With Archimedes and Newton, Gauss ranks as one of the greatest mathematicians of all time. At an early age he overturned the theories and methods of 18th-century mathematics and, following his own revolutionary theory of numbers, opened the way to a mid-19th-century rigorization of analysis. Although he contributed significantly to pure mathematics, he also made practical applications of importance for 20th-century astronomy, geodesy, and electromagnetism. His own dictum, "Mathematics, the queen of the sciences, and arithmetic, the queen of mathematics," aptly conveys his perception of the pivotal role of mathematics in science.

Gauss was the only son of poor parents. Impressed by his ability in mathematics and languages, his teachers and his devoted mother recommended him to the Duke of Brunswick, who granted him financial assistance to continue his education in secondary school and from 1795 to 1798 to study mathematics at the University of Göttingen. In 1799 he obtained his doctorate in absentia from the university at Helmstedt. The subject of his dissertation was a proof of the fundamental theorem of algebra--which was proven only partially before Gauss--which states that every algebraic equation with complex coefficients has complex solutions; moreover, Gauss skillfully formulated and proved this theorem without the use of complex numbers.

At age 24 he published the Disquisitiones Arithmeticae, one of the most brilliant achievements in the history of mathematics, in which he formulated systematic and widely influential concepts and methods of number theory--dealing with relationships and properties of integers (-2, -1, 0, +1, +2, . . . )--which, for him, was of paramount importance in mathematics. He dealt extensively with the theory of congruent numbers--(i.e., those numbers that have the same remainder when they are divided by another number (for example, 7 and 9 are congruent modulo the number 2 since there is a remainder of 1 when each is divided by 2); he gave the first proof of the law of quadratic reciprocity, which has to do with the quadratic residues (a is called quadratic residue with respect to b, if there is an integer x such that when a is divided by b, the remainder is the same as x² divided by b); and he applied this law to special cases of equations in which he was able to bring together algebraic, arithmetic, and geometric ideas. Using number theory, for example, Gauss proposed an algebraic solution to the geometric problem of constructing a regular polygon that has n sides. Euclid had shown that regular polygons, with 3, 4, 5, and 15 sides and those the sides of which result from doubling the above could be constructed geometrically with compass and ruler. No progress had been made in this subject since then. Gauss developed a criterion based on number theory by which it can be decided whether a regular polygon with any given number of corners can be geometrically constructed: these include, for example, the regular polygon with 17 sides, which he inscribed within a circle using only compass and ruler, the first such discovery since the time of Euclid.

This work on number theory contributed to the modern arithmetical theory of algebraic numbers--that is, to the solution of algebraic equations--in which Gauss introduced the first step--that is, the arithmetic of all complex numbers a + b-1, in which a and b are integers. The complex numbers a + b-1 had been introduced only intuitively before Gauss. In the Disquisitiones Arithmeticae Gauss did not hesitate to use complex numbers a + b-1, in which a and b are real numbers. In 1831 (published 1832) he gave a detailed explanation of how an exact theory of complex numbers can be developed with the aid of representation in the x, y plane.

In 1801 Gauss had the opportunity to apply his superior computational skills in a dramatic way and, by so doing, to express gratitude to the Duke for assisting him in obtaining an education. On the first day of the year, a body, subsequently identified as an asteroid and named Ceres, was discovered as it seemed to approach the Sun. Astronomers had been unable to calculate its orbit, although they could observe it for 40 days until lost from view. After only three observations Gauss developed a technique for calculating its orbital components with such accuracy that several astronomers late in 1801 and early in 1802 were able to locate Ceres again without difficulty. As part of his technique, Gauss used his method of least squares, developed about 1794, a method by which the best estimated value is derived from the minimum sums of squared differences in a particular computation. This achievement in astronomy won Gauss prompt recognition. His methods, which he described in his book, in 1809, Theoria Motus Corporum Coelestium, are still in use today, and only a few modifications have been required to adapt his methods for modern computers. He had similar success with the asteroid Pallas, for which he refined his calculations to take into account the perturbations of its orbit by planets.

About 1820 Gauss turned his attention to geodesy--the mathematical determination of the shape and size of the Earth's surface--to which he devoted much time in theoretical studies and field work. To increase the accuracy of surveying he invented the heliotrope, an instrument by which sunlight could be utilized to secure more accurate measurements. By introducing what is now known as the Gaussian error curve, he showed how probability could be represented by a bell-shaped curve, commonly called the normal curve of variation, which is basic to descriptions of statistically distributed data. He also was interested in determining the shape of the Earth by actual geodetic measurements, which led him back to pure theory. Using data from these measurements, he developed a theory of curved surfaces by which characteristics of a surface could be found solely by measuring the lengths of the curves that lie on the surface. This "intrinsic-surface theory" inspired one of his students, Bernhard Riemann, to develop a general intrinsic geometry of spaces with three or more dimensions. It was the subject of Riemann's inaugural lecture at Göttingen in 1854 and is said to have agitated Gauss. About 60 years later Riemann's ideas formed the mathematical basis for Einstein's general theory of relativity.

Gauss was one of the first to dou bt that Euclidean geometry was inherent in nature and thought. Euclid was the first to build a systematic geometry. Certain basic ideas in his model are called axioms; they were the points of departure from which his entire system was constructed through pure logic. Of these, the parallel axiom played a prominent role from the beginning. According to this axiom, only one line can be drawn parallel to a given line through any point not on the given line. From this axiom soon arose the supposition that it can be deduced out of the other axioms and thus can be omitted from the system of axioms. All proofs of it, however, contained errors, and Gauss was one of the first to realize how there might be a geometry in which the parallel axiom does not apply. Gradually he came to the revolutionary conclusion that there is indeed such a geometry that is internally consistent and free of contradiction. Because it ran counter to contemporary views, he feared publication.

When a Hungarian, János Bolyai, and a Russian, Nikolay Lobachevsky, independently published a non-Euclidean geometry about 1830, Gauss announced that he had made the same conclusions approximately 30 years before. Neither did he publish his work on special complex functions, perhaps because he was unable to derive them from more general principles. Thus, this theory had to be reconstructed by other mathematicians from his calculations in work extending over several decades after his death.

Closely related to his interest in gravitation and magnetism was his published paper in 1840 on real analysis. This paper became the starting point for the modern theory of potential. It is probably the only work he did that failed to meet his own high standards. Only at the beginning of the 20th century was it possible for mathematicians to develop potential theory anew, on the basis of different principles or by finding the conditions under which Gauss's conclusions are completely correct.

About 1830, principles of extremals (maximum and minimum quantities) began to assume a substantial role in his mathematical investigations of physical problems, such as the conditions in which a fluid remains at rest. In his treatment of capillary action, he devised mathematical formulations that took into account the mutual actions of all the particles in a fluid system, the force of gravity, and the interaction of its fluid particles and the particles of solid or fluid with which it is in contact. This work contributed to the development of the principle of the conservation of energy. From 1830, Gauss worked closely with the physicist Wilhelm Weber. Because of their interest in terrestrial magnetism, they organized a worldwide system of stations for systematic observations. The most important result of their work in electromagnetism was the development, by other workers, of electric telegraphy. Because their finances were limited, their experiments were on a small scale; Gauss was rather frightened at the thought of worldwide communication.