Gauss' Biography

Overview
Early Years
The Triumphal Decade
Early Göttingen Years
Geodesist
Physicist
Later Years
Mathematical Scientist
A chronolgy/abridged version.

Overview

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Early Years

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. (For the formula, click here.) 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 (for the construction, click here.), 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.

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The Triumphal Decade

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.
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Early Göttingen 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 1816and 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.
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Geodesist

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. See Charts.
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Physicist

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 rogor 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.
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Later Years

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 Ruemann'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.
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Mathematical Scientist

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 (for the construction, click here.). 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 (128 [1931], 341): "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.