Carl Friedrich Gauss

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Carl Friedrich Gauss
Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Born	Johann Carl Friedrich Gauss 30 April 1777 Brunswick, Duchy of Brunswick-Wolfenbüttel, Holy Roman Empire
Died	23 February 1855 (aged 77) Göttingen, Kingdom of Hanover
Residence	Kingdom of Hanover
Nationality	German
Fields	Mathematics and physics
Institutions	University of Göttingen
Alma mater	University of Helmstedt
Doctoral advisor	Johann Friedrich Pfaff
Other academic advisors	Johann Christian Martin Bartels
Doctoral students	Christoph Gudermann Christian Ludwig Gerling Richard Dedekind Johann Listing Bernhard Riemann Christian Peters Moritz Cantor
Other notable students	Johann Encke Peter Gustav Lejeune Dirichlet Gotthold Eisenstein Carl Wolfgang Benjamin Goldschmidt Gustav Kirchhoff Ernst Kummer August Ferdinand Möbius L. C. Schnürlein Julius Weisbach
Known for	See full list
Influenced	Sophie Germain Ferdinand Minding
Notable awards	Lalande Prize (1810) Copley Medal (1838)
Signature

Johann Carl Friedrich Gauss (/ɡaʊs/; German: Gauß, pronounced [ɡaʊs] ( listen); Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician, who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, and optics.

Sometimes referred to as the Princeps mathematicorum^[1] (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.^[2]

Early years (1777–1798)

Statue of Gauss at his birthplace, Brunswick

Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), as the son of poor working-class parents.^[3] Indeed, 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 itself occurs 40 days after Easter. Gauss would later solve 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.^[4] He was christened and confirmed in a church near the school he attended as a child.^[5]

Gauss was a child prodigy. There are many anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, 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,^[2] 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;^[6] his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. 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 by 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.^[7]

The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.^[8] He further advanced modular arithmetic, greatly simplifying manipulations in number theory.^{[citation needed]} 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 famous note: "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". On October 1 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.

Middle years (1799–1830)

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 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 (Latin, Arithmetical Investigations), which, among 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.

Title page of Gauss's Disquisitiones Arithmeticae

In that 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, who was 24 at the time, 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.^[9]

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.^[10]

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.^{[citation needed]} 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 1795.^{[citation needed]}

Gauss's portrait published in Astronomische Nachrichten 1828

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic 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.

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."

Four Gaussian distributions in statistics

This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was "stealing" his idea), but it is now generally taken at face value.^[11] 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 János Bolyai, but that he refused to publish any of it because of his fear of controversy.

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade,^[12] fueled Gauss's interest in differential geometry, a field 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.

Later years and death (1831–1855)

Daguerreotype of Gauss on his deathbed, 1855.

Grave of Gauss at Albanifriedhof in Göttingen, Germany.

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 club in German), 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.

In 1840, Gauss published his influential Dioptrische Untersuchungen,^[13] in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics).^[14] Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points^[15] and he derived the Gaussian lens formula.^[16]

In 1854, Gauss notably selected the topic for Bernhard Riemann's now famous Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen.^[17] On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.^[18]

Gauss died in Göttingen, in the Kingdom of Hanover (now part of Lower Saxony, Germany) in 1855 and is interred in the Albanifriedhof cemetery there. Two individuals 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. His brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams (slightly above average) and the cerebral area equal to 219,588 square millimeters^[19] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.^[20]

Religious views

Gauss biographer G. Waldo Dunnington describes Gauss's religious views on these terms:

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 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.^[21]

Apart from his correspondence, there are not many known details about Gauss' personal creed. Many biographers of Gauss disagree with his religious stance, with Bühler and others who considered him a deist with very unorthodox impressions,^[22][23][24] while Dunnington (though admits while Gauss didn't believe literally in all Christian dogmas and it's unknown if he believe in most doctrinal and confessional questions) points out that he was, at least, a nominal Lutheran.^[25]

In connection to this, there's 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 on 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.^[22][26] 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 Paul Gerhardt, than by Moses;^[27] Other of his religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament.^[28]

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

Gauss' 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.^[29]

Gauss declared he firmly believed in the afterlife, and saw spirituality as something essentially important for human beings.^[30] He was quoted stating: "The world would be nonsense, the whole creation an absurdity without immortality,"^[31] and for this statements he was severely criticized by the atheist Eugen Dühring who judged him as a narrow superstitious man.^[32]

Though he was not a church-goer,^[33] 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."^[2] When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved him saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.^[34]

Family

Gauss's daughter Therese (1816—1864)

Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a depression from which he never fully recovered. He married again, to Johanna's best friend named Friederica Wilhelmine Waldeck but commonly known as Minna. When his second wife died in 1831 after a long illness,^[35] one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1817 until her death in 1839.^[2]

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.^[36] Therese kept house for Gauss until his death, after which she married.

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".^[36] 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. Wilhelm also 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. Mathematical historian Eric Temple Bell estimated that, had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.^[37]

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, Bernhard Riemann, and Friedrich Bessel. Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree.

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.^{[citation needed]} This is justified, if unsatisfactorily, by Gauss in his "Disquisitiones Arithmeticae", where he states that all analysis (i.e., the paths one travelled 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.

Anecdotes

There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances.

Another famous 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's 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^[38] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions); some authors, such as Joseph Rotman in his book A first course in Abstract Algebra, question whether it ever happened.

According to Isaac Asimov, Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done."^[39] This anecdote is briefly discussed in G. Waldo Dunnington's Gauss, Titan of Science where it is suggested that it is an apocryphal story.

He referred to mathematics as "the queen of sciences"^[40] 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.^[41]

Commemorations

German 10-Deutsche Mark Banknote (1993; discontinued) featuring Gauss

Gauss (aged about 26) on East German stamp produced in 1977. Next to him: heptadecagon, compass and straightedge.

From 1989 through 2001, Gauss's portrait, a normal distribution curve and some prominent Göttingen buildings were featured on the German ten-mark banknote. 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.

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World (2006), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer Alexander von Humboldt. A film version directed by Detlev Buck was released in 2012.^[42]

In 2007 a bust of Gauss was placed in the Walhalla temple.^[43]

Things named in honor of Gauss include:

• The Gauss Prize, one of the highest honors in mathematics
• Gauss's Law and Gauss's law for magnetism, two of Maxwell's four equations.
• Degaussing, the process of eliminating a magnetic field
• The CGS unit for magnetic field was named gauss in his honour
• The crater Gauss on the Moon^[44]
• Asteroid 1001 Gaussia
• The ship Gauss, used in the Gauss expedition to the Antarctic
• Gaussberg, an extinct volcano discovered by the above-mentioned expedition
• Gauss Tower, an observation tower in Dransfeld, Germany
• In Canadian junior high schools, an annual national mathematics competition (Gauss Mathematics Competition) administered by the Centre for Education in Mathematics and Computing is named in honour of Gauss
• In University of California, Santa Cruz, in Crown College, a dormitory building is named after him
• The Gauss Haus, an NMR center at the University of Utah
• The Carl-Friedrich-Gauß School for Mathematics, Computer Science, Business Administration, Economics, and Social Sciences of Braunschweig University of Technology
• The Gauss Building at the University of Idaho (College of Engineering)
• The Carl-Friedrich-Gauss Gymnasium (a school for grades 5-13) in Worms, Germany

In 1929 the Polish mathematician Marian Rejewski, who would 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's grave.^[45]

Writings

• 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse ("New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors (i.e., polynomials) of the first or second degree")
• 1801: Disquisitiones Arithmeticae (Latin). A German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 1–453. English translation by Arthur A. Clarke Disquisitiones Arithemeticae (Second, corrected edition). New York: Springer. 1986. ISBN 0-387-96254-9..
• 1808: Theorematis arithmetici demonstratio nova. Göttingen: Comment. Soc. regiae sci, Göttingen XVI.. German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 457–462 [Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity]
• 1809: Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections (English translation by C. H. Davis), reprinted 1963, Dover, New York.
• 1811: Summatio serierun quarundam singularium. Göttingen: Comment. Soc. regiae sci, Göttingen.. German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 463–495 [Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity]
• 1812: Disquisitiones Generales Circa Seriem Infinitam
• 1818: Theorematis fundamentallis in doctrina de residuis quadraticis demonstrationes et amplicationes novae. Göttingen: Comment. Soc. regiae sci, Göttingen.. German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 496–510 [Fifth and sixth proofs of quadratic reciprocity]
• 1821, 1823 and 1826: Theoria combinationis observationum erroribus minimis obnoxiae. Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation) English translation by G. W. Stewart, 1987, Society for Industrial Mathematics.
• 1827: Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146. "General Investigations of Curved Surfaces" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.
• 1828: Theoria residuorum biquadraticorum, Commentatio prima. Göttingen: Comment. Soc. regiae sci, Göttingen 6.. German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 511–533 [Elementary facts about biquadratic residues, proves one of the supplements of the law of biquadratic reciprocity (the biquadratic character of 2)]
• 1832: Theoria residuorum biquadraticorum, Commentatio secunda. Göttingen: Comment. Soc. regiae sci, Göttingen 7.. German translation by H. Maser Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8., pp. 534–586 [Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i]
• 1843/44: Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band, pp. 3–46
• 1846/47: Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band, pp. 3–44
• Mathematisches Tagebuch 1796–1814, Ostwaldts Klassiker, Harri Deutsch Verlag 2005, mit Anmerkungen von Neumamn, ISBN 978-3-8171-3402-1 (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)
• Gauss's collective works are online here This includes German translations of Latin texts and commentaries by various authorities