# Carl Friedrich Gauss

Carl Friedrich Gauss
Portrait of Gauss by Christian Albrecht Jensen, 1840 (copy from Gottlieb Biermann, 1887)[1]
Born
Johann Carl Friedrich Gauss

30 April 1777
Died23 February 1855 (aged 77)
Alma mater
Known forSee full list
Spouses
• Johanna Osthoff
(m. 1805; died 1809)
• Minna Waldeck
(m. 1810; died 1831)
Children
• Joseph
• Wilhelmina
• Louis
• Eugene
• Wilhelm
• Therese
AwardsLalande Prize (1809)
Copley Medal (1838)
Scientific career
FieldsMathematics and sciences
InstitutionsUniversity of Göttingen
ThesisDemonstratio nova... (1799)
Doctoral students
Other notable students
Signature

Johann Carl Friedrich Gauss (/ɡs/; German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ();[2][3] Latin: Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science.[4] Sometimes referred to as the Princeps mathematicorum (Latin for 'the foremost of mathematicians')[5] and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science; he is ranked among history's most influential mathematicians.[6]

He was a child prodigy in mathematics and completed his magnum opus, Disquisitiones Arithmeticae, at age 21. Gauss attended Collegium Carolinum and the University of Göttingen, where he made several mathematical discoveries. In 1807, he became the director of the astronomical observatory at the University of Göttingen, where he remained active in mathematical research. Gauss died of a heart attack on February 23, 1855, in Göttingen.

He had two wives and six children. He had conflicts with his sons over their career choices, as he did not want them to enter mathematics or science, fearing they would not surpass his achievements. Despite being an ardent perfectionist and hard worker, he was not a prolific writer and refused to publish incomplete work. Gauss was known to dislike teaching, but some of his students became influential mathematicians. He supported the monarchy and opposed Napoleon. Gauss believed that the act of learning, not possession of knowledge, granted the greatest enjoyment.

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. He made important contributions to number theory and developed the theories of binary and ternary quadratic forms. Gauss is also credited with inventing the fast Fourier transform algorithm and was instrumental in the discovery of the dwarf planet Ceres. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and the method of least squares, which is still used in all sciences to minimize measurement error.

## Biography

### Early years

Gauss's diary entry related to sum of triangular numbers (1796)
Carl Friedrich Gauß 1803 by Johann Christian August Schwartz

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.[7][8] 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.[9] He was christened and confirmed in a church near the school he attended as a child.[10]

Gauss was a child prodigy in the field of mathematics. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen wrote that when Gauss was barely three years old he corrected a math error his father made; and that when he aged nine in the basic school, solved an arithmetic series problem faster than anyone else in his class of about hundred pupils.[11] There are many versions of this story, with various details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100. (See also under "Anecdotes" below.)[12][13][14] 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, and it was published in 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.

Gauss' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (at that time not equal to a university, now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the Hanoverian University of Göttingen from 1795 to 1798, where he studied mathematics, sciences abd classical languages as well.[15][6] While at university, Gauss independently rediscovered several important theorems.[16] 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.[a] 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.[17]

The year 1796 was productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.[18][19] 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! num = Δ + Δ + Δ". 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.

Gauss graduated in 1799, not in Göttingen as sometimes mentioned,[20] but on the Duke's special request at the University of Helmstedt, the only state university of the Duchy. There Johann Friedrich Pfaff assessed the doctoral thesis, and Gauss got the degree in absentia without further oral examination as usually requested. The next years the Duke granted his costs of living as a scholar in Brunswick. He showed his gratitude to the Duke, when he refused several calls from the Russian Academy of science in St. Peterburg. The Duke of Brunswick had promised him the foundation of an observatory in Brunswick in 1804 and architect Peter Joseph Krahe made preliminary designs, but one of Napoleons wars cancelled those plans :[21] the Duke was deadly wounded in the battle, the Duchy was abolished, and Gauss' financial support stopped. Thus in 1807 he followed a call to the University of Göttingen, then being an institution of the new founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory.

### Later years and death

Gauss on his deathbed (1855)
Gauss' gravesite at Albani Cemetery in Göttingen, Germany

In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture Über die Hypothesen, welche der Geometrie zu Grunde liege [On the Hypotheses which lie at the Bases of Geometry].[22][23] On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.[24]

Gauss was member of many scientific societies, among them the Royal Institute of the Netherlands (1845) and the American Philosophical Society (1853).[25][26]

Gauss remained mentally active into his old age, even while having gout and suffering general unhappiness. At the age of 62, he taught himself Russian.[27] On 23 February 1855, Gauss died of a heart attack in Göttingen;[28] he is interred in the Albani Cemetery there. Two people gave eulogies at his funeral: Gauss' son-in-law Heinrich Ewald, and Wolfgang Sartorius von Waltershausen, who was Gauss' close friend and biographer.

### Gauss' brain

The day after Gauss' death his brain was taken out, preserved and studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams (52.6 oz).[29][30] The cerebral area was determined by Wagners son Hermann in his doctoral thesis to be 219,588 square millimetres (340.362 sq in).[31] Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation for his genius.[32] After various previous investigations, a magnetic resonance study of 1998, done at the Max Planck Institute for Biophysical Chemistry in Göttingen, gave no results which could be used to explain his mathematical abilities.

In 2013, a neurobiologist of the same institute discovered that Gauss' brain had obviously been mixed up, by reason of wrong labelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen in the same year as Gauss.[33] A further investigation showed no remarkable anomalies in the brains of either person. Thus all investigations on Gauss' brain until 1998, except the first ones of Rudolf and Hertmann Wagner, actually refer to the brain of Fuchs.[34]

### Religious views

Gauss was nominally a member of the St. Albans Evangelical Lutheran church in Göttingen.[35] One of his biographers, G. Waldo Dunnington, has described Gauss' 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' 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.

— Dunnington 2004, pp. 298–301

Apart from his correspondence, there are not many known details about Gauss' personal creed. Many biographers of Gauss disagree about his religious stance, with Bühler and others considering him a deist with very unorthodox views,[36][37][38] while Dunnington (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.[b]

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.[36][c] 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.[39] Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament. Two religious works which Gauss read frequently were Braubach's Seelenlehre (Gießen, 1843) and Süssmilch's Göttliche Ordnung, 1756); he also devoted considerable time to the New Testament in the original Greek.[40]

Dunnington further elaborates on Gauss' 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.

— Dunnington 2004, p. 300

Gauss believed in an omniscient source of creation however he claimed that belief or a lack of it did not affect his mathematics.[41]

Though he was not a church-goer,[42] 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."[6] 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.[43]

### Family

Gauss' second wife Wilhelmine Waldeck
Therese Gauss (1834) by Ludwig Becker

On 9 October 1805, Gauss married Johanna Osthoff (1780–1809),[44] and had two sons and a daughter with her : Joseph (1806–1873), Wilhelmina (1808–1840) and Louis (1809–1810). Johanna died on 11 October 1809 one month after Louis' birth, who himself died a few months later. Gauss plunged into a depression from which he never fully recovered. Soon after her death he wrote a last letter to his dead wife in the style of an ancient threnody, the most personal document of Gauss.[45][46]

He then married Wilhelmine (Minna) Waldeck (1788–1831), a friend of his first wife, on 4 August 1810 and had three more children : Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879) and Therese (1816–1864). Minna Gauss died on 12 September 1831 after serious illness for more than a decennium, possibly caused by tuberculosis.[47][48] Then Therese took over the household and cared for Gauss for the rest of his life. His mother Dorothea Gauss lived in his house from 1817 until her death in 1839.[6] The daughter Wilhelmina married the orientalist Heinrich Ewald and died at the age of 42, possibly by tuberculosis.[49] Therese married the actor Constantin Staufenau after her father's death and died at 47, possibly by tuberculosis.

Gauss was never quite the same without his first wife, and just like his father, grew to dominate his children. Gauss eventually had conflicts with his sons, because he did not want any of them to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.

Still being a school boy, the eldest son Joseph helped his father as assistant during his survey campaign in summer 1821. After a short time at university, he joined the Hanoverian army in 1824 and assisted in surveying again in 1829. Later in the 1830's he was responsible for the enlargement of the survey network to the western parts of the Kingdom. But in all these years staying in a low military rank (since 1834 as Premier-Leutnant) with a very small salary, he needed financial support from his father, especially since his marriage in 1840. Thus he left the service and, with the background of his geodetical qualifications, engaged as director of the Royal Hanoverian State Railways with the construction of the railway network. In 1836 Joseph Gauss had studied the railroad system in the USA for some months.[50]

Eugen shared a good measure of Gauss' talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused a scandal in public,[51] he suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. After having wasted the few money he had taken for starting, his father refused further financial support. Thus he joined the army for five years, and thereafter worked for the American Fur Company in the Midwest, where he learned the Sioux language. Later, he moved to Missouri and became a successful businessman.[50] It took many years for Eugene's success to counteract his reputation among Gauss' friends and colleagues.[52]

Wilhelm married a niece of the astronomer Bessel and also moved to Missouri in 1837,[53] starting as a farmer and later becoming wealthy in the shoe business in St. Louis. Eugene and William are progenitors of numerous descendents in America, but the German Gauss issue descends from Joseph as the Gauss daughters had no children.[50]

### Personality

Portrait of Gauss in Volume II of "Carl Friedrich Gauss Werke," 1876

Though he did take in a few students, Gauss was known to dislike teaching. Several of his students became influential mathematicians, among them Richard Dedekind and Bernhard Riemann.

On Gauss' recommendation, Friedrich Wilhelm Bessel was awarded an honorary doctoral degree from Göttingen University in March 1811; they had been friends since 1804.[d] Before she died, Sophie Germain was recommended by Gauss to receive an honorary degree; but she never received it.[54] In 1828, Gauss attended the conference of the Society of German Natural Scientists and Physicians in Berlin as special guest of Alexander von Humboldt; during this occasion he became acquainted with Wilhelm Weber.[55]

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 diary indicates that he had made several important mathematical discoveries years or decades before his contemporaries published them. 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.[56] 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 (in other words, the paths one traveled to reach the solution of a problem) must be suppressed for sake of brevity.

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

Gauss summarized his views on the pursuit of knowledge in a letter[57] 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.

— Dunnington 2004, p. 416

## Career and achievements

### Algebra

Title page of Gauss' magnum opus, Disquisitiones Arithmeticae

In his doctoral thesis from 1799 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' dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss' own attempt is not acceptable, owing to the implicit use of the Jordan 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, which, among other things, introduced the triple bar 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.[58]

In addition, he proved the following conjectured theorems:

He also

### Astronomy

Lithography by Siegfried Bendixen (1828)
German 10-Deutsche Mark Banknote (1993; discontinued) with formula and graph of normal distribution; portrait as mirror image of the Jensen portrait
Back of German 10-Deutsche Mark Banknote (1993; discontinued) with heliotrope and a section of the triangulation network

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres.[59] Piazzi could track Ceres for only 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.[60] This confirmation eventually led to the classification of Ceres as minor-planet designation 1 Ceres: the first asteroid (now dwarf planet) ever discovered.[20]

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.[61]

One such method was the fast Fourier transform. While this method is attributed to a 1965 paper by James Cooley and John Tukey,[62] Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata,[63] was published only posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807.[64]

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.[65] 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.[66] In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares."[67]

### Geodetic survey

Survey marker stone in Garlste (now Garlstedt)

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodetic survey of the Kingdom of Hanover (Gaussian land survey [de]), 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.

In 1828, when studying differences in latitude, Gauss first defined a physical approximation for the figure of the Earth as the surface everywhere perpendicular to the direction of gravity (of which mean sea level makes up a part);[68] later his doctoral student Johann Benedict Listing called this the geoid.[69]

### 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.[70]

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 (1955) 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.[71][6]

### Theorema Egregium

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade,[72] fueled Gauss' 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.[73]

### Magnetism

In 1831, Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber,[28] 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,[74] 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.

### Optics

In 1840, Gauss published his influential Dioptrische Untersuchungen,[75] in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics).[76] Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points[77] and he derived the Gaussian lens formula.[78]

## 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' 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.[79]

## Anecdotes

There are several stories of his early genius. One story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels. Gauss' presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. However, the details of the story are at best uncertain (see[14] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions), and some authors, such as Joseph J. Rotman in his book A First Course in Abstract Algebra (2005), question whether it ever happened.[80]

He referred to mathematics as "the queen of sciences"[81] and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.[82]

## Commemorations

Statue of Gauss in Brunswick

Gauss Monuments were erected in Brunswick and Göttingen (the last together with Weber). busts of Gauss were placed in the Walhalla temple near Regensburg and in the German Research Centre for Geosciences in Potsdam. Several places where Gauss has stayed in Germany are marked with plaques.

From 1989 through 2001, Gauss' portrait, a normal distribution curve and some prominent Göttingen buildings were featured on the front-side of a German ten-mark banknote.[citation needed] The reverse featured the approach for Hanover. Germany has also issued three postage stamps honoring Gauss. One (no. 725) appeared in 1955 on the hundredth anniversary of his death; two others, nos. 1246 and 1811, in 1977, the 200th anniversary of his birth.

The numerous things named in honor of Gauss include:

In 1929 the Polish mathematician Marian Rejewski, who helped to solve the German Enigma cipher machine in December 1932, began studying actuarial statistics at Göttingen. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss' grave.[83]

Daniel Kehlmann's 2005 novel Die Vermessung der Welt explores Gauss as leading figure through a lens of historical fiction, contrasting him with the German explorer Alexander von Humboldt. A film version directed by Detlev Buck was released in 2012.[84]

On 30 April 2018, Google honored Gauss on his would-be 241st birthday with a Google Doodle showcased in Europe, Russia, Israel, Japan, Taiwan, parts of Southern and Central America and the United States.[85]

Carl Friedrich Gauss, who also introduced the so-called Gaussian logarithms, sometimes gets confused with Friedrich Gustav Gauss [de] (1829–1915), a German geodesist, who also published some well-known logarithm tables used up into the early 1980s.[86]

The ″Gauss-Gesellschaft Göttingen″ (Gauss Society) was founded in 1964 for resaerches on life and work of Carl Friedrich Gauss and related persons and edits the ″Mitteilungen der Gauss-Gesellschaft″ (Communications of the Gauss Society).[87]

## Writings

Title page of Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium
Title page to the English Translation of Theoria Motus by Charles Henry Davis (1857)
Title page of Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata

### Physics

Gauss-Weber monument in Göttingen

### Collected Works

• Königlich Preußische Akademie der Wissenschaften, ed. (1863–1933). Carl Friedrich Gauss. Werke (in Latin and German). Vol. 1–12. Göttingen: (diverse publishers).

### Correspondence

• Königlich Preußische Akademie der Wissenschaften, ed. (1880). Briefwechsel zwischen Gauss und Bessel (in German). Leipzig: Wilhelm Engelmann. (letters from December 1804 to August 1844)
• Schwemin, Friedhelm, ed. (2014). Der Briefwechsel zwischen Carl Friedrich Gauß und Johann Elert Bode. Acta Historica Astronomica (in German). Vol. 53. Leipzig: Akademische Verlaganstalt. ISBN 978-3-944913-43-8. (letters from February 1802 to October 1826)
• Schoenberg, Erich; Perlick, Alfons (1955). Unbekannte Briefe von C. F. Gauß und Fr. W. Bessel. Abhandlungen der Bayerischen Akademie der Wissenschaften, Math.-nat. Klasse, Neue Folge, No. 71 (in German). Munich: Verlag der Bayerischen Akademie der Wissenschaften. pp. 5–21. (letters to Boguslawski from February 1835 to January 1848)
• Franz Schmidt, Paul Stäckel, ed. (1899). Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai (in German). Leipzig: B. G. Teubner. (letters from September 1797 to February 1853; added letters of other correspondents)
• Axel Wittmann, ed. (2018). Obgleich und indeßen. Der Briefwechsel zwischen Carl Friedrich Gauss und Johann Franz Encke (in German). Remagen: Verlag Kessel. ISBN 9783945941379. (letters from June 1810 to June 1854)
• Clemens Schaefer, ed. (1927). Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling (in German). Berlin: Otto Elsner. (letters from June 1810 to June 1854)
• Karl Christian Bruhns, ed. (1877). Briefe zwischen A. v. Humboldt und Gauss (in German). Leipzig: Wilhelm Engelmann. (letters from July 1807 to December 1854; added letters of other correspondents)
• Reich, Karin; Roussanova, Elena (2018). Karl Kreil und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext. Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68 (in German). Vienna: Verlag der Österreichischen Akademie der Wissenschaften. (letters from 1835 to 1843)
• Gerardy, Theo, ed. (1959). Briefwechsel zwischen Carl Friedrich Gauß und Carl Ludwig von Lecoq. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 (in German). Göttingen: Vandenhoeck & Ruprecht. pp. 37–63. (letters from February 1799 to September 1800)
• Carl Schilling, ed. (1900). Briefwechsel zwischen Olbers und Gauss: Erste Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. (letters from January 1802 to October 1819)
• Carl Schilling, ed. (1909). Briefwechsel zwischen Olbers und Gauss: Zweite Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. (letters from January 1820 to May 1839; added letters of other correspondents)
• Christian August Friedrich Peters, ed. (1860–1865). Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (in German). Altona: Gustav Esch.
• Poser, Hans, ed. (1987). Briefwechsel zwischen Carl Friedrich Gauß und Eberhard August Zimmermann. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 (in German). Göttingen: Vandenhoeck & Ruprecht. ISBN 9783525821169. (letters from 1795 to 1815)

The Göttingen Academy of Sciences and Humanities provides a complete collection of the yet known letters from and to Carl Friedrich Gauss that is accessible online.[88] Written estate from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.[89]

## References

### Notes

1. ^ Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel. See the Constructible polygon article for further discussion.
2. ^ Dunnington 2004, p. 305 writes "It is not known just what Gauss believed on most doctrinal and confessional questions. He did not believe literally in all Christian dogmas. Officially he was a member of St. Albans Church (Evangelical Lutheran) in Gottingen. All baptisms, burials, and weddings in his family occurred there. It is also not known whether he attended church regularly or contributed financially. A faculty colleague called Gauss a deist, but there is good reason to believe that this label did not fit well. Gauss possessed strong religious tolerance which he carried over to every belief originating in the depths of the human heart. This tolerance is not to be confused with religious indifference. He took a special interest in the religious development of the human race, especially in his own century. With reference to the manifold denominations, which frequently did not agree with his views, he always emphasized that one is not justified in disturbing the faith of others in which they find consolation for earthly sufferings and a safe refuge in days of misfortune"
3. ^ Dunnington 2004, p. 305 quotes: "league, I believe you are more believing in the Bible than I. I am not, and, he added, with the expression of great inner emotion, you are much happier than I. I must say that so often in earlier times when I saw people of the lower classes, simple manual laborers who could believe so rightly with their hearts, I always envied them, and now, he continued, with soft voice and that naive childlike manner peculiar to him, while a tear came to his eye, tell me how does one begin this?..."
4. ^ Bessel never had a university education.

### Citations

1. ^ Axel D. Wittmann, Inna V. Oreshina (2009). "On Jensen's Paintings of C. F. Gauss". Mitteilungen der Gauss-Gesellschaft (46): 57–61.
2. ^ Dudenredaktion; Kleiner, Stefan; Knöbl, Ralf (2015) [First published 1962]. Das Aussprachewörterbuch [The Pronunciation Dictionary] (in German) (7th ed.). Berlin: Dudenverlag. pp. 246, 381, 391. ISBN 978-3-411-04067-4.
3. ^ Krech, Eva-Maria; Stock, Eberhard; Hirschfeld, Ursula; Anders, Lutz Christian (2009). Deutsches Aussprachewörterbuch [German Pronunciation Dictionary] (in German). Berlin: Walter de Gruyter. pp. 402, 520, 529. ISBN 978-3-11-018202-6.
4. ^ "Gauss, Carl Friedrich". Encyclopedia.com. Retrieved 17 September 2018.
5. ^ Zeidler, Eberhard (2004). Oxford Users' Guide to Mathematics. Oxford: Oxford University Press. p. 1188. ISBN 978-0-19-850763-5.
6. Dunnington, Waldo (1927). "The Sesquicentennial of the Birth of Gauss". Scientific Monthly. 24 (5): 402–414. Bibcode:1927SciMo..24..402D. JSTOR 7912. Archived from the original on 26 February 2008. Also available at "The Sesquicentennial of the Birth of Gauss". Retrieved 23 February 2014. Comprehensive biographical article.
7. ^
8. ^ Rudolf Borch (1929) : Ahnentafel des Mathematikers Carl Friedrich Gauß [Genealogical tabel]. Ahnentafeln Berühmter Deutscher, Vol. 1, pp. 63–65. Zentralstelle für Deutsche Personen- und Familiengeschichte (ed.)
9. ^ "Mind Over Mathematics: How Gauss Determined The Date of His Birth". american_almanac.tripod.com.
10. ^ Chamberless, Susan (11 March 2000). "Letter:WORTHINGTON, Helen to Carl F. Gauss – 26 July 1911". Susan D. Chamberless. Retrieved 14 September 2011.
11. ^ Sartorius von Waltershausen, Wolfgang (1856), Gauss zum Gedächtniss (in German), S. Hirzel, p. 12
12. ^ Bruno 2003, p. 178.
13. ^ "Gauss, Carl Friedrich (1777–1855)." (2014). In The Hutchinson Dictionary of scientific biography. Abington, United Kingdom: Helicon.
14. ^ a b Hayes, Brian (2006). "Gauss's Day of Reckoning". American Scientist. 94 (3): 200. doi:10.1511/2006.59.200. Archived from the original on 12 January 2012. Retrieved 30 October 2012.
15. ^ Bruno 2003, p. 178-179.
16. ^
17. ^ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
18. ^ Bruno 2003, p. 179.
19. ^ Carl Friedrich Gauss §§365–366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965.
20. ^ a b Marsden, Brian G. (1 August 1977). "Carl Friedrich Gauss, Astronomer". Journal of the Royal Astronomical Society of Canada. 71: 309–323. Bibcode:1977JRASC..71..309M. ISSN 0035-872X.
21. ^ Michling, Horst (1966). "Zum Projekt einer Gauß-Sternwarte in Braunschweig". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 24.
22. ^ Monastyrsky, Michael (1987). Riemann, Topology, and Physics. Birkhäuser. pp. 21–22. ISBN 978-0-8176-3262-5.
23. ^ On the Hypotheses which lie at the Bases of Geometry. Bernhard Riemann. Translated by William Kingdon Clifford
24. ^ Bühler 1981, p. 154.
25. ^ "C.F. Gauss (1797–1855)". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
26. ^ "APS Member History". search.amphilsoc.org. Retrieved 16 April 2021.
27. ^ Bruno 2003, p. 181.
28. ^ a b Bruno 2003, p. 181.
29. ^
30. ^
31. ^ Wagner, Hermann (1864). Maassbestimmungen der Oberfläche des grossen Gehirns [Measurements of the surface of the large brain] (in German). Cassel & Göttingen: Georg H. Wigand.
32. ^ Bardi, Jason (2008). The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons, Inc. p. 189. ISBN 978-0-470-46736-7.
33. ^ Schweizer, Renate; Wittmann, Axel; Frahm, Jens (2014). "A rare anatomical variation newly identifies the brains of C.F. Gauss and C.H. Fuchs in a collection at the University of Göttingen". Brain. 137 (4): e269. doi:10.1093/brain/awt296. PMID 24163274. (with further references)
34. ^ "Unravelling the true identity of the brain of Carl Friedrich Gauss". Max Planck Society.
35. ^ Dunnington 2004, p. 300.
36. ^ a b Bühler 1981, p. 153.
37. ^ Gerhard Falk (1995). American Judaism in Transition: The Secularization of a Religious Community. University Press of America. p. 121. ISBN 978-0-7618-0016-3. Gauss told his friend Rudolf Wagner, a professor of biology at Gottingen University, that he did not fully believe in the Bible but that he had meditated a great deal on the future of the human soul and speculated on the possibility of the soul being reincarnated on another planet. Evidently, Gauss was a Deist with a good deal of skepticism concerning religion but incorporating a great deal of philosophical interest in the Big Questions, that is. the immortality of the soul, the afterlife and the meaning of man's existence.
38. ^ Bühler 1981, p. 152: "Closely related to Gauss' political and social views were his religious beliefs. Despite his religious beliefs. Despite his strong roots in the Enlightenment, Gauss was not an atheist, rather a deist with very unorthodox convictions, unorthodox even if measured against the very liberal persuasions of the contemporary Protestant church."
39. ^ Dunnington 2004, p. 356: "I must confess that such old theologians and song writers as Paul Gerhard have always made a great impression on me; a song by Paul Gerhard always exerted a wonderful power on me, much more than, for example, Moses, against whom as a man of God I have all sorts of qualms."
40. ^ Dunnington 2004, p. 305.
41. ^ Morris Kline (1982). Mathematics: The Loss of Certainty. Oxford University Press. p. 73. ISBN 978-0-19-503085-3.
42. ^ "Gauss, Carl Friedrich". Complete Dictionary of Scientific Biography. 2008. Retrieved 29 July 2012. In seeming contradiction, his religious and philosophical views leaned toward those of his political opponents. He was an uncompromising believer in the priority of empiricism in science. He did not adhere to the views of Kant, Hegel and other idealist philosophers of the day. He was not a churchman and kept his religious views to himself. Moral rectitude and the advancement of scientific knowledge were his avowed principles.
43. ^ Dunnington 2004, p. 311.
44. ^ "Johanna Osthoff 1780–1809 – Ancestry". www.ancestry.com. Retrieved 10 December 2017.
45. ^ "Letter from Carl Friedrich Gauss to Johanna Gauss, 23. October 1809". Der komplette Briefwechsel von Carl Friedrich Gauss. Akademie der Wissenschaften zu Göttingen. Retrieved 26 March 2023.
46. ^ Dunnington 2004, p. 94–95.
47. ^ Bruno 2003, p. 180.
48. ^ Cajori, Florian (19 May 1899). "Carl Friedrich Gauss and his children". Science. New Series. American Association for the Advancement of Science. 9 (229): 697–704. JSTOR 1626244.
49. ^ Dunnington 2004, p. 206.
50. ^ a b c Gerardy, Theo (1966). "C. F. Gauß und seine Söhne". Mitteilungen der Gauß-Gesellschaft Göttingen (in German) (3): 25–35.
51. ^ "Letter: Charles Henry Gauss to Florian Cajori – 21 December 1898". Retrieved 25 March 2023.
52. ^ See also the letter from Robert Gauss to Felix Klein on 3 September 1912.
53. ^ Weinberger, Joseph (1977). "Carl Friedrich Gauß 1777–1855 und seine Nachkommen". Archiv für Sippenforschung und alle verwandten Gebiete (in German). 43/44 (66): 73–98.
54. ^ Mackinnon, Nick (1990). "Sophie Germain, or, Was Gauss a feminist?". The Mathematical Gazette. The Mathematical Association. 74 (470): 346–351. doi:10.2307/3618130. JSTOR 3618130. See esp. p. 347
55. ^ Dunnington 2004, p. 136.
56. ^ Bell, E.T. (2009). "Ch. 14: The Prince of Mathematicians: Gauss". Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York: Simon and Schuster. pp. 218–269. ISBN 978-0-671-46400-4.
57. ^ Letter Gauss to Bolyai from 2 September 1808
58. ^ "Did Gauss know Dirichlet's class number formula in 1801?". MathOverflow. 10 October 2012.
59. ^
60. ^ Teets, Donald; Karen, Whitehead (1965). "The discovery of Ceres. How Gauss became famous". Mathematical Magazine. 19 (90): 83–91.
61. ^ Klein, Felix; Hermann, Robert (1979). Development of mathematics in the 19th century. Math Sci Press. ISBN 978-0-915692-28-6.
62. ^ Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series". Math. Comput. 19 (90): 297–301. doi:10.2307/2003354. JSTOR 2003354.
63. ^ Gauss, C.F. (1876). Theoria Interpolationis Methodo Nova Tractata. Carl Friedrich Gauss Werke (in Latin). Göttingen: K. Gesellschaft der Wissenschaften zu Göttingen. pp. 265–327.
64. ^ Heideman, M.; Johnson, D.; Burrus, C. (1984). "Gauss and the history of the fast fourier transform" (PDF). IEEE ASSP Magazine. 1 (4): 14–21. doi:10.1109/MASSP.1984.1162257. S2CID 10032502. Archived (PDF) from the original on 19 March 2013.
65. ^ Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926.
66. ^ Oscar Sheynin, History of Statistics, Berlin: NG Verlag Berlin, 2012, p. 81.
67. ^ Stigler, Stephen M. (1981). "Gauss and the Invention of Least Squares". Annals of Statistics. 9 (3): 465–474.
68. ^ Gauß, C.F. (1828). Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector (in German). Vandenhoeck und Ruprecht. p. 73. Retrieved 6 July 2021.
69. ^ Listing, J.B. (1872). Ueber unsere jetzige Kenntniss der Gestalt und Grösse der Erde (in German). Göttingen: Dieterich. p. 9.
70. ^ Steven G. Krantz (1 April 2010). An Episodic History of Mathematics: Mathematical Culture through Problem Solving. MAA. pp. 171–. ISBN 978-0-88385-766-3. Retrieved 9 February 2013.
71. ^ Halsted, G.B. (1912). "Duncan M.Y. Sommerville". American Mathematical Monthly. 19 (1): 1–4. doi:10.2307/2973871. JSTOR 2973871.
72. ^ The Prince of Mathematics Archived 18 September 2020 at the Wayback Machine. The Door to Science by keplersdiscovery.com.
73. ^ "Book of Members, 1780–2010: Chapter G" (PDF). American Academy of Arts and Sciences. Retrieved 8 September 2016.
74. ^ Bruno 2003, p. 181.
75. ^ Bühler 1981, pp. 144–145.
76. ^ Hecht, Eugene (1987). Optics. Addison Wesley. p. 134. ISBN 978-0-201-11609-0.
77. ^ Bass, Michael; DeCusatis, Casimer; Enoch, Jay; Lakshminarayanan, Vasudevan (2009). Handbook of Optics. McGraw Hill Professional. p. 17.7. ISBN 978-0-07-149889-0.
78. ^ Ostdiek, Vern J.; Bord, Donald J. (2007). Inquiry into Physics. Cengage Learning. p. 381. ISBN 978-0-495-11943-2.
79. ^ H.J.S Smith,Presidential Address, Proceedings of the London Math. Soc. VIII, 18.
80. ^ Rotman, Joseph J. (2006). A first course in abstract algebra: with applications (3rd ed.). Upper Saddle River, N.J.: Pearson Prentice Hall. pp. 7–8. ISBN 0-13-186267-7. OCLC 61309485.
81. ^ Quoted in Sartorius von Waltershausen, Wolfgang (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8
82. ^ Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington, DC: Joseph Henry Press. p. 202. ISBN 978-0-309-08549-6.
83. ^ Władysław Kozaczuk, Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two, Frederick, Maryland, University Publications of America, 1984, p. 7, note 6.
84. ^ baharuka (25 October 2012). "Die Vermessung der Welt (2012) – Internet Movie Database". Internet Movie Database.
85. ^ "Johann Carl Friedrich Gauß's 241st Birthday". www.google.com. Retrieved 30 April 2018.
86. ^ Kühn, Klaus (2008). "C.F. Gauß und die Logarithmen" (PDF) (in German). Alling-Biburg, Germany. Archived (PDF) from the original on 14 July 2018. Retrieved 14 July 2018.
87. ^ Website of Gauss Society Göttingen.
88. ^ "Der komplette Briefwechsel von Carl Friedrich Gauss". Akademie der Wissenschaften zu Göttingen. Retrieved 10 March 2023.
89. ^ "Familienarchiv Gauß". Signature : G IX 021. Stadtarchiv Braunschweig. Retrieved 25 March 2023.