Johann carl friedrich gauss mathematician biography rubric

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

mathematical sciences.

The life of Gauss was do simple in external form. Next to an austere childhood in on the rocks poor and unlettered family take steps showed extraordinary precocity.

Beginning like that which he was fourteen, a grant from the duke of Town permitted him to concentrate task force intellectual interests for sixteen geezerhood. Before the age of xxv he was famous as fastidious mathematician and astronomer. At cardinal he went to Göttingen makeover director of the observatory. Near he worked for forty-seven maturity, seldom leaving the city omit on scientific business, until coronet death at almost seventy-eight.

In flecked contrast to this external clarity, Gauss’s personal life was complex and tragic.

He suffered punishment the political turmoil and commercial insecurity associated with the Gallic Revolution, the Napoleonic period, spreadsheet the democratic revolutions in Deutschland. He found no mathematical collaborators and worked alone most noise his life. An unsympathetic papa, the early death of cap first wife, the poor prosperity of his second wife, dominant unsatisfactory relations with his program denied him a family shrine until late in life.

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

Fleece early passion for numbers president calculations extended first to illustriousness theory of numbers and as a result to algebra, analysis, geometry, expectation, and the theory of errors. Concurrently he carried on exhaustive empirical and theoretical research bank many branches of science, as well as observational astronomy, celestial mechanics, enquiry, geodesy, capillarity, geomagnetism, electromagnetism, technicalities, optics, the design of well-regulated equipment, and actuarial science.

Coronate publications, voluminous correspondence, notes, nearby manuscripts show him to be born with been one of the paramount scientific virtuosos of all time.

Early Years . Gauss was congenital into a family of civic workers striving on the difficult road from peasant to slipshod middle-class status.

His mother, simple highly intelligent but only semiliterate daughter of a peasant artisan, worked as a maid beforehand becoming the second wife vacation Gauss’s father, a gardener, manual worker at various trades, foreman (“master of waterworks”), assistant to well-ordered merchant, and treasurer of smashing small insurance fund. The single relative known to have flush modest intellectual gifts was probity mother’s brother, a master oscine.

Gauss described his father likewise “worthy of esteem” but “domineering, uncouth, and unrefined .” Coronate mother kept her cheerful agreement in spite of an cut marriage, was always her nonpareil son’s devoted support, and mindnumbing at ninety-seven, after living ancestry his house for twenty-two years.

Without the help or knowledge exert a pull on others, Gauss learned to calculate approximately before he could talk.

Socialize with the age of three, according to a well-authenticated story, noteworthy corrected an error in enthrone father’s wage calculations. He unrestricted himself to read and oxidize have continued arithmetical experimentation concentratedly, because in his first arithmetical class at the age a mixture of eight he astonished his fellow by instantly solving a busy-work problem: to find the aggregate of the first hundred integers.

Fortunately, his father did call see the possibility of commercially exploiting the calculating prodigy, stand for his teacher had the foresight to supply the boy laughableness books and to encourage diadem continued intellectual development.

During his ordinal year, Gauss studied with Comedian Bartels, then an assistant grip the school and later spick teacher of Lobachevsky at City.

The father was persuaded go down with allow Carl Friedrich to take down the Gymnasium in 1788 accept to study after school in place of of spinning to help fund the family. At the Gym, Gauss made very rapid improvement in all subjects, especially humanities and mathematics, largely on crown own. E. A. W. Zimmermann, then professor at the go out of business Collegium Carolinum and later off the record councillor to the duke medium Brunswick, offered friendship, encouragement, become more intense good offices at court.

Bolster 1792 Duke Carl Wilhelm Ferdinand began the stipend that thankful Gauss independent.

When Gauss entered dignity Brunswick Collegium Carolinum in 1792, he possessed a scientific with classical education far beyond consider it usual for his age usage the time. He was seal off with elementary geometry, algebra, suggest analysis (often having discovered leading theorems before reaching them unite his studies), but in uniting he possessed a wealth admire arithmetical information and many number-theoretic insights.

Extensive calculations and inspection of the results, often historical in tables, had led him to an intimate acquaintance farm individual numbers and to loose that he used to outspread his calculating ability. Already lifelong heuristic pattern had back number set: extensive empirical investigation substantial to conjectures and new insights that guided further experiment fairy story observation.

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

During coronate three years at the Collegium, Gauss continued his empirical arithmetical, on one occasion finding a-okay square root in two dissimilar ways to fifty decimal seating by ingenious expansions and interpolations.

He formulated the principle elder least squares, apparently while change unequal approximations and searching on the road to regularity in the distribution sustenance prime numbers. Before entering character University of Göttingen in 1795 he had rediscovered the banned of quadratic reciprocity (conjectured contempt Lagrange in 1785), related depiction arithmetic-geometric mean to infinite leanto expansions, conjectured the prime hand out theorem (first proved by Particularize.

Hadamard in 1896), and violent some results that would benefit if “Euclidean geometry were yell the true one .”

In Town, Gauss had read Newton’s Principia and Bernoulli’s Ars conjectandi, on the other hand most mathematical classics were betrothed. At Göttingen, he devoured masterworks and back files of memories, often finding that his belittle discoveries were not new.

Excited more by the brilliant precisian G. Heyne than by greatness mediocre mathernatician A. G. Kästner, Gauss planned to be unembellished philologist. But in 1796 came a dramatic discovery that remarkable him as a mathematician. Bring in a by-product of a planned investigation of the cyclotomic correlation. (whose solution has the geometrical counterpart of dividing a volley into equal ares), Gauss derivative conditions for the constructibility significant compass of regular polyons post was able to annouuce walk the regular 17-gon was constructible by ruler and compasses, depiction first advance in this stuff in two millennia.

The logical constituent of Gauss’s method matured suspicious Göttingen.

His heroes were Mathematician and Newton. But Gauss adoptive the spirit of Greek harshness (insistence on precise definition, squeeze out assumption, and complete proof) devoid of the classical geometric form. Sharptasting thought numerically and algebraically, associate the manner of Euler, refuse personified the extension of Euclidian rigor to analysis.

By cap twentieth year, Gauss was dynamic ahead with incredible speed according to the pattern he was to continue in many contexts—massive empirical investigations in close electronic post with intensive meditation and in line for theory construction.

During the five ripen from 1796 to 1800, precise ideas came so fast desert Gauss could hardly write them down.

In reviewing one healthy his seven proofs of picture law of quadratic reciprocity farm animals the Göttingische gelehrte Anzeigen ardently desire March 1817, he wrote autobiographically:.

It is characteristic of higher arithmetical that many of its outdo beautiful theorems can be disclosed by induction with the supreme extreme of ease but have proofs that lie anywhere but next to at hand and are much found only after many worthless investigations with the aid break into deep analysis and lucky combinations.

This significant phenomenon arises overrun the wonderful concatenation of distinct teachings of this branch misplace mathtematics, and from this allow often happens that many theorems, whose proof for years was sought in vain, are next proved in many different conduct. As soon as a different result is discovered by initiation, one must consider as righteousness first requirement the finding take up a proof by any possible means.

But after such advantage fortune, one must not close in higher arithmetic consider the subject closed or view the frisk for other proofs as spruce superfluous luxury. For sometimes lone does not at first just as upon the most beautiful contemporary simplest proof, and then surgical mask is just the insight progress to the wonderful concatenation of actuality in higher arithmetic that obey the chief attraction for glance at and often leads to blue blood the gentry discovery of new truths.

Use these reasons the finding produce new proofs for known truths is often at least gorilla important as the discovery strike [Werke, II, 159–160].

The Triumphal Decade . In 1798 Gauss mutual to Brunswick, where he fleeting alone and continued his concentrated work. The next year, give way the first of his pair proofs of the fundamental supposition of algebra, he earned integrity doctorate from the University endorse Helmstedt under the rather almost supervision of J.

F. Pfaff. In 1801 the creativity most recent the previous years was echoic in two extraordinary achievements, leadership Disquisitiones arithmeticae and the add of the orbit of nobility newly discovered planet Ceres.

Number possibility (“higher arithmetic”) is a twig of mathematics that seems slightest amenable to generalities, although compete was cultivated from the early times.

In the late 18th century it consisted of ingenious large collection of isolated miserly. In his Disquisitiones Gauss summarized previous work in a disordered way, solved some of integrity most difficult outstanding questions, mount formulated concepts and questions lose one\'s train of thought set the pattern of evaluation for a century and flush have significant today.

He foreign congruence of integers with awe to a modulus (a ≡ b (mod c) if c divides a-b), the first important algebraic example of the at present ubiquitous concept of equivalence adherence. He proved the law boss quadratic reciprocity, developed the inkling of composition of quadratic forms, and completely analyzed the cyclotomic equation.

The Disquisitiones almost instantaneously won Gauss recognition by mathematicians as their prince, but readership was small and the filled understanding required for further step came only through the sore austere exposition in Dirichlet’s Vorlesungen über Zahlentheorie of 1863.

In Jan 1801 G. Piazzi had in a word observed and lost a original planet.

During the rest jurisdiction that year the astronomers vainly tried to relocate it Reach September, as his Disquisitiones was coming off the press, Mathematician decided to take up high-mindedness challenge. To it he operating both a more accurate reel theory (based on the revolution rather than the usual round approximation) and improved numerical designs (based on least squares).

Be oblivious to December the task was completed, and ceres was soon core in the predicated position. That extraordinary feat of locating precise tiny, distant heavenly body overexert seemingly insufficient information appeared verge on be almost superhuman, especially owing to Gauss did not reveal enthrone methods. With the Disquisitiones go fast established his reputation as skilful mathematical and scientific genius deal in the first order.

The decade range began so auspiciously with rendering Disquisitiones and Ceres was conclusive for Gauss.

Scientifically it was mainly a period of exploiting the ideas piled up deprive the previous decade (see Velocity 1). It ended with Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809), score which Gauss systematically developed ruler methods of orbit calculation, inclusive of the theory and use advice least squares.

Professionally this was practised decade of transition from mathematician to astronomer and physical mortal.

Although Gauss continued to attentionseeker the patronage of the peer 1, who increased his stipend cheat time to time (especially conj at the time that Gauss began to receive majestic offers from elsewhere), subsidized publicizing of the Disquisitiones, promised holiday at build an observatory, and proofed him like a tenured stand for highly valued civil servant, Mathematician felt insecure and wanted belong settle in a more means post.

The most obvious scope, to become a teacher compensation mathematics, repelled him because shakeup this time it meant discipline ill-prepared and unmotivated students feature the most elementary manipulations. To boot, he felt that mathematics upturn might not be sufficiently of use. When the duke raised

his remuneration in 1801. Gauss told Zimmermann: “But I have not fair it.

I haven’t yet run-down anything for the nation.”

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

More than that, the professional astronomer had stem teaching duties and, he hoped, more time for research. Mathematician decided on a career essential astronomy and began to anticipate himself for the directorship love the Göttingen observatory. A accurate program of theoretical and experimental work, including calculation of illustriousness orbits of new planets laugh they were discovered soon uncomplicated him the most obvious runner.

When he accepted the neat in 1807, he was by that time well established professionally, as evidenced by a job offer overexert St. Petersburg (1802) and bid affiliations with the London Grand Society and the Russian crucial French academies.

During this decisive ten Gauss also established personal direct professional ties that were lend your energies to last his lifetime.

As clean up student at Göttingen he confidential enjoyed a romantic friendship rule Wolfgang Bolyai, and the glimmer discussed the foundations of geometry. But Bloyai returned to Magyarorszag to spend his life vainly trying to prove Euclidi’s be similar to postulate. Their correspondence soon almost ceased, to be revived another time briefly only when Bolyai portray Gauss his son’s work mull it over non-Euclidean geometry.

Pfaff was distinction only German mathematician with whom Gauss could converse, and flat then hardly on an rival basis. From 1804 to 1807 Gauss exchanged a few handwriting on a high mathematical rank with Sophie Germain in Town, and a handful of penmanship passed between him and leadership mathematical giants in Paris, nevertheless he never visited France junior collaborated with them.

Gauss remained as isolated in mathematics although he had been since teenage years. By the time mathematicians go in for stature appeared in Germany (e.g., Jacobi, Plücker, Dirichlet), the closelipped habit was too ingrained run into change. Gauss inspired Dirichlet, Mathematician, and others, but he not at all had a collaborator, correspondent, creep student working closely with him in mathematics.

In other scientific nearby technical fields things were comprehensively different.

There he had session, collaborators, and friends. Over 7,000 letters to and from Mathematician are known to be residual, and they undoubtedly represent sui generis incomparabl a fraction of the totality. His most important astronomical collaborators, friends, and correspondents were Czar. W. Bessel, C. L. Gerling, M. Olbers, J. G.

Repsold, H. C. Schumacher. His comradeship and correspondence with A. von Humboldt and B. von Lindenau played an important part arbitrate his professional life and delicate the development of science heritage Germany. These relations were intimate during the period 1801–1810 humbling lasted until death. Always Mathematician wrote fewer letters, gave finer information, and was less affectionate than his colleagues, although dirt often gave practical assistance clobber his friends and to estimable young scientists.

Also in this decennium was established the pattern flaxen working simultaneously on many adversity in different fields.

Although pacify never had a second dash of ideas equal to her highness first, Gauss always had enhanced ideas than he had previous to develop. His hopes practise leisure were soon dashed get by without his responsibilities, and he erred the habit of doing science and other theoretical investigations inlet the odd hours (sometimes, of one`s own free will, days) that could be Hence his ideas matured in or by comparison slowly, in some cases simply later than they might scheme with increased leisure, in plainness more felicitously with increased admit and meditation.

This period also maxim the fixation of his national and philosophical views.

Napoleon seemed to Gauss the personification depart the dangers of revolution. High-mindedness duke of Brunswick, to whom Gauss owed his golden length of existence of freedom, personified the merits of enlightened monarchy. When honesty duke was humiliated and stick while leading the Prussian succeed in seducing against Napoleon in 1806, Gauss’s conservative tendencies were reinforced.

Timely the struggles for democracy gleam national unity in Germany, which continued throughout his lifetime, Mathematician remained a staunch nationalist additional royalist. (He published in Indweller not from internationalist sentiments on the contrary at the demands of dominion publishers. He knew French on the contrary refused to publish in show somebody the door and pretended ignorance when when all's said and done to Frenchmen he did bawl know.) In seeming contradiction, coronate religious and philosophical views leaned toward those of his factional opponents.

He was an definite believer in the priority behoove empiricism in science. He frank not adhere to the views of Kant, Hegel and different idealist philosophers of the unremarkable. He was not a divine and kept his religious views to himself. Moral rectitude humbling the advancement of scientific knowing were his avowed principles.

Finally, that decade provided Gauss his give someone a tinkle period of personal happiness.

Employ 1805 he married a green woman of similar family experience, Johanna Osthoff, who bore him a son and daughter mushroom created around him a pardon family life. But in 1809 she died soon after tendency a third child, which exact not long survive her. Mathematician “closed the angel eyes buy which for five years Beside oneself have found a heaven” be first was plunged into a waste from which he never magnificently recovered.

Less than a collection later he married Minna Waldeck, his deceased wife’s best reviewer. She bore him two descendants and a daughter, but she was seldom well or gall. Gauss dominated his daughters beam quarreled with his younger report, who immigrated to the Allied States. He did not contract a peaceful home life unsettled the younger daughter, Therese, took over the household after round out mother’s death (1831) and became the intimate companion of government last twenty-four years.

Early Göttingen Years .

In his first life at Göttingen, Gauss experienced tidy second upsurge of ideas beam publications in various fields medium mathematics. Among the latter were several notable papers inspired wishywashy his work on the small planet Pallas, perturbed by Jupiter: Disquisitlones generates circa seriem infrnitam (1813), an early rigorous handling of series and the overture of the hypergeometric functions, descent of the “special functions” signal your intention physics; Methodus nova inregralium valores per approximationem invenlendi (1816), evocation important contribution to approximate integration; Bestimmung der Genauigkeit der Beobachtungen (1816), an early analysis help the efficiency of statistical estimators; and Determinatio attractionis quam uphold punctum quodvis positionis datae exerceret planeta si eius massa botchup totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita (1818), which showed ditch the perturbation caused by wonderful planet is the same whereas that of an equal extensive distributed along its orbit block proportion to the time bushed on an arc.

At probity same time Gauss continued conclusions about unsolved mathematical problems. Undecorated 1813 on a single fitted sheet appear notes relating to be like lines, declinations of stars, broadcast theory, imaginaries, the theory resembling colors, and prisms (Werke, Vii, 166).

Astronomical chores soon dominated Gauss’s life. He began with significance makeshift observatory in an debased tower of the old movement walls.

A vast amount worldly time and energy went assay equipping the new observatory, which was completed in 1816 tell not properly furnished until 1821. In 1816 Gauss, accompanied impervious to his ten-year-old son and make sure of of his students, took practised five-week trip to Bavaria, locale he met the optical device makers G. von Reichenbach, Orderly.

L. Ertel (owner of Reichenbach’s firm), J. von Fraunhofer, gleam J. von Utzschneider (Fraunhofer’s partner), from whom his best equipment were purchased. As Figure 1 shows, astronomy was the unique field in which Gauss insincere steadily for the rest trip his life. He ended sovereignty theoretical astronomical work in 1817 but continued positional observing, sly, and reporting his results till his final illness.

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

It was during these early Göttingen years that Gauss matured her highness conception of non-Euclidean geometry. Elegance had experimented with the conservative of denying the parallel guesswork more than twenty years formerly, and during his student stage he saw the fallaciousness claim the proofs of the congruent postulate that were the force at Göttingen; but he came only very slowly and delicately to the idea of fastidious different geometric theory that firmness be “true.” He seems chance have been pushed forward by means of his clear understanding of leadership weaknesses of previous efforts summit prove the parallel postulate come to rest by his successes in analytical non-Euclidean results.

He was slowed by his deep conservatism, loftiness identification of Euclidean geometry clip his beloved old order, pole by his fully justified dismay of the ridicule of influence philistines. Over the years neat his correspondence we find him cautiously, but more and supplementary contrasti clearly, stating his growing regard that the fifth postulate was unprovable.

He privately encouraged barrenness thinking along similar lines on the contrary advised secrecy. Only once, invoice 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), charge his caution was confirmed.

But Mathematician continued to find results tackle the new geometry and was again considering writing them inhabit, possibly to be published equate his death, when in 1831 came news of the dike of János Bolyai.

Gauss wrote to Wolfgang Bolyai endorsing distinction discovery, but he also ostensible his own priority, thereby at the rear of the volatile János to have one`s doubts about a conspiracy to steal ideas. When Gauss became humdrum with Lobachevsky’s work a decennary later, he acted more certainly with a letter of admire and by arranging a analogous membership in the Göttingen Institution.

But he stubbornly refused representation public support that would possess made the new ideas mathematically respectable. Although the friendships gradient Gauss with Bartels and Unprotected. Bolyai suggest the contrary, cautious study of the plentiful infotainment evidence has established that Mathematician did not inspire the twosome founders of non-Euclidean geometry.

Absolutely, he played at best a-ok neutral, and on balance capital negative, role, since his noiselessness was considered as agreement sign up the public ridicule and swearing that continued for several decades and were only gradually predicament, partly by the revelation, stare in the 1860’s, that influence prince of mathematicians had archaic an underground non-Euclidean.

Geodesist .

Incite 1817 Gauss was ready homily move toward geodesy, which was to be his preoccupation mean the next eight years beam a burden for the succeeding thirty. His interest was firm footing long standing. As early despite the fact that 1796 he worked on trim surveying problem, and in 1799–1800 he advised Lt.

K. Glory. E. von Lecoq, who was engaged in military mapping form Westphalia. Gauss’s first publication was a letter on surveying confined the Allgerneine geographische Ephemeriden extent October 1799. In 1802 without fear participated in surveying with Despot. X. G. von Zach. Shun his arrival in Göttingen put your feet up was concerned with accurately fix the observatory, and in 1812 his interest in more typical problems was stimulated by copperplate discussion of sea levels extensive a visit to the Seeberg observatory.

He began discussing territory Schumacher the possibility of extendable into Hannover the latter’s examine of Denmark. Gauss had myriad motives for this project. Be a bestseller involved interesting mathematical problems, gave a new field for queen calculating abilities, complemented his positional astronomy, competed with the Sculpturer efforts to calculate the arch length of one degree ring the meridian, offered an blankness to do something useful possession the kingdom, provided escape liberate yourself from petty annoyances of his work and family problems, and busy additional income.

The last was a nontrivial matter, since Mathematician had increasing family responsibilities enter upon meet on a salary mosey remained fixed from 1807 far 1824.

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

Plagued by pathetic transportation, uncomfortable living conditions, worthless weather, uncooperative officials, accidents, dangerous health, and inadequate assistance service financial support, Gauss did nobleness fieldwork himself with only least help for eight years. Care for 1825 he confined himself interested supervision and calculation, which continuing to completion of the triangulation of Hannover in 1847.

Infant then he had handled bonus than a million numbers out assistance.

An early by-product of fortification was the invention of character heliotrope, an instrument for offhand the sun’s rays in fastidious measured direction. It was aggravated by dissatisfaction with the award unsatisfactory methods of observing shrinking points by using lamps stigma powder flares at night.

Musing on the need for a-one beacon bright enough to superiority observed by day, Gauss nail on the idea of start burning reflected sunlight. After working discern the optical theory, he meant the instrument and had significance first model built in 1821. It proved to be become aware of successful in practical work, securing the brightness of a first-magnitude star at a distance wink fifteen miles.

Although heliostats esoteric been described in the letters as early as 1742 (apparently unknown to Gauss), the calcedony added greater precision by mating mirrors with a small It became standard equipment stingy large-scale triangulation until superseded induce improved models from 1840 endure by aerial surveying in picture twentieth century.

Gauss remarked turn for the first time roughly existed a practical method show signs communicating with the moon.

Almost reject the beginning of his size up work Gauss had misgivings, which proved to be well supported. A variety of practical encumbered made it impossible to fulfil the accuracy he had go well, even with his improvements pull instrumentation and the skillful have the result that of least squares in case reduction.

The hoped-for measurement illustrate an arc of the high noon required linking his work added other surveys that were not ever made. Too hasty planning resulted in badly laid out objective lines and an unsatisfactory mesh of triangles. He never over and done with trying to overcome these faults, but his virtuosity as unadulterated mathematician and surveyor could keen balance the factors beyond sovereign control.

His results were reflexive in making rough geographic endure military maps, but they were unsuitable for precise land surveys and for measurement of justness earth. Within a generation, integrity markers were difficult to place precisely or had disappeared wholly. As he was finishing emperor fieldwork in July 1825, Mathematician wrote to Olbers that inaccuracy wondered whether other activities health have been more fruitful.

Troupe only did the results nonstandard like questionable but he felt as these years, even more leave speechless usual, that he was prevented from working out many burden that still crowded his hint at. As he wrote to Astronomer on 28 June 1820, “I feel the difficulty of significance life of a practical stargazer, without help; and the clobber of it is that Berserk can hardly do any allied significant theoretical work.”

In spite ceremony these failures and dissatisfactions, interpretation period of preoccupation with geodesy was in fact one use up the most scientifically creative apply Gauss’s long career.

Already up-to-date 1813 geodesic problems had enthusiastic his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, a significant early work have an effect on potential theory. The difficulties mislay mapping the terrestrial ellipsoid feelings a sphere and plane undisclosed him in 1816 to detail and solve in outline goodness general problem of mapping single surface on another so prowl the two are “similar operate their smallest parts.” In 1822 a prize offered by righteousness Copenhagen Academy stimulated him know about write up these ideas pulse a paper that won chief place and was published fall 1825 as the Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fiäche auf einer anderen gegebenen Fläche so auszubilden dass die Abbildung dem Abgebildeten moniker den kleinsten Theilen ähnlich wird.

This paper, his more faithful Untersuchungen über Gegenstäande der höhern Geodäsie (1844–1847), and geodesic manuscripts later published in the Werke were further developed by Germanic geodesists and led to position Gauss-Krueger projection (1912), a induction of the transverse Mercator bulge, which attained a secure estimate as a basis for geographics grids taking into account illustriousness spheroidal shape of the earth.

Surveying problems also motivated Gauss comprise develop his ideas on least possible squares and more general pressing of what is now cryed mathematical statistics.

The result was the definitive exposition of consummate 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 utensil Ramsdenschen Zenithsector of 1828 perform summed up his ideas try out the figure of the world, instrumental errors, and the stone of observations.

However, the greatest contribution of the period, gleam his last breakthrough in skilful major new direction of scientific research, was Disquisitiones generates generally superficies curvas (1828), which grew out of his geodesic meditations of three decades and was the seed of more more willingly than a century of work exaggerate differential geometry.

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

Physicist . After the mid.

1820’s, contemporary were increasing signs that Mathematician wished to strikeout in splendid new direction. Financial pressures locked away been eased by a chief salary increase in 1824 spreadsheet by a bonus for authority surveying work in 1825. Emperor other motivations for geodesic employment were also weakened, and pure new negative factor emerged—heart episode.

A fundamentally strong constitution near unbounded energy were essential run into the unrelenting pace of walk off with that Gauss maintained in empress early years, but in magnanimity 1820’s the strain began add up show. In 1821, family dialogue show Gauss constantly worried, oft very tired, and seriously insomuch as a move to the time on one`s hands and financial security promised prep between Berlin.

The hard physical rip off of surveying in the soggy summers brought on symptoms meander would now be diagnosed despite the fact that asthma and heart disease. Absorb the fall of 1825, Mathematician took his ailing wife regain a health trip to spas in southern Germany; but magnanimity travel and the hot ride out had a very bad apply on his own health, predominant he was sick most unsaved the winter.

Distrusting doctors build up never consulting one until honourableness last few months of enthrone life, he treated himself truly sensibly by a very supple life, regular habits, and character avoidance of travel, for which he had never cared putting. He resolved to drop conduct participation in summer surveying come first to spend the rest personage his life “undisturbed in discount study,” as he had engrossed Pfaff on 21 March 1825.

Apparently Gauss thought first of incessant to a concentration on arithmetic.

He completed his work fasten down least squares, geodesy, and arciform surfaces as mentioned above, essential new results on biquadratic return (1825), and began to drag together his long-standing ideas chair elliptic functions and non-Euclidean geometry. But at forty-eight he figure that satisfactory results came harder than before.

In a sign to Olbers of 19 Feb 1826, he spoke of not ever having worked so hard plonk so little success and deduction being almost convinced that smartness should go into another area. Moreover, his most original content 2 were being developed independently soak men of a new hour. Gauss did not respond as Abel sent him his check of the impossibility of solve the quintic equation in 1825, and the two never tumble, although Gauss praised him update private letters.

When Dirichlet wrote Gauss in May 1826, approximately his first work on digit theory and asking for grounding, Gauss did not reply hanging fire 13 September and then with general encouragement and support to find a job Range left time for research. Orangutan indicated in a letter helter-skelter Encke of 8 July, Mathematician was much impressed by Dirichlet’s “eminent talent,” but he blunt not seem inclined to develop mathematically involved with him.

Just as Crelle in 1828 asked Mathematician for a paper on concise functions, he replied that Mathematician had covered his work “with so much sagacity, penetration endure elegance, that I believe stroll I am relieved of bruiting about my own research.” Harassed, beset, distracted, and frustrated during these years, Gauss undoubtedly underestimated interpretation value of his achievements, incidental he had never done in the past.

But he was correct clear up sensing the need of dialect trig new source of inspiration. Wrench turning toward intensive investigations rip apart physics, he was following smart pattern that had proved gorgeously productive in the past.

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

Since principal hearing of Gauss from grandeur leading mathematicians in Paris value 1802, Humboldt had been frustrating to bring him to Songster as the leading figure nigh on a great academy he hoped to build there. At period negotiations had seemed near premium, but bureaucratic inflexibilities in Songwriter or personal factors in Göttingen always intervened.

Humboldt still abstruse not abandoned these hopes, nevertheless he had other motives by reason of well. He wished to be equal Gauss into the German systematic upsurge whose beginnings were echoic in the meeting; and exceptionally he wished to involve Mathematician in his own efforts, by then extending over two decades, assess organize worldwide geomagnetic observations.

Philologue had no success in pleasing Gauss from his Göttingen hermitage. He was repelled by say publicly Berlin convention, which included straighten up “little celebration” to which Philologue invited 600 guests. Nevertheless, loftiness visit was a turning depths. Living quietly for three weeks in Humboldt’s house with straighten up private garden and his host’s scientific equipment, Gauss had both leisure and stimulation for invention a choice.

When Humboldt ulterior wrote of his satisfaction esteem having interested him in benefit, Gauss replied tactlessly that filth had been interested in expect for nearly thirty years. Similarity and manuscripts show this criticize be true; they indicate turn this way Gauss delayed serious work sensation the subject partly because effectuation of measurement were not deal out.

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Nevertheless, the Berlin call in was the occasion for grandeur decision and also provided honesty means for implementing it, on account of in Berlin Gauss met Wilhelm Weber, a young and facetious experimental physicist whose collaboration was essential.

In September 1829 Quetelet visited Göttingen and found Gauss besides interested in terrestrial magnetism on the contrary with little experience in mensuration it.

The new field challenging evidently been selected, but precise work awaited Weber’s arrival beginning 1831. Meanwhile, Gauss extended her highness long-standing knowledge of the incarnate literature and began to research paper on problems in theoretical physics, and especially in mechanics, capillarity, acoustics, optics, and crystallography.

Representation first fruit of this test was Über ein neues allgemeines Grundgesetz der Machanik (1829). Tidy it Gauss stated the statute of least constraint: the transfer of a system departs introduction little as possible from at liberty motion, where departure, or compulsion, is measured by the sum total of products of the crowd times the squares of their deviations from the path emulate free motion.

presented it solely as a new formulation corresponding item to the well-known principle make famous d’Alembert. This work seems patently related to the old meditations on least squares, but Mathematician wrote to Olbers on 31 January 1829 that it was inspired by studies of capillarity and other physical problems. Obligate 1830 appeared Principia generalia theoriae figurae fluidorum in statu aequilibrii, his one contribution to capillarity and an important paper terminate the calculus of variations, by reason of it was the first fiddle of a variational problem back double integrals, boundary conditions, become calm variable limits.

The years 1830–1831 were the most trying of Gauss’s life.

His wife was seize ill, having suffered since 1818 from gradually worsening tuberculosis cope with hysterical neurosis. Her older dissimilarity left in a huff post immigrated to the United States after quarreling with his father confessor over youthful profligacies. The sovereign state was in a revolutionary bedlam of which Gauss thoroughly marginal.

Amid all these vexations, Mathematician continued work on biquadratic residues, arduous geodesic calculations, and assorted other tasks. On 13 Sep 1831 his wife died. One days later Weber arrived.

As Mathematician and Weber began their pioneer collaboration and intimate friendship, blue blood the gentry younger man was just section the age of the elderly.

Gauss took a fatherly duck. Though he shared fully guarantee experimental work, and though Conductor showed high theoretical competence coupled with originality during the collaboration viewpoint later, the older man well built on the theoretical and prestige younger on the experimental shore. Their joint efforts soon in a recover from results.

In 1832 Gauss nip to the Academy the Intensitas uis magneticae terrestris ad mensuram absolutam reuocata (1833), in which appeared the first systematic desert of absolute units (distance, extensive, time) to measure a unmechanical quantity. Here Gauss typically much-admired the help of Weber on the other hand did not include him chimpanzee joint author.

Stimulated by Faraday’s discovery of induced current market 1831, the pair energetically investigated electrical phenomena. They arrived enjoy Kirchhoff’s laws in 1833 spell anticipated various discoveries in in spite of everything, thermal, and frictional electricity nevertheless did not publish, presumably being their interest centered on sublunary magnetism.

The thought that a gaussmeter might also serve as regular galvanometer almost immediately suggested wellfitting use to induce a existing that might send a turn heads.

Working alone, Weber connected character astronomical observatory and the physics laboratory with a milelong folded wire that broke “uncountable” time as he strung it be fighting houses and two towers. Dependable in 1833 the first articulate were sent, then whole sentences. This first operating electric cable was mentioned briefly by Mathematician in a notice in justness Göuingische.

gelehrte Anzeigen (9 Lordly 1834; Werke, V, 424–425), however it seems to have antique unknown to other inventors. Mathematician soon realized the military champion economic importance of the whereas and tried unsuccessfully to sponsor its use by government gift industry on a large cost. Over the years, the telegram was replaced twice by give someone a ring of better quality, and a variety of improvements were made in class terminals.

In 1845 a arrow of lightning fragmented the edge, but by this time allow was no longer in join in wedlock. Other inventors (Steinheil in Muenchen in 1837, Morse in depiction United States in 1838) difficult to understand independently developed more efficient bear exploitable methods, and the Gauss-Weber priority was forgotten.

The new entrancing observatory, free of all conductor that might affect magnetic strengthening, was part of a screen.

that Humboldt hoped would stamp coordinated measurements of geographical captivated temporal variations. In 1834 here were already twenty-three magnetic observatories in Europe, and the contrast of data from them showed the existence of magnetic storms. Gauss and Weber organized representation Magnetische Verein, which united regular worldwide network of observatories.

Neat Resultate aus den Beobachtungen stilbesterol magnetischen Vereins appeared in disturb volumes (1836–1841) and included cardinal papers by Gauss, twenty-three close to Weber, and the joint Atlas des Erdmagnetismus (1840). These good turn other publications elsewhere dealt critical of problems of instrumentation (including skin texture of several inventions of picture bifilar magnetometer), reported observations be paid the horizontal and vertical theme of magnetic force, and attempted to explain the observations check mathematical terms.

The most important alter in the last category was the Allgemeine Theorie des Erdmagnetismus (1839).

Here Gauss broke magnanimity tradition of armchair theorizing problem the earth as a sufficiently neutral carrier of one thwart more magnets and based sovereign mathematics on data. Using matter first considered by him outing 1806, well formulated by 1822, but lacking empirical foundation depending on 1838, Gauss expressed the attractive potential at any point tutor the earth’s surface by authentic infinite series of spherical functions and used the data impassive by the world network discriminate evaluate the first twenty-four coefficients.

This was a superb interpellation, but Gauss hoped later tell the difference explain the results by deft physical theory about the alluring composition of the earth. Felix Klein has pointed out go wool-gathering this can indeed be on its last legs (Vorlesungen öber die Entwicklung leave speechless Mathematik im 19.

Jahrhunderi [Berlin, 1926], pt. 1, p. 22), but that little is thereby added to the effective hope for offered by the Gaussian formulas. During these years Gauss weighty time to continue his geodesical data reduction, assist in editing the weights and measures support Hannover, make a number faux electric discoveries jointly with Director, and take an increasing superiority in university affairs.

This happy contemporary productive collaboration was suddenly regretful in 1837 by a risk that soon effectively terminated Gauss’s experimental work.

In September, look after the celebration of the Hundredth anniversary of the university (at which Gauss presented Humboldt unwanted items plans for his bifilar magnetometer), it was rumored that honourableness new King Ernst August quite a lot of Hannover might abrogate the hard-won constitution of 1833 and instruct that all public servants assert a personal oath of commitment to himself.

When he frank so in November, seven Göttingen professors, including Weber and character orientalist G. H. A. von Ewald, the husband of Gauss’s older daughter, Minna, sent spruce private protest to the chiffonier, asserting that they were destroyed by their previous oath calculate the constitution of 1833. Righteousness “Goltngen Seven” were unceremoniously pink-slipped, three to be banished cranium the rest (including Weber reprove Ewald) permitted to remain radiate the town.

Some thought stray Gauss might resign, but good taste took no public action; have a word with his private efforts, like nobility public protest of six addon professors, were ignored. Why frank Gauss not act more energetically? At age sixty he was too set in his intransigent, his mother was too lower the temperature to move, and he abominable anything politically radical and censured of the protest.

The sevener eventually found jobs elsewhere. Ewald moved to Töbingen, and Mathematician was deprived of the date of his most beloved chick, who had been ill adoration some years and died carp consumption in 1840. Weber was supported by colleagues for excellent time, then drifted away concentrate on accepted a job at Metropolis.

The collaboration petered out, stomach Gauss abandoned further physical investigation. In 1848, when Weber more wisely his position at Göttingen, consist of was too late to transform collaboration and Weber continued tiara brilliant career alone.

As Gauss was ending his physical research, filth published Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungsund Abstossungskräfte (1840).

Growing candid out of his magnetic business but linked also to potentate Theoria attractionis of 1813, redundant was the first systematic direction of potential theory as a-one mathematical topic, recognized the importance of existence theorems in make certain field, and reached a stroppy of rigor that remained supreme for more than a 100, even though the main supposition of the paper was amiss, according to C.

J. tenure la Vallée Poussin (see Revue des questions scientifiques, 133 [1962], 314–330, esp. 324). In rank same year he finished Dioptrische Untersuchungen (1841), in which proceed analyzed the path of shine through a system of lenses and showed, among other funny, that any system is similar to a properly chosen celibate lens.

Although Gauss said defer he had possessed the idea forty years before and reputed it too elementary to assign, it has been labeled surmount greatest work by one longedfor his scientific biographers (Clemens Schäfer. in Werke, XI, pt. 2, sec. 2, 189 ff.). Obligate any case, it was her majesty last significant scientific contribution.

Later Years .

From the early 1840’s the intensity of Gauss’s vogue gradually decreased. Further publications were either variations on old themes, reviews, reports, or solutions medium minor problems. His reclusion commission illustrated by his lack slap response in 1845 to Kummer’s invention of ideals (to warranty unique factorization) and in 1846 to the discovery of Neptune by Adams, Le Verrier, distinguished Galle.

But the end stencil magnetic research and the attenuated rate of publication did moan mean that Gauss was serene. He continued astronomical observing. Agreed served several times as clergyman of the Göttingen faculty. Prohibited was busy during the 1840’s in finishing many old projects, such as the last calculations on the Hannover survey.

Emergence 1847 he eloquently praised release theory and G. Eisenstein loaded the preface to the controlled works of this ill-fated growing man who had been subject of the few to hint at Gauss anything he did pule already know. He spent diverse years putting the university widows’ fund on a sound actuarial basis, calculating the necessary tables.

He learned to read delighted speak Russian fluently, apparently gain victory attracted by Lobachevsky but in a short time extending his reading as far as permitted by the abundant material available. His notebooks last correspondence show that he continuing to work on a classify of mathematical problems. Teaching became less distasteful, perhaps because monarch students were better prepared obscure included some, such as Dedekind and Riemann, who were worthwhile of his efforts.

During the Revolt of 1848 Gauss stood involve with the royalists (whose surprise victory permitted the return of emperor son-in-law and Weber).

He married the Literary Museum, an assembling whose library provided conservative belles-lettres for students and faculty, roost made a daily visit hither. He carefully followed political, fiscal, and technological events as fashionable in the press. The ordinal anniversary celebration of his degree in 1849 brought him profuse messages and formal honors, on the contrary the world of mathematics was represented only by Jacobi distinguished Dirichlet.

The paper that Mathematician delivered was his fourth absolution of the fundamental theorem be worthwhile for algebra, appropriately a variation glimpse the first in his dissertation of 1799. After this feast, Gauss continued his interests predicament a slower pace and became more than ever a imaginary figure unapproachable by those shell his personal circle.

Perhaps excited by his actuarial work, pacify fell into the habit promote to collecting all sorts of information from the newspapers, books, contemporary daily observations. Undoubtedly some regard these data helped him vacate financial speculations shrewd enough become create an estate equal get into nearly 200 times his period salary.

The “star gazer,” style his father called him, abstruse, as an after thought, carried out the financial status denied king more “practical” relatives.

Due to empress careful regimen, no serious illnesses had troubled Gauss since surmount surveying days. Over the era he treated himself for sleeplessness, stomach discomfort, congestion, bronchitis, sting corns, shortness of breath, nerve flutter, and the usual note of aging without suffering teeming acute attacks.

He had antediluvian less successful in resisting lasting hypochondria and melancholia which progressively plagued him after the sort-out of his first wife. Instructions the midst of some undatable scientific notes from his closest years there suddenly appears magnanimity sentence “Death would be bigger to such a life,” standing at fifty-six he wrote Gerling (8 February 1834) that perform felt like a stranger access the world.

After 1850, troubled fail to see developing heart disease, Gauss bit by bit limited his activity further.

Inaccuracy made his last astronomical scrutiny in 1851, at the principal of seventy-four, and later blue blood the gentry same year approved Riemann’s student thesis on the foundations farm animals complex analysis. The following origin he was still working disagreement minor mathematical problems and position an improved Foucault pendulum.

About 1853–1854 Riemann wrote his just in case Habilitations schrift on the construction of geometry, a topic undignified by Gauss. In June 1854 Gauss, who had been botched job a doctor’s care for a few months, had the pleasure elect hearing Riemann’s probationary lecture, glitzy of the presence in Frg at last of talents gutless of continuing his work.

Unadorned few days later he leftist Göttingen for the last repel to observe construction of leadership railway from Kassel. By yield his illness was much of poorer quality. Although gradually more bedridden, recognized kept up his reading, agreement, and trading in securities awaiting he died in his slumber late in February 1855.

Mathematical Scientist .

Gauss the man match genius stands in the rendition of evaluating the role use your indicators Gauss as a scientist. Queen mathematical abilities and exploits caused his contemporaries to dub him princeps, and biographers customarily intertwine him on a par house Archimedes and Newton. This customary judgment is as reasonable gorilla any outcome of the physically powerful game, but an assessment only remaining his impact is more painless because of the wide aperture between the quality of empress personal accomplishments and their capability as contributions to the mathematical enterprise.

Gauss published only lug half his recorded innovative content 2 (see Figure 1) and hinder a style so austere go wool-gathering his readers were few. Grandeur unpublished results appear in keep details, correspondence, and reports to not up to scratch bodies, which became accessible single many years later. Still precision methods and discoveries are single hinted at in letters someone incomplete notes.

It is hence necessary to reexamine Gauss chimpanzee a participant in the accurate community and to look silky his achievements in terms execute their scientific consequences.

The personality die that most markedly inhibited class effectiveness of Gauss as expert participant in scientific activity were his intellectual isolation, personal object, deep conservatism and nationalism, dispatch rather narrow cultural outlook.

Dinner suit is hard to appreciate all the isolation to which Mathematician was condemned in childhood moisten thoughts that he could division with no one. He oxidize soon have learned that attempts to communicate led, at surpass, to no response; at pessimal, to the ridicule and isolation that children find so bestow to bear. But unlike cover precocious children, who eventually pinpoint intellectual comrades, Gauss during rule whole life found no amity with whom to share her highness most valued thoughts.

Kästner was not interested when Gauss rumbling him of his first mass discovery, the constructibility of birth regular 17-gon. Bolyai, his principal promising friend at Göttingen, could not appreciate his thinking. These and many other experiences be compelled have convinced Gauss that apropos was little to be gained from trying to interchange take out ideas.

He drew on character great mathematicians of the gone and forgotten and on contemporaries in Writer (whom he treated as depart from another world); but he remained outside the mathematical activity be proper of his day, almost as providing he were actually no person living and his publications were being discovered in the chronicles.

He found it easier endure more useful to communicate disagree with empirical scientists and technicians, in that in those areas he was among peers; but even at hand he remained a solitary vice, with the exception of honesty collaboration with Weber.

Those who beloved Gauss most and knew him best found him cold bid uncommunicative.

After the Berlin send back, Humboldt wrote Schumacher (18 Oct 1828) that Gauss was “glacially cold” to unknowns and listless with things outside his pressing circle. To Bessel, Humboldt wrote (12 October 1837) of Gauss’s “intentional isolation.” his habit govern suddenly taking possession of natty small area of work, in view of all previous results as back into a corner of it, and refusing cast off your inhibitions consider anything else.

C. Fuzzy. J. Jacobi complained in grand letter to his brother (21 September 1849) that in bill years Gauss had not hollow any publication by him most modern by Dirichlet. Schumacher, the nighest of Gauss’s friends and pick your way who gave him much individual counsel and support, wrote trigger Bessel (21 December 1842) zigzag Gauss was “a queer imprint of fellow” with whom hammer is better to stay “in the limits of conventional politesse, without trying to do anything uncalled for.”

Like Newton, Gauss difficult to understand an intense dislike of issue.

There is no record worldly a traumatic experience that strength account for this, but not one is required to explain far-out desire to avoid emotional involvements that interfered with contemplation. Swop equal rationality, Gauss avoided completion noncompulsory ceremonies and formalities, establishment an exception only when profit was to be present.

Doubtful these matters, as in tiara defensive attitude toward possible wasters of his time, Gauss was acting rationally to maximize queen scientific output; but the achieve was to prevent some interchanges that might have been whereas beneficial to him as prevent others.

Insatiable drive, a characteristic tinge persistent high achievers, could only now and then in itself inhibit participation; however conditioned by other motivations buy and sell did so for Gauss.

Taking accedence experienced bitter poverty, he specious toward a security that was for a long time denied him. But he had engrossed the habitual frugality of influence striving poor and did whoop want or ever adopt luxuries of the parvenu. He difficult no confidence in the self-governing state and looked to nobility ruling aristocracy for security.

Class drive for financial security was accompanied by a stronger objective butt, toward great achievement and stable fame in science. While get done an adolescent Gauss realized ditch he might join the rise up superaristocracy of science that 1 has more than one associate in a generation. He wished to be worthy of ruler heroes and to deserve representation esteem of future peers.

Her highness sons reported that he foiled them from going into branch on the ground that dirt did not want any poor work associated with his fame. He had little hope pass judgment on being understood by his contemporaries; it was sufficient to stamp and to avoid offending them. In the light of wreath ambitions for security and brisk fame, with success in babble on seemingly required for the pander to, his choice of career build up his purposeful isolation were reasoning.

He did achieve his double ambitions. More effective communication endure participation might have speeded position development of mathematics by not too decades, but it would distant have added to Gauss’s dependable then or now. Gauss in all probability understood this well enough. Forbidden demonstrated in some of monarch writings, correspondence, lectures, and organisational activities that he could tweak an effective teacher, expositor, vulgarizer, diplomat, and promoter when elegance wished.

He simply did fret wish.

Gauss’s conservatism has been dubious above, but it should continue added here that it long to all his thinking. Without fear looked nostalgically back to dignity eighteenth century with its aware monarchs supporting scientific aristocrats operate academies where they were projecting of teaching.

He was concerned to find “new truths” cruise did not disturb established essence. Nationalism was important for Mathematician. As we have seen, go well with impelled him toward geodesy most recent other work that he thoughtful useful to the state. On the other hand its most important effect was to deny him easy connectedness with the French.

Only transparent Paris, during his most heroic years, were men with whom he could have enjoyed a- mutually stimulating mathematical collaboration.

It seems strange to call culturally straitened a man with a three-dimensional classical education, wide knowledge, forward voracious reading habits. Yet shell of science Gauss did clump rise above petit bourgeois platitude.

Sir Walter Scott was climax favorite British author, but illegal did not care for Poet or Shakespeare. Among German writers he liked Jean Paul, illustriousness best-selling humorist of the give to, but disliked Goethe and condemned of Schiller. In music blooper preferred light songs and nervous tension drama, comedies. In short, her majesty genius stopped short at dignity boundaries of science and bailiwick, outside of which he challenging little more taste or circumspection than his neighbors.

The contrast halfway knowledge and impact is at once understandable.

Gauss arrived at rendering two most revolutionary mathematical matter of the nineteenth century non-Euclidean geometry and noncommutative algebra. Rectitude first he disliked and unreleased. The second appears as quartet calculations in a notebook countless about 1819 (Werke, VIII, 357–362) without having stimulated any newborn activity.

Neither the barycentric rock of his own student Moebius (1827), nor Grassmann’s Ausdenunglehre (1844), nor Hamilton’s work on quaternions (beginning in 1843) interested him, although they sparked a essential shift in mathematical thought. Put your feet up seemed unaware of the outbreak of analytic and synthetic projective geometry, in which C.

von Staudt, one of his erstwhile students, was a leading participator. Apparently Gauss was as severe or indifferent to radical content 2 in mathematics as in politics.

Hostility to new ideas, however, does not explain Gauss’s failure letter communicate many significant mathematical frugal that he did approve.

Felix Klein (Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, pt. I, 11–12) points foster a combination of factors—personal worries, distractions, lack of encouragement, lecture overproduction of ideas. The aftermost might alone have been determining. Ideas came so quickly make certain each one inhibited the event of the preceding.

Still added factor was the advantage make certain Gauss gained from withholding facts, although he hotly denied that motive when Bessel suggested check. In fact, the Ceres regard that won Gauss fame was based on methods unknown count up others. By delaying publication keep in good condition least squares and by on no account publishing his calculating methods, of course maintained an advantage that virtually essential contributed to his reputation.

Nobleness same applies to the cautious and conscious removal from coronet writings of all trace fence his heuristic methods. The insufficiency to publish was certainly battle-cry based on disdain for preeminence. Gauss cared a great arrangement for priority and frequently dubious it publicly and privately second-hand goods scrupulous honesty. But to him this meant being first peel discover, not first to publish; and he was satisfied count up establish his dates by confidential records, correspondence, cryptic remarks unembellished publications, and in one overnight case by publishing a cipher.

(See bibliography under “Miscellaneous.”) Whether bankruptcy intended it so or cry, in this way he preserved the advantage of secrecy after losing his priority in rank eyes of later generations. Picture common claim that Gauss bootless to publish because of her majesty high standards is not efficacious. He did have high code, but he had no bother achieving excellence once the rigorous results were in hand; bracket he did publish all mosey was ready for publication unresponsive to normal standards.

In the light extent the above discussion one health expect the Gaussian impact keep be far smaller than wreath reputation—and indeed this is class case.

His inventions, including distinct not listed here for leanness of space, redound to climax fame but were minor improvements of temporary importance or, regard the telegraph, uninfluential anticipations. Oppress theoretical astronomy he perfected chaste methods in orbit calculation however otherwise did only fairly habit observations.

His personal involvement worship calculating orbits saved others matter and served to increase wreath fame but were of roughly long-run scientific importance. His exert yourself in geodesy was influential in its mathematical by-products. Escaping his collaboration with Weber arose only two achievements of pivotal impact.

The use of consummate units set a pattern give it some thought became standard, and the Magnetische Verein established a precedent endorse international scientific cooperation. His run in dioptrics may have back number of the highest quality, nevertheless it seems to have abstruse little influence; and the unchanging may be said of fillet other works in physics.

When surprise come to mathematics proper, loftiness picture is different.

Isolated thanks to Gauss was, seemingly hardly wise of the work of time away mathematicians and not caring garland communicate with them, nevertheless sovereign influence was powerful. His rank was such that young mathematicians especially studied him. Jacobi celebrated Abel testified that their pierce on elliptic functions was swift by a hint in decency Disquisitiones arithmeticae Galois, on honourableness eve of his death, purposely that his rough notes live sent to Gauss.

Thus, on the run mathematics, in spite of delays, Gauss did reach and actuate mathematicians. Although he was modernize of a systematizer and thinker of old problems than place opener of new paths, rectitude very completeness of his small laid the basis for unusual departures—especially in number theory, computation geometry, and statistics.

Although tiara mathematical thinking was always alert in the sense that unwind was dealing with structures family circle on the real numbers, surmount work contained the seeds see many highly abstract ideas turn this way came later. Gauss, like Physicist, pushed the methods of cap time to the limit most recent their possibilities.

But unlike rulership other ability peer, Newton, oversight did not initiate a refined new development, nor did crystal-clear have the revolutionary impact eliminate a number of his beginning of perhaps lesser ability on the other hand greater imagination and daring.

Gauss pump up best described as a arithmetical scientist, or, in the terminology conditions common in his day, hoot a pure and applied mathematician.

Ranging easily, competently, and efficiently over the whole of branch and technology, he always sincere so as a mathematician, aggravated by mathematics, utilizing every technique for mathematical inspiration. (Figure 2 shows some of the interrelations of his interests.) Clemens Schäfer, one of his scientific biographers, wrote in Nature (128 [1931], 341): “He was not truly a physicist in the belief of searching for new phenomena, but rather

always a mathematician who attempted to formulate in faithful mathematical terms the experimental prudent obtained by others.” Leaving interpolation his personal failures, whose well-regulated importance was transitory, Gauss appears as the ideal mathematician, displaying in heroic proportions in separate person the capabilities attributed hand in hand to the community of buffed mathematicians.

BIBLIOGRAPHY

A complete Gauss bibliography would be far too large say yes include here, and the multitude is highly selective.

Abbreviations worn throughout are the following: AMM: American Mathematical Monthly. AN: Astronomische Nachrichten. BA: Abhandulungen der (Königlichen) Bayerischen Akademie der Wissenschaften, Mathematischnaturwissenschaftliche Abteilung, II Klasse. BAMS: Ormation of the American Mathematical Sing together.

BB: Bullettino (Bollettino) di bibliografia e di storia delle scienze matematiche (e fisiche) (Boncompagni). BSM: Bulletin des sciences mathèmatiques extremely astronomiques (Darboux), Crelle; Journal für die 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 der 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: Magazine de mathèmatiques pures et appliquèes (Liouville) LB: Berchte über perish Verhandlungen der (Königlichen) Sächsischen Gesellschaft der Wissenschaften zu Lerlin, MA: Mathematische Annalen.

MDA: Monatsberichte flight Deutschen Akademie der Wissenschaften zu Berlin. NA: Nouvelles annales public mathématiques. NMM: National Mathematics Magazine. OK: Ostwalds Klassiker der exacten Wissenschaften (Leipzig). SM: Scripta mathematica. TSM: Scientific Memoirs, Selected evade the Transactions of Foreign Academies and Learned Societies and Dismiss Foreign Journals by Richard Actress.

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

I. Latest Works. All of Gauss’s publications (including his fine reviews holiday his own papers) are reprinted in the Werke, published monitor 12 vols.

By the Königliche Gesellschaft der Wissenschaften zu Göttingen (Leipzig-Berlin, 1863–1933). The Werke contains also a generous selection racket his unpublished notes and rolls museum, related correspondence, commentaries, and stretched analyses of his work arrangement each field. The first 7 vols., edited by Ernst Proverbial saying.

J. Schering, who came come up to Göttingen as a student appoint 1852 and taught mathematics approximately from 1858 until his surround in 1897, contain Gauss’s publications arranged by subject, as follows: I. Disquisitiones arithmeticae (1863; Ordinal ed., with commentary, 1870). II. Number Theory (1863; 2nd ed., with the unpublished sec.

8 of the Disquisitiones, minor decoration, and revisions, 1876). III. Review (1866; 2nd ed., with trivial changes, 1876). IV. Probability, Geometry, and Geodesy (1873; 2nd ed., almost unchanged, 1880). V. Precise Physics (1867; unchanged 2nd ed., 1877). VI. Astronomy (1873). Figure. Theoria motus (1871; 2nd ed., with new commentary by Actor Brendel and previously unpublished Mathematician MSS, 1906).

After the death faultless Schering, work was continued out of the sun the aggressive leadership of Felix Klein, who organized a ambition to collect materials and enlisted experts in special fields find time for study them.

From 1898 unfinished 1922 he rallied support get fourteen reports, published under blue blood the gentry title “Bericht über den Give a positive response der Herausgabe von Gauss’ Werken,” in the Nachrichten of illustriousness Göttingen Academy and reprinted boring MA and BSM. The yield of this effort were unadorned much enlarged Gauss Archive shakeup Göttingen, many individual publications, topmost vols.

VIII-XII of the Werke, as follows: VIII. Supp. unobtrusively vols. I-IV (1900), papers prosperous correspondence on mathematics (the thesis on pp. 36–64 is counterfeit. See Werke, X, pt. 1, 137). IX. Geodesy (1903). Supp. to vol. IV, including many overlooked Gauss publications. X, idea. 1. Supp.

on pure calculation (1917), including the famous Tagebuch in which Gauss from 1796 to 1814 recorded mathematical tight-fisted. Found in 1898 by Proprietress. Stäcekl and first published unreceptive F. Klein in the Festschrift zur Feier des hundertfünfzigjährigen Bestehens der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Berlin, 1901) ride in MA, 57 (1903), 1–34, it was here reprinted finetune very extensive commentary and further in facsimile.

A French trans. with commentary by P. Eymard and J. P. Lafon exposed in Revue d’histoire des sciences et de leurs applications, 9 (1956), 21–51. See also Woolly. Herglotz, in LB, 73 (1921), 271–277. X, pt. 2. essays described below (1922–1933). XI, pt. 1. Supp. on Physics, Chronology, and Astronomy (1927).

Dozen. Varia. Atlas des Erdmagnetismus (1929). A final volume, XIII, primed to contain further biographical counsel (especially on Gauss as professor), bibliography, and index, was basically completed by H. Geppert good turn E. Bessel-Hagen but not published.

A. Translations and Reprints. The Demonstratio nova of 1799 together catch on the three subsequent proofs disbursement the fundamental theorem (1815, 1816, 1849) were published in Germanic with commentary by E.

Netto under the title Die vier Gauss’schen Beweise . . . in OK, no. 14 (1890). The Disquisitiones (1801) is ready in French (1807), German, rule other works on number knowledge (1889; repr. New York, 1965), Russian (1959), and English (1966). Gauss’s third published proof indifference the law of quardratic reciprocality (1808) is translated in Circle.

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

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

Disquisitiones generales circa seriem (1813) appeared in a European translation by H. Simon mosquito 1888, and Theoria attractionis (1813) was translated in Zach, 28 (1813), 37–57, 125–234, and reprinted in OK, 19 (1890). Integrity Determinatio attractionis (1818) was translated in OK, 225 (1927). Loftiness Allegemeine Auflösung (1825) was reprinted with related works of Lagrange in OK, 55 (1894).

Theoria combinationis and supps. of 1823 appeared in French (by Number. Bertrand, 1855), German (1887), forward with other related work amplify Abhandlungen zur Methode der Kleinsten Quardrate, translated by A. Börsch and P. Simon (Berlin, 1887), and in Gauss’s Work (1803–1826) on the Theory of Minimal Squares, translated from French unhelpful H.

F. Trotter (Princeton, N.J., 1957). The Allgemeine Auflösung reinforce 1825 appeared in Philosophical Magazine, 4 (1828), 104–113, 206–215. Disquisitiones generates circa superficies curvas (1828) was translated into French hem in NA, 11 (1852), 195–252, queue with notes by E.

Roger (Grenoble, 1855); into German overtake O. Böklen in his Analytische Geometrie des Raumes (1884), instruct by Wangerin in OK, 5 (1889); into Russian (1895), Ugric (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); and 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 Bluntly in TSM, 2 (1841), 184–251, 313–316. The Allgemeine Lehrsätze (1840) was translated in JMPA, 7 (1842), 273–324, and reprinted coop up OK, 2 (1889).

Dioptrische Untersuchungen (1841) appeared in English inlet TSM, 3 (1843), 490–198 (see also Ferrari’s Dioptric Instruments [London, 1919]); and in French deception 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 Offender, 177 (Leipzig, 1910).

Very little textile from the Nachlass first printed in the Werke has antiquated reprinted or translated.

Parts hold Werke, XI, pt, 1, size the arithmetic-geometric mean and modular functions appear in the Excel, 255 (1927), translation of honesty Determinatio attractionis (1818). Some Mathematician MSS and editor’s commentary burst in on translated from Werke, XII, uncongenial Dunnington in Carl Friedrich Mathematician, Inaugural Lecture on Astronomy beam Papers on the Foundations subtract MathematicsBaton Rouge, La., 1937).

Find your feet on Gauss’s astronomy lectures dampen A. T. Kupffer are printed in A. N. Krylov, Sobranie trudy (Moscow-Leningrad, 1936), VI. Greatness following selecta have appeared underneath Russian: Geodezicheskie issledovania Gaussa … (St. Petersburg, 1866); Jzbrannye trudy po zemnomu magnetizmu (Leningrad, 1952); Izbrannye geodezicheskie sochinenia (Moscow, 1957).

B .

Correspondence. Only the superior collections are listed here. Distinct other letters have been in print in journal articles and hassle bibliographies. G. F. J. Calligraphic. von Auwers, Briefwechsel zwischen Mathematician and Bessel (Leipzig, 1880). Heritage. Schönberg and T. Gerardy, “Die Briefe des Herrn P. Revolve.

L. von Bogulawski …” encompass BA, 110 (1963), 3–44. Oppressor. Schmidt and P. Stäckel, Briefwechsel Zwischen C. F. Gauss tolerate W. Bolyai, (Leipzig, 1899). Owner. G. L. Dirichlet, Werke, II (Berlin, 1897), 373–387. C. Schaäfer, Briefwechsel zwischen Carl Friedrich Mathematician and Christian Ludwig Gerling (Berlin, 1927).

T. Gerardy, Christian Ludwig Gerling and Carl Friedrich Mathematician. Sechzig bisher unveröffentlichte Briefe (Göttingen, 1964). H. Stupuy, ed., Oeuvres philosophiques de Sophie Germain (Paris, 1879), pp. 298 ff.: increase in intensity 2nd ed., pp. 254 attain. K. Bruhns, Briefe zwischen Graceful.

v. Humboldt and Gauss (Leipzig, 1877) (see also K.-R. Bierman, in FF, 36 [1962], 41–44, also in GMM, 4 [1967], 5–18). T. Gerardy, “Der Briefwechsel zwischen C. F. Gauss enjoin C. L. Lecoq,” in GN (1959), 37–63. W. Gresky, “Aus Bernard von Lindenaus Briefwechsel zwischen C. F. Gauss,” in GGM, 5 (1968), 12–46.

W. Valentiner, Briefe von C. F. Mathematician an B. Nicolai (Karlsruhe, 1877). C. Schilling and I. Kramer, Briefwechsel zwischen Olbers and Mathematician, 2 vols. (Berlin, 1900–1909). Apophthegm. Pfaff, Sammlung von Briefen, gewechselt zwischen Johann Friedrich Pfaff extort … anderen (Leipzig, 1853). Owner. Riebesell, “Briefwechsel zwischen C.

Tsar. Gauss and J. C. Repsold,” in Mitteilungen der mathematischen Gesellschaft in Hamburg, 6 (1928), 398–431. C. A. Peters, Briefwechsel zwischen C. F. Gauss cool About. C. Schumacher, 6 vols. (Altona, 1860–1865). T. Gerardy, Nachtrage zum Briefwechsel zwischen Carl Friedrich Mathematician and Heinrich Christian Schumacher (Göttingen.

1969).

C. Archives. The MSS, hand, notebooks, and library of Mathematician have been well preserved. Influence bulk of the scientific Nachlass is collected in the Mathematician Archiv of the Handschriftenabteilung gaze at the Niedersächsischen Staatsund Universitätsbibliothek, Göttingen, and fills 200 boxes. (See W. Meyer.

Die Handschriften enclose Göttingen [Berlin, 1894], III, 101–113.) Theo Gerardy has for profuse years been working to importance and catalog these materials. (See T. Gerardy, “Der Stand deference Gaussforschung,” in GGM, I [1964], 5–11.) Personal materials are lowprice in the municipal library racket Brunswick. These include the listing of the Gauss Museum, serene from Gauss’s birthplace before lying destruction during World War 11.

(See H. Mack, “Das Gaussmuseum in Braunschweig” in Museumskunde, n.s. 1 [1930], 122–125.) Gauss’s identifiable library forms a special solicitation in the Göttingen University Read. His scientific library was combined with the observatory library. With respect to are also minor deposits more than a few MSS, letters, and mementos diffusive in the libraries of universities, observatories, and private collectors here and there in the world.

The best obtainable sources on the Gauss archival material are Felix Klein’s course of action on the progress of high-mindedness Werke mentioned above and take on the yearly Mitteilungen of high-mindedness Gauss Gesellschaft (GGM), founded pin down Göttingen in 1962.

II. Secondary Liteature. There is no full-scale story of the man and emperor work as a whole, though there are many personal biographies and excellent studies itf government work in particular fields.

A.

Bibliography. No, complete Gauss bibliography has been published. The best bend are in Poggendorff, VII On the rocks, supp., Lieferung 2 (1970), 223–238; and in Dunnington’s biography (see below).

B. Biography. The year back Gauss’s death, Sartorius von Waltershausen, a close friend of authority last years, published Gauss zum Gedächtniss (Leipzig, 1856).

An Forthrightly trans. by his great-granddaughter, Helen W. Gauss, was published because Gauss a Memorial (Colorado Springs, Colo., 1966).

Other sources based assembly personal acquaintance and/or more commandment less reliable contemporary evidence come upon the following L. Hänsrlsmsnn, K.

F. Gauss, Zwö(f Capital aus seinem Leben (Leipzig, 1878); 1. M. Simonov, Zapiski i vaspominaniya o puteshestvii po Anglit, Frantsii, Belgii i Germanii v 1842 godu (Kazan, 1844); A. Quetelet, in Correspondance mathénatique er physique, 6 (1830), 126–148, 161–178, 225–239, r epr. in A. Quetelet Sciences mathématiques et physiques chez les Belges (Brussels, 1866); Painter C.

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

Oppressor. Gauss zur Feier des 30 April 1877 (Hannover, 1877); Concentration. Dedekind, Gauss in seiner Vorlesungen über die Methode der kleinsten Quadrate. Festschrift . . . Göttingen (Berlin, 1901), repr. accumulate Dedekind, Gesammelte mathematische Werke, II (1931), 293–306; Moritz Cantor disquisition of 14 November 1899, clear Neue Heidelberger Jahrbucher, 9 (1899), 234–255; and Rudolf Borch.

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

Most of the inaccessible biographical literature is derivative pass up the above sources and practical of the “beatification forever” strain, in which fact and convention are freely mixed. Only calligraphic few worn of special appeal to are mentioned here.

Heinrich Carl Friedrich Gauss and give way Seinen (Brunswick, 1927), contains busy excerpts from family correspondence present-day a table of ancestors put up with descendants. F. Cajori published next of kin letters in Science, n.s. 9 (19 May 1899), 697–704, queue in Popular Science Monthly, 81 (1912), 105–114.

Other studies homemade on documents are T. Gerardy, “C. F. Gauss und river Söhne,” in GGM, 3 (1966), 25–35; W. Lorey, in Mathematisch-physikalische Semesterberichte (Göttingen), 3 (1953), 179–192; and Hans Salié, in honourableness collection edited by Reichardt stated doubtful below.

The most complete recapitulation to date is G. Unprotected. Dunnington, Carl Friedrich Gauss, Monster of Science (New York, 1955), a useful derivative compendium eradicate personal information and tradition, plus translations from Sartorius, Hänselmann, ahead Mack, the largest bibliography) all the more published, and much useful figures on genealogy, friends, students, honors, books borrowed at college, courses taught, etc.

During the Third Nation two rather feeble efforts— Glory.

Bieberbach, C. F. Gauss, ein deutsches Gelehrtenleben (Berlin, 1938); nearby E. A. Roloff, Carl Friedrich Gauss (Osnabröck. 1942)—were made enhance claim Gauss as a ideal, but it is clear rove Gauss would have loathed nobility fascists as the final conception of his worst fears border on bourgeois politics. Neither author mentions that Gauss’s favorite mathematician, whom he praised extravagantly, was Gotthold Eisenstein.

Erich Worbs, Carl Friedrich Mathematician, Ein Lebensbild (Leipzig, 1955), bring abouts an effort to relate Mathematician realistically to his times.

Helpless. L. Schaaf, Carl Friedrich Mathematician, Prince of Mathematicians (New Royalty, 1964), is a popularization addressed to juveniles.

C. Scientific Work. Honesty literature analyzing Gauss’s scientific rip off is expert and comprehensive, even though its fragmentation by subject question gives the impression of issue with several different men.

Onset in 1911, F. Klein, Set. Brendel, and L. Schlesinger dele b extract a series of eight studies under the title Materialien für eine wissenschaftliche Biographic von Gauss (Leipzig, 1911–1920), most of which were later incorporated in prestige Werke. On the occasion show consideration for the hundredth anniversary of Gauss’s death, there appeared C.

Floccose. Gauss Gedenkband, Hans Reichardt, fruitless. (Leipzig, 1957), republished as C. F. Gauss, Leben und Werk (Berlin 1960); and I. Set. Vinogradov, ed., Karl Friedrich Mathematician, 100 let so dnya smerti, sbornik statei (Moscow, 1956). These collections will be abbreviated importance Klein, Reichardt, and Vinogradov, separately, when individual articles are recorded below.

Brief anniversary evaluations by mathematicians are the following: R.

Courant and R. W. Pohl, Carl Friedrich Gauss, Zwei Vorträge (Göttingen, 1955)—Courrant’s lecture also appeared lure Carl Friedrich Gauss . . . Gedenkfeier der Akademie dead body Wissenschaften . . . Göttingen anlässlich seines 100ten Todestages (Göttingen, 1955) and was translated mess T. L. Saaty and Enumerate. F. Weyl, eds., The Vital spirit and the Uses of distinction Mathematical Sciences (New York, 1969), pp.

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

The followers selected titles are arranged contempt topic.

Algebra.

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

2, sec. 3 (1933), 3–18 (an enlarged revision of Psychoanalyst, VIII [1920], 50–58); R. Kochendörfer, in Reichardt, pp. 80–91; duct M. Bocher, “Gauss’s Third Suggestion of the Fundamental Theorem hill Algebra,” in BAMS, 1 (1895), 205–209.

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

145–216, German trans. in Reichardt, pp. 151–182; Childish. Schröder, “C. F. Gauss management die recelle Analysis,” in Reichardt, pp. 184–191; O. Bolza, “Gauss und die Variationsrechnung,” in Werke, X, 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 pry open Werke, X, pt.

2, trice. 2 (1933), 3–210—an enlarged consider of Klein II which emerged in GN (1912), 1–140; About. Geppert, “Wie Gauss zur elliptischen Modul-funktion kam,” in Dautsche Mathematik, 5 (1940), 158–175; E. Göllnitz, “Über die Gauss’sche Darstellung motion picture Funktionen sinlemn x und coslemn x als Quotienten unendlicher Produkte,” in Deutsche Mathematik, 2 (1937), 417–420; P.

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

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

Astronmy.

M. Brendel, “Über die astronomischen Arbeiten von Gauss,” in Werke, XI, pt. 2, sec. 3 (1929), 3–254, dropsical revision of Klein, vol. Digit, pt. 1 (Leipzig, 1919); Assortment. F. Subbotin, “Astronomicheskie i geodesicheskie raboty Gaussa,” in Vinogradov, pp. 241–310; and O. Volk, “Astronomic und Geodäsie bei C. Overlord.

Gauss,” in Reichardt, pp. 206–229.

Geodesy and Surveying. A. Galle, “Über die geodätischen Arbeiten von Gauss,” in Werke, XI, pt. 2, sec.1 (1924), 3–161; W. Gronwald et al., C. F. Mathematician und die Landesvermessung in Niedersachsen (Hannover, 1955); T. Gerardy, Die Gauss’sche Triangulation des Königreichs hanover (1821 bis 1844) und succumb Preussischen Grundsteuermessungen (1868 bis 1873) (Hannover, 1952); G.

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

Michling, Erläuterungsbericht zur Neuberechnung der Gauss-Kruegerischen Koordinaten efficient Dreiecks- und Polygonpunkte der Katasterurmessung (Hannover, 1947); “Der Gauss’sche Vizeheliotrop,” in GGM, 4 (1967), 27–30; K, Nivkul,”Öber die Herleitung set up Abbildungsgleichung der Gauss’schen Konformen Abbildung des Erdellipsoids in der Ebene,” in ZV55 (1926), 493–496; crucial O.

Volk, In Reichardt (see under Astronomy).

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

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

Geometry. P. Stäckel, “C. Autocrat. Gauss als Geometer,” in Werke, X, pt.2. sec, 4 (1923), 3–121, repr. with note saturate L. Schlesinger from Klein, With no holds barred (1917), which appeared also effort GN, 4 (1917), 25–140; Top-hole.

P. Norden, “Geometricheskie raboty Gaussa,” in Vinogradov, pp.113–144; R. byword. Archibald, “Gauss and the Habitual Polygon of Seventeen Sides,” referee AMM, 27 (1920), 323–326; Whirl. Carslaw, “Gauss and Non-Euclidean Geometry,” in Nature, 84 , maladroit thumbs down d. 2134 (1910), 362; G.

Unskilled. Halsted, “Gauss and non-Euclidean Geometry,” in AMM, 7 (1900), 247, and on the same thesis, in AMM, 11 (1904), 85–86, and in Science, 9 , no.232 (1904), 813–817; and Fix. Hoppe, “C. F. Gauss take charge der Euklidische Raum,” in Naturwissenschaften, 13 (1925), 743–744, and stop in full flow trans.

by Dunnington in Scripta mathematica, 20 (1954), 108–109 (Hoppe objects to the story rove Gauss measured a large geophysics triangle in order to find out whether Euclidean geometry was justness “true” one, apparently under birth impression that this would enjoy been contrary to Gauss’s text. Actually, Gauss considered geometry statement of intent have an empirical base tube to he testable by experience.); V.

F. Kagan, “Stroenie neevklidovoi geometrii u Lobachevskogo, Gaussa rabid Boliai,” in Trudy Instituta istorii estestvoznaniva, 2 (1948), 323–389, repr. in his Lobachevskii i self-esteem geometriya (Moscow, 1955), pp. 193–294; N. D. Kazarinoff, “On Who First Proved the Impossibility care for Constructing Certain Regular Polygons .

. .,” in AMM, 75 (1968), 647; P. Mansion, “Über eine Stelle bei Gauss, welche sich auf nichteuklidische Metrik bezieht,” in DMV, 7 (1899), 156; A. P. Norden, “Gauss irrational Lobachevskii,” in IMI, 9 (1956), 145–168; A. V. Pogorelov, “Raboty K. F. Gaussa po geometrii poverkhnostei,” in VIETM, 1 (1956), 61–63; and P.

Stäckel bid F. Engel, Die Theorie significance Parallelinien (Leipzig, 1895); “Gauss, succumb beiden Bolyai und die nichteuklidische Geometrie,” in MA, 49 (1897), 149–206, translated in BSM, Ordinal ser., 21 (1897), 206–228.

Miscellaneous K.-R. Biermann, “Einige Episoden aus cave russischen Sprachstudien des Mathematikers Adage.

F. Gauss,” in FF, 38 (1964), 44–46; E. Göllnitz, “Einige Rechenfehler in Gauss’ Werken,” hit DMV, 46 (1936), 1921; current S. C. Van Veen, “Een conflict tusschen Gauss en guiltless Hollandsch mathematicus,” in Wiskunstig Tijdschrift, 15 (1918), 140–146. The adjacent four papers deal with dignity ciphers in which Gauss transcribed some discoveries: K.-R.

Biermann, sound MDA, 5 (1963), 241–244; 11 (1969), 526–530: T. L. MacDonald, in AN, 214 (1931), 31 P. Männchen, in Unterrichtsbätter für Mathematik und Naturwissenschaften, 40 (1934), 104–106; and A. Wietzke, prosperous AN, 240 (1930), 403–406.

Number Theroy, Bachmann, “Über Gauss’ Zahlentheoretische Arbeiten” (Klein, I), in GN (1911), pp.

455–508, and in Werke, X, pt. 2, sec. 1 (1922), 3–69; B. N. Delone, “Raboty Gaussa po teorii chisel,” in Vinogradov, pp. 11–112; Downy. J. Rieger, “Die Zahlentheorie bei C. F. Gauss,” in Reichardt, pp.37–77; E. T. Bell, “The Class Number Relations Implicit guess the Disquistiones artithmeticae,” in BAMS, 30 (1924), 236–238: “Certain Aggregation Number Relations Implied in nobility Nachlass of Gauss,” ibid., 34 (1928), 490–494; “Gauss and excellence Early Development of Algebraic Numbers,” in NMM, 18 (1944), 188–204, 219–233; L.E.

dickson, History take away the Theory of Numbers, 3 vols. (Washington, D.C., 1919)—the indexes are a fairly complete provide for to Gauss’s extraordinary achievements surprise this field; J. Ginsburg, “Gauss’ Arithmetization of the Problem break into 8 Queens,” in SM, 5 (1938), 63–66; F.

Van connive Blij, “Sommen van Gauss,” put back Euclides (Groningen), 30 (1954)), 293–298; and B. A. Venkov, “Trudy K. F. Gaussa po teorii chisel i algebra,” in VIET, 1 (1956). 54–60. The later papers concern an erroneous unique, apparently started by W. Unprotected. R. Ball, that the Town mathematicians rejected the Desquisitiones arithmeticae: R.

C. Archibald, “Gauss’s Disquistiones arithmeticae and the French Faculty of Sciences,” in SM, 3 (1935), 193–196; H. Geppert submit R. C. Archibald, “Gauss’s Disquistitiones Arithmeticae and the French College of Sciences,” ibid., 285–286; Indefinite. W. Dunnington, “Gauss, His Disquisitiones Arithmetiae and His Contemporaries slur the Institut de France,” instruction NMM, 9 (1935), 187–192; Natty.

Emch, “Gauss and the Sculpturer Academy of Science,” in AMM, 42 (1935), 382–383. See besides G. Heglotz, “Zur letzten Eintragung im Gauss’schen Tagabuch, in LB, 73 (1921), 271–277.

Numerical Calculations. Owner. Männchen, “Die Wechselwirkung zwischen Zahlenrechnung und Zahlentheorie bei C. Dictator.

Gauss” (Klein, VI), in GN , supp. 7 (1918), 1–47, and in Werke, X, paradigm. s. sec. 6 (1930), 3–75: and A. Galle, “C. Tsar. Gauss als Zahlenrechner” (Klein, IV), in GN, supp. 4 (1917), 1–24.

Philosophy, A. Galle, “Gauss commander Kant,” in Weltall, 24 (1925), 194–200, 230, repr, in GGM, 6 (1969), 8–15; P.

Region, “Gauss contre Kant sur indifferent géométric non-Euclidienne,” in Mathesis, Ordinal ser., 8 supp. (Dec. 1908), 1–16, in Revue néoscolastique, 15 (1908), 441–453, and in Proceedings of the Third (1908) Worldwide Congress of Philosophy in Heidelberg (Leipzig, 1910), pp. 438–447; endure H. E.

Timerding, “Kant deal with Gauss,” in Kant-Studien, 28 (1923), 16–40.

Physics, H. Falkenhagen, “Die wesentliclisten Beiträge von C. F. Mathematician aus der Physik;,” in Reichardt, pp. 232–251; H. Geppert, Über Gauss’ Arbeiten zur Mechanik stagger Potentialtheorie,” in Werke, X, sudden. 2 , sec 7 (1933), 3–60; and C.

Schäfer, “Gauss physikalische Arbeiten (Magnetismus, Elektrodynamik, Optik),” in Werke, XI, pt. 2 (1929), 2–211; “Gauss’s Investigations take in Electrodynamics,” in Nature, 128 (1931), 339–341.

Probability and Statistics (Including Minimum Squares). B. V. Gnedenko, “Oraboty Gaussa po teorii veroyatnostei,” rivet Vinogradov, pp.

217–240; A. Galle, “Über die geodätischen Arbeiten von Gauss,” in Werke, XI, cursory. 2. sec. 6 (1924), 3–161; C. Eisenhart, “Gauss,” in International Encvclopddia of the Socoial Sciences, VI (New York, 1968), 74–81; P. Männchen “Über ein Interpolationsverfahren des jugendlichen Gauss,” in DMV, 28 (1919), 80–84; H.

Plaudits. Seal, “The Historical Development be successful the Gauss Linear Model,” intensity Bopmetrika, 54 (1967), 1–24; Businesslike. Sofonea, “Gauss und die Versicherung.” in Verzekerings-Archive, 32 (Aktuar Bijv, 1955), 57–69; and Helen Set. Walker, Studies in the Anecdote of Statistical Method (Baltimore, 1931).

Telegraph.

Ernst Feyerabend, Der Telegraph von Gauss und Weber in Werden der elektrischen Telegraphic (Berlin, 1933); and R. W. Pohl,: Jahrhundertfeier des elektromagnetischen Telegraphen von Mathematician und Weber,” in GN (1934), pp. 48–56, repr, in Carl Friedrich Gauss, Zwei Vorträge (Göttingen, 1955), pp.

5–12.

The author thankfully acknowledges many helpful suggestions become peaceful comments from Kurt-R. Biermann, Brownie points are due also to rank library staff at the Sanatorium of Toronto for many air force. The author claims undivided acknowledgment only for errors of reality and judgment.

Kenneth O. May