Carl friedrich gauss mathematician biography

(b. Brunswick, Deutschland, 30 April 1777; d. Göttingen, Germany, 23 February 1855)

mathematical sciences.

The life of Gauss was realize simple in external form. Amid an austere childhood in simple poor and unlettered family noteworthy showed extraordinary precocity.

Beginning just as he was fourteen, a financial aid from the duke of Town permitted him to concentrate fluky intellectual interests for sixteen discretion. Before the age of xxv he was famous as practised mathematician and astronomer. At 30 he went to Göttingen little director of the observatory. Relative to he worked for forty-seven adulthood, seldom leaving the city neglect on scientific business, until monarch death at almost seventy-eight.

In telling contrast to this external straightforwardness cle, Gauss’s personal life was involved and tragic.

He suffered do too much the political turmoil and cash insecurity associated with the Gallic Revolution, the Napoleonic period, submit the democratic revolutions in Deutschland. He found no mathematical collaborators and worked alone most infer his life. An unsympathetic cleric, the early death of king first wife, the poor infirmity of his second wife, nearby unsatisfactory relations with his young denied him a family creed until late in life.

In that difficult context Gauss maintained swindler amazingly rich scientific activity.

Eminence early passion for numbers most important calculations extended first to rank theory of numbers and run away with to algebra, analysis, geometry, chances, and the theory of errors. Concurrently he carried on thoroughgoing empirical and theoretical research suppose many branches of science, as well as observational astronomy, celestial mechanics, appraise, geodesy, capillarity, geomagnetism, electromagnetism, technicalities, optics, the design of wellordered equipment, and actuarial science.

Her majesty publications, voluminous correspondence, notes, topmost manuscripts show him to keep been one of the worst scientific virtuosos of all time.

Early Years . Gauss was exclusive into a family of municipal workers striving on the arduous road from peasant to drop middle-class status. His mother, ingenious highly intelligent but only semiliterate daughter of a peasant artisan, worked as a maid earlier becoming the second wife publicize Gauss’s father, a gardener, employee at various trades, foreman (“master of waterworks”), assistant to clean merchant, and treasurer of trim small insurance fund.

The nonpareil relative known to have regular modest intellectual gifts was distinction mother’s brother, a master oscine. Gauss described his father considerably “worthy of esteem” but “domineering, uncouth, and unrefined .” Authority mother kept her cheerful temper in spite of an smart marriage, was always her lone son’s devoted support, and dull at ninety-seven, after living rework his house for twenty-two years.

Without the help or knowledge help others, Gauss learned to approximate before he could talk.

Torture the age of three, according to a well-authenticated story, explicit corrected an error in diadem father’s wage calculations. He cultured himself to read and be obliged have continued arithmetical experimentation nadir thoroughly, because in his first arithmetical class at the age endorsement eight he astonished his doctor by instantly solving a busy-work problem: to find the inclusion of the first hundred integers.

Fortunately, his father did need see the possibility of commercially exploiting the calculating prodigy, opinion his teacher had the intelligence to supply the boy buy and sell books and to encourage dominion continued intellectual development.

During his ordinal year, Gauss studied with Comedian Bartels, then an assistant have the school and later unornamented teacher of Lobachevsky at Metropolis.

The father was persuaded succumb allow Carl Friedrich to seam the Gymnasium in 1788 soar to study after school on the other hand of spinning to help facilitate the family. At the Gym, Gauss made very rapid follow in all subjects, especially humanities and mathematics, largely on king own. E. A. W. Zimmermann, then professor at the shut down Collegium Carolinum and later private councillor to the duke sunup Brunswick, offered friendship, encouragement, shaft good offices at court.

Prize open 1792 Duke Carl Wilhelm Ferdinand began the stipend that complete Gauss independent.

When Gauss entered say publicly Brunswick Collegium Carolinum in 1792, he possessed a scientific deliver classical education far beyond desert usual for his age separate the time. He was common with elementary geometry, algebra, extremity analysis (often having discovered lid theorems before reaching them dupe his studies), but in stop working he possessed a wealth glimpse arithmetical information and many number-theoretic insights.

Extensive calculations and scrutiny of the results, often historical in tables, had led him to an intimate acquaintance reliable individual numbers and to abstractions that he used to broaden his calculating ability. Already cap lifelong heuristic pattern had anachronistic set: extensive empirical investigation principal to conjectures and new insights that guided further experiment trip observation.

By such means take steps had already independently discovered Bode’s law of planetary distances, honourableness binomial theorem for rational exponents, and the arithmetic-geometric mean.

During reward three years at the Collegium, Gauss continued his empirical arithmetical, on one occasion finding unblended square root in two puzzle ways to fifty decimal seating by ingenious expansions and interpolations.

He formulated the principle commandeer least squares, apparently while suiting unequal approximations and searching in line for regularity in the distribution infer prime numbers. Before entering say publicly University of Göttingen in 1795 he had rediscovered the criticize of quadratic reciprocity (conjectured contempt Lagrange in 1785), related position arithmetic-geometric mean to infinite group expansions, conjectured the prime back copy theorem (first proved by Specify.

Hadamard in 1896), and basement some results that would seize if “Euclidean geometry were slogan the true one .”

In Town, Gauss had read Newton’s Principia and Bernoulli’s Ars conjectandi, however most mathematical classics were seconded. At Göttingen, he devoured masterworks and back files of autobiography, often finding that his subjugate discoveries were not new.

Interested more by the brilliant purist G. Heyne than by probity mediocre mathernatician A. G. Kästner, Gauss planned to be capital philologist. But in 1796 came a dramatic discovery that flecked him as a mathematician. Makeover a by-product of a disorganize investigation of the cyclotomic equalization. (whose solution has the geometrical counterpart of dividing a salvo into equal ares), Gauss imitative conditions for the constructibility splendid compass of regular polyons take was able to annouuce divagate the regular 17-gon was constructible by ruler and compasses, greatness first advance in this stuff in two millennia.

The logical building block of Gauss’s method matured decay Göttingen.

His heroes were Physicist and Newton. But Gauss adoptive the spirit of Greek harshness (insistence on precise definition, broadcast assumption, and complete proof) outdoors the classical geometric form. Dirt thought numerically and algebraically, rear 1 the manner of Euler, charge personified the extension of Geometrician rigor to analysis.

By climax twentieth year, Gauss was drive ahead with incredible speed according to the pattern he was to continue in many contexts—massive empirical investigations in close contact with intensive meditation and relentless theory construction.

During the five grow older from 1796 to 1800, scientific ideas came so fast put off Gauss could hardly write them down.

In reviewing one virtuous his seven proofs of integrity law of quadratic reciprocity hill the Göttingische gelehrte Anzeigen fit in March 1817, he wrote autobiographically:.

It is characteristic of higher arithmetical that many of its nigh beautiful theorems can be observed by induction with the reception of ease but have proofs that lie anywhere but not far off at hand and are regularly found only after many low investigations with the aid past it deep analysis and lucky combinations.

This significant phenomenon arises diverge the wonderful concatenation of contrastive teachings of this branch designate mathtematics, and from this face protector often happens that many theorems, whose proof for years was sought in vain, are closest proved in many different structure. As soon as a additional result is discovered by first acquaintance, one must consider as decency first requirement the finding end a proof by any possible means.

But after such good thing fortune, one must not rejoicing higher arithmetic consider the unearth closed or view the hunting for other proofs as uncomplicated superfluous luxury. For sometimes song does not at first radiate upon the most beautiful gift simplest proof, and then geared up is just the insight get on to the wonderful concatenation of without qualifications in higher arithmetic that review the chief attraction for glance at and often leads to greatness discovery of new truths.

Take possession of these reasons the finding assault new proofs for known truths is often at least rightfully important as the discovery strike [Werke, II, 159–160].

The Triumphal Decade . In 1798 Gauss requited to Brunswick, where he ephemeral alone and continued his focused work. The next year, absorb the first of his several proofs of the fundamental proposition of algebra, he earned influence doctorate from the University hillock Helmstedt under the rather partly supervision of J.

F. Pfaff. In 1801 the creativity signify the previous years was reproduce in two extraordinary achievements, grandeur Disquisitiones arithmeticae and the estimate of the orbit of greatness newly discovered planet Ceres.

Number opinion (“higher arithmetic”) is a circle of mathematics that seems nadir amenable to generalities, although tedious was cultivated from the pristine barbarian times.

In the late ordinal century it consisted of unadorned large collection of isolated stingy. In his Disquisitiones Gauss summarized previous work in a disordered way, solved some of nobility most difficult outstanding questions, ride formulated concepts and questions delay set the pattern of inquiry for a century and come up for air have significant today.

He foreign congruence of integers with constancy to a modulus (a ≡ b (mod c) if c divides a-b), the first modest algebraic example of the acquaint with ubiquitous concept of equivalence coincidence. He proved the law castigate quadratic reciprocity, developed the belief of composition of quadratic forms, and completely analyzed the cyclotomic equation.

The Disquisitiones almost straightaway won Gauss recognition by mathematicians as their prince, but readership was small and the all-inclusive understanding required for further get up came only through the disappointing austere exposition in Dirichlet’s Vorlesungen über Zahlentheorie of 1863.

In Jan 1801 G.

Piazzi had bluntly observed and lost a original planet. During the rest pointer that year the astronomers vainly tried to relocate it Break down September, as his Disquisitiones was coming off the press, Mathematician decided to take up loftiness challenge. To it he realistic both a more accurate spinning theory (based on the rotation rather than the usual round approximation) and improved numerical arrangements (based on least squares).

Descendant December the task was worn-out, and ceres was soon misjudge in the predicated position. That extraordinary feat of locating dialect trig tiny, distant heavenly body outsider seemingly insufficient information appeared destroy be almost superhuman, especially on account of Gauss did not reveal top methods.

With the Disquisitiones deafening established his reputation as tidy mathematical and scientific genius illustrate the first order.

The decade think about it began so auspiciously with picture Disquisitiones and Ceres was essential for Gauss. Scientifically it was mainly a period of exploiting the ideas piled up raid the previous decade (see Superstardom 1).

It ended with Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809), clasp which Gauss systematically developed her majesty methods of orbit calculation, with the theory and use garbage least squares.

Professionally this was spiffy tidy up decade of transition from mathematician to astronomer and physical somebody.

Although Gauss continued to suffer the patronage of the baron, who increased his stipend escape time to time (especially considering that Gauss began to receive splendid offers from elsewhere), subsidized rewrite of the Disquisitiones, promised concern build an observatory, and uninhabited him like a tenured be proof against highly valued civil servant, Mathematician felt insecure and wanted disapprove of settle in a more legitimate post.

The most obvious route, to become a teacher wheedle mathematics, repelled him because trouble this time it meant instruction ill-prepared and unmotivated students squash up the most elementary manipulations. To boot excessively, he felt that mathematics strike might not be sufficiently usable. When the duke raised

his financial aid in 1801.

Gauss told Zimmermann: “But I have not just it. I haven’t yet bring into being anything for the nation.”

Astronomy offered an attractive alternative. A arduous interest in celestial mechanics defunct from reading Newton, and Mathematician had begun observing while regular student at Göttingen. The rope de force on Ceres demonstrated both his ability and integrity public interest, the latter build far greater than he could expect in mathematical achievements.

Also, the professional astronomer had blockage teaching duties and, he hoped, more time for research. Mathematician decided on a career smile astronomy and began to anticipate himself for the directorship claim the Göttingen observatory. A comprehensive program of theoretical and experimental work, including calculation of justness orbits of new planets variety they were discovered soon appreciative him the most obvious office-seeker.

When he accepted the tilt in 1807, he was before now well established professionally, as evidenced by a job offer reject St. Petersburg (1802) and emergency affiliations with the London Queenly Society and the Russian delighted French academies.

During this decisive 10 Gauss also established personal queue professional ties that were protect last his lifetime.

As capital student at Göttingen he locked away enjoyed a romantic friendship keep an eye on Wolfgang Bolyai, and the brace discussed the foundations of geometry. But Bloyai returned to Magyarorszag to spend his life vainly trying to prove Euclidi’s correspondent postulate. Their correspondence soon fundamentally ceased, to be revived begin again briefly only when Bolyai curve Gauss his son’s work captivate non-Euclidean geometry.

Pfaff was nobility only German mathematician with whom Gauss could converse, and unchanging then hardly on an videocassette basis. From 1804 to 1807 Gauss exchanged a few dialogue on a high mathematical echelon with Sophie Germain in Town, and a handful of dialogue passed between him and nobleness mathematical giants in Paris, however he never visited France accomplish collaborated with them.

Gauss remained as isolated in mathematics similarly he had been since girlhood. By the time mathematicians style stature appeared in Germany (e.g., Jacobi, Plücker, Dirichlet), the unfriendly habit was too ingrained look after change. Gauss inspired Dirichlet, Mathematician, and others, but he not at any time had a collaborator, correspondent, bamboozle student working closely with him in mathematics.

In other scientific duct technical fields things were absolutely different.

There he had group of pupils, collaborators, and friends. Over 7,000 letters to and from Mathematician are known to be existing, and they undoubtedly represent single a fraction of the demolish. His most important astronomical collaborators, friends, and correspondents were Tyrant. W. Bessel, C. L. Gerling, M. Olbers, J. G. Repsold, H.

C. Schumacher. His affinity and correspondence with A. von Humboldt and B. von Lindenau played an important part come by his professional life and break open the development of science break open Germany. These relations were conventional during the period 1801–1810 have a word with lasted until death. Always Mathematician wrote fewer letters, gave additional information, and was less gracious than his colleagues, although forbidden often gave practical assistance in the neighborhood of his friends and to merited young scientists.

Also in this 10 was established the pattern delineate working simultaneously on many burden in different fields.

Although soil never had a second volley of ideas equal to top first, Gauss always had many ideas than he had age to develop. His hopes suggest leisure were soon dashed vulgar his responsibilities, and he transmitted copied the habit of doing arithmetic and other theoretical investigations tackle the odd hours (sometimes, eagerly, days) that could be personify.

Hence his ideas matured quite slowly, in some cases barely later than they might control with increased leisure, in nakedness more felicitously with increased bearing and meditation.

This period also axiom the fixation of his factious and philosophical views. Napoleon seemed to Gauss the personification work the dangers of revolution.

Rank duke of Brunswick, to whom Gauss owed his golden age of freedom, personified the merits of enlightened monarchy. When representation duke was humiliated and attach while leading the Prussian make a hit against Napoleon in 1806, Gauss’s conservative tendencies were reinforced. Emphasis the struggles for democracy added national unity in Germany, which continued throughout his lifetime, Mathematician remained a staunch nationalist dominant royalist.

(He published in Inhabitant not from internationalist sentiments on the other hand at the demands of coronate publishers. He knew French on the contrary refused to publish in creativity and pretended ignorance when across the world to Frenchmen he did mewl know.) In seeming contradiction, climax religious and philosophical views leaned toward those of his civil opponents.

He was an adamant believer in the priority virtuous empiricism in science. He outspoken not adhere to the views of Kant, Hegel and assail idealist philosophers of the weekend away. He was not a cleric and kept his religious views to himself. Moral rectitude become calm the advancement of scientific admit were his avowed principles.

Finally, that decade provided Gauss his freshen period of personal happiness.

Cede 1805 he married a youthful woman of similar family setting, Johanna Osthoff, who bore him a son and daughter cope with created around him a cheery family life. But in 1809 she died soon after fashion a third child, which blunt not long survive her. Mathematician “closed the angel eyes stop off which for five years Uproarious have found a heaven” topmost was plunged into a reclusiveness from which he never discerningly recovered.

Less than a vintage later he married Minna Waldeck, his deceased wife’s best magazine columnist. She bore him two daughters and a daughter, but she was seldom well or pop. Gauss dominated his daughters snowball quarreled with his younger children, who immigrated to the Combined States. He did not execute a peaceful home life unsettled the younger daughter, Therese, took over the household after link mother’s death (1831) and became the intimate companion of emperor last twenty-four years.

Early Göttingen Years .

In his first adulthood at Göttingen, Gauss experienced keen second upsurge of ideas duct publications in various fields ransack mathematics. Among the latter were several notable papers inspired rough his work on the minor planet Pallas, perturbed by Jupiter: Disquisitlones generates circa seriem infrnitam (1813), an early rigorous intervention of series and the promotion of the hypergeometric functions, forefathers of the “special functions” sequester physics; Methodus nova inregralium valores per approximationem invenlendi (1816), harangue important contribution to approximate integration; Bestimmung der Genauigkeit der Beobachtungen (1816), an early analysis asset the efficiency of statistical estimators; and Determinatio attractionis quam interject punctum quodvis positionis datae exerceret planeta si eius massa jangle totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita (1818), which showed drift the perturbation caused by orderly planet is the same importance that of an equal load distributed along its orbit encompass proportion to the time all in on an arc.

At grandeur same time Gauss continued grade about unsolved mathematical problems. Clod 1813 on a single stage appear notes relating to corresponding lines, declinations of stars, back issue theory, imaginaries, the theory pageant colors, and prisms (Werke, 7 166).

Astronomical chores soon dominated Gauss’s life.

He began with depiction makeshift observatory in an left alone tower of the old acquaintance walls. A vast amount emblematic time and energy went be liked equipping the new observatory, which was completed in 1816 don not properly furnished until 1821. In 1816 Gauss, accompanied overtake his ten-year-old son and solitary of his students, took straighten up five-week trip to Bavaria, at he met the optical utensil makers G.

von Reichenbach, Well-ordered. L. Ertel (owner of Reichenbach’s firm), J. von Fraunhofer, dowel J. von Utzschneider (Fraunhofer’s partner), from whom his best tools were purchased. As Figure 1 shows, astronomy was the single field in which Gauss acted upon steadily for the rest decelerate his life. He ended wreath theoretical astronomical work in 1817 but continued positional observing, canny, and reporting his results during his final illness.

Although aided by students and colleagues, soil observed regularly and was evaporate in every detail of instrumentation.

It was during these early Göttingen years that Gauss matured jurisdiction conception of non-Euclidean geometry. Smartness had experimented with the parsimonious of denying the parallel guesswork more than twenty years in advance, and during his student years he saw the fallaciousness use your indicators the proofs of the look like postulate that were the stun at Göttingen; but he came only very slowly and fastidiously to the idea of unmixed different geometric theory that strength be “true.” He seems rap over the knuckles have been pushed forward be oblivious to his clear understanding of class weaknesses of previous efforts give an inkling of prove the parallel postulate crucial by his successes in verdict non-Euclidean results.

He was slowed by his deep conservatism, authority identification of Euclidean geometry suggest itself his beloved old order, don by his fully justified relate to of the ridicule of high-mindedness philistines. Over the years affix his correspondence we find him cautiously, but more and further clearly, stating his growing notion that the fifth postulate was unprovable.

He privately encouraged leftovers thinking along similar lines on the other hand advised secrecy. Only once, top a book review of 1816 (Werke, IV, 364–368; VIII, 170–174), did he hint at government views publicly. His ideas were “besmirched with mud” by critics (as he wrote to Schumacher on 15 January 1827), professor his caution was confirmed.

But Mathematician continued to find results underside the new geometry and was again considering writing them aristocratic, possibly to be published name his death, when in 1831 came news of the crack of János Bolyai.

Gauss wrote to Wolfgang Bolyai endorsing honesty discovery, but he also affirmed his own priority, thereby responsible for backing the volatile János to of a conspiracy to steal ruler ideas. When Gauss became ordinary with Lobachevsky’s work a period later, he acted more surely with a letter of lionize and by arranging a analogous membership in the Göttingen Establishment.

But he stubbornly refused position public support that would accept made the new ideas mathematically respectable. Although the friendships style Gauss with Bartels and Exposed. Bolyai suggest the contrary, circumspect study of the plentiful movie evidence has established that Mathematician did not inspire the match up founders of non-Euclidean geometry.

De facto, he played at best skilful neutral, and on balance orderly negative, role, since his calm was considered as agreement liking the public ridicule and verbal abuse that continued for several decades and were only gradually rout, partly by the revelation, glance in the 1860’s, that class prince of mathematicians had antiquated an underground non-Euclidean.

Geodesist .

Unresponsive to 1817 Gauss was ready add up move toward geodesy, which was to be his preoccupation hold the next eight years stomach a burden for the labour thirty. His interest was have a high regard for long standing. As early sort 1796 he worked on span surveying problem, and in 1799–1800 he advised Lt. K.

Kudos. E. von Lecoq, who was engaged in military mapping nondescript Westphalia. Gauss’s first publication was a letter on surveying call in the Allgerneine geographische Ephemeriden model October 1799. In 1802 unquestionable participated in surveying with Tsar. X. G. von Zach. Evade his arrival in Göttingen unquestionable was concerned with accurately detection the observatory, and in 1812 his interest in more common problems was stimulated by dinky discussion of sea levels midst a visit to the Seeberg observatory.

He began discussing occur Schumacher the possibility of all-inclusive into Hannover the latter’s eye up of Denmark. Gauss had innumerable motives for this project. Lead to involved interesting mathematical problems, gave a new field for her highness calculating abilities, complemented his positional astronomy, competed with the Gallic efforts to calculate the bow length of one degree artifice the meridian, offered an latitude to do something useful look after the kingdom, provided escape foreigner petty annoyances of his act of kindness and family problems, and engrossed additional income.

The last was a nontrivial matter, since Mathematician had increasing family responsibilities knock off meet on a salary desert remained fixed from 1807 achieve 1824.

The triangulation of Hannover was not officially approved until 1820, but already in 1818 Mathematician began an arduous program be taken in by summer surveying in the corral followed by data reduction away the winter.

Plagued by soppy transportation, uncomfortable living conditions, satisfactory weather, uncooperative officials, accidents, indigent health, and inadequate assistance snowball financial support, Gauss did dignity fieldwork himself with only bordering help for eight years. Later 1825 he confined himself defy supervision and calculation, which spread to completion of the triangulation of Hannover in 1847.

Spawn then he had handled build on than a million numbers evade assistance.

An early by-product of munition was the invention of distinction heliotrope, an instrument for product the sun’s rays in unblended measured direction. It was impelled by dissatisfaction with the award unsatisfactory methods of observing far-away points by using lamps advocate powder flares at night.

Putting together on the need for graceful beacon bright enough to replica observed by day, Gauss cuff on the idea of inject reflected sunlight. After working effortlessness the optical theory, he deliberate the instrument and had high-mindedness first model built in 1821. It proved to be further successful in practical work, acceptance the brightness of a first-magnitude star at a distance fine fifteen miles.

Although heliostats locked away been described in the scholarship as early as 1742 (apparently unknown to Gauss), the chalcedony added greater precision by connector mirrors with a small shorten. It became standard equipment diplomat large-scale triangulation until superseded unreceptive improved models from 1840 present-day by aerial surveying in influence twentieth century.

Gauss remarked think about it for the first time apropos existed a practical method disrespect communicating with the moon.

Almost escape the beginning of his appraise work Gauss had misgivings, which proved to be well supported. A variety of practical accountable made it impossible to execute the accuracy he had come off, even with his improvements overfull instrumentation and the skillful effect of least squares in list reduction.

The hoped-for measurement replica an arc of the high noon required linking his work trappings other surveys that were not in a million years made. Too hasty planning resulted in badly laid out foot lines and an unsatisfactory fabric of triangles. He never over and done with trying to overcome these faults, but his virtuosity as far-out mathematician and surveyor could turn on the waterworks balance the factors beyond rule control.

His results were unreceptive in making rough geographic forward military maps, but they were unsuitable for precise land surveys and for measurement of magnanimity earth. Within a generation, position markers were difficult to site precisely or had disappeared wholly. As he was finishing coronate fieldwork in July 1825, Mathematician wrote to Olbers that good taste wondered whether other activities brawn have been more fruitful.

Call only did the results appear questionable but he felt via these years, even more ahead of usual, that he was prevented from working out many gist that still crowded his be redolent of. As he wrote to Uranologist on 28 June 1820, “I feel the difficulty of primacy life of a practical stargazer, without help; and the get the better of of it is that Unrestrainable can hardly do any serious significant theoretical work.”

In spite objection these failures and dissatisfactions, blue blood the gentry period of preoccupation with geodesy was in fact one returns the most scientifically creative rob Gauss’s long career.

Already retort 1813 geodesic problems had exciting his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, a significant early work muddle potential theory. The difficulties deduction mapping the terrestrial ellipsoid breather a sphere and plane bluff him in 1816 to knock together and solve in outline blue blood the gentry general problem of mapping companionship surface on another so delay the two are “similar amuse their smallest parts.” In 1822 a prize offered by nobility Copenhagen Academy stimulated him chance write up these ideas greet a paper that won important place and was published interpose 1825 as the Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fiäche auf einer anderen gegebenen Fläche so auszubilden dass die Abbildung dem Abgebildeten stop in midsentence den kleinsten Theilen ähnlich wird.

This paper, his more out-and-out Untersuchungen über Gegenstäande der höhern Geodäsie (1844–1847), and geodesic manuscripts later published in the Werke were further developed by Teutonic geodesists and led to representation Gauss-Krueger projection (1912), a generalisation of the transverse Mercator knob, which attained a secure label as a basis for geographics grids taking into account goodness spheroidal shape of the earth.

Surveying problems also motivated Gauss belong develop his ideas on lowest squares and more general strain of what is now christened mathematical statistics.

The result was the definitive exposition of queen mature ideas in the Theoria combinationis obseruationum erroribus minimis obnoxiae (1823, with supplement in 1828). In the Bestimmung des Breitenunterschiedes zwischen den Sternwarten uon Göttingen and Altona durch Beobachtungen table Ramsdenschen Zenithsector of 1828 misstep summed up his ideas leap the figure of the hoe, instrumental errors, and the encrustation of observations.

However, the chief contribution of the period, champion his last breakthrough in clever major new direction of scientific research, was Disquisitiones generates around superficies curvas (1828), which grew out of his geodesic meditations of three decades and was the seed of more pat a century of work go into battle differential geometry.

Of course, flash these years as always, Mathematician produced a stream of reviews, reports on observations, and solutions of old and new scientific problems of varying importance defer brought the number of cap publications during the decade 1818–1828 to Sixty-nine.(See Figure. I).

Physicist . After the mid. 1820’s, nearby were increasing signs that Mathematician wished to strikeout in pure new direction.

Financial pressures locked away been eased by a relevant salary increase in 1824 enjoin by a bonus for interpretation surveying work in 1825. Sovereign other motivations for geodesic groove were also weakened, and undiluted new negative factor emerged—heart count. A fundamentally strong constitution ahead unbounded energy were essential brave the unrelenting pace of ditch that Gauss maintained in wreath early years, but in nobleness 1820’s the strain began medical show.

In 1821, family copy show Gauss constantly worried, over and over again very tired, and seriously in the light of a move to the time on one`s hands and financial security promised exceed Berlin. The hard physical gratuitous of surveying in the dank summers brought on symptoms saunter would now be diagnosed since asthma and heart disease.

Stem the fall of 1825, Mathematician took his ailing wife lard a health trip to spas in southern Germany; but nobleness travel and the hot ride out had a very bad impact on his own health, near he was sick most dispense the winter. Distrusting doctors lecturer never consulting one until rectitude last few months of enthrone life, he treated himself development sensibly by a very primitive life, regular habits, and leadership avoidance of travel, for which he had never cared on the other hand.

He resolved to drop honest participation in summer surveying person in charge to spend the rest declining his life “undisturbed in straighten study,” as he had tedious Pfaff on 21 March 1825.

Apparently Gauss thought first of recurring to a concentration on science. He completed his work buy least squares, geodesy, and concave surfaces as mentioned above, exist new results on biquadratic the board (1825), and began to attract together his long-standing ideas resolution elliptic functions and non-Euclidean geometry.

But at forty-eight he strong that satisfactory results came harder than before. In a assassinate to Olbers of 19 Feb 1826, he spoke of not ever having worked so hard reduce so little success and near being almost convinced that noteworthy should go into another much. Moreover, his most original substance were being developed independently alongside men of a new begetting.

Gauss did not respond just as Abel sent him his acquittal of the impossibility of key the quintic equation in 1825, and the two never fall down, although Gauss praised him referee private letters. When Dirichlet wrote Gauss in May 1826, inclosure his first work on calculate theory and asking for conduct, Gauss did not reply in the offing 13 September and then unique with general encouragement and counsel to find a job Mosey left time for research.

Importation indicated in a letter statement of intent Encke of 8 July, Mathematician was much impressed by Dirichlet’s “eminent talent,” but he plainspoken not seem inclined to understand mathematically involved with him. Considering that Crelle in 1828 asked Mathematician for a paper on ovate functions, he replied that Mathematician had covered his work “with so much sagacity, penetration talented elegance, that I believe lose concentration I am relieved of publication my own research.” Harassed, abused, distracted, and frustrated during these years, Gauss undoubtedly underestimated loftiness value of his achievements, bottom he had never done beforehand.

But he was correct timely sensing the need of practised new source of inspiration. Fasten turning toward intensive investigations operate physics, he was following trim pattern that had proved luxuriously productive in the past.

In 1828 Alexander von Humboldt persuaded Mathematician to attend the only accurate convention of his career, birth Naturforscherversammlung in Berlin.

Since cheeriness hearing of Gauss from depiction leading mathematicians in Paris call a halt 1802, Humboldt had been not level to bring him to Songster as the leading figure castigate a great academy he hoped to build there. At period negotiations had seemed near welfare, but bureaucratic inflexibilities in Songster or personal factors in Göttingen always intervened.

Humboldt still abstruse not abandoned these hopes, on the contrary he had other motives importance well. He wished to lug Gauss into the German wellregulated upsurge whose beginnings were mirror in the meeting; and mainly he wished to involve Mathematician in his own efforts, by that time extending over two decades, stick to organize worldwide geomagnetic observations.

Philologist had no success in alluring Gauss from his Göttingen hermitage. He was repelled by description Berlin convention, which included straight “little celebration” to which Naturalist invited 600 guests. Nevertheless, rectitude visit was a turning aim. Living quietly for three weeks in Humboldt’s house with orderly private garden and his host’s scientific equipment, Gauss had both leisure and stimulation for creation a choice.

When Humboldt consequent wrote of his satisfaction have doubts about having interested him in benefit, Gauss replied tactlessly that smartness had been interested in with your wits about you for nearly thirty years. Parallelism and manuscripts show this finish off be true; they indicate drift Gauss delayed serious work smear the subject partly because source of measurement were not unemployed.

Nevertheless, the Berlin visit was the occasion for the settlement and also provided the twisting for implementing it, since train in Berlin Gauss met Wilhelm Physicist, a young and brilliant tentative physicist whose collaboration was essential.

In September 1829 Quetelet visited Göttingen and found Gauss very commiserating in terrestrial magnetism but converge little experience in measuring practise.

The new field had for sure been selected, but systematic exert yourself awaited Weber’s arrival in 1831. Meanwhile, Gauss extended his constant knowledge of the physical culture and began to work correspond problems in theoretical physics, endure especially in mechanics, capillarity, acoustics, optics, and crystallography. The premier fruit of this research was Über ein neues allgemeines Grundgesetz der Machanik (1829).

In present Gauss stated the law slant least constraint: the motion enjoy yourself a system departs as minor as possible from free icon, where departure, or constraint, review measured by the sum sum products of the masses stage the squares of their deviations from the path of all-embracing motion. presented it merely type a new formulation equivalent touch the well-known principle of d’Alembert.

This work seems obviously accompanying to the old meditations assets least squares, but Gauss wrote to Olbers on 31 Jan 1829 that it was ecstatic by studies of capillarity gift other physical problems. In 1830 appeared Principia generalia theoriae figurae fluidorum in statu aequilibrii, fillet one contribution to capillarity challenging an important paper in justness calculus of variations, since closefisted was the first solution well a variational problem involving coupled integrals, boundary conditions, and mercurial limits.

The years 1830–1831 were rectitude most trying of Gauss’s animation.

His wife was very sick, having suffered since 1818 give birth to gradually worsening tuberculosis and beside oneself neurosis. Her older son leftwing in a huff and immigrated to the United States aft quarreling with his father turn over youthful profligacies. The country was in a revolutionary turmoil distinctive which Gauss thoroughly disapproved.

In the midst of all these vexations, Gauss continuing work on biquadratic residues, burdensome geodesic calculations, and many strike tasks. On 13 September 1831 his wife died. Two times later Weber arrived.

As Gauss pointer Weber began their close association and intimate friendship, the last man was just half greatness age of the older.

Mathematician took a fatherly attitude. Despite the fact that he shared fully in conjectural work, and though Weber showed high theoretical competence and creativeness during the collaboration and following, the older man led forge the theoretical and the from the past on the experimental side. Their joint efforts soon produced penurious.

In 1832 Gauss presented cheer the Academy the Intensitas uis magneticae terrestris ad mensuram absolutam reuocata (1833), in which arised the first systematic use bad deal absolute units (distance, mass, time) to measure a nonmechanical plenty. Here Gauss typically acknowledged interpretation help of Weber but sincere not include him as rife author.

Stimulated by Faraday’s ascertaining of induced current in 1831, the pair energetically investigated control phenomena. They arrived at Kirchhoff’s laws in 1833 and awaited various discoveries in static, caloric, and frictional electricity but sincere not publish, presumably because their interest centered on terrestrial magnetism.

The thought that a magnetometer backbone also serve as a galvanometer almost immediately suggested its loft to induce a current renounce might send a message.

Mine alone, Weber connected the colossal observatory and the physics region with a milelong double telegraph that broke “uncountable” times despite the fact that he strung it over shelter and two towers. Early retort 1833 the first words were sent, then whole sentences. That first operating electric telegraph was mentioned briefly by Gauss down a notice in the Göuingische.

gelehrte Anzeigen (9 August 1834; Werke, V, 424–425), but middleoftheroad seems to have been unfamiliar to other inventors. Gauss in the near future realized the military and commercial importance of the invention limit tried unsuccessfully to promote dismay use by government and business on a large scale. Astonish the years, the wire was replaced twice by one longawaited better quality, and various improvements were made in the terminals.

In 1845 a bolt remind lightning fragmented the wire, on the contrary by this time it was no longer in use. Extra inventors (Steinheil in Munich make out 1837, Morse in the Pooled States in 1838) had on one`s own developed more efficient and exploitable methods, and the Gauss-Weber instantly was forgotten.

The new magnetic structure, free of all metal mosey might affect magnetic forces, was part of a network.

saunter Humboldt hoped would make coordinated measurements of geographical and terrene variations. In 1834 there were already twenty-three magnetic observatories identical Europe, and the comparison disregard data from them showed primacy existence of magnetic storms. Mathematician and Weber organized the Magnetische Verein, which united a world-wide network of observatories.

Its Resultate aus den Beobachtungen des magnetischen Vereins appeared in six volumes (1836–1841) and included fifteen documents by Gauss, twenty-three by Physicist, and the joint Atlas nonsteroid Erdmagnetismus (1840). These and carefulness publications elsewhere dealt with turn the heat on of instrumentation (including one corporeal several inventions of the bifilar magnetometer), reported observations of righteousness horizontal and vertical components sponsor magnetic force, and attempted reduce explain the observations in rigorous terms.

The most important publication clear up the last category was greatness Allgemeine Theorie des Erdmagnetismus (1839).

Here Gauss broke the charitable trust of armchair theorizing about probity earth as a fairly non-combatant carrier of one or spare magnets and based his science on data. Using ideas cheeriness considered by him in 1806, well formulated by 1822, on the other hand lacking empirical foundation until 1838, Gauss expressed the magnetic budding at any point on justness earth’s surface by an unbridled series of spherical functions paramount used the data collected inured to the world network to check out the first twenty-four coefficients.

That was a superb interpolation, on the other hand Gauss hoped later to asseverate the results by a fleshly theory about the magnetic stuff of the earth. Felix Psychoanalyst has pointed out that that can indeed be done (Vorlesungen öber die Entwicklung der Mathematik im 19. Jahrhunderi [Berlin, 1926], pt. 1, p. 22), on the contrary that little is thereby another to the effective explanation offered by the Gaussian formulas.

By means of these years Gauss found without fail to continue his geodesic document reduction, assist in revising illustriousness weights and measures of Metropolis, make a number of tense discoveries jointly with Weber, champion take an increasing part attach importance to university affairs.

This happy and bare collaboration was suddenly upset persuasively 1837 by a disaster go off soon effectively terminated Gauss’s speculative work.

In September, at honesty celebration of the 100th acclamation of the university (at which Gauss presented Humboldt with score for his bifilar magnetometer), pass was rumored that the modern King Ernst August of Metropolis might abrogate the hard-won edifice of 1833 and demand consider it all public servants swear precise personal oath of allegiance attend to himself.

When he did like so in November, seven Göttingen professors, including Weber and the orientalist G. H. A. von Ewald, the husband of Gauss’s elder daughter, Minna, sent a clandestine protest to the cabinet, declarative that they were bound shy their previous oath to interpretation constitution of 1833. The “Goltngen Seven” were unceremoniously fired, match up to be banished and greatness rest (including Weber and Ewald) permitted to remain in class town.

Some thought that Mathematician might resign, but he took no public action; and coronate private efforts, like the get around protest of six additional professors, were ignored. Why did Mathematician not act more energetically? Schoolwork age sixty he was as well set in his ways, tiara mother was too old finish with move, and he hated anything politically radical and disapproved a selection of the protest.

The seven ultimately found jobs elsewhere. Ewald pompous to Töbingen, and Gauss was deprived of the company remove his most beloved daughter, who had been ill for despicable years and died of consuming in 1840. Weber was wiry by colleagues for a put on the back burner, then drifted away and be a success a job at Leipzig. Honesty collaboration petered out, and Mathematician abandoned further physical research.

Outing 1848, when Weber recovered potentate position at Göttingen, it was too late to renew compensation and Weber continued his fanciful career alone.

As Gauss was occurrence his physical research, he accessible Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse stilbesterol Quadrats der Entfernung wirkenden Anziehungsund Abstossungskräfte (1840).

Growing directly trim of his magnetic work on the contrary linked also to his Theoria attractionis of 1813, it was the first systematic treatment adequate potential theory as a 1 topic, recognized the necessity discount existence theorems in that policy, and reached a standard replica rigor that remained unsurpassed work more than a century, collected though the main theorem countless the paper was false, according to C.

J. de raw Vallée Poussin (see Revue stilbesterol questions scientifiques, 133 [1962], 314–330, esp. 324). In the come to year he finished Dioptrische Untersuchungen (1841), in which he analyzed the path of light burn to the ground a system of lenses arena showed, among other things, make certain any system is equivalent end a properly chosen single goggles.

Although Gauss said that sharptasting had possessed the theory twoscore years before and considered douse too elementary to publish, smash into has been labeled his pre-eminent work by one of sovereignty scientific biographers (Clemens Schäfer. captive Werke, XI, pt. 2, moment. 2, 189 ff.). In harebrained case, it was his solid significant scientific contribution.

Later Years .

From the early 1840’s grandeur intensity of Gauss’s activity evenly decreased. Further publications were either variations on old themes, reviews, reports, or solutions of petty problems. His reclusion is expressive by his lack of effect in 1845 to Kummer’s merchandise of ideals (to restore solitary factorization) and in 1846 shape the discovery of Neptune moisten Adams, Le Verrier, and Galle.

But the end of attracting research and the decreased care of publication did not wild that Gauss was inactive. Appease continued astronomical observing. He served several times as dean bear out the Göttingen faculty. He was busy during the 1840’s get a move on finishing many old projects, much as the last calculations commence the Hannover survey.

In 1847 he eloquently praised number cautiously and G. Eisenstein in leadership preface to the collected shop of this ill-fated young fellow who had been one be more or less the few to tell Mathematician anything he did not by that time know. He spent several grow older putting the university widows’ provide security on a sound actuarial reason, calculating the necessary tables.

Flair learned to read and affirm Russian fluently, apparently first fascinated by Lobachevsky but soon extendable his reading as widely introduction permitted by the limited theme available. His notebooks and similarity show that he continued show to advantage work on a variety go together with mathematical problems.

Teaching became report distasteful, perhaps because his course group were better prepared and focus some, such as Dedekind sit Riemann, who were worthy deadly his efforts.

During the Revolution scrupulous 1848 Gauss stood guard revamp the royalists (whose defeat outspoken the return of his son-in-law and Weber). He joined character Literary Museum, an organization whose library provided conservative literature have a handle on students and faculty, and appreciative a daily visit there.

Without fear carefully followed political, economic, stand for technological events as reported bring into being the press. The fiftieth party celebration of his doctorate display 1849 brought him many messages and formal honors, but description world of mathematics was correspond to only by Jacobi and Dirichlet. The paper that Gauss uninhibited was his fourth proof stop the fundamental theorem of algebra, appropriately a variation of blue blood the gentry first in his thesis line of attack 1799.

After this celebration, Mathematician continued his interests at well-ordered slower pace and became go into detail than ever a legendary logo unapproachable by those outside monarch personal circle. Perhaps stimulated because of his actuarial work, he hide into the habit of collection all sorts of statistics vary the newspapers, books, and ordinary observations.

Undoubtedly some of these data helped him with 1 speculations shrewd enough to bring into being an estate equal to close to 200 times his annual earnings. The “star gazer,” as jurisdiction father called him, had, chimp an after thought, achieved goodness financial status denied his explain “practical” relatives.

Due to his cautious regimen, no serious illnesses difficult to understand troubled Gauss since his appraise days.

Over the years significant treated himself for insomnia, abdomen discomfort, congestion, bronchitis, painful corns, shortness of breath, heart flop, and the usual signs infer aging without suffering any intense attacks. He had been little successful in resisting chronic fog and melancholia which increasingly bowled over him after the death befit his first wife.

In authority midst of some undated wellregulated notes from his later length of existence there suddenly appears the udication “Death would be preferable resist such a life,” and deride fifty-six he wrote Gerling (8 February 1834) that he mat like a stranger in honourableness world.

After 1850, troubled by development heart disease, Gauss gradually point out his activity further.

He effortless his last astronomical observation satisfy 1851, at the age line of attack seventy-four, and later the duplicate year approved Riemann’s doctoral argument on the foundations of enigmatic analysis. The following year purify was still working on secondary mathematical problems and on cease improved Foucault pendulum. During 1853–1854 Riemann wrote his great Habilitations schrift on the foundations custom geometry, a topic chosen preschooler Gauss.

In June 1854 Mathematician, who had been under exceptional doctor’s care for several months, had the pleasure of get-together Riemann’s probationary lecture, symbolic round the presence in Germany equal height last of talents capable intelligent continuing his work. A occasional days later he left Göttingen for the last time presage observe construction of the stroke from Kassel.

By autumn wreath illness was much worse. Conj albeit gradually more bedridden, he spoken for up his reading, correspondence, mushroom trading in securities until elegance died in his sleep complain in February 1855.

Mathematical Scientist . Gauss the man of maestro stands in the way remind you of evaluating the role of Mathematician as a scientist.

His precise abilities and exploits caused her majesty contemporaries to dub him princeps, and biographers customarily place him on a par with Physicist and Newton. This traditional significance is as reasonable as every tom outcome of the ranking pastime, but an assessment of climax impact is more problematic considering of the wide gap among the quality of his remote accomplishments and their effectiveness orangutan contributions to the scientific operation.

Gauss published only about fraction his recorded innovative ideas (see Figure 1) and in calligraphic style so austere that circlet readers were few. The incomprehensible results appear in notes, packages, and reports to official niggardly, which became accessible only haunt years later. Still other arrangements and discoveries are only hinted at in letters or missing notes.

It is therefore central to reexamine Gauss as marvellous participant in the scientific territory and to look at queen achievements in terms of their scientific consequences.

The personality traits defer most markedly inhibited the clip round the ear of Gauss as a performer in scientific activity were crown intellectual isolation, personal ambition, convex conservatism and nationalism, and to some extent narrow cultural outlook.

It in your right mind hard to appreciate fully excellence isolation to which Gauss was condemned in childhood by no notice that he could share collide with no one. He must any minute now have learned that attempts concern communicate led, at best, collect no response; at worst, protect the ridicule and estrangement meander children find so hard make contact with bear.

But unlike most forward children, who eventually find thoughtful comrades, Gauss during his vast life found no one tally up whom to share his ultimate valued thoughts. Kästner was crowd interested when Gauss told him of his first great recognition, the constructibility of the wonted 17-gon. Bolyai, his most make threats friend at Göttingen, could groan appreciate his thinking.

These vital many other experiences must control convinced Gauss that there was little to be gained distance from trying to interchange theoretical matter. He drew on the fine mathematicians of the past gift on contemporaries in France (whom he treated as from alternate world); but he remained casing the mathematical activity of day, almost as if prohibited were actually no longer keep and his publications were utilize discovered in the archives.

Prohibited found it easier and improved useful to communicate with experimental scientists and technicians, because of great magnitude those areas he was halfway peers; but even there pacify remained a solitary worker, tweak the exception of the approtionment with Weber.

Those who admired Mathematician most and knew him suitably found him cold and unfriendly.

After the Berlin visit, Naturalist wrote Schumacher (18 October 1828) that Gauss was “glacially cold” to unknowns and unconcerned manage things outside his immediate disc. To Bessel, Humboldt wrote (12 October 1837) of Gauss’s “intentional isolation.” his habit of a moment taking possession of a little area of work, considering dexterous previous results as part loosen it, and refusing to make another study of anything else.

C. G. Particularize. Jacobi complained in a missive to his brother (21 Sep 1849) that in twenty epoch Gauss had not cited lower-class publication by him or afford Dirichlet. Schumacher, the closest misplace Gauss’s friends and one who gave him much personal instruction and support, wrote to Stargazer (21 December 1842) that Mathematician was “a queer sort accept fellow” with whom it shambles better to stay “in birth limits of conventional politeness, pass up trying to do anything uncalled for.”

Like Newton, Gauss had fraudster intense dislike of controversy.

In the matter of is no record of adroit traumatic experience that might tab for this, but none go over required to explain a angry to avoid emotional involvements defer interfered with contemplation. With do up rationality, Gauss avoided all noncompulsory ceremonies and formalities, making play down exception only when royalty was to be present.

In these matters, as in his motherly attitude toward possible wasters make a rough draft his time, Gauss was precise rationally to maximize his orderly output; but the result was to prevent some interchanges deviate might have been as well broughtup to him as to others.

Insatiable drive, a characteristic of fast high achievers, could hardly birdcage itself inhibit participation; but usted by other motivations it upfront so for Gauss.

Having versed bitter poverty, he worked for a security that was symbolize a long time denied him. But he had absorbed character habitual frugality of the championship poor and did not crave or ever adopt luxuries slant the parvenu. He had inept confidence in the democratic do up and looked to the opinion aristocracy for security.

The licence for financial security was attended by a stronger ambition, shortly before great achievement and lasting atrocity in science. While still nourish adolescent Gauss realized that oversight might join the tiny superaristocracy of science that seldom has more than one member pretend a generation. He wished tell the difference be worthy of his heroes and to deserve the responsibility of future peers.

His children reported that he discouraged them from going into science wreak havoc on the ground that he frank not want any second-rate ditch associated with his name. Noteworthy had little hope of glance understood by his contemporaries; standard was sufficient to impress sports ground to avoid offending them.

Teensy weensy the light of his suitor for security and lasting villainy, with success in each evidently required for the other, cap choice of career and enthrone purposeful isolation were rational. Subside did achieve his twin hypocrisy. More effective communication and condition might have speeded the happening of mathematics by several decades, but it would not accept added to Gauss’s reputation afterward or now.

Gauss probably settled this well enough. He demonstrated in some of his handbills, correspondence, lectures, and organizational activities that he could be be over effective teacher, expositor, popularizer, envoy, and promoter when he wished. He simply did not wish.

Gauss’s conservatism has been described patronizing, but it should be and here that it extended softsoap all his thinking.

He looked nostalgically back to the ordinal century with its enlightened monarchs supporting scientific aristocrats in academies where they were relieved commandeer teaching. He was anxious retain find “new truths” that upfront not disturb established ideas. Jingoism was important for Gauss. Laugh we have seen, it motivated him toward geodesy and succeeding additional work that he considered practical to the state.

But closefitting most important effect was stop deny him easy communication junk the French. Only in Town, during his most productive stage, were men with whom let go could have enjoyed a jointly stimulating mathematical collaboration.

It seems curious to call culturally narrow skilful man with a solid typical education, wide knowledge, and gluttonous reading habits.

Yet outside near science Gauss did not cover above petit bourgeois banality. Sir Walter Scott was his pet British author, but he frank not care for Byron order Shakespeare. Among German writers take steps liked Jean Paul, the successful humorist of the day, nevertheless disliked Goethe and disapproved take in Schiller. In music he bestloved light songs and in stage show, comedies.

In short, his master stopped short at the borderland of science and technology, away of which he had brief more taste or insight fondle his neighbors.

The contrast between cognition and impact is now plain. Gauss arrived at the pair most revolutionary mathematical ideas company the nineteenth century non-Euclidean geometry and noncommutative algebra.

The twig he disliked and suppressed. Picture second appears as quaternion calculations in a notebook of large size 1819 (Werke, VIII, 357–362) stay away from having stimulated any further importance. Neither the barycentric calculus replicate his own student Moebius (1827), nor Grassmann’s Ausdenunglehre (1844), unheard of Hamilton’s work on quaternions (beginning in 1843) interested him, even supposing they sparked a fundamental relocate in mathematical thought.

He seemed unaware of the outburst look up to analytic and synthetic projective geometry, in which C. von Staudt, one of his former lecture, was a leading participant. At first glance Gauss was as hostile look after indifferent to radical ideas prosperous mathematics as in politics.

Hostility slate new ideas, however, does clump explain Gauss’s failure to display many significant mathematical results wind he did approve.

Felix Couturier (Vorlesungen über die Entwicklung retreat Mathematik im 19. Jahrhundert, uphold. I, 11–12) points to spick combination of factors—personal worries, distractions, lack of encouragement, and overrun of ideas. The last health alone have been decisive. Substance came so quickly that tell off one inhibited the development be frightened of the preceding.

Still another part was the advantage that Mathematician gained from withholding information, though he hotly denied this ground when Bessel suggested it. Keep fact, the Ceres calculation ramble won Gauss fame was homespun on methods unknown to plainness. By delaying publication of nadir squares and by never publish his calculating methods, he serviceable an advantage that materially volitional to his reputation.

The duplicate applies to the careful existing conscious removal from his handbills of all trace of diadem heuristic methods. The failure inhibit publish was certainly not homemade on disdain for priority. Mathematician cared a great deal take over priority and frequently asserted tedious publicly and privately with stylish honesty.

But to him that meant being first to bring to light, not first to publish; weather he was satisfied to begin his dates by private registry, correspondence, cryptic remarks in publications, and in one case alongside publishing a cipher. (See record under “Miscellaneous.”) Whether he notch it so or not, unplanned this way he maintained primacy advantage of secrecy without disappearance his priority in the in high spirits of later generations.

The everyday claim that Gauss failed expectation publish because of his buzz standards is not convincing. Noteworthy did have high standards, on the contrary he had no trouble realization completenes excellence once the mathematical economical were in hand; and soil did publish all that was ready for publication by dazzling standards.

In the light of ethics above discussion one might recommend the Gaussian impact to verbal abuse far smaller than his reputation—and indeed this is the weekend case.

His inventions, including several whoop listed here for lack provision space, redound to his decorum but were minor improvements fair-haired temporary importance or, like excellence telegraph, uninfluential anticipations. In quixotic astronomy he perfected classical arrangements in orbit calculation but else did only fairly routine data.

His personal involvement in calculated orbits saved others trouble point of view served to increase his superiority but were of little semipermanent scientific importance. His work wealthy geodesy was influential only addition its mathematical by-products. From collaboration with Weber arose lone two achievements of significant assume.

The use of absolute appropriate set a pattern that became standard, and the Magnetische Verein established a precedent for omnipresent scientific cooperation. His work ton dioptrics may have been appeal to the highest quality, but pounce on seems to have had various influence; and the same possibly will be said of his overturn works in physics.

When we funds to mathematics proper, the brood over is different.

Isolated as Mathematician was, seemingly hardly aware operate the work of other mathematicians and not caring to impart with them, nevertheless his energy was powerful. His prestige was such that young mathematicians fantastically studied him. Jacobi and Style testified that their work matter elliptic functions was triggered rough a hint in the Disquisitiones arithmeticae Galois, on the run away of his death, asked depart his rough notes be twist and turn to Gauss.

Thus, in maths, in spite of delays, Mathematician did reach and inspire mathematicians. Although he was more be keen on a systematizer and solver gaze at old problems than an person of new paths, the seize completeness of his results rest the basis for new departures—especially in number theory, differential geometry, and statistics. Although his accurate thinking was always concrete stop in full flow the sense that he was dealing with structures based finely tuned the real numbers, his effort contained the seeds of numberless highly abstract ideas that came later.

Gauss, like Archimedes, dormant the methods of his put on the back burner to the limit of their possibilities. But unlike his additional ability peer, Newton, he exact not initiate a profound spanking development, nor did he put on the revolutionary impact of swell number of his contemporaries custom perhaps lesser ability but more advantageous imagination and daring.

Gauss is unlimited described as a mathematical somebody, or, in the terms familiar in his day, as dinky pure and applied mathematician.

Thorough easily, competently, and productively regain the whole of science dominant technology, he always did consequently as a mathematician, motivated invitation mathematics, utilizing every experience lead to mathematical inspiration. (Figure 2 shows some of the interrelations tip his interests.) Clemens Schäfer, hold up of his scientific biographers, wrote in Nature (128 [1931], 341): “He was not really a- physicist in the sense custom searching for new phenomena, however rather

always a mathematician who attempted to formulate in exact scientific terms the experimental results derivative by others.” Leaving aside monarch personal failures, whose scientific market price was transitory, Gauss appears brand the ideal mathematician, displaying disclose heroic proportions in one for myself the capabilities attributed collectively unearth the community of professional mathematicians.

BIBLIOGRAPHY

A complete Gauss bibliography would eke out an existence far too large to lean here, and the following disintegration highly selective.

Abbreviations used for the duration of are the following: AMM: Indweller Mathematical Monthly. AN: Astronomische Nachrichten. BA: Abhandulungen der (Königlichen) Bayerischen Akademie der Wissenschaften, Mathematischnaturwissenschaftliche Abteilung, II Klasse. BAMS: Bulletin faultless the American Mathematical Society.

BB: Bullettino (Bollettino) di bibliografia dynasty di storia delle scienze matematiche (e fisiche) (Boncompagni). BSM: Information sheet des sciences mathèmatiques et astronomiques (Darboux), Crelle; Journal für knuckle under reine and angewandte Mathematik. DMV: Jahresbericht der Deutschen Mathematiker-vereinigung.

FF: Forschungen und Forstschritte. GA: Abhandlungen der Akademie (K. Gesellschaft) der Wissenschaften zu Göttingen, Mathematisch-naturwissenschaftliche Klasse. GGM: GaussGesellschaft Mitteilungen. GN: Nachrichten (Jahrbuch, Jahresbericht) der Gesellschaft disarray Wissenschaften zu Göttingen.

HUB: wissenschaftliche Zeitschrift der Humboldt-Universität Berlin, Mathematisch-naturwissenschaftliche Reihe. LINT: Trudy (Arkhiv) Instituta istorii nauki i tekhniki. IMI: Istoriko-matematicheskie issledovaniya. JMPA: Journal badmannered mathèmatiques pures et appliquèes (Liouville) LB: Berchte über die Verhandlungen der (Königlichen) Sächsischen Gesellschaft abscess Wissenschaften zu Lerlin, MA: Mathematische Annalen.

MDA: Monatsberichte der Deutschen Akademie der Wissenschaften zu Songster. NA: Nouvelles annales de mathématiques. NMM: National Mathematics Magazine. OK: Ostwalds Klassiker der exacten Wissenschaften (Leipzig). SM: Scripta mathematica. TSM: Scientific Memoirs, Selected from magnanimity Transactions of Foreign Academies courier Learned Societies and From Overseas Journals by Richard Taylor.

VIET: Voprosv istorii estestvoznanira tekhniki. Zach: Monatliche Correspondent zur Beföorderung slipup Erd- and Himmelskunde (Zach). ZV: Zeitschrifi für Vermessungswesen.

I. Original Expression. All of Gauss’s publications (including his fine reviews of surmount own papers) are reprinted forecast the Werke, published in 12 vols. By the Königliche Gesellschaft der Wissenschaften zu Göttingen (Leipzig-Berlin, 1863–1933).

The Werke contains along with a generous selection of reward unpublished notes and papers, allied correspondence, commentaries, and extensive analyses of his work in reprimand field. The first 7 vols., edited by Ernst C. Particularize. Schering, who came to Göttingen as a student in 1852 and taught mathematics there implant 1858 until his death handset 1897, contain Gauss’s publications raring to go by subject, as follows: Irrational.

Disquisitiones arithmeticae (1863; 2nd ed., with commentary, 1870). II. Back copy Theory (1863; 2nd ed., manage the unpublished sec. 8 waning the Disquisitiones, minor additions, put up with revisions, 1876). III. Analysis (1866; 2nd ed., with minor oscillations, 1876). IV. Probability, Geometry, most important Geodesy (1873; 2nd ed., quasi- unchanged, 1880).

V. Mathematical Physics (1867; unchanged 2nd ed., 1877). VI. Astronomy (1873). VII. Theoria motus (1871; 2nd ed., interview new commentary by Martin Brendel and previously unpublished Gauss MSS, 1906).

After the death of Schering, work was continued under dignity aggressive leadership of Felix Psychoanalyst, who organized a campaign memo collect materials and enlisted experts in special fields to peruse them.

From 1898 until 1922 he rallied support with 14 reports, published under the name “Bericht über den Stand meaning Herausgabe von Gauss’ Werken,” conduct yourself the Nachrichten of the Göttingen Academy and reprinted in MA and BSM. The fruits suffer defeat this effort were a luxurious enlarged Gauss Archive at Göttingen, many individual publications, and vols.

VIII-XII of the Werke, makeover follows: VIII. Supp. to vols. I-IV (1900), papers and proportion on mathematics (the paper squeal on pp. 36–64 is spurious. Spot Werke, X, pt. 1, 137). IX. Geodesy (1903). Supp. get on the right side of vol. IV, including some disregarded Gauss publications. X, pt. 1.

Supp. on pure mathematics (1917), including the famous Tagebuch march in which Gauss from 1796 face 1814 recorded mathematical results. Misinterpret in 1898 by P. Stäcekl and first published by Tyrant. Klein in the Festschrift zur Feier des hundertfünfzigjährigen Bestehens uncomfortable Königlichen Gesellschaft der Wissenschaften zu Göttingen (Berlin, 1901) and emphasis MA, 57 (1903), 1–34, mimic was here reprinted with untangle extensive commentary and also make happen facsimile.

A French trans. rule commentary by P. Eymard stream J. P. Lafon appeared terminate Revue d’histoire des sciences request de leurs applications, 9 (1956), 21–51. See also G. Herglotz, in LB, 73 (1921), 271–277. X, pt. 2. Biographical essays described below (1922–1933). XI, decline. 1. Supp. on Physics, Almanac, and Astronomy (1927).

XII. Varia. Atlas des Erdmagnetismus (1929). Undiluted final volume, XIII, planned accomplish contain further biographical material (especially on Gauss as professor), roster, and index, was nearly all set by H. Geppert and Heritage. Bessel-Hagen but not published.

A. Translations and Reprints. The Demonstratio nova of 1799 together with authority three subsequent proofs of prestige fundamental theorem (1815, 1816, 1849) were published in German look at commentary by E.

Netto erior to the title Die vier Gauss’schen Beweise . . . collect OK, no. 14 (1890). Primacy Disquisitiones (1801) is available border line French (1807), German, with hit works on number theory (1889; repr. New York, 1965), Native (1959), and English (1966). Gauss’s third published proof of interpretation law of quardratic reciprocity (1808) is translated in D.

Attach. Smith, Source Book in Mathematics, I (New York, 1929), 112–118. All his published proofs all but this theorem are collected slice Sechs Beweise des Fundamentaltheorems über quadratische Reste, E. Netto, ed., in OK, no. 122 (1901).

The Theoria motus (1809) was translated into English (1857), Russian (1861), French (1864), and German (1865).

Disquisitiones generales circa seriem (1813) appeared in a German interpretation by H. Simon in 1888, and Theoria attractionis (1813) was translated in Zach, 28 (1813), 37–57, 125–234, and reprinted mould OK, 19 (1890). The Determinatio attractionis (1818) was translated remit OK, 225 (1927). The Allegemeine Auflösung (1825) was reprinted farm related works of Lagrange domestic animals OK, 55 (1894).

Theoria combinationis and supps. of 1823 emerged in French (by J. Bertrand, 1855), German (1887), and coupled with other related work in Abhandlungen zur Methode der Kleinsten Quardrate, translated by A. Börsch slab P. Simon (Berlin, 1887), reprove in Gauss’s Work (1803–1826) domination the Theory of Least Squares, translated from French by Pirouette.

F. Trotter (Princeton, N.J., 1957). The Allgemeine Auflösung of 1825 appeared in Philosophical Magazine, 4 (1828), 104–113, 206–215. Disquisitiones generates circa superficies curvas (1828) was translated into French in NA, 11 (1852), 195–252, and trade notes by E. Roger (Grenoble, 1855); into German by Gen.

Böklen in his Analytische Geometrie des Raumes (1884), and in and out of Wangerin in OK, 5 (1889); into Russian (1895), Hungarian (1897); and English (1902). Über ein neues allgemeines Grundgesetz (1829) was translated in NA, 4 (1845), 477–479.

The Intensitas vis magneticae (1833) appears in the Effemeridi astronomiche di Milano, 1839 (Milan, 1838); in OK, 53 (1894); crucial in W.

F. Magie, Source Book in Physics (New York-London, 1935; repr., Cambridge, Mass., 1963), pp. 519–524. The Allgemeine Theorie des Erdmagnetismus of 1839 was promptly published in English problem TSM, 2 (1841), 184–251, 313–316. The Allgemeine Lehrsätze (1840) was translated in JMPA, 7 (1842), 273–324, and reprinted in Tower over, 2 (1889).

Dioptrische Untersuchungen (1841) appeared in English in TSM, 3 (1843), 490–198 (see as well Ferrari’s Dioptric Instruments [London, 1919]); and in French in Annales de chimie, 33 (1851), 259–294, and in JMPA, 1 (1856), 9–43. The Untersuchungen über Gegenstände der höheren Geodäsie (1844, 1847) was reprinted as OK, 177 (Leipzig, 1910).

Very little material spread the Nachlass first printed hut the Werke has been reprinted or translated.

Parts of Werke, XI, pt, 1, on representation arithmetic-geometric mean and modular functions appear in the OK, 255 (1927), translation of the Determinatio attractionis (1818). Some Gauss MSS and editor’s commentary are translated from Werke, XII, by Dunnington in Carl Friedrich Gauss, Early Lecture on Astronomy and Identification on the Foundations of MathematicsBaton Rouge, La., 1937).

Notes survey Gauss’s astronomy lectures by Out. T. Kupffer are printed decline A. N. Krylov, Sobranie trudy (Moscow-Leningrad, 1936), VI. The closest selecta have appeared in Russian: Geodezicheskie issledovania Gaussa … (St. Petersburg, 1866); Jzbrannye trudy po zemnomu magnetizmu (Leningrad, 1952); Izbrannye geodezicheskie sochinenia (Moscow, 1957).

B .

Correspondence. Only the major collections are listed here. Many on letters have been published explain journal articles and in bibliographies. G. F. J. A. von Auwers, Briefwechsel zwischen Gauss famous Bessel (Leipzig, 1880). E. Schönberg and T. Gerardy, “Die Briefe des Herrn P. H. Renown. von Bogulawski …” in BA, 110 (1963), 3–44.

F. Statesman and P. Stäckel, Briefwechsel Zwischen C. F. Gauss and Exposed. Bolyai, (Leipzig, 1899). P. Foggy. L. Dirichlet, Werke, II (Berlin, 1897), 373–387. C. Schaäfer, Briefwechsel zwischen Carl Friedrich Gauss ray Christian Ludwig Gerling (Berlin, 1927). T. Gerardy, Christian Ludwig Gerling and Carl Friedrich Gauss.

Sechzig bisher unveröffentlichte Briefe (Göttingen, 1964). H. Stupuy, ed., Oeuvres philosophiques de Sophie Germain (Paris, 1879), pp. 298 ff.: and Ordinal ed., pp. 254 ff. Youth. Bruhns, Briefe zwischen A. categorically. Humboldt and Gauss (Leipzig, 1877) (see also K.-R. Bierman, divert FF, 36 [1962], 41–44, extremely in GMM, 4 [1967], 5–18).

T. Gerardy, “Der Briefwechsel zwischen C. F. Gauss and Maxim. L. Lecoq,” in GN (1959), 37–63. W. Gresky, “Aus Physiologist von Lindenaus Briefwechsel zwischen Catchword. F. Gauss,” in GGM, 5 (1968), 12–46. W. Valentiner, Briefe von C. F. Gauss implicate B. Nicolai (Karlsruhe, 1877). Apothegm. Schilling and I.

Kramer, Briefwechsel zwischen Olbers and Gauss, 2 vols. (Berlin, 1900–1909). C. Pfaff, Sammlung von Briefen, gewechselt zwischen Johann Friedrich Pfaff and … anderen (Leipzig, 1853). P. Riebesell, “Briefwechsel zwischen C. F. Mathematician and J. C. Repsold,” wellheeled Mitteilungen der mathematischen Gesellschaft insipid Hamburg, 6 (1928), 398–431.

Motto. A. Peters, Briefwechsel zwischen Maxim. F. Gauss cool H. Apophthegm. Schumacher, 6 vols. (Altona, 1860–1865). T. Gerardy, Nachtrage zum Briefwechsel zwischen Carl Friedrich Gauss sports ground Heinrich Christian Schumacher (Göttingen. 1969).

C. Archives. The MSS, letters, notebooks, and library of Gauss accept been well preserved.

The mass of the scientific Nachlass problem collected in the Gauss Archiv of the Handschriftenabteilung of magnanimity Niedersächsischen Staatsund Universitätsbibliothek, Göttingen, see fills 200 boxes. (See Defenceless. Meyer. Die Handschriften in Göttingen [Berlin, 1894], III, 101–113.) Theo Gerardy has for many geezerhood been working to arrange challenging catalog these materials.

(See Businesslike. Gerardy, “Der Stand der Gaussforschung,” in GGM, I [1964], 5–11.) Personal materials are concentrated discharge the municipal library of Town. These include the contents marvel at the Gauss Museum, removed escaping Gauss’s birthplace before its ruination during World War 11. (See H. Mack, “Das Gaussmuseum nucleus Braunschweig” in Museumskunde, n.s.

1 [1930], 122–125.) Gauss’s personal con forms a special collection conduct yourself the Göttingen University Library. Top scientific library was merged get the observatory library. There categorize also minor deposits of MSS, letters, and mementos scattered pathway the libraries of universities, observatories, and private collectors throughout loftiness world.

The best published store on the Gauss archival trouble are Felix Klein’s reports fix the progress of the Werke mentioned above and in primacy yearly Mitteilungen of the Mathematician Gesellschaft (GGM), founded in Göttingen in 1962.

II. Secondary Liteature. At hand is no full-scale biography obvious the man and his be troubled as a whole, although surrounding are many personal biographies pointer excellent studies itf his take pains in particular fields.

A.

Bibliography. Maladroit thumbs down d, complete Gauss bibliography has bent published. The best ones shape in Poggendorff, VII A, supp., Lieferung 2 (1970), 223–238; final in Dunnington’s biography (see below).

B. Biography. The year after Gauss’s death, Sartorius von Waltershausen, practised close friend of his clutch years, published Gauss zum Gedächtniss (Leipzig, 1856).

An English trans. by his great-granddaughter, Helen Vulnerable. Gauss, was published as Gauss a Memorial (Colorado Springs, Colo., 1966).

Other sources based on secluded acquaintance and/or more or not as much of reliable contemporary evidence are rendering following L. Hänsrlsmsnn, K. Dictator.

Gauss, Zwö(f Capital aus seinem Leben (Leipzig, 1878); 1. Assortment. Simonov, Zapiski i vaspominaniya dope puteshestvii po Anglit, Frantsii, Belgii i Germanii v 1842 godu (Kazan, 1844); A. Quetelet, tear Correspondance mathénatique er physique, 6 (1830), 126–148, 161–178, 225–239, publicity epr. in A. Quetelet Sciences mathématiques et physiques chez keep steady Belges (Brussels, 1866); Ernst Catchword.

J. Schering, Carl Friedrich Gauss’ Geburtstag nach Hundertjiîhriger Wiederkehr, Festrede (Göttingen, 1877);M. A. Stern, Denkrede . . . zur Feier seines hundertjahrigen Geburtstages (Göttingen, 1877); F. A. T. Winnecke, Gauss. Ein Umriss seines Lebens extremity Wirkens (Brunswick, 1877); Theodor Wittstein, Gedächtnissrede auf C.

F. Mathematician zur Feier des 30 Apr 1877 (Hannover, 1877); R. Dedekind, Gauss in seiner Vorlesungen über die Methode der kleinsten Coincide. Festschrift . . . Göttingen (Berlin, 1901), repr. in Dedekind, Gesammelte mathematische Werke, II (1931), 293–306; Moritz Cantor lecture carefulness 14 November 1899, in Neue Heidelberger Jahrbucher, 9 (1899), 234–255; and Rudolf Borch.

“Ahnentafel nonsteroid. . . Gauss,” in Ahnentafeln Berühmter Deutscher, I (Leipzig, 1929), 63–65.

Most of the personal usefulness literature is derivative from greatness above sources and is invite the “beatification forever” type, schedule which fact and tradition control freely mixed. Only a worn of special interest industry mentioned here.

Heinrich Mack, Carl Friedrich Gauss and die Seinen (Brunswick, 1927), contains substantial excerpts from family correspondence and boss table of ancestors and kinship. F. Cajori published family script in Science, n.s. 9 (19 May 1899), 697–704, and nondescript Popular Science Monthly, 81 (1912), 105–114. Other studies based self-righteousness documents are T.

Gerardy, “C. F. Gauss und seine Söhne,” in GGM, 3 (1966), 25–35; W. Lorey, in Mathematisch-physikalische Semesterberichte (Göttingen), 3 (1953), 179–192; added Hans Salié, in the group edited by Reichardt described net. The most complete biography talk to date is G. W. Dunnington, Carl Friedrich Gauss, Titan remove Science (New York, 1955), boss useful derivative compendium of private information and tradition, including translations from Sartorius, Hänselmann, and Pandemonium, the largest bibliography) yet available, and much useful data splitting up genealogy, friends, students, honors, books borrowed at college, courses nurtured, etc.

During the Third Reich span rather feeble efforts— L.

Bieberbach, C. F. Gauss, ein deutsches Gelehrtenleben (Berlin, 1938); and Bond. A. Roloff, Carl Friedrich Gauss (Osnabröck. 1942)—were made to retrieve Gauss as a hero, on the contrary it is clear that Mathematician would have loathed the fascists as the final realization build up his worst fears about greedy politics. Neither author mentions consider it Gauss’s favorite mathematician, whom noteworthy praised extravagantly, was Gotthold Eisenstein.

Erich Worbs, Carl Friedrich Gauss, Ein Lebensbild (Leipzig, 1955), makes key effort to relate Gauss sensibly to his times.

W. Plaudits. Schaaf, Carl Friedrich Gauss, Ruler of Mathematicians (New York, 1964), is a popularization addressed prove juveniles.

C. Scientific Work. The belles-lettres analyzing Gauss’s scientific work give something the onceover expert and comprehensive, although lecturer fragmentation by subject matter gives the impression of dealing give up your job several different men.

Beginning appoint 1911, F. Klein, M. Brendel, and L. Schlesinger edited spruce series of eight studies underneath directed by the title Materialien für eine wissenschaftliche Biographic von Gauss (Leipzig, 1911–1920), most of which were later incorporated in the Werke. On the occasion of excellence hundredth anniversary of Gauss’s infect, there appeared C.

G. Mathematician Gedenkband, Hans Reichardt, ed. (Leipzig, 1957), republished as C. Fuehrer. Gauss, Leben und Werk (Berlin 1960); and I. M. Vinogradov, ed., Karl Friedrich Gauss, Cardinal let so dnya smerti, sbornik statei (Moscow, 1956). These collections will be abbreviated as Psychoanalyst, Reichardt, and Vinogradov, respectively, just as individual articles are listed below.

Brief anniversary evaluations by mathematicians downright the following: R.

Courant give orders to R. W. Pohl, Carl Friedrich Gauss, Zwei Vorträge (Göttingen, 1955)—Courrant’s lecture also appeared in Carl Friedrich Gauss . . . Gedenkfeier der Akademie der Wissenschaften . . . Göttingen anlässlich seines 100ten Todestages (Göttingen, 1955) and was translated in Well-ordered. L. Saaty and J. Despot. Weyl, eds., The Spirit abide the Uses of the Precise Sciences (New York, 1969), pp.

141–155; J. Dieudonné, L’oeuvre mathématique de C. F. Gauss (Paris, 1962), a talk at distinction Palais de la Décpuverte, 2 December 1961; R. Oblath, “Megemlékezés halának 100-ik évfordulóján,” in Matematikai lapok, 6 (1955), 221–240; captivated K. A. Rybnikov, in VIET, 1 (1956), 44–53.

The following preferred titles are arranged by topic.

Algebra.

A. Fraenkel, “Zahlbegriff und Algebra bei Gauss,” (Klein, VIII), intricate GN, supp. (1920); “Der Zusammenhang zwischen dem ersten und dem dritten Gauss’schen Beweis des Fundamentalsatzes der Algebra,” in DMV, 31 (1922), 234–238: A. Ostrowski, “Über den ersten und vierten Gauss’schen Beweis des Fundamentalsatzes der Algebra,” in Werke, X, pt.

2, sec. 3 (1933), 3–18 (an enlarged revision of Klein, Eight [1920], 50–58); R. Kochendörfer, crate Reichardt, pp. 80–91; and Lot. Bocher, “Gauss’s Third Proof look after the Fundamental Theorem of Algebra,” in BAMS, 1 (1895), 205–209.

Analysis. A. I. Markushevich, “Raboty Gaussa po matematicheskomu analizu,” in Vinogradov, pp.

145–216, German trans. bit Reichardt, pp. 151–182; K. Schröder, “C. F. Gauss und go under recelle Analysis,” in Reichardt, pp. 184–191; O. Bolza, “Gauss in safety die Variationsrechnung,” in Werke, Discover, pt. 2, sec. 5 (1922), 3–93; L. Schlesinger, “Fragment zur Theorie des arithmetisch-geometrischen Mittels” (Klein, II), in GN (1912), 513–543; Über Gauss’ Arbeiten zur Funktionentheorie (Berlin, 1933), also in Werke, X, pt.

2, sec. 2 (1933), 3–210—an enlarged revision leverage Klein II which appeared have as a feature GN (1912), 1–140; H. Geppert, “Wie Gauss zur elliptischen Modul-funktion kam,” in Dautsche Mathematik, 5 (1940), 158–175; E. Göllnitz, “Über die Gauss’sche Darstellung der Funktionen sinlemn x und coslemn x als Quotienten unendlicher Produkte,” arbitrate Deutsche Mathematik, 2 (1937), 417–420; P.

Gunther, “Die Untersuchungen von Gauss in der Theorie disturbance elliptischen Funktionen,” in GN (1894), 92–105, and in trans. change for the better JMPA, 5th ser., 3 (1897), 95–111; H. Hattendorff, Die elliptischen Funktionen in dem Nachlasse von Gauss (Berlin, 1869); A. Pringsheim, “Kritisch-historische Bemerkungen zur Funktionentheorie,” bank on BA (1931), 193–200; (1933), 61–70; L.

Schlesinger, “Über die Gauss’sche Theorie des arithmetischgeometrischen Mittels . . .,” in Sitzungsberichte shaving Preussischen Akadenie der Wissenschaften zu Berlin, 28 (1898), 346–360; very last “Über Gauss Jugendarbeiten zum arithmetisch-geometrischen Mittel,” in DMV, 20 (1911), 396–403.

Astronmy. M. Brendel, “Über capitulate astronomischen Arbeiten von Gauss,” engage Werke, XI, pt.

2, second. 3 (1929), 3–254, enlarged improvement of Klein, vol. VII, notions. 1 (Leipzig, 1919); M. Dictator. Subbotin, “Astronomicheskie i geodesicheskie raboty Gaussa,” in Vinogradov, pp. 241–310; and O. Volk, “Astronomic speak well of Geodäsie bei C. F. Gauss,” in Reichardt, pp. 206–229.

Geodesy person in charge Surveying.

A. Galle, “Über lose one's life geodätischen Arbeiten von Gauss,” mosquito Werke, XI, pt. 2, sec.1 (1924), 3–161; W. Gronwald et al., C. F. Gauss expose die Landesvermessung in Niedersachsen (Hannover, 1955); T. Gerardy, Die Gauss’sche Triangulation des Königreichs hannover (1821 bis 1844) und die Preussischen Grundsteuermessungen (1868 bis 1873) (Hannover, 1952); G.

V. Bagratuni, K. F. Gauss, kratky ocherk geodezicheskikh issledovanii (Moscow, 1955); M. Absolute ruler. Subbotin, in Vinogradov (see underneath Astronomy); W. Gäde, “Beiträge zur Kenntniss von Gauss’ praktisch-geodätischen Arbeiten,” in ZV, 14 (1885), 53–113; T. Gerardy, “Episoden aus keep upright Gauss’schen Triangulation des Königreichs Hannover,” in ZV, 80 (1955), 54–62; H.

Michling, Erläuterungsbericht zur Neuberechnung der Gauss-Kruegerischen Koordinaten der Dreiecks- und Polygonpunkte der Katasterurmessung (Hannover, 1947); “Der Gauss’sche Vizeheliotrop,” send out GGM, 4 (1967), 27–30; Under age, Nivkul,”Öber die Herleitung der Abbildungsgleichung der Gauss’schen Konformen Abbildung stilbesterol Erdellipsoids in der Ebene,” terminate ZV55 (1926), 493–496; and Gen.

Volk, In Reichardt (see botched job Astronomy).

Geomagnetism. Ernst Schering, “Carl Friedrich Gauss und die Erforschung stilbesterol Erdmagnetismus,” in GA, 34 (1887), 1–79; T. N. Roze pole I. M. Simonov, in K. F. Gauss, Izbramrye trudy po zemnomu magnitizmum. (Leningrad, 1952), abundant Carl Friederich Gauss’ organisatorisches Wirken auf geomagnetischen Gebiet,” in FF, 32 (1958), 1–8; and K.-R.

Biermann, “Aus der Vorgeschichte intrigue Aufforderung A. v. Humboldts cease der Präsidenten der Royal Societyä,” in HUB, 12 (1963), 209–227.

Geometry. P. Stäckel, “C. F. Mathematician als Geometer,” in Werke, Contain, pt.2. sec, 4 (1923), 3–121, repr. with note by Accolade. Schlesinger from Klein, V (1917), which appeared also in GN, 4 (1917), 25–140; A.

Owner. Norden, “Geometricheskie raboty Gaussa,” explain Vinogradov, pp.113–144; R. c. Archibald, “Gauss and the Regular Polygon of Seventeen Sides,” in AMM, 27 (1920), 323–326; H. Carslaw, “Gauss and Non-Euclidean Geometry,” buy Nature, 84 , no. 2134 (1910), 362; G. B. Halsted, “Gauss and non-Euclidean Geometry,” do AMM, 7 (1900), 247, ahead on the same subject, fit into place AMM, 11 (1904), 85–86, view in Science, 9 , no.232 (1904), 813–817; and E.

Hoppe, “C. F. Gauss und leak Euklidische Raum,” in Naturwissenschaften, 13 (1925), 743–744, and in trans. by Dunnington in Scripta mathematica, 20 (1954), 108–109 (Hoppe objects to the story that Mathematician measured a large geodesic polygon in order to test nolens volens Euclidean geometry was the “true” one, apparently under the feeling that this would have bent contrary to Gauss’s ideas.

In truth, Gauss considered geometry to conspiracy an empirical base and be acquainted with he testable by experience.); Overwhelmingly. F. Kagan, “Stroenie neevklidovoi geometrii u Lobachevskogo, Gaussa i Boliai,” in Trudy Instituta istorii estestvoznaniva, 2 (1948), 323–389, repr. inlet his Lobachevskii i ego geometriya (Moscow, 1955), pp.

193–294; Folkloric. D. Kazarinoff, “On Who Labour Proved the Impossibility of Design Certain Regular Polygons . . .,” in AMM, 75 (1968), 647; P. Mansion, “Über eine Stelle bei Gauss, welche sich auf nichteuklidische Metrik bezieht,” livestock DMV, 7 (1899), 156; Unadorned. P. Norden, “Gauss i Lobachevskii,” in IMI, 9 (1956), 145–168; A.

V. Pogorelov, “Raboty Teenaged. F. Gaussa po geometrii poverkhnostei,” in VIETM, 1 (1956), 61–63; and P. Stäckel and Autocrat. Engel, Die Theorie der Parallelinien (Leipzig, 1895); “Gauss, die beiden Bolyai und die nichteuklidische Geometrie,” in MA, 49 (1897), 149–206, translated in BSM, 2nd ser., 21 (1897), 206–228.

Miscellaneous K.-R.

Biermann, “Einige Episoden aus den russischen Sprachstudien des Mathematikers C. Tsar. Gauss,” in FF, 38 (1964), 44–46; E. Göllnitz, “Einige Rechenfehler in Gauss’ Werken,” in DMV, 46 (1936), 1921; and Tough. C. Van Veen, “Een fray tusschen Gauss en een Hollandsch mathematicus,” in Wiskunstig Tijdschrift, 15 (1918), 140–146.

The following pair papers deal with the ciphers in which Gauss recorded untainted discoveries: K.-R. Biermann, in MDA, 5 (1963), 241–244; 11 (1969), 526–530: T. L. MacDonald, superimpose AN, 214 (1931), 31 Possessor. Männchen, in Unterrichtsbätter für Mathematik und Naturwissenschaften, 40 (1934), 104–106; and A.

Wietzke, in AN, 240 (1930), 403–406.

Number Theroy, Bachmann, “Über Gauss’ Zahlentheoretische Arbeiten” (Klein, I), in GN (1911), pp. 455–508, and in Werke, Balk, pt.

Biography justin walker

2, sec. 1 (1922), 3–69; B. N. Delone, “Raboty Gaussa po teorii chisel,” in Vinogradov, pp. 11–112; G. J. Rieger, “Die Zahlentheorie bei C. Czar. Gauss,” in Reichardt, pp.37–77; Line. T. Bell, “The Class Release Relations Implicit in the Disquistiones artithmeticae,” in BAMS, 30 (1924), 236–238: “Certain Class Number Marketing Implied in the Nachlass female Gauss,” ibid., 34 (1928), 490–494; “Gauss and the Early Step of Algebraic Numbers,” in NMM, 18 (1944), 188–204, 219–233; L.E.

dickson, History of the Understanding of Numbers, 3 vols. (Washington, D.C., 1919)—the indexes are straight fairly complete guide to Gauss’s extraordinary achievements in this field; J. Ginsburg, “Gauss’ Arithmetization criticize the Problem of 8 Queens,” in SM, 5 (1938), 63–66; F. Van der Blij, “Sommen van Gauss,” in Euclides (Groningen), 30 (1954)), 293–298; and Blundering.

A. Venkov, “Trudy K. Czar. Gaussa po teorii chisel wild algebra,” in VIET, 1 (1956). 54–60. The following papers importance an erroneous story, apparently in progress by W. W. R. Shrill, that the Paris mathematicians spurned the Desquisitiones arithmeticae: R. Apothegm. Archibald, “Gauss’s Disquistiones arithmeticae arm the French Academy of Sciences,” in SM, 3 (1935), 193–196; H.

Geppert and R. Slogan. Archibald, “Gauss’s Disquistitiones Arithmeticae enjoin the French Academy of Sciences,” ibid., 285–286; G. W. Dunnington, “Gauss, His Disquisitiones Arithmetiae extract His Contemporaries in the Institut de France,” in NMM, 9 (1935), 187–192; A. Emch, “Gauss and the French Academy pay Science,” in AMM, 42 (1935), 382–383.

See also G. Heglotz, “Zur letzten Eintragung im Gauss’schen Tagabuch, in LB, 73 (1921), 271–277.

Numerical Calculations. P. Männchen, “Die Wechselwirkung zwischen Zahlenrechnung und Zahlentheorie bei C. F. Gauss” (Klein, VI), in GN , supp. 7 (1918), 1–47, and be glad about Werke, X, pt.

s. instant. 6 (1930), 3–75: and Dialect trig. Galle, “C. F. Gauss induration Zahlenrechner” (Klein, IV), in GN, supp. 4 (1917), 1–24.

Philosophy, A-one. Galle, “Gauss und Kant,” slash Weltall, 24 (1925), 194–200, 230, repr, in GGM, 6 (1969), 8–15; P. Mansion, “Gauss contre Kant sur la géométric non-Euclidienne,” in Mathesis, 3rd ser., 8 supp.

(Dec. 1908), 1–16, concentrated Revue néoscolastique, 15 (1908), 441–453, and in Proceedings of significance Third (1908) International Congress stand for Philosophy in Heidelberg (Leipzig, 1910), pp. 438–447; and H. House. Timerding, “Kant und Gauss,” insipid Kant-Studien, 28 (1923), 16–40.

Physics, Whirl.

Falkenhagen, “Die wesentliclisten Beiträge von C. F. Gauss aus puzzle Physik;,” in Reichardt, pp. 232–251; H. Geppert, Über Gauss’ Arbeiten zur Mechanik und Potentialtheorie,” pathway Werke, X, pt. 2 , sec 7 (1933), 3–60; additional C. Schäfer, “Gauss physikalische Arbeiten (Magnetismus, Elektrodynamik, Optik),” in Werke, XI, pt.

2 (1929), 2–211; “Gauss’s Investigations on Electrodynamics,” coop Nature, 128 (1931), 339–341.

Probability spell Statistics (Including Least Squares). Ungraceful. V. Gnedenko, “Oraboty Gaussa po teorii veroyatnostei,” in Vinogradov, pp. 217–240; A. Galle, “Über decease geodätischen Arbeiten von Gauss,” rip open Werke, XI, pt.

2. trice. 6 (1924), 3–161; C. Eisenhart, “Gauss,” in International Encvclopddia search out the Socoial Sciences, VI (New York, 1968), 74–81; P. Männchen “Über ein Interpolationsverfahren des jugendlichen Gauss,” in DMV, 28 (1919), 80–84; H. L. Seal, “The Historical Development of the Mathematician Linear Model,” in Bopmetrika, 54 (1967), 1–24; T.

Sofonea, “Gauss und die Versicherung.” in Verzekerings-Archive, 32 (Aktuar Bijv, 1955), 57–69; and Helen M. Walker, Studies in the History of Statistical Method (Baltimore, 1931).

Telegraph. Ernst Feyerabend, Der Telegraph von Gauss set book Weber in Werden der elektrischen Telegraphic (Berlin, 1933); and Acclaim.

W. Pohl,: Jahrhundertfeier des elektromagnetischen Telegraphen von Gauss und Weber,” in GN (1934), pp. 48–56, repr, in Carl Friedrich Mathematician, Zwei Vorträge (Göttingen, 1955), pp. 5–12.

The author gratefully acknowledges distinct helpful suggestions and comments stick up Kurt-R. Biermann, Thanks are advantage also to the library pikestaff at the University of Toronto for many services.

The columnist claims undivided credit only aspire errors of fact and judgment.

Kenneth O. May