German mathematician who is sometimes called the "prince of mathematics." He was a prodigious child, at the age of three informing his father of an arithmetical error in a complicated payroll calculation and stating the correct answer. In school, when his teacher gave the problem of summing the integers from 1 to 100 (an arithmetic series ) to his students to keep them busy, Gauss immediately wrote down the correct answer 5050 on his slate. At age 19, Gauss demonstrated a method for constructing a heptadecagon using only a straightedge and compass which had eluded the Greeks. (The explicit construction of the heptadecagon was accomplished around 1800 by Erchinger.) Gauss also showed that only regular polygons of a certain number of sides could be in that manner (a heptagon, for example, could not be constructed.)

Gauss proved the fundamental theorem of algebra, which states that every polynomial has a root of the form a+bi. In fact, he gave four different proofs, the first of which appeared in his dissertation. In 1801, he proved the fundamental theorem of arithmetic, which states that every natural number can be represented as the product of primes in only one way.

At age 24, Gauss published one of the most brilliant achievements in mathematics, Disquisitiones Arithmeticae (1801). In it, Gauss systematized the study of number theory (properties of the integers ). Gauss proved that every number is the sum of at most three triangular numbers and developed the algebra of congruences.

In 1801, Gauss developed the method of least squares fitting, 10 years before Legendre, but did not publish it. The method enabled him to calculate the orbit of the asteroid Ceres, which had been discovered by Piazzi from only three observations. However, after his independent discovery, Legendre accused Gauss of plagiarism. Gauss published his monumental treatise on celestial mechanics Theoria Motus in 1806. He became interested in the compass through surveying and developed the magnetometer and, with Wilhelm Weber measured the intensity of magnetic forces. With Weber, he also built the first successful telegraph.

Gauss is reported to have said "There have been only three epoch-making mathematicians: Archimedes, Newton and Eisenstein" (Boyer 1968, p. 553). Most historians are puzzled by the inclusion of Eisenstein in the same class as the other two. There is also a story that in 1807 he was interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment 'til I'm through" (Asimov 1972, p. 280).

Gauss arrived at important results on the parallel postulate, but failed to publish them. Credit for the discovery of non-Euclidean geometry therefore went to Janos Bolyai and Lobachevsky. However, he did publish his seminal work on differential geometry in Disquisitiones circa superticies curvas. The Gaussian curvature (or "second" curvature) is named for him. He also discovered the Cauchy integral theorem

for analytic functions, but did not publish it. Gauss solved the general problem of making a conformal map of one surface onto another.

Unfortunately for mathematics, Gauss reworked and improved papers incessantly, therefore publishing only a fraction of his work, in keeping with his motto "pauca sed matura" (few but ripe). Many of his results were subsequently repeated by others, since his terse diary remained unpublished for years after his death. This diary was only 19 pages long, but later confirmed his priority on many results he had not published. Gauss wanted a heptadecagon placed on his gravestone, but the carver refused, saying it would be indistinguishable from a circle. The heptadecagon appears, however, as the shape of a pedestal with a statue erected in his honor in his home town of Braunschweig.

Bolyai (Janos), Eisenstein, Kovalevskaya, Legendre, Weber (Wilhelm)

Additional biographies: MacTutor (St. Andrews), Bonn