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Carl Friedrich Gauss

From the outside, Gauss' life was very simple. Having brought up in an austere childhood in a poor and uneducated family he showed extraordinary precocity. He received a stipend from the duke of Brunswick starting at the age of 14 which allowed him to devote his time to his studies for 16 years. Before his 25th birthday, he was already famous for his work in mathematics and astronomy. When he became 30 he went to Göttingen to become director of the observatory. He rarely left the city except on scientific business. From there, he worked for 47 years until his death at almost 78. In contrast to his external simplicity, Gauss' personal life was tragic and complicated. Due to the French Revolution, Napoleonic period and the democratic revolutions in Germany, he suffered from political turmoil and financial insecurity. He found no fellow mathematical collaborators and worked alone for most of his life. An unsympathetic father, the early death of his first wife, the poor health of his second wife, and terrible relations with his sons denied him a family sanctuary until late in life.

Even with all of these troubles, Gauss kept an amazingly rich scientific activity. An early passion for numbers and calculations extended first to the theory of numbers, to algebra, analysis, geometry, probability, and the theory of errors. At the same time, he carried on intensive empirical and theoretical research in many branches of science, including observational astronomy, celestial mechanics, surveying, geodesy, capillarity, geomagnetism, electromagnetism, mechanism optics, actuarial science. His publications, abundant correspondence, notes, and manuscripts show him to have been one of the greatest scientific virtuosos of all time.

Gauss was born into a family of town workers ranging from the peasant to the lower middle-class status. Gauss' mother, a highly intelligent, but only semiliterate daughter of a peasant stonemason, worked as a maid before becoming the second wife of Gauss' father. Gauss' father, a gardener, laborer at various trades, foreman ("master of waterworks"), assistant to a merchant and treasurer of a small insurance fund. The mother's brother was the only relative to have even the modest intellectual gifts, a master weaver. Gauss described his father as "worthy of esteem," but "domineering, uncouth, and unrefined." His mother was always optimistic even through an unhappy marriage. She was her son's only devoted support, and died at 97, after living in his house for 22 years.

It is said, that without any help, Gauss was able to calculate before he could even talk. According to a well-authenticated story, Gauss corrected his father's error in calculating wages. He taught himself to read, and must have continued his arithmetical experimentation intensively, because in his first arithmetic class at the age of eight, he astonished his teacher by instantly solving a busy-work problem: to find the sum of the first hundred integers. Fortunately, his father did not see the possibility of commercially exploiting the calculating prodigy, and his teacher had the insight to supply the boy with books and to encourage his continued intellectual development.

During his 11th year, Gauss studied with Martin Bartels, then an assistant in the school and later a teacher of Lobachevsky at Kazan. The father was persuaded to allow Gauss to enter the Gymnasium (a college preparatory school) in 1788 and to study after school instead of spinning to help support the family. At the Gymnasium, Gauss made rapid progress in all subjects, the classics and mathematics in particular, mostly on his own. E. A. W. Zimmerman, then professor at the local Collegium Carolinium and alter privy counselor to the duke of Brunswick, offered friendship, encouragement and good offices at court. In 1792, Duke Carl Wilhelm Ferdinand began the stipend that made Gauss independent.

When Gauss entered the Brunswick Collegium Carolinum in 1792, he possessed a scientific and classical education far beyond that usual for his age at the time. He was familiar with elementary geometry, algebra, and analysis. He often "discovered" important theorems before reaching them in his studies). In addition, he possessed a wealth of arithmetical information and many number-theoretic insights. Extensive calculations and observation of the results often recorded in tables, had led him to a intimate acquaintance with individual numbers and to generalizations that he used to extend his calculating ability. By then, his lifelong heuristic pattern had been set: extensive empirical investigation leading to conjectures and new insights that guided further experiment and observation. With this, he alone "discovered" Bode's law of planetary distances, the binomial theorem for rational exponents, and the arithmetic-geometric mean.

Gauss spent three years at the Collegium, in which he continued his empirical arithmetic, once finding a square root in two different ways to fifty decimal places by ingenious expansions and interpolations. He formulated the principle of least squares, apparently while adjusting unequal approximations and searching for regularity in the distribution of prime numbers. Before entering the University of Göttingen in 1795, he had rediscovered the law of quadratic reciprocity (conjectured by Lagrange in 1785), related the arithmetic-geometric mean to infinite series expansions, conjectured the prime number theorem (first proved by J. Hadamard in 1896), and found some results that would hold if "Euclidean geometry (plane geometry) were not the true one."

In Brunswick, Gauss had read Newton's Principia and Bernoulli's Ares conjectandi, but most mathematical classics were unavailable. At Göttingen, he devoured masterworks and back files of journals, often finding that his own "discoveries" were not new. Attracted more by the brilliant classist, G. Heyne, than by the mediocre mathematician, A. G. Kästner, Gauss planned to be a philologist. However, in 1796 came a dramatic discovery that marked him as a mathematician. As a by-product of a systematic investigation of the cycoltomic equation (whose solution has the geometric counterpart of dividing a circle into equal arcs), Gauss obtained conditions for the constructibility by ruler and compass of regular polygons and was able to announce that the regular 17-gon was constructible by ruler and compasses, the first advance in this matter in two millennia.

The logical component of Gauss' method matured at Göttingen. Even though his heroes were Archmedes and Newton, Gauss adopted the spirit of Greek rigor (insistence on precise definition, explicit assumption and complete proof) without the classical geometric form. Thinking numerically algebraically, after the manner in Euler, and personified the extension of Euclidean rigor to analysis. By the time he was twenty Gauss was thrusting with incredible speed according to the patter he was to continue in many contexts - massive empirical investigations in close interaction with intensive meditation and rigorous theory construction.

From 1796 to 1800, mathematical ideas came so fast that Gauss had difficulty writing them down. In reviewing, one of his seven proofs of the law of quadratic reciprocity in the Göttingischegelehrie Anzeigen for March 1817, he wrote autobiography:

It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations. This significant phenomenon arises from the wonderful concatenation of different teachings of this branch of mathematics, and from this it often happens that many theorems, whose proof for years was sought in vain, are later proved in many different ways. As soon as a new result is discovered by induction, one must consider as the first requirement the finding of a proof by any possible means. But after such good fortune, one must not in higher arithmetic consider the investigation closed or view the search for other proofs as a superfluous luxury. For sometimes one does not at first come upon the most beautiful and simplest proof, and then it is just the insight into the wonderful concatenation of truth in higher arithmetic that is the chief attraction for study and often leads to the discovery of new truths. For these reasons the finding of new proofs for known truths is often at least as important as the discovery itself.

In 1798, Gauss returned to Brunswick, where he lived alone and continued his intensive work. The next year, with the first of his four proofs of the fundamental theorem of algebra, he earned his doctorate from the University of Helmstedt under the rather nominal supervision of J. F. Pfaff. In 1801, the creativity of the previous years was shown through two extraordinary achievements, the Disquisitiones arithmetucae and the calculation of the orbit of the newly discovered planet Ceres.

Number theory (coined "higher arithmetic") is a branch of mathematics that seems least amenable to generalities. In the late eighteenth century, it consisted of a large collection of isolated results. In his Disquisitiones arithmetucae, Gauss summarized previous work in a systematic way, solved some of the most difficult outstanding questions, and formulated concepts and questions that set the pattern of research that is still in effect today. He introduced congruence of integers with respect to a modulus, the first significant algebraic example of the now ubiquitous concept of equivalence relation. He proved the law of quadratic reciprocity, developed the theory of composition of quadratic forms, and completely analyzed the cyclotomic equation. Disquisitiones arithmetucae almost instantly won Gauss recognition by mathematicians as their prince, but readership was small and the full understanding required for further development came only through the less austere exposition in Dirichlet's Vorlesungenüber Zahlentheorie (Lecture on Counted Theory) of 1863.

In January 1801, G. Piazzi had briefly observed and lost a new planet. During the rest of that year, astronomers tried in vain to relocate it. In September, as Disquisitiones arithmetucae was coming off the press, Gauss decided to take up the challenge. He applied both a more accurate orbit theory (based on the ellipse rather than the usual circular approximation) and improved numerical methods (based on least squares). By December, Ceres was soon found in the predicted location. This remarkable feat of locating a tiny distant body from seemingly insufficient information appeared to be almost superhuman, especially since Gauss did not reveal his methods. With Disquisitiones arithmetucae, it established his reputation as a mathematical and scientific genius of the highest order.

The decade began auspiciously with Disquisitiones arithmetucae and Ceres was decisive for Gauss. Scientifically, it was mainly a period of exploiting the ideas piled up from the previous decade. It ended with Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809), in which Gauss systematically developed his methods of orbit calculation, including the theory and use of least squares. Professionally, this was a decade of transition from being a mathematician to an astronomer and physical scientist. Although Gauss continued to enjoy the patronage of the duke, who increased his stipend from time to time (especially when Gauss began to receive attractive offers from elsewhere), subsidized publication of Disquisitiones arithmetucae, promised to build an observatory, and treated him like a tenured and highly valued civil servant, Gauss felt insecure and wanted to settle in a more established post. The most obvious course, to become a teacher of mathematics, repelled him because at this time it meant drilling ill-prepared and unmotivated students in the most elementary manipulations. Moreover, he felt that mathematics itself might not be sufficiently useful. When the duke raised his stipend in 1801, Gauss told Zimmermann: "But I have not earned it. I haven't yet done anything for the nation." Astronomy offered an attractive alternative. A strong interest in celestial mechanics dated from reading Newton, Gauss had begun observing while a student in Göttingen. The ingenuity on Ceres demonstrated both his ability and the public interest, the latter being far greater than he could expect in mathematical achievements. Moreover, the professional astronomer had light teaching duties and more time for research. Gauss decided on a career in astronomy and began to groom himself for the directorship of the Göttingen observatory. A systematic program of theoretical and observatory. A systematic program of theoretical and observational work, including calculation of the orbits of new planets as they were discovered, soon made him the most obvious candidate. When he accepted the position in 1807, he was already well established professionally, as supported by a job offer from St. Petersburg (1802) and by affiliations with the London Royal Society and the Russian and French academies.

During this decisive decade, Gauss also established personal and professional ties that were to last his lifetime. As a student at Göttingen he had enjoyed a romantic friendship with Wolfgang Bolyai, and the two discussed the foundations of geometry. But Bolyai returned to Hungary to spend his life vainly trying to prove Euclid's parallel postulate. Their correspondence soon practically ceased, to be revived again briefly only when Bolyai sent Gauss his son's work on non-Euclidean geometry. Pfaff was the only German mathematician with whom Gauss could converse, and even then hardly on an equal basis. From 1804 to 1807, Gauss exchanged a few letters on a high mathematical level with Sophie Germain in Paris, and a handful of letters passed between him and the mathematical giants in Paris, but he never visited France or collaborated with them. Gauss remained as isolated in mathematics as he had been since boyhood. By the time mathematicians of stature appeared in Germany (e.g. Jacobi, Plücker, Dirichlet), the uncommunicative habit was too ingrained to change. Gauss inspired Dirichlet, Riemann, and others, but he had never had a collaborator, correspondent, or student working closely with him in mathematics.

Things were quite different in the scientific and technical fields. There he had students, collaborators, and friends. Over 7,000 letters to and from Gauss are known to exist, and they undoubtedly represent only a fraction of the total. His most important astronomical collaborators, friends, and correspondents were F. W. Bessel, C. L. Gerling, M. Olbers, J. G. Repsold, H. C. Schumacher. His friendship and correspondence with A. von Humboldt and B. von Lindenau played an important part in his professional life and in the development of science in Germany. These relations were established during the period 1801- 1810 and lasted until death. Always Gauss wrote fewer letters, gave more information, and was less cordial than his colleagues, although he often gave practical assistance to his friends and to deserving young scientists.

This decade also established the pattern of working simultaneously on many problems in different fields. Although he never had a second burst of ideas equal to his first, Gauss always had more ideas than he had time to develop. His hopes for leisure were soon dashed by his responsibilities, and he acquired the habit of doing mathematics and other theoretical investigations in the odd hours (sometimes even happily) during days that could be spared. Hence, his ideas matured rather slowly, in some cases merely later than they might have with increased leisure, in others more felicitously with increased knowledge and meditation.

This period also saw the fixation of his political and philosophical views. Napoleon seemed to Gauss the personification of the dangers of revolution. The duke of Brunswick, to whom Gauss owed his golden tears of freedom, personified the merits of enlightened monarchy. When the duke was humiliated and killed while leading the Prussian armies against Napoleon in 1806, Gauss' conservative tendencies were reinforced. In the struggles for democracy and national unity in Germany, which continued throughout his lifetime, Gauss remained a staunch nationalist and royalist. (he published in Latin not from internationalist sentiments but at the demands of his publishers. He knew French but refused to publish in it and pretended ignorance when speaking to Frenchmen he did not know.) In seeming contradiction, his religious and philosophical views leaned toward those of his political opponents. He was an uncompromising believer in the priority of empiricism in science. He did not adhere to the views of Kant, Hegel, and other idealist philosophers of the day. He was not a churchman and kept his religious views to himself. Moral rectitude and the advancement of scientific knowledge were his avowed principles.

Lastly, this decade also provided Gauss with his one period of personal happiness. In 1805, he married a young woman of similar family background, Johanna Osthoff, who bore him a son and a daughter, creating a cheerful family life. However, in 1809 she died soon after giving birth to a third child, which did not long survive her. Gauss "closed the angel eyes in which for five years I have found a heaven" and was then plunged into a loneliness from which he never fully recovered. Less than a year later, he remarried Minna Waldeck, his deceased wife's best friend. She bore him two sons and a daughter, but she was seldom well or happy. Gauss dominated his daughters and quarreled with his younger sons, who immigrated to the United States. He did not achieve a peaceful home life until the younger daughter, Therese, took over the household after her mother's death (1831) and became the intimate companion of his last twenty-four years.

In his first years at Göttingen, Gauss experienced a second upsurge of ideas and publications in various fields of mathematics. Among the latter were several notable papers inspired by his work on the tiny planet Pallas, perturbed by Jupiter: Disquisitiones generales circa seriem infinitam (1913), an early rigorous treatment of series and the introduction of the hypergeometric functions, ancestors of the "special function" of physics; Methodus nova intergralium valores per approximationem inveniendi (1816), an important contribution to approximate intergration; Bestimmung der Genauigkeit der Beobachtungen (1816), an early analysis of the efficiency of statistical estimators; and Determinatio attractionis quam in punctum quodvis positionis datae exercerer planeta si eius massa per tatam orbitam ratione temporis quo singulae partes describuntur uniformiter essest dispertita (1818), which showed that the perturbation caused by a planet is the same as that of an equal mass disturbed along its orbit in proportion to the time spent on an arc. At the same time, Gauss continued thinking about unsolved mathematical problems. In 1813, a single sheep appear notes relating to parallel lines, declination of stars, number theory, imaginaries, the theory of colors, and prisms.

Astronomical chores soon dominated Gauss' life. He began with the makeshift observatory in an abandoned tower of the old city walls. A vast amount of time and energy went into equipping the new observatory, which was completed in 1816 and not properly furnished until 1821. In 1816, Gauss accompanied by his ten-year-old son and one of his students, took a five-week trip to Bavaria, where he met the optical instrument makers G. von Reichnbach, T. L. Ertel (owner of Reichenbach's firm), J. con Fraunhofer, and J. con Utzschneider (Fraunhofer's partner), from whom his best instruments were purchased. As shown in the pictorial charts astronomy was the only field in which Gauss worked on steadily for the rest of his life. He ended his theoretical astronomical work in 1817, but continued positional observing, calculating, and reporting his results until his death. Although assisted by students and colleagues, he observed regularly and was involved in every detail of instrumentation.

It was during there early Göttingen years that Gauss matured his conception of non-Euclidean geometry. He had experimented with the consequences of denying the parallel postulate more than twenty years before, and during his student days he saw the fallaciousness of the proofs of the parallel postulate that were the rage at Göttingen; but he came only very slowly and reluctantly to the idea of a different geometric theory that might be "true." He seems to have been pushed forward by his clear understanding of the weaknesses of previous efforts to prove the parallel postulate and by his successes in finding non-Euclidean results. He was slowed by his deep conservatism, the identification of Euclidean geometry with his beloved old order, and by his fully justified fear of the ridicule of the philistines. Over the years in his correspondence, we find him cautiously, but more and more clearly, stating his growing belief that the fifth postulate was unprovable. He privately encouraged others thinking along similar lines but advised secrecy. Only once, in a book review of 1816, he did hint at his views publicly. His ideas were "besmirched with mud" by critics (as he wrote to Shumacher on January 15, 1827), and his caution was confirmed.

Nonetheless, Gauss continued to find results in the new geometry and was again considering writing them up, possibly to be published after his death, when in 1831 came news of the work of János Bolyai. Gauss wrote to Wolfgang Bolyai endorsing the discovery, but he also asserted his own priority, thereby causing the volatile János to suspect a conspiracy to steal his ideas. When Gauss became familiar with Lobachevsky's work a decade later, he acted more positively with a letter of praise and by arranging a corresponding membership in the Göttingen Academy. But he stubbornly refused the public support that would have made the new ideas mathematically respectable. Although the friendships of Gauss with Bartels and W. Bolyai suggest the contrary, careful study of the plentiful documentary evidence has established that Gauss did not inspire the two founders of non-Euclidean geometry. Indeed, he played at best a neutral, and on balance a negative, role, since his silence was considered as agreement with the public ridicule and neglect that continued for several decades and were only gradually overcome, partly by the revelation, beginning in the 1860's, that the prince of mathematicians had been an underground non-Euclidean.

By 1817, Gauss was ready to move toward geodesy, which was to be his preoccupation for the next eight years and a burden for the next thirty. His interest was of long standing. As early as 1796, he worked on a surveying problem, and in 1799-1800 he advised Lt. K. L. E. con Lecoq, who was engaged in military mapping in Westphalia, Gauss' first publication was a letter on surveying in the Allegmeine geographische Ephermeriden (General geographical ?????????) of October 1799. In 1802, he participated in surveying with F. X. G. con Zach. From his arrival in Göttingen , he was concerned with the accuracy of locating the observatory, and in 1812, his interest in more general problems was stimulated by a discussion of sea levels during a visit to the Seeberg observatory. He began discussing with Schumacher the possibility of extending into Hannover the latter's survey of Denmark. Gauss had many motives for this project. It involved interesting mathematical problems, gave a new field for his calculating abilities, complemented his positional astronomy, competed with the French efforts to calculate the arc length of one degree on the meridian, offered an opportunity to do something useful for the kingdom, provided escape from petty annoyances of his job and family problems, and promised additional income. The last was a nontrivial matter, since Gauss had increasing family responsibilities to meet on a salary that remained fixed from 1807 to 1824.

The triangulation of Hannover was not officially approved until 1820, but already in 1818 Gauss began an arduous program of summer surveying in the field followed by data reduction during the winter. Plagued by poor transpiration, uncomfortable living conditions, bad weather, uncooperative officials, accidents, poor health, and inadequate assistance and financial support, Gauss did the fieldwork himself with only minimal help for eight years. After 1825, he confined himself to supervision and calculation, which continued to completion of the triangulation of Hannover in 1947. By then he had handled more than million numbers without assistance.

An early by-product of fieldwork was the invention of the heliotrope, an instrument for reflecting the sun's rays in a measured direction. It was motivated by dissatisfaction with the existing unsatisfactory methods of observing distant points by using lamps or powder flares at night. Meditating on the need for a beacon bright enough to be observed by day, Gauss hit on the idea of using reflected sunlight. After working out the optical theory, he designed the instrument and had the first model built in 1821. It proved to be very successful in practical work, having the brightness of a first-magnitude star at a distance of fifteen miles (24.14 km). Although heliostats had been described in the literature as early as 1742 (apparently unknown to Gauss), the heliotrope added greater precision by coupling mirrors with a small telescope. It became, standard equipment for large-scale triangulation until superseded by improved models from 1840 and by aerial surveying in the twentieth century. Gauss remarked that for the first time there existed a practical method of communicating with the moon.

Almost from the beginning of his surveying work, Gauss had misgivings, which proved to be well founded. A variety of practical difficulties made it impossible to achieve the accuracy he had expected, even with his improvements in instrumentation and the skillful use of least squares in data reduction. The hoped-for measurement of an arc of the meridian required linking his work with other surveys that were never made. Too hasty planning resulted in badly laid out base lines and an unsatisfactory network of triangles. He never ceased trying to overcome these faults, but his virtuosity as a mathematician and surveyor could not balance the factors beyond his control. His results were used in making rough geographic and military maps, but they were unsuitable for precise land surveys and for measurement of the earth. Within a generation, the markers were difficult to locate precisely of had disappeared altogether. As he was finishing his fieldwork in July 1825, Gauss wrote to Olbers that he wondered whether other activities might have been more fruitful. Not only did the results seem questionable, but he felt during these years, even more that usual, tat he was prevented from working out many ideas that still crowded his mind. As he wrote to Bessell on June 28, 1820, "I feel the difficulty of the life of a practical astronomer, without help; and the worst of it is that I can hardly do any connected significant theoretical work.

In spite of these failures and dissatisfactions, the period of preoccupation with geodesy was in fact one of the most scientifically creative of Gauss' long career. Already in 1814 geodesic problems had inspired his Theoria attrationis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, a significant early work on potential theory. 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 are "similar in their smallest parts." In 1822, a prize offered by the Copenhagen Academy stimulated him to write up these ideas in a paper that won first place and was published in 1825 as the Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche so auszubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird. This paper, his more detailed Untersuchen über Gegenständ der höhern Geodäsie (Examination over higher Geodesy) (1844-1847), and geodesic manuscripts later published in the Werke. were further developed by German geodesists and led to the Gauss-Krueger projection (1912), a generalization of the transverse Mercator projection, which attained a secure position as a basis of topographical grids taking into account the spherical shape of the earth.

Surveying problems also motivated Gauss to develop his ideas on least squares and more general problems of what is now called mathematical statistics. the result was the definitive exposition of his mature ideas in the Theoria combinations observationum erroribus minnimis obnoxiae (1823, with supplement in 1828). In the Bestimming des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector (Regulating the wide rule between the Starwaiting from Göttingen and Altona through observation on Ramsdenschen Zenith sector) of 1828 he summed up his ideas on the figure of the earth, instrumental errors, and the calculus of observations. However, the crowning contribution of the period, and his last breakthrough in a major new direction of mathematics was Disquistiones generales circa superficies curvas (1828), which grew out of his geodesic meditations of three decades and was the seed of more than a century of work on differential geometry. Of course,, in these years as always, Gauss produced a stream of reviews, reports on observations, and solutions of old and new mathematical problems of varying importance that brought the number of his publications during the decade 1818-1828 to 69.

After the mid 1820's, there were increasing signs that Gauss wished to strike out in a new direction. Financial pressures had been eased by a substantial salary increase in 1824 and by a bonus for the surveying work in 1825. His other motivations for geodesic work were also weakened, and a new negative factor emerged - heart trouble. A fundamentally strong constitution and unbounded energy were essential to the unrelenting pace of work that Gauss maintained in his early years, but in the 1820's the strain began to show. In 1821, family letters show Gauss constantly worried, often very tired, and seriously considering a move to the leisure and financial security promised by Berlin. The hard physical work of surveying in the humid summers brought on symptoms that would now be diagnosed as asthma and heart disease. In the fall of 1825, Gauss took his ailing wife on a health trip to spas in southern Germany; but the travel and the hot weather had a very bad effect on his own health, and was sick most of the winter. Distrusting doctors and never consulting one until the last few months of his life, he treated himself very sensibly by a simple life, regular habits, and the avoidance of travel, for which he had never cared anyway. He resolved to drop direct participation in summer surveying and to spend the rest of his life "undisturbed in my study," as he had written to Pfaff on March 21, 1825.

Apparently Gauss thought first of returning to a concentration on mathematics. He completed his work on least squares, geodesy, and curved surfaces, found new results on biquadratic reciprocity (1825), and began to pull together his long-standing ideas on elliptic functions and non-Euclidean geometry. But at forty-eight, he found that satisfactory results came harder than before. In a letter to Olbers of February 19, 1826, he spoke of never having worked so hard with so little success and of being almost convinced that he should go into another field. Moreover, his most original ideas were being developed independently by men of a new generation. Gauss did not respond when Abel sent him proof of the impossibility of solving the quintic equation in 1825, and the two never met, although Gauss praised him in private letters. When Dirichlet wrote Gauss in May 1826, enclosing his first work on number theory and asking for guidance, Gauss did not reply until September 13 and then only with general encouragement and advice to find a job that left time for research. As indicated in a letter to Encke of July 8, Gauss was much impressed by Dirichlet's "eminent talents," but he did not seem inclined to become mathematically involved with him. When Crelle in 1828 asked Gauss for a paper on elliptic functions, he replied that Jacobi had covered his work "with so much sagacity, penetration, and elegance, that I believe that I am relieved of publishing my own research." Harassed, overworked, distracted, and frustrated during these years, Gauss undoubtedly underestimated the value of his achievement, something he had never done before. But he was correct in sensing the need of a new source of inspiration. In turning toward intensive investigations in physics, he was following a pattern that had proved richly productive in the past.

In 1828, Alexmander von Humboldt persuaded Gauss to attend the only scientific convention of his career, the Naturforscherversammlung (Nature Research Collection) in Berlin. Since first hearing of Gauss from the leading mathematicians in Paris in 1802, Humboldt had been trying to bring him to Berlin as the leading figure of a great academy he hoped to build there. At times negotiations had seemed near success, but bureaucratic inflexibilities in Berlin or personal factors in Göttingen always intervened. Humboldt still had not abandoned these hopes, but he had other motives as well. He wished to draw Gauss into the German scientific upsurge whose beginnings were reflected in the meeting; and especially he wished to involve Gauss in his own efforts, already extending over two decade, to organize worldwide geomagnetic observations. Humboldt had no success in luring Gauss from his Göttingen hermitage. He was repelled by the Berlin convention which included a "little celebration" to which Humboldt invited 600 guests. nevertheless, the visit was a turning point. Living quietly for three weeks in Humboldt's house with a private garden and his host's scientific equipment, Gauss had both leisure and stimulation for making a choice. When Humboldt later wrote of his satisfaction at having interested him in magnetism, Gauss replied tactlessly that he had been interested in it for nearly thirty years. Correspondence and manuscripts show this to be true; they indicate that Gauss delayed serious work on the subject partly because means of measurement were not available. Nevertheless, the Berlin visit was the occasion for the decision and also provided the means for implementing it, since in Berlin, Gauss met Wilhelm Weber, a young and brilliant experimental physicist whose collaboration was essential.

In September 1829, Quetelet visited Göttingen and found Gauss very interested in terrestrial magnetism but with little experience in measuring it. The new field had evidently been selected, but with little experience magnetism but with little experience in measuring it. The new field had evidently been selected, but systematic work awaited Weber's arrival in 1831. Meanwhile, Gauss extended his long-standing knowledge of the physical literature and began to work on problems in theoretical physics, and especially in mechanics, capillarity, acoustics, optics, and crystallography. The first fruit of this research was Uber ein neues allgemeines Grundgesiz der Mechanik (1829). In it Gauss stated the law of least constraint: the motion of a system departs as little as possible from free motion, where departure, or constraint, is measured by the sum of product of the masses times the square of their deviations from the path of free motion, where departure, or constraint, is measured by sum of products of the masses times the square of their deviations from the path of free motion. He presented it merely as a new formulation equivalent to the well-known principle of d'Alembert. This work seems obviously related to the old meditation on least squares, but Gauss wrote to Olbers on January 31, 1829 that it was inspired by studies of capillarity and an important paper in the calculus of variations, since it was the first solution of a variational problems involving double integrals, boundary conditions, and variable limits.

As Gauss and Weber began to their close collaboration and intimate friendship, the younger man was just half the age of the older. Gauss took a fatherly attitude. though he shared fully in experimental work, and though Weber showed high theoretical competence and originality during the collaboration and later, the older man led on the theoretical and the younger on the experimental side. Their joint efforts soon produced results. In 1832, Gauss presented to the Academy the Ubtebsitas vis magneticae terrestris ad mensuram absolutam revocata (1833), in which appeared the systematic use of absolute units (distance, mass, time) to measure a nonmechanical quantity. Here Gauss typically acknowledged the help of Weber but did not include him as joint author. Stimulated by Faraday's discovered of induced current 1831, the pair energetically investigated electrical phenomena. They arrived at Kirchhoff's laws in 1833 and anticipated various discoveries in static, thermal, and frictional electricity but did not publish, presumably because their interest centered on terrestrial magnetism.

The thought that a magnetometer might also serve as a galvanometer almost immediately suggested its use to induce a current that might send a message. Working alone, Weber connected the astronomical observatory and the physics laboratory with a mile long double wire that broke "uncountable" times as long double wire that broke "uncountable" times as he strung it over houses and two towers. Early in 1833 the first words were sent, then whole sentences. This first operating electric telegraph was mentioned briefly by Gauss in a notice in Göttingische gelehrte Anzeigen (Göttingische learned announcement), but it seems to have been unknown to other inventors. Gauss soon realized the military and economic importance of the invention and tried unsuccessfully to promote its use by government and industry on a large scale. Over the years, the wire was replaced twice by one of better quality, and various improvement were made in the terminals. In 1845 a bolt if lightning-fragmented the wire, but by this time it was no longer in use. Other investors (Steinheil in Munich in 1837, Morse in the United States in 1828) had independently developed more efficient and exploitable methods, and the Gauss-Weber priority was forgotten.

The most important publication in the last category was the Allgemeine Theorie des Erdmagnetismus (1839). Here, Gauss broke the tradition of armchair theorizing about the earth as a fairly neutral carrier of one or more magnets and based on his mathematics on data. Using ideas first considered by him in 1806, well formulated by 1822, but lacking empirical foundation until 1838, Gauss expressed the magnetic potential at any point on the earth's surface by an infinite series of spherical functions and used the data collected by the world network to evaluate the first twenty-four coefficients. This was a superb interpolation, but Gauss hoped later to explain the results by a physical theory about the magnetic composition of the earth. Felix Klein has pointed out that this can indeed be done, but little is thereby added to the effective explanation offered by Gaussian formulas. During these years, Gauss found time to continue his geodesic data reduction, assist in revising the weights and measures of Hannover, make a number of electric discoveries jointly with Weber, and take an increasing part in university affairs.

This happy and productive collaboration was suddenly upset in 1837 by a disaster that soon effectively terminated Gauss' experimental work. In September, at the celebration of the 100th anniversary of the of the university (at which Gauss presented Humboldt with plans for his bifilar magnetometer), it was rumored that the new King Ernst August of Hannover might abrogate the hard-won constitution 1833 and demand that all public servents swears a personal oath of allegiance to himself. When he did so in November, seven Göttingen professors, including Weber and the orientalist G. H. A. von Ewald, the husband of Gauss' older daughter, Minna, sent a private protest to the cabinet, asserting that they were bound by their previous oath to the constitution of 1833. The "Göttingen Seven" were unceremoniously fired, three to be banished and the rest (including Weber and Ewald) permitted to remain in the town. Some thought that Gauss might resign, but he took no public action; and his private efforts, like the public protest of six additional professors, were ignored. Why did Gauss not act more energetically? At age sixty he was too set in his ways, his ways, his mother was too old to move, and he hated anything politically radical and disapproved of the protest. The seven eventually found jobs elsewhere. Ewald moved to Tübingen, and Gauss was deprived of the company of his most beloved daughter, who had been ill for some years and died of consumption in 1840. Weber was supported by colleagues for a time, then drifted away and accepted a job at Leipzig. The collaboration petered out, and Gauss abandoned further physical research. In 1848, when Weber recovered his position at Göttingen , it was too late to renew collaboration and Weber continued his brilliant career alone.

As Gauss was ending his physical research, he published Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1840). Growing directly out of his magnetic work but linked also to his Theoria attractionis of 1813, it was the first systematic treatment of potential theory as a mathematical topic, recognized the necessity of existence theorems in that field, and reached a standard of rigor that remained unsurpassed for more than a century, even though the main theorem of the paper was false, according to C. J. de la Vallèe Poussin. In the same year he finished Dioptrische Untersuchungen (1841), in which he analyzed the path of light through a system of lenses and showed, among other things, that any system is equivalent to a properly chosen single lens. Although Gauss said that he had possessed the theory forty years before and considered it too elementary to publish, it has been labeled his greatest work by one if his scientific biographers. In any case, it was his last significant scientific contribution.

From the early 1840's, the intensity of Gauss' activity gradually decreased. Further publications were either variations of old themes, reviews, reports, or solutions of minor problems. His reclusion is illustrated by his lack of response in 1845 to Kummer's invention ideas (to the discovery of Neptune by Adams, Le Verrier, and Galle. But the end of magnetic research and the decreased rate of publications did not mean that Gauss was inactive. He continued astronomical observing. He served several times as dean of the Göttingen faculty. He was busy during the 1840's in finishing many old projects, such as the last calculations on the Hannover survey. In 1847, he eloquently praised number theory and G. Eisenstein in the preface to the collected works of this ill-fated young man who had been one of the few to tell Gauss anything he did not already know. He spent several years putting the university’s widows' fund on a sound actuarial basis, calculating the necessary tables. He learned to read and speak Russian fluently, apparently first attracted by Lobachevsky but soon extending his reading as widely as permitted by the limited material available. His notebooks and correspondence show that he continued to work on a variety of mathematical problems. Teaching became less distasteful, perhaps because his students were better prepared and included some, such as Dedekind and Riemann, who were worthy of his efforts.

During the Revolution of 1848, Gauss stood guard with the royalists (whose defeat permitted the return of his son-in-law and Weber). He joined the Literary Museum, an organization whose library provided conservative literature for students and faculty, and made a daily visit there. He carefully followed politically, economically, and technological events as reported in the press. The fiftieth anniversary celebration of his doctorate in 1849 brought him many messages and formal honors, but the world of mathematics was represented only by Jacobi Dirichlet. The paper that Gauss delivered was his fourth of the fundamental theorem of algebra, appropriately a variation of the first in his thesis of 1799. After this celebration, Gauss continued his interests at a slower pace and became more than ever a legendary figure unapproachable by those outside his personal circle. Perhaps stimulated by his actuarial work, he fell into the habit of collecting all sorts of statistics from the newspapers, books, and daily observations. undoubtedly some of these data helped him with financial speculations shrewd enough to create an estate equal to nearly 200 times his annual salary. The "star gazer," as his father called him, had, as an afterthought, achieved the financial status denied his more "practical" relatives.

Due to his careful regimen, no serious illnesses had troubled Gauss since his surveying days. Over the years he treated himself for insomnia, stomach discomfort, congestion, bronchitis, painful corns, shortness of breath, heart flutter, and the usual signs of aging without suffering any acute attacks. He had been less successful in resisting chronic hypochondria and melancholia which increasingly plagued him after the death of his first wife. In the midst of some undated scientific notes from his later years there suddenly appears the sentence "Death would be preferable to such a life," and at fifty-six he wrote to Gerling on February 8, 1834, that he felt like a stranger in the world.

After 1850, troubled by developing heart disease, Gauss gradually limited his activity further. He made his last astronomical observation in 1851, at the age of 74., and later the same year approved Riemann's doctoral thesis on the foundations of complex analysis. The following year he was still working on minor mathematical problems and on an improved Foucault pendulum. During 1853-4, Riemann wrote his great Habilitationsschrift on the foundations of geometry, a topic chosen by Gauss. In June 1854, Gauss who had been under a doctor's care for several months, had the pleasure of hearing Riemann's probationary lecture, symbolic of the presence in Germany at last of talents capable of continuing his work. A few days later, he left Göttingen for the last time to observe construction of the railway from Kassel. By autumn, his illness was much worse. Although gradually more bedridden, he kept up his reading, correspondence, and trading in securities until he died in his sleep late in February 1855.

Gauss the man of genius stands in the way of evaluating the role of Gauss as a scientist. His mathematical abilities and exploits caused his contemporaries to dub him princeps, and biographers customarily place him on a par with Archimedes and Newton. This traditional judgment is as reasonable as any outcome of the ranking game, but an assessment of his impact is more problematic because of the wide gay between the quality of his personal accomplishments and their effectiveness as contributions to the scientific enterprise. Gauss published only about half of his recorded innovative ideas and in a style so austere that his readers were few. The unpublished results appear in notes, correspondence, and reports to official bodies, which became accessible only many years later. Still other methods and discoveries are only hinted at in letters or incomplete notes. It is therefore necessary to reexamine Gauss as a participant in the scientific community and to look at his achievements in terms of their scientific consequences.

The personality traits that most markedly inhibited the effectiveness of Gauss as a participant in scientific activity were his intellectual isolation, personal ambition, deep conservatism and nationalism, and rather narrow cultural outlook. It is hard to appreciate fully the isolation to which Gauss was condemned in childhood by thoughts that he could share with no one. He must soon have learned that attempts to communicate led, at best, to no response; at worst, to the ridicule and estrangement that children find so hard to bear. But unlike most precocious children, who eventually find intellectual comrades, Gauss during his whole life found no one with whom to share his most valued thoughts. Kästner was not interested when Gauss told him of his first great discovery, the constructibility of the regular 17-gon. Bolyai, his most promising friend at Göttingen could not appreciate his thinking. These and many other experiences must have convinced Gauss that there was little to be gained from trying to interchange theoretical ideas. He drew on the great mathematicians of the past and on contemporaries in France (whom he treated as from another world); but he remained outside the mathematical activity of his day, almost as if he were actually no longer living and his publications were being discovered in the archives. He found it easier and more useful to communicate with empirical scientists and technicians, because in those areas he was among peers; but even there he remained a solitary worker, with the exception of the collaboration with Weber.

Those who admired Gauss most and knew him best found him cold and uncommunicative. After the Berlin visit, Humboldt wrote Shumacher (October 18, 1828) that Gauss was "glacially cold" to unknowns and unconcerned with things outside his immediate circle. To Bessel. Humboldt wrote (October 12, 1837) to Gauss' "intentional isolation," his habit of suddenly taking possession of a small area of work, considering all previous results as part of it, and refusing to consider anything else. C. G. J. Jacobi complained in a letter to his brother (September 21, 1849) that in twenty years Gauss had not cited any publication by him or by Dirichlet. Schumacher, the closest of Gauss' friends and one who gave him much personal counsel and support, wrote to Bessel (December 21 1842) that Gauss was "a queer sort of fellow" with whom it is better to stay "in the limits of conventional politeness, without trying to do anything uncalled for."

Like Newton, Gauss had an intense dislike of controversy. There is no record of a traumatic experience that might account for this, but none is required to explain a desire to avoid emotional involvements that interfered with contemplation. With equal rationality, Gauss avoided all noncompulsory ceremonies and formalities, making an exception only when royalty was to be present. In there matters, as in his defensive attitude toward possible wasters of his time, Gauss was acting rationally to maximize his scientific output; but the result was to prevent some interchanges that might have been as beneficial to him as to others.

Insatiable drive, a characteristic of persistent high achievers, could hardly in itself inhibit participation; but conditioned by other motivations it did so for Gauss. having experienced poverty, he worked toward a security that was for a long time denied him. But he had absorbed the habitual frugality of the striving poor and did not want or ever adopt luxuries of the parvenu. He had no confidence in the democratic state and looked to the ruling aristocracy for security. The drive for financial security was accompanied by a stronger ambition, toward great achievement and lasting fame in science. While still an adolescent, Gauss realized that he might join the tiny superaristocracy of science that seldom has more than one member in a generation. He wished to be worthy of his heroes and to deserve the esteem of future peers. His sons reported that he discouraged them from going into science on the ground that he did not want any second-rate work associated with his name. He had little hope of being understood by his contemporaries; it was sufficient to impress and to avoid offending them. In the light of his ambitions for security and fame, with success in each seemingly required for the other, his choice of career and his purposeful isolation were rational. He did achieve twin ambitions, More effective communication and participation might have speeded the development of mathematics by several decades, but it would not have added to Gauss' reputation then or now. Gauss probably understood this well enough. He demonstrated in some of his writings, correspondence, lectures, and organizational activities that he could be an effective teacher, expositor, popularizer, diplomat, and promoter when he wished. He simply did not wish.

Gauss' conservatism has been described above, but it should be added here that it extended to all his thinking. He looking nostalgically back to the eighteenth century with its enlightened monarchs supporting scientific aristocrats in academies where they were relieved of teaching. He was anxious to find "new truths" that did not disturb established ideas. Nationalism was important for Gauss. As we have seen, it impelled him toward geodesy and other work that he considered useful to the state. But its most important effect was to deny him easy communication with the French. Only in Paris, during his most productive years, were men with whom he could have enjoyed a mutually stimulating mathematical collaboration.

It seems strange to call culturally narrow a man with a solid classical education, wide knowledge, and voracious reading Gauss' conservatism has been described above, but it should be added here that it extended to all his thinking. He looking nostalgically back to the eighteenth century with its enlightened monarchs supporting scientific aristocrats in academies where they were relieved of teaching. He was anxious to find "new truths" that did not disturb established ideas. Nationalism was important for Gauss. As we have seen, it impelled him toward geodesy and other work that he considered useful to the state. But its most important effect was to deny him easy communication with the French. Only in Paris, during his most productive years, were men with whom he could have enjoyed a mutually stimulating mathematical collaboration.

It seems strange to call culturally narrow a man with solid classical education, wide knowledge, and voracious reading habits. Yet outside of science, Gauss did not rise about petit bourgeois banality. Sir Walter Scott was his favorite British author, but he did not care for Byron or Shakespeare. Among German writers he like Jean Paul, the best-selling humorist of the day, but disliked Goethe and disapproved of Schiller. In music, he preferred light songs and in drama, comedies. In short, his genius stopped short at the boundaries of science and technology, outside of which he had little more taste or insight than his neighbors.

The contrast between knowledge and impact is now understandable. Gauss arrived at the two most revolutionary mathematical ideas of the nineteenth century: non-Euclidean geometry and noncommutative algebra. The first he disliked and suppressed. the second appears as quaternion calculations in a notebook of about 1819 without having stimulated any further activity. Neither the barycentric calculus of his own student Moebius (1827), nor Grassmann's Ausdenunglehre (1844), nor Hamilton's work on quaternions (beginning in 1843) interested him, although they sparked a fundamental shift in mathematical thought. He seemed unaware of the outburst of analytic and synthetic projective geometry, in which C. von Staudt, one of his former students, was a leading participant. Apparently Gauss was as hostile or indifferent to radical ideas in mathematics as in politics.

Hostility to new ideas, however, does not explain Gauss' failure to communicate many significant mathematical results that he did approve. Felix Klein points to a combination of factors - personal worries, distractions, lack of encouragement and overproduction of ideas. The last might alone have been decisive. Ideas came so quickly that each one inhibited the development of the preceding. Still another factor was the advantage that Gauss gained from withholding information, although he hotly denied this motive when Bessel suggested it. In fact, the Ceres calculation that won Gauss fame was based on methods unknown to others. By delaying publication of least squares and by never publishing his calculating methods, he maintained an advantage that materially contributed to his reputation. The same applies to the careful and conscious removal from his writings of all trace of his heuristic methods. The failure to publish was certainly not based on disdain for priority. Gauss cared a great deal for priority and frequently asserted it publicly and privately with scrupulous honesty. But to him this meant being first to discover, not first to publish; and he was satisfied to establish his dates by private records, correspondence, cryptic remarks in publications, and in one case by publishing a cipher. Whether he intended it so or not, in this way he maintained the advantage of secrecy without losing his priority in the eyes of later generations. The common claim that Gauss failed to publish because of his high standards is not convincing. he did have high standards, but he had no trouble achieving excellence once the mathematical results were in hand; and he did publish all that was ready for publications by normal standards.

In the light of the above discussion one might expect the Gaussian impact to be far smaller than his reputation - and indeed this is the case. His inventions, including several not listed here for lack of space, redound to his fame by were minor improvements of temporary importance or, like the telegraph uninfluential anticipations. In theoretical astronomy he perfected classical methods in orbit calculation but otherwise did only fair routine observations. His personal involvement in calculating orbits saved others the trouble and served to increase his fame but were little long-run scientific importance. His work in geodesy was influential only in its mathematical by-products. From his collaboration with Weber arose only two achievements of significant impact. The use 9of absolute units set a pattern that became standard, and the Magnetische Verein established a precedent for international scientific cooperation. His work in dioptrics may have been of the highest quality, but it seems to have had little influence; and the same may be said of his other works in physics.

However, on mathematics proper, the picture is quite different. Isolated as Gauss was, seemingly hardly aware of the work of other mathematicians and not caring to communicate with them, nevertheless his influence was powerful. His prestige was such that young mathematicians especially studied him. Jacobi and Abel testified that their work on elliptic functions was triggered by a hint in Disquisitines arithmeticae. Galois, on the eve of his death, asked that his rough notes be sent to Gauss. Thus, in mathematics, in spite of delays, Gauss did reach and inspire mathematicians. Although he was more of a systematizer and solver of old problems than an opener of new paths, the very completeness of his results laid the basis for new departures - especially in number theory, differential geometry, and statistics. Although his mathematical thinking was always concrete in the sense that he was dealing with structures based on the real numbers, his work contained the seeds of many highly abstract ideas that came later. Gauss, like Archimedes, pushed the methods of his time to the limit of their possibilities. But unlike his other ability peer, Newton, he did not initiate a profound new development, nor did he have the revolutionary impact of a number of his contemporaries of perhaps lesser ability but greater imagination and daring.

Gauss is best described as a mathematical scientist or, in the terms common in his day, as a pure and applied mathematician. Ranging easily, competently, and productively over the whole of science and technology, he always did so as a mathematician, motivated by mathematics, utilizing every experience for mathematical inspiration. Clemens Schäfer one of his scientific biographers wrote in Nature

"He was not really a physicist in the sense of searching for new phenomena, but rather always a mathematician who attempted to formulate in exact mathematical terms the experimental results obtained by others."

Leaving aside his personal failures, whose scientific importance was transitory, in heroic proportions in one person the capabilities attributed collectively to the community of professional mathematicians.