered the law of quadratic reciprocity and, independently of Legendre, the method of least squares. He showed that a regular polygon of *n* sides can be constructed using only compass and straight edge only if *n* is of the form 2^{p}(2^{q}+1)(2^{r}+1) … , where 2^{q} + 1, 2^{r} + 1, … are prime numbers.

In 1801, following the discovery of the asteroid Ceres by Piazzi, Gauss calculated its orbit on the basis of very few accurate observations, and it was rediscovered the following year in the precise location he had predicted for it. He tested his method again successfully on the orbits of other asteroids discovered over the next few years and finally presented in his *Theoria motus corporum celestium* (1809) a complete treatment of the calculation of the orbits of planets and comets from observational data. From 1821, Gauss was engaged by the governments of Hanover and Denmark in connection with geodetic survey work. This led to his extensive investigations in the theory of space curves and surfaces and his important contributions to differential geometry as well as to such practical results as his invention of the heliotrope, a device used to measure distances by means of reflected sunlight.

Gauss was also interested in electric and magnetic phenomena and after about 1830 was involved in research in collaboration with Wilhelm Weber. In 1833 he invented the electric telegraph. He also made studies of terrestrial magnetism and electromagnetic theory. During the last years of his life Gauss was concerned with topics now falling under the general heading of topology, which had not yet been developed at that time, and he correctly predicted that this subject would become of great importance in mathematics.

### Bibliography

See biography by T. Hall (tr. 1970).

## Carl Friedrich Gauss

(person)Gauss was something of a child prodigy; the most commonly told story relates that when he was 10 his teacher, wanting a rest, told his class to add up all the numbers from 1 to 100. Gauss did it in seconds, having noticed that 1+...+100 = 100+...+1 = (101+...+101)/2.

He did important work in almost every area of mathematics. Such eclecticism is probably impossible today, since further progress in most areas of mathematics requires much hard background study.

Some idea of the range of his work can be obtained by noting the many mathematical terms with "Gauss" in their names. E.g. Gaussian elimination (linear algebra); Gaussian primes (number theory); Gaussian distribution (statistics); Gauss [unit] (electromagnetism); Gaussian curvature (differential geometry); Gaussian quadrature (numerical analysis); Gauss-Bonnet formula (differential geometry); Gauss's identity (hypergeometric functions); Gauss sums (number theory).

His favourite area of mathematics was number theory. He conjectured the Prime Number Theorem, pioneered the theory of quadratic forms, proved the quadratic reciprocity theorem, and much more.

He was "the first mathematician to use complex numbers in a really confident and scientific way" (Hardy & Wright, chapter 12).

He nearly went into architecture rather than mathematics; what decided him on mathematics was his proof, at age 18, of the startling theorem that a regular N-sided polygon can be constructed with ruler and compasses if and only if N is a power of 2 times a product of distinct Fermat primes.

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