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Johann Carl Friedrich Gauss was a __German mathematician__ and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum (__Latin__ for "the foremost of mathematicians") and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.

Early years

Johann Carl Friedrich Gauss was born on 30 April 1777 in __Brunswick (Braunschweig)__, in the __Duchy of Brunswick-Wolfenbüttel__ (now part of __Lower Saxony__, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the __Feast of the Ascension__ (which occurs 39 days after Easter). Gauss later solved this puzzle about his birthdate in the context of __finding the date of Easter__, deriving methods to compute the date in both past and future years. He was christened and __confirmed__ in a church near the school he attended as a child.

Gauss was a __child prodigy__. In his memorial on Gauss, __Wolfgang Sartorius von Waltershausen__ says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an __arithmetic series__ problem faster than anyone else in his class of 100 students. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his __magnum opus__, __Disquisitiones Arithmeticae__, in 1798, at the age of 21—though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.

Gauss's intellectual abilities attracted the attention of the __Duke of Brunswick__, who sent him to the Collegium Carolinum (now __Braunschweig University of Technology__), which he attended from 1792 to 1795, and to the __University of Göttingen__ from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular __polygon__ can be constructed by __compass and straightedge__ if the number of its sides is the product of distinct __Fermat primes__ and a __power__ of 2. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the __Ancient Greeks__, and the discovery ultimately led Gauss to choose mathematics instead of __philology__ as a career. Gauss was so pleased with this result that he requested that a regular __heptadecagon__ be inscribed on his tombstone. The __stonemason__ declined, stating that the difficult construction would essentially look like a circle.

The year 1796 was more productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced __modular arithmetic__, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the __quadratic reciprocity__ law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The __prime number theorem__, conjectured on 31 May, gives a good understanding of how the __prime numbers__ are distributed among the integers.

Gauss also discovered that every positive integer is representable as a sum of at most three __triangular numbers__ on 10 July and then jotted down in __his diary__ the note: "__ΕΥΡΗΚΑ__! ". On 1 October he published a result on the number of solutions of polynomials with coefficients in __finite fields__, which 150 years later led to the __Weil conjectures__.

Later years and death

Gauss remained mentally active into his old age, even while suffering from __gout__ and general unhappiness. For example, at the age of 62, he taught himself Russian.

In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a __paraxial approximation__ (__Gaussian optics__). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its __cardinal points__ and he derived the Gaussian lens formula.

In 1845, he became an associated member of the Royal Institute of the Netherlands; when that became the __Royal Netherlands Academy of Arts and Sciences__ in 1851, he joined as a foreign member.

In 1854, Gauss selected the topic for __Bernhard Riemann__'s inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (About the hypotheses that underlie Geometry). On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.

On 23 February 1855, Gauss died of a heart attack in Göttingen (then __Kingdom of Hanover__ and now __Lower Saxony__); he is interred in the __Albani Cemetery__ there. Two people gave eulogies at his funeral: Gauss's son-in-law __Heinrich Ewald__, and __Wolfgang Sartorius von Waltershausen__, who was Gauss's close friend and biographer. Gauss's brain was preserved and was studied by __Rudolf Wagner__, who found its mass to be slightly above average, at 1,492 grams, and the cerebral area equal to 219,588 square millimeters (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius.

Religious views

Gauss was a __Lutheran__ __Protestant__, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had previously defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by the grace of the Lord." One of his biographers, __G. Waldo Dunnington__, described Gauss's religious views as follows:

For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength, it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life, it brought him confidence. Gauss's God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report the truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.

Apart from his correspondence, there are not many known details about Gauss's personal creed. Many biographers of Gauss disagree about his religious stance, with Bühler and others considering him a __deist__ with very unorthodox views, while Dunnington (though admitting that Gauss did not believe literally in all Christian dogmas and that it is unknown what he believed on most doctrinal and confessional questions) points out that he was, at least, a nominal __Lutheran__.

In connection to this, there is a record of a conversation between __Rudolf Wagner__ and Gauss, in which they discussed __William Whewell__'s book Of the Plurality of Worlds. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe. This later led them to discuss the topic of __faith__, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Lutheran minister __Paul Gerhardt__ than by __Moses__. Other religious influences included Wilhelm Braubach, __Johann Peter Süssmilch__, and the __New Testament__.

Dunnington further elaborates on Gauss's religious views by writing:

Gauss's religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.

Gauss declared he firmly believed in the __afterlife__, and saw spirituality as something essentially important for human beings. He was quoted stating: "The world would be nonsense, the whole creation an absurdity without immortality," and for this statement he was severely criticized by the atheist __Eugen Dühring__ who judged him as a narrow superstitious man.

Though he was not a church-goer, Gauss strongly upheld __religious tolerance__, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble." When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved of this, saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.

Family

On 9 October 1805, Gauss married Johanna Osthoff (1780–1809), and had a son and a daughter with her. Johanna died on 11 October 1809, and her most recent child, Louis, died the following year. Gauss plunged into a depression from which he never fully recovered. He then married Minna Waldeck (1788–1831) on 4 August 1810, and had three more children. Gauss was never quite the same without his first wife, and he, just like his father, grew to dominate his children. Minna Waldeck died on 12 September 1831.

Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene shared a good measure of Gauss's talent in languages and computation. After his second wife's death in 1831 Therese took over the household and cared for Gauss for the rest of his life. His mother lived in his house from 1817 until her death in 1839.

Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements. Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and, in about 1832, emigrated to the United States, where he was quite successful. While working for the American Fur Company in the Midwest, he learned the Sioux language. Later, he moved to __Missouri__ and became a successful businessman. Wilhelm also moved to America in 1837 and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in __St. Louis__. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues. See also __the letter from Robert Gauss to Felix Klein__ on 3 September 1912.

Personality

Carl Gauss was an ardent __perfectionist__ and a hard worker. He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism. This was in keeping with his personal motto pauca sed matura ("few, but ripe"). His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Scottish-American mathematician and writer __Eric Temple Bell__ said that if Gauss had published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.

Though he did take in a few students, Gauss was known to dislike teaching. It is said that he attended only a single scientific conference, which was in __Berlin__ in 1828. However, several of his students became influential mathematicians, among them __Richard Dedekind__ and __Bernhard Riemann__.

On Gauss's recommendation, __Friedrich Bessel__ was awarded an honorary doctor degree from Göttingen in March 1811. Around that time, the two men engaged in an epistolary correspondence. However, when they met in person in 1825, they quarrelled; the details are unknown.

Before she died, __Sophie Germain__ was recommended by Gauss to receive her honorary degree; she never received it.

Gauss usually declined to present the intuition behind his often very elegant proofs—he preferred them to appear "out of thin air" and erased all traces of how he discovered them. This is justified, if unsatisfactorily, by Gauss in his __Disquisitiones Arithmeticae__, where he states that all analysis (i.e., the paths one traveled to reach the solution of a problem) must be suppressed for sake of brevity.

Gauss supported the monarchy and opposed __Napoleon__, whom he saw as an outgrowth of revolution.

Gauss summarized his views on the pursuit of knowledge in a letter to __Farkas Bolyai__ dated 2 September 1808 as follows:

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

Career and achievements

Algebra

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the __fundamental theorem of algebra__ which states that every non-constant single-variable __polynomial__ with complex coefficients has at least one complex __root__. Mathematicians including __Jean le Rond d'Alembert__ had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to the implicit use of the __Jordan ____HYPERLINK "https://en.wikipedia.org/wiki/Jordan_curve_theorem"____curve theorem__. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Gauss also made important contributions to __number theory__ with his 1801 book __Disquisitiones Arithmeticae__ (__Latin__, Arithmetical Investigations), which, among other things, introduced the symbol ≡ for __congruence__ and used it in a clean presentation of __modular arithmetic__, contained the first two proofs of the law of __quadratic reciprocity__, developed the theories of binary and ternary __quadratic forms__, stated the __class number problem__ for them, and showed that a regular __heptadecagon__ (17-sided polygon) can be __constructed with straightedge and compass__. It appears that Gauss already knew the __class number formula__ in 1801.

In addition, he proved the following conjectured theorems:

Astronomy

In the same year, Italian astronomer __Giuseppe Piazzi__ discovered the __dwarf planet__ __Ceres__. Piazzi could only track Ceres for somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit. Gauss heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by __Franz Xaver von Zach__ on 31 December at __Gotha__, and one day later by __Heinrich Olbers__ in __Bremen__.

__Gauss's method__ involved determining a __conic section__ in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by __Kepler's Second Law__). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.

One such method was the __fast Fourier transform__. While this method is traditionally attributed to a 1965 paper by __J.W. Cooley__and __J.W. Tukey__, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by __Joseph Fourier__ on the subject in 1807.

Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical __observatory in Göttingen__, a post he held for the remainder of his life.

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the __Gaussian gravitational constant__, and contained an influential treatment of the __method of least squares__, a procedure used in all sciences to this day to minimize the impact of __measurement error__.

Gauss proved the method under the assumption of __normally distributed__ errors (see __Gauss–Markov theorem__; see also __Gaussian__). The method had been described earlier by __Adrien-Marie Legendre__ in 1805, but Gauss claimed that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares."

Geodetic survey

In 1818 Gauss, putting his calculation skills to practical use, carried out a __geodetic survey__ of the __Kingdom of Hanover__, linking up with previous Danish surveys. To aid the survey, Gauss invented the __heliotrope__, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.

Non-Euclidean geometries

Gauss also claimed to have discovered the possibility of __non-Euclidean geometries__ but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

Research on these geometries led to, among other things, __Einstein__'s theory of general relativity, which describes the universe as non-Euclidean. His friend __Farkas Wolfgang Bolyai__ with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

Bolyai's son, __János Bolyai__, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

This unproved statement put a strain on his relationship with Bolyai who thought that Gauss was "stealing" his idea.

Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. __Waldo Dunnington__, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.

Theorema Egregium

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade, fueled Gauss's interest in __differential geometry__ and __topology__, fields of mathematics dealing with __curves__ and __surfaces__. Among other things, he came up with the notion of __Gaussian curvature__. This led in 1828 to an important theorem, the __Theorema Egregium__ (remarkable theorem), establishing an important property of the notion of __curvature__. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring __angles__ and __distances__ on the surface.

That is, curvature does not depend on how the surface might be __embedded__ in 3-dimensional space or 2-dimensional space.

In 1821, he was made a foreign member of the __Royal Swedish Academy of Sciences__. Gauss was elected a Foreign Honorary Member of the __American Academy of Arts and Sciences__ in 1822.

Magnetism

In 1831, Gauss developed a fruitful collaboration with the physics professor __Wilhelm Weber__, leading to new knowledge in __magnetism__ (including finding a representation for the unit of magnetism in terms of mass, charge, and time) and the discovery of __Kirchhoff's circuit laws__ in electricity. It was during this time that he formulated his namesake __law__. They constructed the first __electromechanical telegraph__ in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic __observatory__ to be built in the garden of the observatory, and with Weber founded the "Magnetischer Verein" (magnetic association), which supported measurements of Earth's magnetic field in many regions of the world. He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer (__magnetospheric__) sources of Earth's magnetic field.

Appraisal

The British mathematician __Henry John Stephen Smith__ (1826–1883) gave the following appraisal of Gauss:

If we except the great name of __Newton__ it is probable that no mathematicians of any age or country have ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in demonstration, which the ancient Greeks themselves might have envied. It may seem paradoxical, but it is probably nevertheless true that it is precisely the efforts after logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and unnecessary difficulty. Gauss says more than once that, for brevity, he gives only the synthesis, and suppresses the analysis of his propositions. If, on the other hand, we turn to a memoir of __Euler__'s, there is a sort of free and luxuriant gracefulness about the whole performance, which tells of the quiet pleasure which Euler must have taken in each step of his work. It is not the least of Gauss's claims to the admiration of mathematicians, that, while fully penetrated with a sense of the vastness of the science, he exacted the utmost rigorousness in every part of it, never passed over a difficulty, as if it did not exist, and never accepted a theorem as true beyond the limits within which it could actually be demonstrated.

__http://www.nat-geo.ru/ludi-planety/1192233-chem-znamenit-iogann-karl-fridrikh-gauss/__

__http://spacegid.com/biografiya-karla-gaussa.html__

__https://to-name.ru/biography/karl-fridrih-gauss.htm__

__https://obrazovaka.ru/carl-friedrich-gauss.html__

__http://www.univer.omsk.su/omsk/Edu/Math/ggauss.htm__

__http://biografiivsem.ru/gauss-karl-fridrih__

__http://math4school.ru/gauss.html__

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