VI.26 Carl Friedrich Gauss
b. Brunswick, Germany, 1777; d. Göttingen, Germany, 1855
Algebra; astronomy; complex function theory including elliptic
function theory; differential equations; differential geometry;
land surveying; number theory; potential theory; statistics
Gauss’s prodigious mathematical abilities brought him to the attention of the duke of Brunswick when he was fifteen, when the duke paid for his further education, lifting him out of near poverty. For the rest of his life Gauss felt a loyalty to the state and a strong desire to do useful work, which led him to become a professional astronomer. In 1801 he was the first person to manage to reobserve Ceres, the first asteroid to be discovered, after it had disappeared behind the Sun. Gauss produced a novel statistical analysis of the original observations, using the method of least squares, which he had invented but not published, to predict where Ceres would reappear. Gauss then assisted for many years in the analysis of the orbits of several more asteroids. He also wrote extensively on celestial mechanics and cartography, and did important work on telegraphy.
Nonetheless, it is as a pure mathematician that Gauss will always be remembered. In 1801 he published his Disquisitiones Arithmeticae, the book that created modern algebraic number theory. In it he gave the first rigorous proof of the law of QUADRATIC RECIPROCITY [V.28], going on to find seven more proofs over the years. Later he extended the theorem to higher powers, introducing the Gaussian integers for the purpose in 1831 (Gaussian integers are numbers of the form m + ni, where m and n are integers and 
. He did major work on differential equations, chiefly the hypergeometric equation, which is a second-order linear differential equation depending on three parameters and having two singular points, with the property that many of the familiar functions of analysis are related to its solutions. He showed that this equation played a significant role in the new theory of ELLIPTIC FUNCTIONS [V.31], but because most of this work was unpublished it had no influence on the dramatic and rapidly advancing publications of ABEL [VI.33] and JACOBI [VI.35]. This unpublished work showed that he was the first mathematician to see the need to create a theory of complex functions of a complex variable. He also gave four proofs of THE FUNDAMENTAL THEOREM OF ALGEBRA [V.13]. By the 1820s he was persuaded that physical space might not be Euclidean, but he confined his opinion to his circle of friends, most of them astronomers and sympathetic to the idea; the much more detailed accounts of BOLYAI [VI.34] and LOBACHEVSKII [VI.31] were published independently in the early 1830s. Credit for the first detailed, mathematical descriptions of a non-Euclidean space therefore rightly attaches to Bolyai and Lobachevskii (for further discussion of this, see GEOMETRY [II.2 §7]). In 1827 Gauss wrote his Disquisitiones Generales Circa Superficies Curvas, in which the concept of intrinsic (Gaussian) curvature of a surface was put forward for the first time, thus reformulating differential geometry.
Carl Friedrich Gauss
In statistics, he was one of the two or three discoverers of the NORMAL DISTRIBUTION [III.71 §5], and he was an expert in error analysis, bringing the levels of accuracy in astronomy to land surveying. In that context he invented the heliotrope, which couples a mirror to a telescope in order to transmit a precise beam of light, to improve precision measurement.
The sheer volume of Gauss’s work is overwhelming. The Werke run to twelve volumes, and there are several books, of which the Disquisitiones Arithmeticae stands out.
A truly original mathematician and scientist, Gauss was otherwise a conservative in his tastes and views. His first marriage ended after only four years with the death of his wife in 1809; he then married again. A number of Gauss’s descendants may now be found in the United States.
Gauss was the last great mathematician to be called the “Prince of mathematicians,” and he has been admired as much for his breadth as for the depth of his insights and the fertility of his ideas. His own view of mathematics and its importance is captured both in the much-quoted remark that “mathematics is the queen of the sciences and arithmetic the queen of mathematics” (which he did say) and in the apocryphal remark that “mathematics is the queen and the servant of science.”
Further Reading
Dunnington, G. W. 2003. Gauss: Titan of Science, new edition with additional material by J. J. Gray. Washington, DC: Mathematical Association of America.