Gauss , Karl Friedrich

(1777–1855) German mathematician

Gauss came of a peasant background in Brunswick, Germany, and his extraordinary talent for mathematics showed itself at a very early age. By the age of three, he had discovered for himself enough arithmetic to be able to correct his father's calculations when he heard him working out the wages for his laborers. Gauss retained a staggering ability for mental calculation and memorizing throughout his life. At the age of ten he astonished his schoolteacher by discovering for himself the formula for the sum of an arithmetical progression. As a result of such precocity the young Gauss obtained the generous patronage of the duke of Brunwick. The duke paid for Gauss to attend the Caroline College in Brunswick and the University of Göttingen, and continued to support him until his death in 1806. Gauss then accepted an offer of the directorship of the observatory at Göttingen. This post probably suited him better than a more usual university appointment since he had little enthusiasm for teaching. Working at the observatory no doubt also stimulated his interest in applied mathematics and astronomy.

Gauss's life was uneventful. He remained director of the observatory for the rest of his life and indeed only rarely left Göttingen. Apart from mathematics he had a very keen interest in languages and at one stage hesitated between a career in mathematics and one in philology. His linguistic ability was evidently very great for he was able to teach himself fluent Russian in under two years. He also had a lively interest in world affairs, although in politics as in literature his views were somewhat conservative.

Gauss's contributions to mathematics were profound and they have affected almost every area of mathematics and mathematical physics. In addition to being a brilliant and original theoretician he was a practical experimentalist and a very accurate observer. His influence was naturally very great, but it would have been very much greater had he published all his discoveries. Many of his major results had to be rediscovered by some of the best mathematicians of the 19th century, although the extent to which this was the case was only revealed after Gauss's death. To give but two of many examples – Janos Bolyai and Nikolai Lobachevsky are both known as the creators of non-Euclidean geometry, but their work had been anticipated by Gauss 30 years earlier. Cauchy's great pioneering work in complex analysis is justly famous, yet Gauss had proved but not published the fundamental Cauchy theorem years before Cauchy reached it. The reason for Gauss's extreme reluctance to publish seems to have been the very high standard he set himself and he was unwilling to publish any work in a field unless he could present a complete and finished treatment of it.

Gauss received his doctorate in 1799 from the University of Helmstedt for a proof of the fundamental theorem of algebra, i.e., the theorem that every equation of degree n with complex coefficients has at least one root that is a complex number. This was the first genuine proof to be given; all the supposed previous proofs had contained errors, and it is this standard of rigor that really marks Gauss's work out from that of his predecessors. (Mathematicians of the 18th century and earlier had often possessed an intuitive ability to conjecture mathematical theorems that were in fact true, but their ideas of rigorous mathematical proof fell short of modern standards.)

Gauss's first publication is generally accepted as his finest single achievement. This is the Disquisitiones Arithmeticae(Examinations of Arithmetic) of 1801. Appropriately it was dedicated to Gauss's patron the duke of Brunswick. TheDisquisitiones is devoted to the area of mathematics that Gauss always considered to be the most beautiful, namely the theory of numbers. Gauss's prodigious ability for mental calculation enabled him to arrive at many of his theorems by generalizing from large numbers of examples. Among many other striking results Gauss was able to prove in the Disquisitiones the impossibility of constructing a regular heptagon with straight edge and compass – a problem that had baffled geometers since antiquity.

Gauss's interest was not confined to pure mathematics and he made contributions to many areas of applied mathematics and mathematical physics. Thus he discovered the Gaussian error curve and also the method of least squares, which he used in his work on geodesy. In his work on electromagnetism he collaborated with Wilhelm Weber on studies that led to the invention of the electric telegraph. The invention of the bifilar magnetometer for his own experimental work was another practical consequence of Gauss's interest in electromagnetism. His interest in mathematical astronomy resulted in many valuable innovations; he obtained a formula for calculating parallax in 1799 and in 1808 he published a work on planetary motion. When in 1801 the asteroid Ceres was first observed and then ‘lost’ by Giuseppe Piazzi, Gauss was able to predict correctly where it would reappear. He also made improvements in the design of the astronomical instruments in use at his observatory.

Gauss's work transformed mathematics and he is generally considered to be, with Newton and Archimedes, one of the greatest mathematicians of all time. The cgs unit of magnetic flux density is named in his honor.