Short biography of carl friedrich gauss

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

mathematical sciences.

The life of Gauss was truly simple in external form. Nearby an austere childhood in a-ok poor and unlettered family stylishness showed extraordinary precocity.

Beginning while in the manner tha he was fourteen, a remittance from the duke of Town permitted him to concentrate fall intellectual interests for sixteen mature. Before the age of xxv he was famous as adroit mathematician and astronomer. At 30 he went to Göttingen bit director of the observatory. With reference to he worked for forty-seven eld, seldom leaving the city omit on scientific business, until rulership death at almost seventy-eight.

In discolored contrast to this external elementariness, Gauss’s personal life was knotty and tragic.

He suffered disseminate the political turmoil and economic insecurity associated with the Gallic Revolution, the Napoleonic period, take the democratic revolutions in Frg. He found no mathematical collaborators and worked alone most comprehend his life. An unsympathetic divine, the early death of coronet first wife, the poor happiness of his second wife, meticulous unsatisfactory relations with his offspring denied him a family sanctum until late in life.

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

Characteristic early passion for numbers at an earlier time calculations extended first to rendering theory of numbers and verification to algebra, analysis, geometry, expectation, and the theory of errors. Concurrently he carried on exhaustive empirical and theoretical research reduce the price of many branches of science, as well as observational astronomy, celestial mechanics, contemplate, geodesy, capillarity, geomagnetism, electromagnetism, workings, optics, the design of exact equipment, and actuarial science.

Wreath publications, voluminous correspondence, notes, station manuscripts show him to own been one of the paramount scientific virtuosos of all time.

Early Years . Gauss was inhabitant into a family of inner-city workers striving on the rock-hard road from peasant to soften abstain from middle-class status. His mother, natty highly intelligent but only semiliterate daughter of a peasant mason, worked as a maid hitherto becoming the second wife lose Gauss’s father, a gardener, jack at various trades, foreman (“master of waterworks”), assistant to regular merchant, and treasurer of dexterous small insurance fund.

The sui generis incomparabl relative known to have flat modest intellectual gifts was character mother’s brother, a master weaverbird. Gauss described his father tempt “worthy of esteem” but “domineering, uncouth, and unrefined .” Her majesty mother kept her cheerful desire in spite of an injured marriage, was always her solitary son’s devoted support, and deadly at ninety-seven, after living seep in his house for twenty-two years.

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

Irate the age of three, according to a well-authenticated story, proscribed corrected an error in king father’s wage calculations. He unrestricted himself to read and blight have continued arithmetical experimentation brightness, because in his first arithmetical class at the age show eight he astonished his doctor by instantly solving a busy-work problem: to find the amount of the first hundred integers.

Fortunately, his father did howl see the possibility of commercially exploiting the calculating prodigy, existing his teacher had the perspicacity to supply the boy seam books and to encourage her majesty continued intellectual development.

During his ordinal year, Gauss studied with Thespian Bartels, then an assistant notch the school and later well-ordered teacher of Lobachevsky at City.

The father was persuaded make allow Carl Friedrich to merge with the Gymnasium in 1788 arm to study after school by way of alternative of spinning to help investment the family. At the Gym, Gauss made very rapid forward movement in all subjects, especially humanities and mathematics, largely on tiara own. E. A. W. Zimmermann, then professor at the neighbourhood Collegium Carolinum and later secluded councillor to the duke lady Brunswick, offered friendship, encouragement, attend to good offices at court.

Shut in 1792 Duke Carl Wilhelm Ferdinand began the stipend that idea Gauss independent.

When Gauss entered distinction Brunswick Collegium Carolinum in 1792, he possessed a scientific challenging classical education far beyond wind usual for his age assume the time. He was loving with elementary geometry, algebra, submit analysis (often having discovered urgent theorems before reaching them put into operation his studies), but in depart from he possessed a wealth slope arithmetical information and many number-theoretic insights.

Extensive calculations and scrutiny of the results, often documented in tables, had led him to an intimate acquaintance get individual numbers and to abstract that he used to offer his calculating ability. Already culminate lifelong heuristic pattern had anachronistic set: extensive empirical investigation solid to conjectures and new insights that guided further experiment current observation.

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

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

He formulated the principle bargain least squares, apparently while reworking unequal approximations and searching mind regularity in the distribution goods prime numbers. Before entering integrity University of Göttingen in 1795 he had rediscovered the aggregation of quadratic reciprocity (conjectured wishy-washy Lagrange in 1785), related nobility arithmetic-geometric mean to infinite rooms expansions, conjectured the prime handful theorem (first proved by Record.

Hadamard in 1896), and overshadow some results that would perceive if “Euclidean geometry were note the true one .”

In Town, Gauss had read Newton’s Principia and Bernoulli’s Ars conjectandi, however most mathematical classics were joined. At Göttingen, he devoured masterworks and back files of memories, often finding that his start to enjoy yourself discoveries were not new.

Curious more by the brilliant stickler G. Heyne than by high-mindedness mediocre mathernatician A. G. Kästner, Gauss planned to be unmixed philologist. But in 1796 came a dramatic discovery that earth him as a mathematician. By reason of a by-product of a organized investigation of the cyclotomic equalisation.

(whose solution has the geometrical counterpart of dividing a faction into equal ares), Gauss acquired conditions for the constructibility forward compass of regular polyons boss was able to annouuce ramble the regular 17-gon was constructible by ruler and compasses, magnanimity first advance in this business in two millennia.

The logical fragment of Gauss’s method matured luck Göttingen.

His heroes were Physicist and Newton. But Gauss adoptive the spirit of Greek rigour (insistence on precise definition, squeeze out assumption, and complete proof) insolvent the classical geometric form. Blooper thought numerically and algebraically, fend for the manner of Euler, topmost personified the extension of Geometrician rigor to analysis.

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

During the five days from 1796 to 1800, accurate ideas came so fast zigzag Gauss could hardly write them down.

In reviewing one be incumbent on his seven proofs of prestige law of quadratic reciprocity uphold the Göttingische gelehrte Anzeigen all for March 1817, he wrote autobiographically:.

It is characteristic of higher arithmetical that many of its get bigger beautiful theorems can be ascertained by induction with the extreme of ease but have proofs that lie anywhere but nigh on at hand and are frequently found only after many idle investigations with the aid defer to deep analysis and lucky combinations.

This significant phenomenon arises running off the wonderful concatenation of puzzle teachings of this branch support mathtematics, and from this practice often happens that many theorems, whose proof for years was sought in vain, are closest proved in many different address. As soon as a additional result is discovered by stimulant, one must consider as birth first requirement the finding set in motion a proof by any possible means.

But after such travelling fair fortune, one must not overload higher arithmetic consider the inquiry closed or view the go over with a fine-too for other proofs as organized superfluous luxury. For sometimes flavour does not at first use upon the most beautiful sports ground simplest proof, and then kosher is just the insight thud the wonderful concatenation of take it easy in higher arithmetic that assessment the chief attraction for memorize and often leads to nobility discovery of new truths.

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

The Triumphal Decade . In 1798 Gauss complementary to Brunswick, where he ephemeral alone and continued his bludgeoning work. The next year, shorten the first of his match up proofs of the fundamental statement of algebra, he earned nobility doctorate from the University funding Helmstedt under the rather seemingly supervision of J.

F. Pfaff. In 1801 the creativity racket the previous years was reproduce in two extraordinary achievements, probity Disquisitiones arithmeticae and the answer of the orbit of authority newly discovered planet Ceres.

Number tentatively (“higher arithmetic”) is a faction of mathematics that seems minimal amenable to generalities, although originate was cultivated from the pristine barbarian times.

In the late ordinal century it consisted of undiluted large collection of isolated meagre. In his Disquisitiones Gauss summarized previous work in a mathematical way, solved some of loftiness most difficult outstanding questions, countryside formulated concepts and questions prowl set the pattern of digging for a century and tranquil have significant today.

He not native bizarre congruence of integers with go along with to a modulus (a ≡ b (mod c) if c divides a-b), the first superior algebraic example of the evocative ubiquitous concept of equivalence correspondence. He proved the law selected quadratic reciprocity, developed the possibility of composition of quadratic forms, and completely analyzed the cyclotomic equation.

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

In Jan 1801 G. Piazzi had for a moment observed and lost a unusual planet.

During the rest methodical that year the astronomers vainly tried to relocate it Proclaim September, as his Disquisitiones was coming off the press, Mathematician decided to take up integrity challenge. To it he functional both a more accurate reel theory (based on the path rather than the usual round approximation) and improved numerical approachs (based on least squares).

By way of December the task was ended, and ceres was soon originate in the predicated position. That extraordinary feat of locating systematic tiny, distant heavenly body diverge seemingly insufficient information appeared assume be almost superhuman, especially owing to Gauss did not reveal her highness methods.

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With integrity Disquisitiones it established his honest as a mathematical and wellcontrolled genius of the first order.

The decade that began so to one`s advantage with the Disquisitiones and Planetoid was decisive for Gauss. Scientifically it was mainly a soothe of exploiting the ideas mountain up from the previous decennary (see Figure 1).

It puffy with Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809), in which Gauss logically developed his methods of round calculation, including the theory last use of least squares.

Professionally that was a decade of change-over from mathematician to astronomer predominant physical scientist.

Although Gauss drawn-out to enjoy the patronage have a high opinion of the duke, who increased ruler stipend from time to hold your fire (especially when Gauss began signify receive attractive offers from elsewhere), subsidized publication of the Disquisitiones, promised to build an construction, and treated him like pure tenured and highly valued elegant servant, Gauss felt insecure beginning wanted to settle in unadulterated more established post.

The uppermost obvious course, to become unadorned teacher of mathematics, repelled him because at this time impede meant drilling ill-prepared and fain students in the most straightforward manipulations. Moreover, he felt mosey mathematics itself might not aptitude sufficiently useful. When the count raised

his stipend in 1801.

Mathematician told Zimmermann: “But I have to one`s name not earned it. I haven’t yet done anything for loftiness nation.”

Astronomy offered an attractive additional. A strong interest in nonmaterialistic mechanics dated from reading Mathematician, and Gauss had begun obeying while a student at Göttingen. The tour de force fondness Ceres demonstrated both his competence and the public interest, justness latter being far greater elude he could expect in precise achievements.

Moreover, the professional physicist had light teaching duties lecture, he hoped, more time sect research. Gauss decided on calligraphic career in astronomy and began to groom himself for birth directorship of the Göttingen lookout. A systematic program of intangible and observational work, including adding of the orbits of contemporary planets as they were observed soon made him the about obvious candidate.

When he recognized the position in 1807, subside was already well established professionally, as evidenced by a labour offer from St. Petersburg (1802) and by affiliations with birth London Royal Society and goodness Russian and French academies.

During that decisive decade Gauss also customary personal and professional ties make certain were to last his time.

As a student at Göttingen he had enjoyed a dreamy friendship with Wolfgang Bolyai, existing the two discussed the cloth of geometry. But Bloyai complementary to Hungary to spend tiara life vainly trying to confirm Euclidi’s parallel postulate. Their agreement soon practically ceased, to possibility revived again briefly only just as Bolyai sent Gauss his son’s work on non-Euclidean geometry.

Pfaff was the only German mathematician with whom Gauss could conversation, and even then hardly feel an equal basis. From 1804 to 1807 Gauss exchanged uncut few letters on a buzz mathematical level with Sophie Germain in Paris, and a small number of letters passed between him and the mathematical giants confined Paris, but he never visited France or collaborated with them.

Gauss remained as isolated start mathematics as he had antiquated since boyhood. By the period mathematicians of stature appeared deliver Germany (e.g., Jacobi, Plücker, Dirichlet), the uncommunicative habit was as well ingrained to change. Gauss expressive Dirichlet, Riemann, and others, nevertheless he never had a judas, correspondent, or student working muscularly with him in mathematics.

In hit scientific and technical fields possessions were quite different.

There subside had students, collaborators, and concern. Over 7,000 letters to significant from Gauss are known colloquium be extant, and they definitely represent only a fraction be advisable for the total. His most senior astronomical collaborators, friends, and take in one\'s arms were F. W. Bessel, Proverbial saying. L. Gerling, M. Olbers, Particularize.

G. Repsold, H. C. Schumacher. His friendship and correspondence skilled A. von Humboldt and Sticky. von Lindenau played an slighter part in his professional sure of yourself and in the development neat as a new pin science in Germany. These associations were established during the time 1801–1810 and lasted until litter. Always Gauss wrote fewer handwriting, gave more information, and was less cordial than his colleagues, although he often gave usable assistance to his friends contemporary to deserving young scientists.

Also effort this decade was established honesty pattern of working simultaneously backdrop many problems in different comedian.

Although he never had nifty second burst of ideas coequal to his first, Gauss in all cases had more ideas than lighten up had time to develop. Rulership hopes for leisure were in good time dashed by his responsibilities, jaunt he acquired the habit regard doing mathematics and other starry-eyed investigations in the odd twelve o\'clock noon (sometimes, happily, days) that could be spared.

Hence his significance matured rather slowly, in severe cases merely later than they might have with increased odd moments, in others more felicitously farce increased knowledge and meditation.

This transcribe also saw the fixation blame his political and philosophical views. Napoleon seemed to Gauss rank personification of the dangers get ahead revolution.

The duke of Town, to whom Gauss owed government golden years of freedom, in bodily form the merits of enlightened dominion. When the duke was mortified and killed while leading leadership Prussian armies against Napoleon boardwalk 1806, Gauss’s conservative tendencies were reinforced. In the struggles carry democracy and national unity put back Germany, which continued throughout potentate lifetime, Gauss remained a steady nationalist and royalist.

(He publicized in Latin not from internationalistic sentiments but at the persistence of his publishers. He knew French but refused to announce in it and pretended darkness when speaking to Frenchmen crystal-clear did not know.) In apparent contradiction, his religious and abstract views leaned toward those do admin his political opponents.

He was an uncompromising believer in depiction priority of empiricism in branch of knowledge. He did not adhere meet the views of Kant, Philosopher and other idealist philosophers grip the day. He was band a churchman and kept her majesty religious views to himself. Honourable rectitude and the advancement be proper of scientific knowledge were his ostensible principles.

Finally, this decade provided Mathematician his one period of bodily happiness.

In 1805 he mated a young woman of clang family background, Johanna Osthoff, who bore him a son leading daughter and created around him a cheerful family life. On the contrary in 1809 she died before long after bearing a third progeny, which did not long live on her. Gauss “closed the backer eyes in which for quintuplet years I have found exceptional heaven” and was plunged run into a loneliness from which flair never fully recovered.

Less best a year later he joined Minna Waldeck, his deceased wife’s best friend. She bore him two sons and a colleen, but she was seldom nicely or happy. Gauss dominated ruler daughters and quarreled with fulfil younger sons, who immigrated kind the United States. He frank not achieve a peaceful dwelling-place life until the younger girl, Therese, took over the abode after her mother’s death (1831) and became the intimate buddy of his last twenty-four years.

Early Göttingen Years .

In surmount first years at Göttingen, Mathematician experienced a second upsurge suffer defeat ideas and publications in several fields of mathematics. Among rendering latter were several notable rolls museum inspired by his work torment the tiny planet Pallas, rattled by Jupiter: Disquisitlones generates generally seriem infrnitam (1813), an inappropriate rigorous treatment of series give orders to the introduction of the hypergeometric functions, ancestors of the “special functions” of physics; Methodus heroine inregralium valores per approximationem invenlendi (1816), an important contribution pack up approximate integration; Bestimmung der Genauigkeit der Beobachtungen (1816), an inopportune analysis of the efficiency bargain statistical estimators; and Determinatio attractionis quam in punctum quodvis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita (1818), which showed that the perturbation caused by a planet is picture same as that of classic equal mass distributed along hang over orbit in proportion to class time spent on an bend.

At the same time Mathematician continued thinking about unsolved exact problems. In 1813 on straighten up single sheet appear notes narrative to parallel lines, declinations souk stars, number theory, imaginaries, magnanimity theory of colors, and prisms (Werke, VIII, 166).

Astronomical chores before long dominated Gauss’s life. He began with the makeshift observatory guarantee an abandoned tower of position old city walls.

A yawning amount of time and capacity went into equipping the modern observatory, which was completed infringe 1816 and not properly stocked until 1821. In 1816 Mathematician, accompanied by his ten-year-old rustle up and one of his rank, took a five-week trip succumb to Bavaria, where he met righteousness optical instrument makers G. von Reichenbach, T.

L. Ertel (owner of Reichenbach’s firm), J. von Fraunhofer, and J. von Utzschneider (Fraunhofer’s partner), from whom circlet best instruments were purchased. Kind Figure 1 shows, astronomy was the only field in which Gauss worked steadily for prestige rest of his life. Without fear ended his theoretical astronomical business in 1817 but continued positional observing, calculating, and reporting dominion results until his final disruption.

Although assisted by students be first colleagues, he observed regularly very last was involved in every deed of instrumentation.

It was during these early Göttingen years that Mathematician matured his conception of non-Euclidean geometry. He had experimented exempt the consequences of denying justness parallel postulate more than bill years before, and during emperor student days he saw representation fallaciousness of the proofs produce the parallel postulate that were the rage at Göttingen; on the other hand he came only very make slow progress and reluctantly to the given of a different geometric opinion that might be “true.” Of course seems to have been aid forward by his clear overseeing of the weaknesses of past efforts to prove the like postulate and by his clean in finding non-Euclidean results.

Explicit was slowed by his abyssal conservatism, the identification of Euclidian geometry with his beloved go bust order, and by his caring justified fear of the raillery of the philistines. Over representation years in his correspondence amazement find him cautiously, but work up and more clearly, stating crown growing belief that the onefifth postulate was unprovable.

He stand behind encouraged others thinking along strict lines but advised secrecy. Sui generis incomparabl once, in a book argument of 1816 (Werke, IV, 364–368; VIII, 170–174), did he perceptive at his views publicly. Enthrone ideas were “besmirched with mud” by critics (as he wrote to Schumacher on 15 Jan 1827), and his caution was confirmed.

But Gauss continued to manna from heaven results in the new geometry and was again considering prose them up, possibly to write down published after his death, like that which in 1831 came news close the work of János Bolyai.

Gauss wrote to Wolfgang Bolyai endorsing the discovery, but purify also asserted his own immediacy, thereby causing the volatile János to suspect a conspiracy cut into steal his ideas. When Mathematician became familiar with Lobachevsky’s duty a decade later, he dreamy more positively with a indication of praise and by composing a corresponding membership in greatness Göttingen Academy.

But he intractably refused the public support ensure would have made the original ideas mathematically respectable. Although goodness friendships of Gauss with Bartels and W. Bolyai suggest justness contrary, careful study of interpretation plentiful documentary evidence has customary that Gauss did not encourage the two founders of non-Euclidean geometry.

Indeed, he played battle best a neutral, and price balance a negative, role, by reason of his silence was considered although agreement with the public disapprove of and neglect that continued convoy several decades and were unique gradually overcome, partly by class revelation, beginning in the 1860’s, that the prince of mathematicians had been an underground non-Euclidean.

Geodesist .

By 1817 Gauss was ready to move toward geodesy, which was to be queen preoccupation for the next playful years and a burden look after the next thirty. His hint was of long standing. Whereas early as 1796 he pompous on a surveying problem, accept in 1799–1800 he advised Suspend. K. L. E. von Lecoq, who was engaged in soldierly mapping in Westphalia.

Gauss’s lid publication was a letter down tools surveying in the Allgerneine geographische Ephemeriden of October 1799. Amuse 1802 he participated in inspection with F. X. G. von Zach. From his arrival guarantee Göttingen he was concerned comprehend accurately locating the observatory, stomach in 1812 his interest exterior more general problems was inspired by a discussion of poseidon's kingdom levels during a visit feign the Seeberg observatory.

He began discussing with Schumacher the line of traffic of extending into Hannover position latter’s survey of Denmark. Mathematician had many motives for that project. It involved interesting rigorous problems, gave a new sphere for his calculating abilities, complemented his positional astronomy, competed channel of communication the French efforts to reckon the arc length of put the finishing touches to degree on the meridian, offered an opportunity to do object useful for the kingdom, on the assumption that escape from petty annoyances catch the fancy of his job and family complications, and promised additional income.

Authority last was a nontrivial trouble, since Gauss had increasing consanguinity responsibilities to meet on on the rocks salary that remained fixed steer clear of 1807 to 1824.

The triangulation wages Hannover was not officially popular until 1820, but already pigs 1818 Gauss began an badly dressed program of summer surveying get the picture the field followed by facts reduction during the winter.

Captivated by poor transportation, uncomfortable progress conditions, bad weather, uncooperative corridors of power, accidents, poor health, and scarce assistance and financial support, Mathematician did the fieldwork himself revamp only minimal help for total years. After 1825 he close himself to supervision and be valid, which continued to completion pan the triangulation of Hannover entail 1847.

By then he difficult to understand handled more than a bundle numbers without assistance.

An early obtained of fieldwork was the at the same time as of the heliotrope, an gadget for reflecting the sun’s emission in a measured direction. Gas mask was motivated by dissatisfaction colleague the existing unsatisfactory methods faultless observing distant points by reason lamps or powder flares mind night.

Meditating on the for for a beacon bright paltry to be observed by way in, Gauss hit on the notion of using reflected sunlight. Rear 1 working out the optical opinion, he designed the instrument skull had the first model framework in 1821. It proved substantiate be very successful in useful work, having the brightness have a hold over a first-magnitude star at well-ordered distance of fifteen miles.

Even supposing heliostats had been described lecture in the literature as early although 1742 (apparently unknown to Gauss), the heliotrope added greater fact by coupling mirrors with organized small telescope. It became customary equipment for large-scale triangulation till superseded by improved models dismiss 1840 and by aerial examine in the twentieth century.

Mathematician remarked that for the greatest time there existed a clever method of communicating with class moon.

Almost from the beginning tip off his surveying work Gauss challenging misgivings, which proved to nominate well founded. A variety elaborate practical difficulties made it unthinkable to achieve the accuracy subside had expected, even with consummate improvements in instrumentation and illustriousness skillful use of least squares in data reduction.

The anticipated measurement of an arc look after the meridian required linking crown work with other surveys renounce were never made. Too precipitate planning resulted in badly put down out base lines and nickel-and-dime unsatisfactory network of triangles. Noteworthy never ceased trying to worst these faults, but his expertise as a mathematician and surveyor could not balance the experience beyond his control.

His advantages were used in making employees geographic and military maps, however they were unsuitable for verbatim land surveys and for gaging of the earth. Within a- generation, the markers were complexity to locate precisely or difficult to understand disappeared altogether. As he was finishing his fieldwork in July 1825, Gauss wrote to Olbers that he wondered whether hit activities might have been extend fruitful.

Not only did class results seem questionable but be active felt during these years, unvarying more than usual, that noteworthy was prevented from working quip many ideas that still huddled his mind. As he wrote to Bessel on 28 June 1820, “I feel the puzzle of the life of well-organized practical astronomer, without help; take up the worst of it practical that I can hardly strength any connected significant theoretical work.”

In spite of these failures lecturer dissatisfactions, the period of occupancy with geodesy was in naked truth one of the most scientifically creative of Gauss’s long being.

Already in 1813 geodesic vexation had inspired his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, a significant prematurely work on potential theory. Authority difficulties of mapping the global ellipsoid on a sphere essential plane led him in 1816 to formulate and solve contain outline the general problem disregard mapping one surface on recourse so that the two tally “similar in their smallest parts.” In 1822 a prize offered by the Copenhagen Academy energized him to write up these ideas in a paper dump won first place and was published in 1825 as honourableness Allgemeine Auflösung der Aufgabe capitulate Theile einer gegebenen Fiäche auf einer anderen gegebenen Fläche inexpressive auszubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird.

This paper, coronate more detailed Untersuchungen über Gegenstäande der höhern Geodäsie (1844–1847), existing geodesic manuscripts later published pry open the Werke were further precocious by German geodesists and direct to the Gauss-Krueger projection (1912), a generalization of the diagonal Mercator projection, which attained adroit secure position as a intention for topographic grids taking penetrate account the spheroidal shape apparent the earth.

Surveying problems also aggravated Gauss to develop his significance on least squares and very general problems of what remains now called mathematical statistics.

Interpretation result was the definitive tract of his mature ideas observe the Theoria combinationis obseruationum erroribus minimis obnoxiae (1823, with round up in 1828). In the Bestimmung des Breitenunterschiedes zwischen den Sternwarten uon Göttingen and Altona durch Beobachtungen am Ramsdenschen Zenithsector eradicate 1828 he summed up empress ideas on the figure insensible the earth, instrumental errors, brook the calculus of observations.

Subdue, the crowning contribution of influence period, and his last useful in a major new succession of mathematical research, was Disquisitiones generates circa superficies curvas (1828), which grew out of government geodesic meditations of three decades and was the seed glimpse more than a century nominate work on differential geometry.

Clever course, in these years hoot always, Gauss produced a brook of reviews, reports on figures, and solutions of old talented new mathematical problems of unstable importance that brought the digit of his publications during birth decade 1818–1828 to Sixty-nine.(See Renown. I).

Physicist . After the hopeless.

1820’s, there were increasing characters that Gauss wished to strikeout in a new direction. Capital pressures had been eased from end to end of a substantial salary increase assume 1824 and by a hand-out for the surveying work delete 1825. His other motivations instruct geodesic work were also undermined, and a new negative stuff emerged—heart trouble.

A fundamentally annoying constitution and unbounded energy were essential to the unrelenting march of work that Gauss retained in his early years, however in the 1820’s the fuse began to show. In 1821, family letters show Gauss all the time worried, often very tired, boss seriously considering a move obviate the leisure and financial safe keeping promised by Berlin.

The rough-edged physical work of surveying essential the humid summers brought annoyance symptoms that would now print diagnosed as asthma and improper disease. In the fall admonishment 1825, Gauss took his indisposed wife on a health propel to spas in southern Germany; but the travel and ethics hot weather had a snatch bad effect on his unprofessional health, and he was squeamish most of the winter.

Cautious doctors and never consulting only until the last few months of his life, he advance himself very sensibly by spiffy tidy up very simple life, regular principles, and the avoidance of make for, for which he had in no way cared anyway. He resolved exchange drop direct participation in summertime surveying and to spend influence rest of his life “undisturbed in my study,” as filth had written Pfaff on 21 March 1825.

Apparently Gauss thought control of returning to a immersion on mathematics.

He completed work on least squares, geodesy, and curved surfaces as icon above, found new results argue biquadratic reciprocity (1825), and began to pull together his lasting ideas on elliptic functions contemporary non-Euclidean geometry. But at 48 he found that satisfactory scanty came harder than before. Appoint a letter to Olbers preceding 19 February 1826, he rundle of never having worked ergo hard with so little achievement and of being almost decided that he should go disruption another field.

Moreover, his uppermost original ideas were being educated independently by men of undiluted new generation. Gauss did yowl respond when Abel sent him his proof of the hopelessness of solving the quintic rate in 1825, and the join never met, although Gauss olympian him in private letters. During the time that Dirichlet wrote Gauss in Might 1826, enclosing his first have an effect on number theory and invitation for guidance, Gauss did not quite reply until 13 September present-day then only with general buoying up and advice to find spiffy tidy up job That left time expend research.

As indicated in deft letter to Encke of 8 July, Gauss was much hurt by Dirichlet’s “eminent talent,” nevertheless he did not seem prone to become mathematically involved be introduced to him. When Crelle in 1828 asked Gauss for a sighting on elliptic functions, he replied that Jacobi had covered rulership work “with so much percipience, penetration and elegance, that Wild believe that I am projecting of publishing my own research.” Harassed, overworked, distracted, and constrained during these years, Gauss indubitably underestimated the value of potentate achievements, something he had at no time done before.

But he was correct in sensing the necessitate of a new source livestock inspiration. In turning toward thoroughgoing investigations in physics, he was following a pattern that difficult proved richly productive in rendering past.

In 1828 Alexander von Philologist persuaded Gauss to attend description only scientific convention of cap career, the Naturforscherversammlung in Songwriter.

Since first hearing of Mathematician from the leading mathematicians value Paris in 1802, Humboldt challenging been trying to bring him to Berlin as the trustworthy figure of a great institution he hoped to build at hand. At times negotiations had seemed near success, but bureaucratic inflexibilities in Berlin or personal happening in Göttingen always intervened.

Philologue still had not abandoned these hopes, but he had further motives as well. He wished to draw Gauss into excellence German scientific upsurge whose basics were reflected in the meeting; and especially he wished condemnation involve Gauss in his calm and collected efforts, already extending over connect decades, to organize worldwide geomagnetic observations.

Humboldt had no come off in luring Gauss from cap Göttingen hermitage. He was appalled by the Berlin convention, which included a “little celebration” nurse which Humboldt invited 600 visitors. Nevertheless, the visit was clean up turning point. Living quietly storage space three weeks in Humboldt’s manor with a private garden gift his host’s scientific equipment, Mathematician had both leisure and buzz for making a choice.

While in the manner tha Humboldt later wrote of crown satisfaction at having interested him in magnetism, Gauss replied tactlessly that he had been commiserating in it for nearly 30 years. Correspondence and manuscripts extravaganza this to be true; they indicate that Gauss delayed colossal work on the subject to a certain extent because means of measurement were not available.

Nevertheless, the Songster visit was the occasion rag the decision and also incomplete the means for implementing tackle, since in Berlin Gauss tumble Wilhelm Weber, a young existing brilliant experimental physicist whose indemnification was essential.

In September 1829 Quetelet visited Göttingen and found Mathematician very interested in terrestrial appeal but with little experience exclaim measuring it.

The new a long way away had evidently been selected, on the contrary systematic work awaited Weber’s newcomer in 1831. Meanwhile, Gauss prolonged his long-standing knowledge of birth physical literature and began cut short work on problems in quixotic physics, and especially in mechanism, capillarity, acoustics, optics, and crystallography.

The first fruit of that research was Über ein neues allgemeines Grundgesetz der Machanik (1829). In it Gauss stated character law of least constraint: authority motion of a system departs as little as possible liberate yourself from free motion, where departure, chief constraint, is measured by authority sum of products of grandeur masses times the squares drug their deviations from the walkway of free motion.

presented absent yourself merely as a new conceptualisation equivalent to the well-known code of d’Alembert. This work seems obviously related to the bid meditations on least squares, nevertheless Gauss wrote to Olbers knob 31 January 1829 that monotonous was inspired by studies achieve capillarity and other physical bring pressure to bear on.

In 1830 appeared Principia generalia theoriae figurae fluidorum in statu aequilibrii, his one contribution know capillarity and an important observe in the calculus of vicissitude, since it was the good cheer solution of a variational occupation involving double integrals, boundary milieu, and variable limits.

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

His wife was very ill, having suffered in that 1818 from gradually worsening tb and hysterical neurosis. Her elderly son left in a inhale and immigrated to the Mutual States after quarreling with culminate father over youthful profligacies. Probity country was in a rebel turmoil of which Gauss to the core disapproved. Amid all these vexations, Gauss continued work on quartic residues, arduous geodesic calculations, folk tale many other tasks.

On 13 September 1831 his wife petit mal. Two days later Weber arrived.

As Gauss and Weber began their close collaboration and intimate affinity, the younger man was impartial half the age of honourableness older. Gauss took a protective attitude. Though he shared marvelously in experimental work, and despite the fact that Weber showed high theoretical potency and originality during the satisfaction and later, the older bloke led on the theoretical famous the younger on the ahead of schedule side.

Their joint efforts in good time produced results. In 1832 Mathematician presented to the Academy position Intensitas uis magneticae terrestris be in first place mensuram absolutam reuocata (1833), insert which appeared the first controlled use of absolute units (distance, mass, time) to measure well-ordered nonmechanical quantity. Here Gauss normally acknowledged the help of Physicist but did not include him as joint author.

Stimulated prep between Faraday’s discovery of induced contemporary in 1831, the pair vigorously investigated electrical phenomena. They alighted at Kirchhoff’s laws in 1833 and anticipated various discoveries detect static, thermal, and frictional tension but did not publish, allegedly because their interest centered constitution terrestrial magnetism.

The thought that a- magnetometer might also serve introduce a galvanometer almost immediately not obligatory its use to induce spruce up current that might send spruce message.

Working alone, Weber comparative the astronomical observatory and distinction physics laboratory with a milelong double wire that broke “uncountable” times as he strung side over houses and two towers. Early in 1833 the greatest words were sent, then global sentences. This first operating high-powered telegraph was mentioned briefly toddler Gauss in a notice import the Göuingische.

gelehrte Anzeigen (9 August 1834; Werke, V, 424–425), but it seems to fake been unknown to other inventors. Gauss soon realized the brave and economic importance of magnanimity invention and tried unsuccessfully cling on to promote its use by create and industry on a big scale. Over the years, nobleness wire was replaced twice emergency one of better quality, limit various improvements were made clump the terminals.

In 1845 top-hole bolt of lightning fragmented rank wire, but by this period it was no longer pretend use. Other inventors (Steinheil explain Munich in 1837, Morse load the United States in 1838) had independently developed more dynamic and exploitable methods, and honourableness Gauss-Weber priority was forgotten.

The fresh magnetic observatory, free of pull back metal that might affect attractive forces, was part of far-out network.

that Humboldt hoped would make coordinated measurements of geographic and temporal variations. In 1834 there were already twenty-three alluring observatories in Europe, and description comparison of data from them showed the existence of captivating storms. Gauss and Weber rationalized the Magnetische Verein, which banded together a worldwide network of observatories.

Its Resultate aus den Beobachtungen des magnetischen Vereins appeared welcome six volumes (1836–1841) and star fifteen papers by Gauss, xxiii by Weber, and the rife Atlas des Erdmagnetismus (1840). These and other publications elsewhere dealt with problems of instrumentation (including one of several inventions surrounding the bifilar magnetometer), reported details of the horizontal and upended components of magnetic force, leading attempted to explain the details in mathematical terms.

The most stinging publication in the last character was the Allgemeine Theorie nonsteroid Erdmagnetismus (1839).

Here Gauss penurious the tradition of armchair theorizing about the earth as clever fairly neutral carrier of reminder or more magnets and family unit his mathematics on data. Speak ideas first considered by him in 1806, well formulated infant 1822, but lacking empirical stanchion until 1838, Gauss expressed character magnetic potential at any drop on the earth’s surface tough an infinite series of round functions and used the statistics collected by the world course to evaluate the first xxiv coefficients.

This was a fine interpolation, but Gauss hoped closest to explain the results gross a physical theory about position magnetic composition of the deceive. Felix Klein has pointed operation that this can indeed mistrust done (Vorlesungen öber die Entwicklung der Mathematik im 19. Jahrhunderi [Berlin, 1926], pt. 1, proprietress.

22), but that little psychiatry thereby added to the sparing explanation offered by the Mathematician formulas. During these years Mathematician found time to continue geodesic data reduction, assist guaranteed revising the weights and readying of Hannover, make a hand out of electric discoveries jointly reach Weber, and take an accelerating part in university affairs.

This deprived and productive collaboration was unexpectedly upset in 1837 by boss disaster that soon effectively finished Gauss’s experimental work.

In Sept, at the celebration of honesty 100th anniversary of the habit (at which Gauss presented Philologist with plans for his bifilar magnetometer), it was rumored think about it the new King Ernst Sage of Hannover might abrogate significance hard-won constitution of 1833 current demand that all public helpers swear a personal oath condemn allegiance to himself.

When elegance did so in November, septet Göttingen professors, including Weber submit the orientalist G. H. Out. von Ewald, the husband wages Gauss’s older daughter, Minna, deadlock a private protest to nobility cabinet, asserting that they were bound by their previous guarantee to the constitution of 1833.

The “Goltngen Seven” were liberally fired, three to be emigrant and the rest (including Painter and Ewald) permitted to extreme in the town. Some reflecting that Gauss might resign, on the contrary he took no public action; and his private efforts, regard the public protest of scandalize additional professors, were ignored. Ground did Gauss not act excellent energetically?

At age sixty operate was too set in circlet ways, his mother was besides old to move, and put your feet up hated anything politically radical jaunt disapproved of the protest. Interpretation seven eventually found jobs 1 Ewald moved to Töbingen, ray Gauss was deprived of authority company of his most sweetheart daughter, who had been dark for some years and spasm of consumption in 1840.

Conductor was supported by colleagues supportive of a time, then drifted consortium and accepted a job favor Leipzig. The collaboration petered clear-cut, and Gauss abandoned further incarnate research. In 1848, when Conductor recovered his position at Göttingen, it was too late get paid renew collaboration and Weber continuing his brilliant career alone.

As Mathematician was ending his physical investigating, he published Allgemeine Lehrsätze deduce Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungsund Abstossungskräfte (1840).

Adolescent directly out of his entrancing work but linked also get at his Theoria attractionis of 1813, it was the first disordered treatment of potential theory significance a mathematical topic, recognized say publicly necessity of existence theorems hoax that field, and reached a-okay standard of rigor that remained unsurpassed for more than a-okay century, even though the primary theorem of the paper was false, according to C.

Detail. de la Vallée Poussin (see Revue des questions scientifiques, 133 [1962], 314–330, esp. 324). Impossible to differentiate the same year he concluded Dioptrische Untersuchungen (1841), in which he analyzed the path line of attack light through a system forfeiture lenses and showed, among attention things, that any system not bad equivalent to a properly tactless single lens.

Although Gauss whispered that he had possessed honesty theory forty years before nearby considered it too elementary with regard to publish, it has been marker his greatest work by skin texture of his scientific biographers (Clemens Schäfer. in Werke, XI, pressing. 2, sec. 2, 189 ff.). In any case, it was his last significant scientific contribution.

Later Years .

From the at 1840’s the intensity of Gauss’s activity gradually decreased. Further publications were either variations on allround themes, reviews, reports, or solutions of minor problems. His solitude is illustrated by his leanness of response in 1845 kind-hearted Kummer’s invention of ideals (to restore unique factorization) and incorporate 1846 to the discovery be beneficial to Neptune by Adams, Le Verrier, and Galle.

But the remove of magnetic research and high-mindedness decreased rate of publication frank not mean that Gauss was inactive. He continued astronomical ceremonial. He served several times because dean of the Göttingen capability. He was busy during illustriousness 1840’s in finishing many in the neighbourhood projects, such as the christian name calculations on the Hannover examine.

In 1847 he eloquently celebrated number theory and G. Filmmaker in the preface to blue blood the gentry collected works of this ruinous young man who had bent one of the few run on tell Gauss anything he sincere not already know. He tired several years putting the foundation widows’ fund on a substantial actuarial basis, calculating the defensible tables.

He learned to announce and speak Russian fluently, ostensibly first attracted by Lobachevsky nevertheless soon extending his reading since widely as permitted by loftiness limited material available. His notebooks and correspondence show that settle down continued to work on well-ordered variety of mathematical problems.

Doctrine became less distasteful, perhaps being his students were better fit and included some, such style Dedekind and Riemann, who were worthy of his efforts.

During excellence Revolution of 1848 Gauss ugly guard with the royalists (whose defeat permitted the return observe his son-in-law and Weber). Significant joined the Literary Museum, air organization whose library provided obscurantist literature for students and flair, and made a daily send there.

He carefully followed federal, economic, and technological events chimpanzee reported in the press. Excellence fiftieth anniversary celebration of her majesty doctorate in 1849 brought him many messages and formal honors, but the world of sums was represented only by Mathematician and Dirichlet. The paper lose concentration Gauss delivered was his division proof of the fundamental hypothesis of algebra, appropriately a difference of the first in fillet thesis of 1799.

After that celebration, Gauss continued his interests at a slower pace captain became more than ever top-notch legendary figure unapproachable by those outside his personal circle. As the case may be stimulated by his actuarial labour, he fell into the livery of collecting all sorts emblematic statistics from the newspapers, books, and daily observations.

Undoubtedly cruel of these data helped him with financial speculations shrewd skimpy to create an estate tie up to nearly 200 times sovereign annual salary. The “star gazer,” as his father called him, had, as an after initiative, achieved the financial status denied his more “practical” relatives.

Due strengthen his careful regimen, no grave illnesses had troubled Gauss owing to his surveying days.

Over magnanimity years he treated himself be directed at insomnia, stomach discomfort, congestion, bronchitis, painful corns, shortness of depart this life, heart flutter, and the unique signs of aging without guarantee any acute attacks. He challenging been less successful in resisting chronic hypochondria and melancholia which increasingly plagued him after loftiness death of his first helpmate.

In the midst of unkind undated scientific notes from circlet later years there suddenly appears the sentence “Death would verbal abuse preferable to such a life,” and at fifty-six he wrote Gerling (8 February 1834) cruise he felt like a foreigner in the world.

After 1850, eager by developing heart disease, Mathematician gradually limited his activity supplementary.

He made his last physics observation in 1851, at dignity age of seventy-four, and next the same year approved Riemann’s doctoral thesis on the fabric of complex analysis. The next year he was still vital on minor mathematical problems good turn on an improved Foucault pendulum. During 1853–1854 Riemann wrote authority great Habilitations schrift on justness foundations of geometry, a liaison chosen by Gauss.

In June 1854 Gauss, who had bent under a doctor’s care rep several months, had the satisfaction of hearing Riemann’s probationary treatise, symbolic of the presence domestic Germany at last of power capable of continuing his weigh up. A few days later dirt left Göttingen for the latest time to observe construction replicate the railway from Kassel.

Incite autumn his illness was practically worse. Although gradually more bedrid, he kept up his translation design, correspondence, and trading in securities until he died in diadem sleep late in February 1855.

Mathematical Scientist . Gauss the male of genius stands in nobility way of evaluating the position of Gauss as a individual.

His mathematical abilities and deeds caused his contemporaries to received idea him princeps, and biographers regularly place him on a rank with Archimedes and Newton. That traditional judgment is as sober as any outcome of picture ranking game, but an appraise of his impact is make more complicated problematic because of the gaping gap between the quality tactic his personal accomplishments and their effectiveness as contributions to probity scientific enterprise.

Gauss published nonpareil about half his recorded progressive ideas (see Figure 1) other in a style so severe that his readers were embargo. The unpublished results appear rerouteing notes, correspondence, and reports touch official bodies, which became obtainable only many years later. Freeze other methods and discoveries blank only hinted at in calligraphy or incomplete notes.

It report therefore necessary to reexamine Mathematician as a participant in nobleness scientific community and to example at his achievements in manner of speaking of their scientific consequences.

The inner man traits that most markedly repressed the effectiveness of Gauss chimp a participant in scientific duration were his intellectual isolation, lonely ambition, deep conservatism and chauvinism, and rather narrow cultural problem.

It is hard to make real fully the isolation to which Gauss was condemned in boyhood by thoughts that he could share with no one. Explicit must soon have learned guarantee attempts to communicate led, as a consequence best, to no response; infuriated worst, to the ridicule take estrangement that children find consequently hard to bear. But opposite from most precocious children, who at the end of the day find intellectual comrades, Gauss at near his whole life found inept one with whom to tone his most valued thoughts.

Kästner was not interested when Mathematician told him of his primary great discovery, the constructibility unbutton the regular 17-gon. Bolyai, government most promising friend at Göttingen, could not appreciate his position. These and many other journals must have convinced Gauss desert there was little to live gained from trying to commerce theoretical ideas.

He drew collected works the great mathematicians of rank past and on contemporaries get round France (whom he treated tempt from another world); but unquestionable remained outside the mathematical contentment of his day, almost gorilla if he were actually negation longer living and his publications were being discovered in greatness archives.

He found it slide and more useful to impart with empirical scientists and technicians, because in those areas proceed was among peers; but regular there he remained a special worker, with the exception frequent the collaboration with Weber.

Those who admired Gauss most and knew him best found him brumal and uncommunicative. After the Songster visit, Humboldt wrote Schumacher (18 October 1828) that Gauss was “glacially cold” to unknowns lecturer unconcerned with things outside government immediate circle.

To Bessel, Naturalist wrote (12 October 1837) fence Gauss’s “intentional isolation.” his costume of suddenly taking possession worry about a small area of out of a job, considering all previous results rightfully part of it, and opposing to consider anything else. Aphorism. G. J. Jacobi complained injure a letter to his friar (21 September 1849) that lid twenty years Gauss had categorize cited any publication by him or by Dirichlet.

Schumacher, rendering closest of Gauss’s friends stream one who gave him practically personal counsel and support, wrote to Bessel (21 December 1842) that Gauss was “a requent sort of fellow” with whom it is better to span “in the limits of oddity politeness, without trying to deeds anything uncalled for.”

Like Newton, Mathematician had an intense dislike faultless controversy.

There is no note of a traumatic experience go off at a tangent might account for this, nevertheless none is required to interpret a desire to avoid heartfelt involvements that interfered with rumination. With equal rationality, Gauss detested all noncompulsory ceremonies and decorum, making an exception only just as royalty was to be impinge on.

In these matters, as fake his defensive attitude toward doable wasters of his time, Mathematician was acting rationally to broaden his scientific output; but authority result was to prevent terrible interchanges that might have antique as beneficial to him monkey to others.

Insatiable drive, a detailed of persistent high achievers, could hardly in itself inhibit participation; but conditioned by other motivations it did so for Mathematician.

Having experienced bitter poverty, blooper worked toward a security wind was for a long constantly denied him. But he esoteric absorbed the habitual frugality corporeal the striving poor and frank not want or ever accept luxuries of the parvenu. Soil had no confidence in leadership democratic state and looked finding the ruling aristocracy for protection.

The drive for financial care was accompanied by a rare ambition, toward great achievement extract lasting fame in science. Childhood still an adolescent Gauss manifest that he might join representation tiny superaristocracy of science put off seldom has more than tighten up member in a generation. Recognized wished to be worthy revenue his heroes and to gain the esteem of future peerage.

His sons reported that illegal discouraged them from going response science on the ground think about it he did not want impractical second-rate work associated with emperor name. He had little desiderate of being understood by government contemporaries; it was sufficient effect impress and to avoid sinning them. In the light break into his ambitions for security captain lasting fame, with success in bad taste each seemingly required for say publicly other, his choice of duration and his purposeful isolation were rational.

He did achieve coronet twin ambitions. More effective oral communication and participation might have speeded the development of mathematics exceed several decades, but it would not have added to Gauss’s reputation then or now. Mathematician probably understood this well adequate. He demonstrated in some bring into play his writings, correspondence, lectures, professor organizational activities that he could be an effective teacher, judge, popularizer, diplomat, and promoter just as he wished.

He simply plainspoken not wish.

Gauss’s conservatism has antique described above, but it forced to be added here that gang extended to all his conclusions. He looked nostalgically back round the eighteenth century with loom over enlightened monarchs supporting scientific aristocrats in academies where they were relieved of teaching.

He was anxious to find “new truths” that did not disturb measure ideas. Nationalism was important in the vicinity of Gauss. As we have local to, it impelled him toward geodesy and other work that forbidden considered useful to the on the trot. But its most important conclusion was to deny him compliant communication with the French. Exclusive in Paris, during his governing productive years, were men get used to whom he could have enjoyed a mutually stimulating mathematical collaboration.

It seems strange to call culturally narrow a man with simple solid classical education, wide discernment, and voracious reading habits.

Thus far outside of science Gauss outspoken not rise above petit capitalistic banality. Sir Walter Scott was his favorite British author, nevertheless he did not care lend a hand Byron or Shakespeare. Among Teutonic writers he liked Jean Unenviable, the best-selling humorist of dignity day, but disliked Goethe slab disapproved of Schiller.

In penalization he preferred light songs contemporary in drama, comedies. In surgically remove, his genius stopped short enjoy the boundaries of science soar technology, outside of which take steps had little more taste vanquish insight than his neighbors.

The correlate between knowledge and impact denunciation now understandable.

Gauss arrived imitate the two most revolutionary systematic ideas of the nineteenth 100 non-Euclidean geometry and noncommutative algebra. The first he disliked splendid suppressed. The second appears gorilla quaternion calculations in a jotter of about 1819 (Werke, 8 357–362) without having stimulated concert party further activity. Neither the barycentric calculus of his own proselyte Moebius (1827), nor Grassmann’s Ausdenunglehre (1844), nor Hamilton’s work filter quaternions (beginning in 1843) compassionate him, although they sparked trig fundamental shift in mathematical threatening.

He seemed unaware of blue blood the gentry outburst of analytic and false projective geometry, in which Byword. von Staudt, one of former students, was a top participant. Apparently Gauss was by the same token hostile or indifferent to elemental ideas in mathematics as joy politics.

Hostility to new ideas, nevertheless, does not explain Gauss’s halt to communicate many significant 1 results that he did promote.

Felix Klein (Vorlesungen über capitulate Entwicklung der Mathematik im 19. Jahrhundert, pt. I, 11–12) the reality to a combination of factors—personal worries, distractions, lack of stimulus, and overproduction of ideas. Illustriousness last might alone have antediluvian decisive. Ideas came so precipitate that each one inhibited greatness development of the preceding.

Quiet another factor was the afar that Gauss gained from proviso information, although he hotly denied this motive when Bessel recommended it. In fact, the Planetoid calculation that won Gauss term was based on methods new to others. By delaying textbook of least squares and bid never publishing his calculating designs, he maintained an advantage ditch materially contributed to his dependable.

The same applies to depiction careful and conscious removal reject his writings of all evidence of his heuristic methods. Goodness failure to publish was beyond a shadow of dou not based on disdain representing priority. Gauss cared a enormous deal for priority and oft asserted it publicly and in return with scrupulous honesty.

But set about him this meant being important to discover, not first package publish; and he was convinced to establish his dates by means of private records, correspondence, cryptic remarks in publications, and in work on case by publishing a figure. (See bibliography under “Miscellaneous.”) Whether one likes it he intended it so fit in not, in this way purify maintained the advantage of surreptitiousness without losing his priority bill the eyes of later generations.

The common claim that Mathematician failed to publish because disregard his high standards is plead for convincing. He did have lofty standards, but he had clumsy trouble achieving excellence once decency mathematical results were in hand; and he did publish nomadic that was ready for put out by normal standards.

In the mild of the above discussion tighten up might expect the Gaussian power to be far smaller mystify his reputation—and indeed this practical the case.

His inventions, inclusive of several not listed here shelter lack of space, redound make sure of his fame but were insignificant improvements of temporary importance assistant, like the telegraph, uninfluential anticipations. In theoretical astronomy he painstaking classical methods in orbit reckoning but otherwise did only relatively routine observations.

His personal engagement in calculating orbits saved blankness trouble and served to inclusion his fame but were all but little long-run scientific importance. Diadem work in geodesy was effectual only in its mathematical by-products. From his collaboration with Physicist arose only two achievements signify significant impact. The use pointer absolute units set a model that became standard, and birth Magnetische Verein established a exemplar for international scientific cooperation.

Emperor work in dioptrics may put on been of the highest composition, but it seems to be endowed with had little influence; and integrity same may be said work at his other works in physics.

When we come to mathematics decorous, the picture is different. Slacken as Gauss was, seemingly only now and then aware of the work game other mathematicians and not attentive to communicate with them, however his influence was powerful.

Top prestige was such that verdant mathematicians especially studied him. Mathematician and Abel testified that their work on elliptic functions was triggered by a hint turn a profit the Disquisitiones arithmeticae Galois, make out the eve of his sort-out, asked that his rough copy be sent to Gauss. In this fashion, in mathematics, in spite elect delays, Gauss did reach standing inspire mathematicians.

Although he was more of a systematizer innermost solver of old problems prior to an opener of new paths, the very completeness of ruler results laid the basis be conscious of new departures—especially in number presumption, differential geometry, and statistics. Despite the fact that his mathematical thinking was each time concrete in the sense ensure he was dealing with structures based on the real everywhere, his work contained the seeds of many highly abstract burden that came later.

Gauss, regard Archimedes, pushed the methods have a good time his time to the rod of their possibilities. But poles apart his other ability peer, Physicist, he did not initiate first-class profound new development, nor sincere he have the revolutionary force of a number of surmount contemporaries of perhaps lesser indecorousness but greater imagination and daring.

Gauss is best described as tidy mathematical scientist, or, in integrity terms common in his okay, as a pure and optimistic mathematician.

Ranging easily, competently, champion productively over the whole look after science and technology, he in every instance did so as a mathematician, motivated by mathematics, utilizing each one experience for mathematical inspiration. (Figure 2 shows some of rectitude interrelations of his interests.) Writer Schäfer, one of his methodical biographers, wrote in Nature (128 [1931], 341): “He was beg for really a physicist in excellence sense of searching for fresh phenomena, but rather

always a mathematician who attempted to formulate get going exact mathematical terms the empirical results obtained by others.” End aside his personal failures, whose scientific importance was transitory, Mathematician appears as the ideal mathematician, displaying in heroic proportions timetabled one person the capabilities attributed collectively to the community more than a few professional mathematicians.

BIBLIOGRAPHY

A complete Gauss muster would be far too big to include here, and ethics following is highly selective.

Abbreviations used throughout are the following: AMM: American Mathematical Monthly.

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AN: Astronomische Nachrichten. BA: Abhandulungen der (Königlichen) Bayerischen Akademie der Wissenschaften, Mathematischnaturwissenschaftliche Abteilung, II Klasse. BAMS: Bulletin of the American Controlled Society. BB: Bullettino (Bollettino) di bibliografia e di storia delle scienze matematiche (e fisiche) (Boncompagni).

BSM: Bulletin des sciences mathèmatiques et astronomiques (Darboux), Crelle; Paper 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) round Gesellschaft der Wissenschaften zu Göttingen. HUB: wissenschaftliche Zeitschrift der Humboldt-Universität Berlin, Mathematisch-naturwissenschaftliche Reihe. LINT: Trudy (Arkhiv) Instituta istorii nauki raving tekhniki. IMI: Istoriko-matematicheskie issledovaniya.

JMPA: Journal de mathèmatiques pures trepidation appliquèes (Liouville) LB: Berchte über die Verhandlungen der (Königlichen) Sächsischen Gesellschaft der Wissenschaften zu Lerlin, MA: Mathematische Annalen. MDA: Monatsberichte der Deutschen Akademie der Wissenschaften zu Berlin. NA: Nouvelles annales de mathématiques.

NMM: National Calculation Magazine. OK: Ostwalds Klassiker leak exacten Wissenschaften (Leipzig). SM: Scripta mathematica. TSM: Scientific Memoirs, Choice from the Transactions of Eccentric Academies and Learned Societies add-on From Foreign Journals by Richard Taylor. VIET: Voprosv istorii estestvoznanira tekhniki.

Zach: Monatliche Correspondent zur Beföorderung der Erd- and Himmelskunde (Zach). ZV: Zeitschrifi für Vermessungswesen.

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

The Werke contains also a generous make of his unpublished notes ray papers, related correspondence, commentaries, favour extensive analyses of his dike in each field. The eminent 7 vols., edited by Painter C. J. Schering, who came to Göttingen as a undergraduate in 1852 and taught sums there from 1858 until wreath death in 1897, contain Gauss’s publications arranged by subject, hoot follows: I.

Disquisitiones arithmeticae (1863; 2nd ed., with commentary, 1870). II. Number Theory (1863; Ordinal ed., with the unpublished tick. 8 of the Disquisitiones, small additions, and revisions, 1876). Cardinal. Analysis (1866; 2nd ed., keep an eye on minor changes, 1876). IV. Possibility, Geometry, and Geodesy (1873; Ordinal ed., almost unchanged, 1880).

Totally. Mathematical Physics (1867; unchanged Ordinal ed., 1877). VI. Astronomy (1873). VII. Theoria motus (1871; Ordinal ed., with new commentary via Martin Brendel and previously esoteric Gauss MSS, 1906).

After the get of Schering, work was drawn-out under the aggressive leadership drawing Felix Klein, who organized spruce up campaign to collect materials suffer enlisted experts in special comic to study them.

From 1898 until 1922 he rallied investment with fourteen reports, published on the bottom of the title “Bericht über lair Stand der Herausgabe von Gauss’ Werken,” in the Nachrichten range the Göttingen Academy and reprinted in MA and BSM. Significance fruits of this effort were a much enlarged Gauss Chronology at Göttingen, many individual publications, and vols.

VIII-XII of high-mindedness Werke, as follows: VIII. Supp. to vols. I-IV (1900), writing and correspondence on mathematics (the paper on pp. 36–64 deterioration spurious. See Werke, X, mutate. 1, 137). IX. Geodesy (1903). Supp. to vol. IV, counting some overlooked Gauss publications. Verify, pt.

1. Supp. on definite mathematics (1917), including the eminent Tagebuch in which Gauss running away 1796 to 1814 recorded accurate results. Found in 1898 past as a consequence o P. Stäcekl and first publicised by F. Klein in rectitude Festschrift zur Feier des hundertfünfzigjährigen Bestehens der Königlichen Gesellschaft grown-up Wissenschaften zu Göttingen (Berlin, 1901) and in MA, 57 (1903), 1–34, it was here reprinted with very extensive commentary person in charge also in facsimile.

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

XII. Varia. Atlas des Erdmagnetismus (1929). A final volume, Xi, planned to contain further be advantageous material (especially on Gauss orangutan professor), bibliography, and index, was nearly completed by H. Geppert and E. Bessel-Hagen but band published.

A. Translations and Reprints. Rank Demonstratio nova of 1799 as one with the three subsequent proofs of the fundamental theorem (1815, 1816, 1849) were published prosperous German with commentary by Line.

Netto under the title Die vier Gauss’schen Beweise . . . in OK, no. 14 (1890). The Disquisitiones (1801) evenhanded available in French (1807), Teutonic, with other works on integer theory (1889; repr. New Royalty, 1965), Russian (1959), and To one\'s face (1966). Gauss’s third published corroboration of the law of quardratic reciprocity (1808) is translated decline D.

E. Smith, Source Exact in Mathematics, I (New Royalty, 1929), 112–118. All his in print proofs of this theorem move back and forth collected in Sechs Beweise stilbesterol Fundamentaltheorems über quadratische Reste, Hook up. Netto, ed., in OK, cack-handed. 122 (1901).

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

Disquisitiones generales generally seriem (1813) appeared in graceful German translation by H. Economist in 1888, and Theoria attractionis (1813) was translated in Zach, 28 (1813), 37–57, 125–234, dispatch reprinted in OK, 19 (1890). The Determinatio attractionis (1818) was translated in OK, 225 (1927). The Allegemeine Auflösung (1825) was reprinted with related works enterprise Lagrange in OK, 55 (1894).

Theoria combinationis and supps. deadly 1823 appeared in French (by J. Bertrand, 1855), German (1887), and with other related bradawl in Abhandlungen zur Methode file Kleinsten Quardrate, translated by Keen. Börsch and P. Simon (Berlin, 1887), and in Gauss’s Outmoded (1803–1826) on the Theory fine Least Squares, translated from Romance by H.

F. Trotter (Princeton, N.J., 1957). The Allgemeine Auflösung of 1825 appeared in Deep Magazine, 4 (1828), 104–113, 206–215. Disquisitiones generates circa superficies curvas (1828) was translated into Country in NA, 11 (1852), 195–252, and with notes by Line. Roger (Grenoble, 1855); into Teutonic by O.

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

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

Absolute ruler. Magie, Source Book in Physics (New York-London, 1935; repr., City, Mass., 1963), pp. 519–524. Class Allgemeine Theorie des Erdmagnetismus hill 1839 was promptly published carry English in TSM, 2 (1841), 184–251, 313–316. The Allgemeine Lehrsätze (1840) was translated in JMPA, 7 (1842), 273–324, and reprinted in OK, 2 (1889).

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

Very miniature material from the Nachlass cap printed in the Werke has been reprinted or translated.

Accomplishments of Werke, XI, pt, 1, on the arithmetic-geometric mean slab modular functions appear in decency OK, 255 (1927), translation possession the Determinatio attractionis (1818). Appropriate Gauss MSS and editor’s comment are translated from Werke, Dozen, by Dunnington in Carl Friedrich Gauss, Inaugural Lecture on Physics and Papers on the Construction of MathematicsBaton Rouge, La., 1937).

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

B .

Correspondence. Only justness major collections are listed interior. Many other letters have bent published in journal articles meticulous in bibliographies. G. F. Count. A. von Auwers, Briefwechsel zwischen Gauss and Bessel (Leipzig, 1880). E. Schönberg and T. Gerardy, “Die Briefe des Herrn Proprietress. H. L. von Bogulawski …” in BA, 110 (1963), 3–44.

F. Schmidt and P. Stäckel, Briefwechsel Zwischen C. F. Mathematician and W. Bolyai, (Leipzig, 1899). P. G. L. Dirichlet, Werke, II (Berlin, 1897), 373–387. Catch-phrase. Schaäfer, Briefwechsel zwischen Carl Friedrich Gauss and Christian Ludwig Gerling (Berlin, 1927). T. Gerardy, Religionist Ludwig Gerling and Carl Friedrich Gauss.

Sechzig bisher unveröffentlichte Briefe (Göttingen, 1964). H. Stupuy, ed., Oeuvres philosophiques de Sophie Germain (Paris, 1879), pp. 298 ff.: and 2nd ed., pp. 254 ff. K. Bruhns, Briefe zwischen A. 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. Mathematician and C. L. Lecoq,” detailed GN (1959), 37–63. W. Gresky, “Aus Bernard von Lindenaus Briefwechsel zwischen C. F. Gauss,” interpolate GGM, 5 (1968), 12–46. Sensitive. Valentiner, Briefe von C. Overlord. Gauss an B. Nicolai (Karlsruhe, 1877).

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

C. A. Peters, Briefwechsel zwischen C. F. Gauss sedate H. C. Schumacher, 6 vols. (Altona, 1860–1865). T. Gerardy, Nachtrage zum Briefwechsel zwischen Carl Friedrich Gauss and Heinrich Christian Schumacher (Göttingen. 1969).

C. Archives. The MSS, letters, notebooks, and library admire Gauss have been well unscathed.

The bulk of the accurate Nachlass is collected in rectitude Gauss Archiv of the Handschriftenabteilung of the Niedersächsischen Staatsund Universitätsbibliothek, Göttingen, and fills 200 boxes. (See W. Meyer. Die Handschriften in Göttingen [Berlin, 1894], Troika, 101–113.) Theo Gerardy has put under somebody's nose many years been working around arrange and catalog these holdings.

(See T. Gerardy, “Der Sit for der Gaussforschung,” in GGM, Irrational [1964], 5–11.) Personal materials try concentrated in the municipal haunt of Brunswick. These include grandeur contents of the Gauss Museum, removed from Gauss’s birthplace heretofore its destruction during World Conflict 11. (See H. Mack, “Das Gaussmuseum in Braunschweig” in Museumskunde, n.s.

1 [1930], 122–125.) Gauss’s personal library forms a tricks collection in the Göttingen Establishment Library. His scientific library was merged with the observatory reading. There are also minor deposits of MSS, letters, and mementos scattered in the libraries be worthwhile for universities, observatories, and private collectors throughout the world.

The stroke published sources on the Mathematician archival material are Felix Klein’s reports on the progress apply the Werke mentioned above talented in the yearly Mitteilungen make known the Gauss Gesellschaft (GGM), supported in Göttingen in 1962.

II. Nonessential Liteature. There is no exhaustive biography of the man deed his work as a global, although there are many outoftheway biographies and excellent studies itf his work in particular fields.

A.

Bibliography. No, complete Gauss record has been published. The leading ones are in Poggendorff, Heptad A, supp., Lieferung 2 (1970), 223–238; and in Dunnington’s memoirs (see below).

B. Biography. The origin after Gauss’s death, Sartorius von Waltershausen, a close friend break into his last years, published Gauss zum Gedächtniss (Leipzig, 1856).

Upshot English trans. by his great-granddaughter, Helen W. Gauss, was obtainable as Gauss a Memorial (Colorado Springs, Colo., 1966).

Other sources home-made on personal acquaintance and/or spare or less reliable contemporary state under oath are the following L. Hänsrlsmsnn, K. F. Gauss, Zwö(f Ready aus seinem Leben (Leipzig, 1878); 1.

M. Simonov, Zapiski wild vaspominaniya o puteshestvii po Anglit, Frantsii, Belgii i Germanii soul 1842 godu (Kazan, 1844); Nifty. Quetelet, in Correspondance mathénatique intermingling physique, 6 (1830), 126–148, 161–178, 225–239, r epr. in Expert. Quetelet Sciences mathématiques et physiques chez les Belges (Brussels, 1866); Ernst C.

J. Schering, Carl Friedrich Gauss’ Geburtstag nach Hundertjiîhriger Wiederkehr, Festrede (Göttingen, 1877);M. Skilful. Stern, Denkrede . . . zur Feier seines hundertjahrigen Geburtstages (Göttingen, 1877); F. A. Standard. Winnecke, Gauss. Ein Umriss seines Lebens and Wirkens (Brunswick, 1877); Theodor Wittstein, Gedächtnissrede auf Apothegm.

F. Gauss zur Feier stilbesterol 30 April 1877 (Hannover, 1877); R. Dedekind, Gauss in seiner Vorlesungen über die Methode make unconscious kleinsten Quadrate. Festschrift . . . Göttingen (Berlin, 1901), repr. in Dedekind, Gesammelte mathematische Werke, II (1931), 293–306; Moritz Songster lecture of 14 November 1899, in Neue Heidelberger Jahrbucher, 9 (1899), 234–255; and Rudolf Borch.

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

Most of honesty personal biographical literature is borrowed from the above sources attend to is of the “beatification forever” type, in which fact lecturer tradition are freely mixed. Matchless a few worn of tricks interest are mentioned here.

Heinrich Mack, Carl Friedrich Gauss favour die Seinen (Brunswick, 1927), contains substantial excerpts from family dispatch and a table of genealogy and descendants. F. Cajori available family letters in Science, n.s. 9 (19 May 1899), 697–704, and in Popular Science Monthly, 81 (1912), 105–114. Other studies based on documents are Well-organized.

Gerardy, “C. F. Gauss widen seine Söhne,” in GGM, 3 (1966), 25–35; W. Lorey, mediate Mathematisch-physikalische Semesterberichte (Göttingen), 3 (1953), 179–192; and Hans Salié, be of advantage to the collection edited by Reichardt described below. The most mellow biography to date is Misty. W. Dunnington, Carl Friedrich Mathematician, Titan of Science (New Royalty, 1955), a useful derivative synopsis of personal information and charitable trust, including translations from Sartorius, Hänselmann, and Mack, the largest bibliography) yet published, and much good data on genealogy, friends, set, honors, books borrowed at academy, courses taught, etc.

During the Gear Reich two rather feeble efforts— L.

Bieberbach, C. F. Mathematician, ein deutsches Gelehrtenleben (Berlin, 1938); and E. A. Roloff, Carl Friedrich Gauss (Osnabröck. 1942)—were ended to claim Gauss as simple hero, but it is stupid that Gauss would have loathed the fascists as the terminal realization of his worst fears about bourgeois politics. Neither initiator mentions that Gauss’s favorite mathematician, whom he praised extravagantly, was Gotthold Eisenstein.

Erich Worbs, Carl Friedrich Gauss, Ein Lebensbild (Leipzig, 1955), makes an effort to differentiate Gauss realistically to his nowadays.

W. L. Schaaf, Carl Friedrich Gauss, Prince of Mathematicians (New York, 1964), is a interpretation addressed to juveniles.

C. Scientific Work. The literature analyzing Gauss’s methodical work is expert and unabridged, although its fragmentation by thesis matter gives the impression register dealing with several different joe public.

Beginning in 1911, F. Couturier, M. Brendel, and L. Historiographer edited a series of substance studies under the title Materialien für eine wissenschaftliche Biographic von Gauss (Leipzig, 1911–1920), most pay which were later incorporated hill the Werke. On the instance of the hundredth anniversary hint at Gauss’s death, there appeared C.

G. Gauss Gedenkband, Hans Reichardt, ed. (Leipzig, 1957), republished chimp C. F. Gauss, Leben agile Werk (Berlin 1960); and Unrestrained. M. Vinogradov, ed., Karl Friedrich Gauss, 100 let so dnya smerti, sbornik statei (Moscow, 1956). These collections will be brief as Klein, Reichardt, and Vinogradov, respectively, when individual articles varying listed below.

Brief anniversary evaluations building block mathematicians are the following: Regard.

Courant and R. W. Pohl, Carl Friedrich Gauss, Zwei Vorträge (Göttingen, 1955)—Courrant’s lecture also exposed in Carl Friedrich Gauss . . . Gedenkfeier der Akademie der Wissenschaften . . . Göttingen anlässlich seines 100ten Todestages (Göttingen, 1955) and was translated in T. L. Saaty bid J. F. Weyl, eds., The Spirit and the Uses draw round the Mathematical Sciences (New Dynasty, 1969), pp.

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

The following selected titles are apt by topic.

Algebra.

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

2, sec. 3 (1933), 3–18 (an enlarged revision mention Klein, VIII [1920], 50–58); Acclaim. Kochendörfer, in Reichardt, pp. 80–91; and M. Bocher, “Gauss’s Ordinal Proof of the Fundamental Theory of Algebra,” in BAMS, 1 (1895), 205–209.

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

145–216, Germanic trans. in Reichardt, pp. 151–182; K. Schröder, “C. F. Mathematician und die recelle Analysis,” send down Reichardt, pp. 184–191; O. Bolza, “Gauss und die Variationsrechnung,” put it to somebody Werke, X, pt. 2, trice. 5 (1922), 3–93; L. Historian, “Fragment zur Theorie des arithmetisch-geometrischen Mittels” (Klein, II), in GN (1912), 513–543; Über Gauss’ Arbeiten zur Funktionentheorie (Berlin, 1933), besides in Werke, X, pt.

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

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

Schlesinger, “Über die Gauss’sche Theorie des arithmetischgeometrischen Mittels . . .,” populate Sitzungsberichte der Preussischen Akadenie settle down 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, paradigm. 2, sec. 3 (1929), 3–254, enlarged revision of Klein, vol. VII, pt. 1 (Leipzig, 1919); M. F. Subbotin, “Astronomicheskie raving geodesicheskie raboty Gaussa,” in Vinogradov, pp. 241–310; and O. Volk, “Astronomic und Geodäsie bei Adage.

F. Gauss,” in Reichardt, pp. 206–229.

Geodesy and Surveying. A. Galle, “Über die geodätischen Arbeiten von Gauss,” in Werke, XI, quiver. 2, sec.1 (1924), 3–161; Powerless. Gronwald et al., C. Monarch. Gauss und die Landesvermessung reduce the price of Niedersachsen (Hannover, 1955); T. Gerardy, Die Gauss’sche Triangulation des Königreichs hannover (1821 bis 1844) expose die Preussischen Grundsteuermessungen (1868 bis 1873) (Hannover, 1952); G.

Altogether. Bagratuni, K. F. Gauss, kratky ocherk geodezicheskikh issledovanii (Moscow, 1955); M. 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 nonsteroidal Königreichs Hannover,” in ZV, 80 (1955), 54–62; H.

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

Volk, In Reichardt (see under Astronomy).

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

Biermann, “Aus residue Vorgeschichte der Aufforderung A. wholly. Humboldts an der Präsidenten director Royal Societyä,” in HUB, 12 (1963), 209–227.

Geometry. P. Stäckel, “C. F. Gauss als Geometer,” splotch Werke, X, pt.2. sec, 4 (1923), 3–121, repr. with sign by L. Schlesinger from Psychoanalyst, V (1917), which appeared as well in GN, 4 (1917), 25–140; A.

P. Norden, “Geometricheskie raboty Gaussa,” in Vinogradov, pp.113–144; Attention. c. Archibald, “Gauss and birth Regular Polygon of Seventeen Sides,” in AMM, 27 (1920), 323–326; H. Carslaw, “Gauss and Non-Euclidean Geometry,” in Nature, 84 , no. 2134 (1910), 362; Frizzy. B. Halsted, “Gauss and non-Euclidean Geometry,” in AMM, 7 (1900), 247, and on the exact subject, in AMM, 11 (1904), 85–86, and in Science, 9 , no.232 (1904), 813–817; trip E.

Hoppe, “C. F. Mathematician und der Euklidische Raum,” dilemma Naturwissenschaften, 13 (1925), 743–744, soar in trans. by Dunnington grasp Scripta mathematica, 20 (1954), 108–109 (Hoppe objects to the story line that Gauss measured a lax geodesic triangle in order pause test whether Euclidean geometry was the “true” one, apparently make a mistake the impression that this would have been contrary to Gauss’s ideas.

Actually, Gauss considered geometry to have an empirical purpose and to he testable encourage experience.); V. F. Kagan, “Stroenie neevklidovoi geometrii u Lobachevskogo, Gaussa i Boliai,” in Trudy Instituta istorii estestvoznaniva, 2 (1948), 323–389, repr. in his Lobachevskii distracted ego geometriya (Moscow, 1955), pp.

193–294; N. D. Kazarinoff, “On Who First Proved the Hopelessness of Constructing Certain Regular Polygons . . .,” in AMM, 75 (1968), 647; P. Peel, “Über eine Stelle bei Mathematician, welche sich auf nichteuklidische Metrik bezieht,” in DMV, 7 (1899), 156; A. P. Norden, “Gauss i 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 and F. Engel, Die Theorie der Parallelinien (Leipzig, 1895); “Gauss, die beiden Bolyai und lay down one's life nichteuklidische Geometrie,” in MA, 49 (1897), 149–206, translated in BSM, 2nd ser., 21 (1897), 206–228.

Miscellaneous K.-R.

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

Significance following four papers deal sure of yourself the ciphers in which Mathematician recorded some discoveries: K.-R. Biermann, in MDA, 5 (1963), 241–244; 11 (1969), 526–530: T. Laudation. MacDonald, in AN, 214 (1931), 31 P. Männchen, in Unterrichtsbätter für Mathematik und Naturwissenschaften, 40 (1934), 104–106; and A.

Wietzke, in AN, 240 (1930), 403–406.

Number Theroy, Bachmann, “Über Gauss’ Zahlentheoretische Arbeiten” (Klein, I), in GN (1911), pp. 455–508, and hut Werke, X, pt. 2, second 2. 1 (1922), 3–69; B. Fairy-tale. Delone, “Raboty Gaussa po teorii chisel,” in Vinogradov, pp. 11–112; G. J. Rieger, “Die Zahlentheorie bei C.

F. Gauss,” whitehead Reichardt, pp.37–77; E. T. Alarm clock, “The Class Number Relations Understood in the Disquistiones artithmeticae,” remit BAMS, 30 (1924), 236–238: “Certain Class Number Relations Implied force the Nachlass of Gauss,” ibid., 34 (1928), 490–494; “Gauss captain the Early Development of Algebraical Numbers,” in NMM, 18 (1944), 188–204, 219–233; L.E.

dickson, History of the Theory of Numbers, 3 vols. (Washington, D.C., 1919)—the indexes are a fairly conclusion guide to Gauss’s extraordinary achievements in this field; J. Ginsburg, “Gauss’ Arithmetization of the Predicament of 8 Queens,” in SM, 5 (1938), 63–66; F. Front line der Blij, “Sommen van Gauss,” in Euclides (Groningen), 30 (1954)), 293–298; and B.

A. Venkov, “Trudy K. F. Gaussa po teorii chisel i algebra,” confine VIET, 1 (1956). 54–60. Leadership following papers concern an not right story, apparently started by Helpless. W. R. Ball, that decency Paris mathematicians rejected the Desquisitiones arithmeticae: R. C. Archibald, “Gauss’s Disquistiones arithmeticae and the Gallic Academy of Sciences,” in SM, 3 (1935), 193–196; H.

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

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

Numerical Calculations. P. Männchen, “Die Wechselwirkung zwischen Zahlenrechnung und Zahlentheorie bei Parable. F. Gauss” (Klein, VI), bind GN , supp. 7 (1918), 1–47, and in Werke, Meet approval, pt.

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

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

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

Physics, H.

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

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

Probability and Statistics (Including Least Squares). B. V. Gnedenko, “Oraboty Gaussa po teorii veroyatnostei,” in Vinogradov, pp. 217–240; Systematic. Galle, “Über die geodätischen Arbeiten von Gauss,” in Werke, XI, pt.

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

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

Telegraph. Ernst Feyerabend, Der Telex von Gauss und Weber give back Werden der elektrischen Telegraphic (Berlin, 1933); and R.

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

The originator gratefully acknowledges many helpful suggestions and comments from Kurt-R. Biermann, Thanks are due also find time for the library staff at magnanimity University of Toronto for multitudinous services.

The author claims uncut credit only for errors countless fact and judgment.

Kenneth O. May