Carl Friedrich Gauss

German scientist

• Born: April 30, 1777
• Birthplace: Brunswick, Duchy of Brunswick (now in Germany)
• Died: February 23, 1855
• Place of death: Göttingen, Hanover (now in Germany)

A great scientific thinker who is often ranked with Archimedes and Isaac Newton, Gauss made significant contributions in many branches of science. His most notable achievement was the articulation of the two most revolutionary mathematical ideas of the nineteenth century: non-Euclidean geometry and noncommutative algebra.

Early Life

Carl Friedrich Gauss (gowz) was born into a family of town workers who were struggling to achieve lower-middle-class status. Without assistance, Gauss learned to calculate before he could talk; he also taught himself to read. At the age of three, he corrected an error in his father’s wage calculations. In his first arithmetic class, at the age of eight, he astonished his teacher by instantly solving a word problem that involved finding the sum of the first hundred integers. However, his teacher had the insight to furnish the child with books and encourage his intellectual development.

When he was eleven, Gauss studied with Martin Bartels, then an assistant in the school and later a teacher of Nikolay Ivanovich Lobachevsky at Kazan. Gauss’s father was persuaded to allow his son to enter the gymnasium in 1788. At the gymnasium, Gauss made rapid progress in all subjects, especially in classics and mathematics, largely on his own. E. A. W. Zimmermann, then professor at the local Collegium Carolinum and later privy councillor to the duke of Brunswick, encouraged Gauss; in 1792, Duke Carl Wilhelm Ferdinand began the stipend that would assure Gauss’s independence.

When Gauss entered the Brunswick Collegium Carolinum in 1792, he possessed a scientific and classical education far beyond his years. He was acquainted with elementary geometry, algebra, and analysis (often having discovered important theorems before reaching them in his books), but he also possessed much arithmetical information and number-theoretic insights. His lifelong pattern of research had become established: Extensive empirical investigation led to conjectures, and new insights guided further experiment and observation. By such methods, he had already discovered Johann Elert Bode’s law of planetary distances, the binomial theorem for rational exponents, and the arithmetic-geometric mean.

During his three years at the Collegium, among other things, Gauss formulated the principle of least squares. Before entering the University of Göttingen in 1795, he had rediscovered the law of quadratic reciprocity, related the arithmetic-geometric mean to infinite series expansions, and conjectured the prime number theorem (first proved by Jacques-Salomon Hadamard in 1896).

While Gauss was in Brunswick, most mathematical classics had been unavailable to him. At Göttingen, however, he devoured masterworks and back issues of journals and often found that his discoveries were not new. Attracted more by the brilliant classicist Christian Gottlob Heyne than by the mediocre mathematician A. G. Kästner, Gauss planned to be a philologist, but in 1796 he made a dramatic discovery that marked him as a mathematician. As a result of a systematic investigation of the cyclotomic equation (whose solution has the geometric counterpart of dividing a circle into equal arcs), Gauss declared that the regular seventeen-sided polygon was constructible by ruler and compasses, the first advance on this subject in two thousand years.

The logical aspect of Gauss’s method matured at Göttingen. Although he adopted the spirit of Greek rigor, it was without the classical geometric form; Gauss, rather, thought numerically and algebraically, in the manner of Leonhard Euler. By the age of twenty, Gauss was conducting large-scale empirical investigations and rigorous theoretical constructions, and during the years from 1796 to 1800 mathematical ideas came so quickly that Gauss could hardly write them down.

Life’s Work

In 1798, Gauss returned to Brunswick, and the next year, with the first of his four proofs of the fundamental theorem of algebra, earned a doctorate from the University of Helmstedt. In 1801, the creativity of the previous years was reflected in two extraordinary achievements, the Disquisitiones arithmeticae (1801; Arithmetical Inquisitions , 1966) and the calculation of the orbit of the newly discovered planet Ceres.

Although number theory was developed from the earliest times, during the late eighteenth century it consisted of a large collection of isolated results. In Arithmetical Inquisitions, Gauss systematically summarized previous work, solved some of the most difficult outstanding questions, and formulated concepts and questions that established the pattern of research for a century. The work almost instantly won for Gauss recognition by mathematicians, although readership was small.

In January, 1801, Giuseppi Piazzi had briefly discovered but lost track of a new planet he had observed, and during the rest of that year astronomers unsuccessfully attempted to relocate it. Gauss decided to pursue the matter. Applying both a more accurate orbit theory and improved numerical methods, he accomplished the task by December. Ceres was soon found in the predicted position. This feat of locating a distant, tiny planet from apparently insufficient information was astonishing, especially because Gauss did not reveal his methods. Along with Arithmetical Inquisitions, it established his reputation as a first-rate mathematical and scientific genius.

The decade of these achievements (1801-1810) was decisive for Gauss. Scientifically it was a period of exploiting ideas accumulated from the previous decade, and it ended with a work in which Gauss systematically developed his methods of orbit calculation, including a theory of and use of least squares. Professionally this decade was one of transition from mathematician to astronomer and physical scientist. Gauss accepted the post of director of the Göttingen Observatory in 1807.

This decade also provided Gauss with his one period of personal happiness. In 1805, he married Johanna Osthoff, with whom he had a son and a daughter. She created a happy family life around him. When she died in 1809, Gauss was plunged into a loneliness from which he never fully recovered. Less than a year later, he married Minna Waldeck, his deceased wife’s best friend. Although she bore him two sons and a daughter, she was unhealthy and often unhappy. Gauss did not achieve a peaceful home life until his youngest daughter, Therese, assumed management of the household after her mother’s death in 1831 and became his companion for the last twenty-four years of his life.

In his first years as director of the Göttingen Observatory, Gauss experienced a second burst of ideas and publications in various fields of mathematics and matured his conception of non-Euclidean geometry. However, astronomical tasks soon dominated Gauss’s life.

By 1817, Gauss moved toward geodesy, which was to be his preoccupation for the next eight years. The invention of the heliotrope, an instrument for reflecting the sun’s rays in a measured direction, was an early by-product of fieldwork. The invention was motivated by dissatisfaction with the existing methods of observing distant points by using lamps or powder flares at night. In spite of failures and dissatisfactions, the period of geodesic investigation was one of the most scientifically creative of Gauss’s long career. The difficulties of mapping the terrestrial ellipsoid on a sphere and plane led him, in 1816, to formulate and solve in outline the general problem of mapping one surface on another so that the two were “similar in their smallest parts.” In 1822, the chance of winning a prize offered by the Copenhagen Academy motivated him to write these ideas in a paper that won for him first place and was published in 1825.

Surveying problems also inspired Gauss to develop his ideas on least squares and more general problems of what is now called mathematical statistics. His most significant contribution during this period, and his last breakthrough in a major new direction of mathematical research, was Disquisitiones generales circa superficies curvas (1828; General Investigations of Curved Surfaces , 1902), which was the result of three decades of geodesic investigations and that drew upon more than a century of work on differential geometry.

After the mid-1820’s, Gauss, feeling harassed and overworked and suffering from asthma and heart disease, turned to investigations in physics. Gauss accepted an offer from Alexander von Humboldt to come to Berlin to work. An incentive was his meeting in Berlin with Wilhelm Eduard Weber, a young and brilliant experimental physicist with whom Gauss would eventually collaborate on many significant discoveries. They were also to organize a worldwide network of magnetic observatories and to publish extensively on magnetic force. From the early 1840’s, the intensity of Gauss’s activity gradually decreased. Increasingly bedridden as a result of heart disease, he died in his sleep in late February, 1855.

Significance

Carl Friedrich Gauss’s impact as a scientist falls far short of his reputation. His inventions were usually minor improvements of temporary importance. In theoretical astronomy, he perfected classical methods in orbit calculation but otherwise made only fairly routine observations. His personal involvement in calculating orbits saved others work but was of little long-lasting scientific importance. His work in geodesy was influential only in its mathematical by-products. Furthermore, his collaboration with Weber led to only two achievements of significant impact: The use of absolute units set a pattern that became standard, and the worldwide network of magnetic observatories established a precedent for international scientific cooperation. Also, his work in physics may have been of the highest quality, but it seems to have had little influence.

In the area of mathematics, however, his influence was powerful. Carl Gustav Jacobi and Niels Henrik Abel testified that their work on elliptic functions was triggered by a hint in the Arithmetical Inquisitions. Évariste Galois, on the eve of his death, asked that his rough notes be sent to Gauss. Thus, in mathematics, in spite of delays, Gauss reached and inspired countless mathematicians. Although he was more of a systematizer and solver of old problems than a creator of new paths, the completeness of his results laid the basis for new departures—especially in number theory, differential geometry, and statistics.

Bibliography

Bell, Eric T. Men of Mathematics. Reprint. New York: Simon & Schuster, 1961. Historical account of the major figures in mathematics from the Greeks to Georg Cantor, written in an interesting, if at times exaggerated, style. In a lengthy chapter devoted to Gauss titled “The Prince of Mathematicians,” Bell describes the life and work of Gauss, focusing almost exclusively on the mathematical contributions. No bibliography.

Boyer, Carl B. A History of Mathematics. New York: John Wiley & Sons, 1968. In “The Time of Gauss and Cauchy,” chapter 23 of this standard history of mathematics, Boyer briefly discusses biographical details of Gauss’s life before summarizing the proofs of Gauss’s major theorems. Boyer also discusses Gauss’s work in the context of the leading contemporary figures in mathematics of the day. Includes charts, an extensive bibliography, and student exercises.

Buhler, W. K. Gauss: A Biographical Study. New York: Springer-Verlag, 1981. The author’s purpose is not to write a definitive life history but to select from Gauss’s life and work those aspects that are interesting and comprehensible to a lay reader. Contains quotations from Gauss’s writings, illustrations, a bibliography, lengthy footnotes, appendixes on his collected works, a useful survey of the secondary literature, and an index to Gauss’s works.

Dunnington, Guy Waldo. Carl Friedrich Gauss, Titan of Science. With additional material by Jeremy Gray and Fritz-Egbert Dohse. Washington, D.C.: Mathematical Association of America, 2004. Originally published in 1955, this is an expanded version of a biography describing Gauss’s life and times to reveal the man as well as the scientist. The new edition includes introductory remarks, an updated bibliography, and a commentary on Gauss’s mathematical diary. It also reprints the features contained in the original edition, including appendixes on honors and diplomas, children, genealogy, a chronology, books borrowed from the college library, courses taught, and views and opinions.

Goldman, Jay. The Queen of Mathematics: An Historically Motivated Guide to Number Theory. Wellesley, Mass.: A. K. Peters, 1998. A history of number theory, described by Gauss as the “queen of mathematics,” describing how number theory developed from the seventeenth through the nineteenth centuries. Includes a chapter on Gauss and his work, and another chapter on the ideas contained in Disquisitiones arithmeticae. Aimed at readers with a knowledge of mathematics.

Turnbull, H. W. The Great Mathematicians. New York: New York University Press, 1962. Useful as a quick reference guide to the lives and works of the major figures in mathematics from the Greeks to the twentieth century.