GAUSS GAUSS 679 ; and Pierre Larousse, ed ., Grand dictionnaire universe!, fined ." His mother kept her cheerful disposition in 15 vols . (Paris, 1866-1876), VIII, 1086 . spite of an unhappy marriage, was always her only ALEX BERMAN son's devoted support, and died at ninety-seven, after living in his house for twenty-two years . GAUSS, CARL FRIEDRICH (b. Brunswick, Ger- Without the help or knowledge of others, Gauss many, 30 April 1777 ; d. Gottingen, Germany, 23 learned to calculate before he could talk . At the age February 1855), mathematical sciences. of three, according to a well-authenticated story, he The life of Gauss was very simple in external form . corrected an error in his father's wage calculations . During an austere childhood in a poor and unlettered He taught himself to read and must have continued family he showed extraordinary precocity . Beginning arithmetical experimentation intensively, because in when he was fourteen, a stipend from the duke of his first arithmetic class at the age of eight he aston- Brunswick permitted him to concentrate on intellec- ished his teacher by instantly solving a busy-work tual interests for sixteen years . Before the age of problem : to find the sum of the first hundred integers . twenty-five he was famous as a mathematician and Fortunately, his father did not see the possibility of astronomer. At thirty he went to Gottingen as director commercially exploiting the calculating prodigy, and of the observatory . There he worked for forty-seven his teacher had the insight to supply the boy with years, seldom leaving the city except on scientific books and to encourage his continued intellectual business, until his death at almost seventy-eight . development . In marked contrast to this external simplicity, During his eleventh year, Gauss studied with Gauss's personal life was complicated and tragic . He Martin Bartels, then an assistant in the school and suffered from the political turmoil and financial in- later a teacher of Lobachevsky at Kazan . The father security associated with the French Revolution, the was persuaded to allow Carl Friedrich to enter the Napoleonic period, and the democratic revolutions Gymnasium in 1788 and to study after school instead in Germany . He found no mathematical collaborators of spinning to help support the family . At the Gym- and worked alone most of his life . An unsympathetic nasium, Gauss made very rapid progress in all sub- father, the early death of his first wife, the poor health jects, especially classics and mathematics, largely on of his second wife, and unsatisfactory relations with his own . E. A. W. Zimmermann, then professor at the his sons denied him a family sanctuary until late in local Collegium Carolinum and later privy councillor life. to the duke of Brunswick, offered friendship, en- In this difficult context Gauss maintained an couragement, and good offices at court. In 1792 Duke amazingly rich scientific activity . An early passion for Carl Wilhelm Ferdinand began the stipend that made numbers and calculations extended first to the theory Gauss independent . of numbers and then to algebra, analysis, geometry, When Gauss entered the Brunswick Collegium probability, and the theory of errors . Concurrently he Carolinum in 1792, he possessed a scientific and clas- carried on intensive empirical and theoretical research sical education far beyond that usual for his age at in many branches of science, including observational the time . He was familiar with elementary geometry, astronomy, celestial mechanics, surveying, geodesy, algebra, and analysis (often having discovered im- capillarity, geomagnetism, electromagnetism, me- portant theorems before reaching them in his studies), chanics, optics, the design of scientific equipment, and but in addition he possessed a wealth of arithmetical actuarial science. His publications, voluminous corre- information and many number-theoretic insights . spondence, notes, and manuscripts show him to have Extensive calculations and observation of the results, been one of the greatest scientific virtuosos of all time . often recorded in tables, had led him to an intimate Early Years . Gauss was born into a family of town acquaintance with individual numbers and to gener- workers striving on the hard road from peasant to alizations that he used to extend his calculating abil- lower middle-class status . His mother, a highly intel- ity . Already his lifelong heuristic pattern had been ligent but only semiliterate daughter of a peasant set : extensive empirical investigation leading to con- stonemason, worked as a maid before becoming the jectures and new insights that guided further experi- second wife of Gauss's father, a gardener, laborer at ment and observation . By such means he had already various trades, foreman ("master of waterworks"), independently discovered Bode's law of planetary assistant to a merchant, and treasurer of a small distances, the binomial theorem for rational expo- insurance fund. The only relative known to have even nents, and the arithmetic-geometric mean . modest intellectual gifts was the mother's brother, a During his three years at the Collegium, Gauss master weaver . Gauss described his father as "worthy continued his empirical arithmetic, on one occasion of esteem" but "domineering, uncouth, and unre- finding a square root in two different ways to fifty 298 GAUSS GAUSS decimal places by ingenious expansions and interpo- non arises from the wonderful concatenation of differ- lations. He formulated the principle of least squares, ent teachings of this branch of mathematics, and from apparently while adjusting unequal approximations this it often happens that many theorems, whose proof and searching for regularity in the distribution of for years was sought in vain, are later proved in many prime numbers . Before entering the University of different ways . As soon as a new result is discovered Gottingen in 1795 he had rediscovered the law of by induction, one must consider as the first requirement the finding of a proof by any possible means . But after quadratic reciprocity (conjectured by Lagrange in such good fortune, one must not in higher arithmetic 1785), related the arithmetic-geometric mean to in- consider the investigation closed or view the search for finite series expansions, conjectured the prime num- other proofs as a superfluous luxury . For sometimes one ber theorem (first proved by J . Hadamard in 1896), does not at first come upon the most beautiful and and found some results that would hold if "Euclidean simplest proof, and then it is just the insight into the geometry were not the true one ." wonderful concatenation of truth in higher arithmetic In Brunswick, Gauss had read Newton's Principia that is the chief attraction for study and often leads to and Bernoulli's Ars conjectandi, but most mathemati- the discovery of new truths . For these reasons the find- cal classics were unavailable . At Gottingen, he de- ing of new proofs for known truths is often at least as voured masterworks and back files of journals, often important as the discovery itself [Werke, II, 159-1601 . finding that his own discoveries were not new . At- The Triumphal Decade. In 1798 Gauss returned to tracted more by the brilliant classicist G . Heyne than Brunswick, where he lived alone and continued his by the mediocre mathematician A . G. Kastner, intensive work . The next year, with the first of his Gauss planned to be a philologist . But in 1796 came four proofs of the fundamental theorem of algebra, a dramatic discovery that marked him as a mathe- he earned the doctorate from the University of matician. As a by-product of a systematic investi- Helmstedt under the rather nominal supervision of gation of the cyclotomic equation (whose solution has J. F. Pfaff. In 1801 the creativity of the previous years the geometric counterpart of dividing a circle into was reflected in two extraordinary achievements, the equal arcs), Gauss obtained conditions for the con- Disquisitiones arithmeticae and the calculation of the structibility by ruler and compass of regular polygons orbit of the newly discovered planet Ceres . and was able to announce that the regular 17-gon Number theory ("higher arithmetic") is a branch was constructible by ruler and compasses, the first of mathematics that seems least amenable to general- advance in this matter in two millennia . ities, although it was cultivated from the earliest The logical component of Gauss's method ma- times . In the late eighteenth century it consisted of tured at Gottingen . His heroes were Archimedes and a large collection of isolated results. In his Disquisi- Newton . But Gauss adopted the spirit of Greek rigor tiones Gauss summarized previous work in a system- (insistence on precise definition, explicit assumption, atic way, solved some of the most difficult outstanding and complete proof) without the classical geometric questions, and formulated concepts and questions form. He thought numerically and algebraically, after that set the pattern of research for a century and still the manner of Euler, and personified the extension have significance today . He introduced congruence of Euclidean rigor to analysis . By his twentieth year, of integers with respect to a modulus (a - b (mod c) Gauss was driving ahead with incredible speed ac- if c divides a-b), the first significant algebraic example cording to the pattern he was to continue in many of the now ubiquitous concept of equivalence rela- contexts-massive empirical