Carl Friedrich Gauss

Gauss was born on April 30, 1777, in Brunswick in the Duchy of Brunswick (now in Germany), and was baptized Johann Friedrich Carl; he later dropped his first name and reversed the order of the other two. He was the only child of his father, Gebhard Dietrich Gauss, and his father’s second wife, Dorothea Benze; his father worked as a bricklayer and gardener and kept accounts for a local insurance fund, and his mother, the daughter of a stonemason, could barely read.⁠[1][2][2][2][2]

He showed a talent for calculation early: before he was three, while watching his father calculate wages, he pointed out an error. After entering primary school, he instantly found the sum of the integers from 1 to 100 by treating them as 50 pairs, each summing to 101, amazing his teacher Büttner and the assistant Martin Bartels; he and Bartels afterward studied algebra and the rudiments of calculus together outside school hours.⁠[2][1][2]

In 1788, his father agreed to let Gauss enter the Gymnasium and study after school instead of spinning to help support the family. At the Gymnasium he made rapid progress in classics and mathematics, largely on his own; E. A. W. Zimmermann, a professor at the local Collegium Carolinum, introduced him at court.⁠[2][2][2]

In early 1791, Carl Wilhelm Ferdinand, Duke of Brunswick, learned of Gauss’s talents and became his patron, paying for his studies at the Collegium Carolinum from 1792 to 1795 and at the University of Göttingen from 1795 to 1798, and continuing to support him until the duke’s death in 1806.⁠[2][2]

While at the Collegium Carolinum, he independently discovered, through extensive calculation, Bode’s law of planetary distances, the binomial theorem for rational exponents, and the arithmetic-geometric mean, and in March 1795 he rediscovered the law of quadratic reciprocity; he also conjectured the prime number theorem, which was not proved until 1896, by J. Hadamard.⁠[2][2][2]

In October 1795, Gauss entered the University of Göttingen, still undecided whether to make mathematics or philology his career; he was more attracted by the classicist G. Heyne and thought little of the mathematics teacher A. G. Kästner. His only recorded friend among the students was the Hungarian Farkas Bolyai, with whom he corresponded for many years afterward.⁠[2][2][1]

In the same year, he conceived of fitting observational data by choosing the parameter values that minimize the sum of squared residuals, the earliest form of the method of least squares; according to his diary, he completed the derivation of the method on June 17, 1798.⁠[2][2][2]

On March 30, 1796, Gauss discovered that a regular heptadecagon (17-gon) can be constructed with straightedge and compass, the first progress on the problem in two thousand years; by June 1 of that year, he had further found the condition for a regular polygon with an odd number of sides to be constructible: the number of sides must be a Fermat prime or a product of distinct Fermat primes. The discovery led him to make mathematics his career and was also the first entry in his “mathematical diary.”⁠[2][2][3]

In 1798, Gauss left Göttingen without a diploma and returned to Brunswick to pursue his research alone. In 1799, with a dissertation giving the first rigorous proof of the fundamental theorem of algebra (every nonconstant polynomial has a root), he received a doctorate in absentia from the University of Helmstedt under the nominal supervision of J. F. Pfaff. The duke continued to pay his stipend, allowing him to devote himself to research without needing to find a job.⁠[1][2][2][2][1]

In August 1800, Gauss published a numerical algorithm for calculating the date of Easter, reducing the complicated reckoning of the church calendar to a few simple formulas; it was also his first astronomical publication. The original version gave wrong dates for certain years, and he corrected it in 1807.⁠[3][2][2]

Disquisitiones Arithmeticae was completed in the autumn of 1798 and went to press in 1799, but because the original print shop was sold, printing was not finished until September 1801. The book has seven sections, all devoted to number theory except the seventh, which deals with the construction of regular polygons. It introduced the congruence of integers with respect to a modulus, proved the law of quadratic reciprocity, developed the theory of composition of quadratic forms, and completely analyzed the cyclotomic equation, and it is regarded as marking the beginning of number theory as a separate, systematic branch.⁠[2][1][2][2][2]

On January 1, 1801, the Italian astronomer G. Piazzi discovered the small body Ceres, but after observing only about 9 degrees of its orbit he lost track of it as it came close to the Sun. In September of that year, Gauss calculated its orbit using an orbit theory based on the ellipse and numerical methods based on least squares, arriving at a predicted position that differed greatly from those of others; on December 7, F. X. von Zach found Ceres again almost exactly where Gauss had predicted. Gauss did not make his method public at the time, and this achievement, together with the Disquisitiones Arithmeticae, made him famous.⁠[1][2][2][1][2]

In 1802, Gauss was elected a corresponding member of the Imperial Academy of Arts and Sciences in St. Petersburg, and in the same year he received a job offer from St. Petersburg; in 1804 he was elected a fellow of the Royal Society of London. Seeking a more secure position, he decided on a career in astronomy and began preparing to become director of the Göttingen observatory.⁠[2][2][2][2]

On October 9, 1805, Gauss married Johanna Osthoff, a Brunswick woman of similar family background. The following year, his patron the Duke of Brunswick was fatally wounded while leading the Prussian armies against Napoleon.⁠[1][2][2]

In 1807, the Hanoverian government appointed Gauss professor of astronomy and director of the observatory of the University of Göttingen, and he took up the post at the end of that year. Construction of the new observatory outside the city walls had begun in 1803, with funds initially provided by King George III of England, but it was interrupted by the Napoleonic Wars and not completed until 1816; until then, he worked in a makeshift observatory in an abandoned tower of the old city walls. He worked in Göttingen for the rest of his life and seldom left the city.⁠[3][1][4][4][2][2]

In 1809, Gauss published the astronomical treatise Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, which systematically set out methods for calculating the motion of celestial bodies along conic sections and for determining the orbit of a comet or planet from three observations; its third section (articles 172–189) gives a detailed exposition of the method of least squares. The German text had been completed in 1806, but amid the turmoil following Prussia’s defeat, the publisher would accept it only if it were translated into Latin, so publication was delayed until 1809.⁠[1][2][2][2][2]

In the book he used inverse probability to derive the probability distribution of observational errors, the curve later known as the Gaussian (normal) distribution, and from it derived the method of least squares, weighting methods, and the elimination procedure for solving the normal equations. The French mathematician Legendre had independently published the method of least squares in 1805, without considering probability or the precision of observations; a priority dispute between the two followed, bitter on Legendre’s side.⁠[3][2][2][2][2][2]

In the same year, his wife Johanna died soon after giving birth to their third child, who also did not survive. Less than a year later, Gauss married Minna Waldeck, a close friend of his late wife.⁠[2][2]

In 1810, Gauss became a full member of the Royal Academy of Sciences in Berlin. Berlin repeatedly invited him to take up a post there, and Alexander von Humboldt long hoped to bring him to Berlin to lead the work of the academy, but he ultimately remained in Göttingen.⁠[2][2][1]

In his first years at Göttingen, Gauss published a series of papers arising from the problem of the perturbation of the minor planet Pallas by Jupiter: Disquisitiones generales circa seriem infinitam (1813) was an early rigorous treatment of series and introduced the hypergeometric functions; papers of 1816 dealt respectively with numerical integration and with the efficiency of statistical estimators; and an 1813 paper on the attraction of ellipsoids was an important early work in potential theory.⁠[2][2][2][2][2]

The new observatory was completed in 1816 but not fully equipped until 1821. In the same year, Gauss made a five-week trip to Bavaria with his ten-year-old son and one of his students, meeting optical instrument makers such as Reichenbach and Fraunhofer and buying his best instruments from them. In 1817 he ended his theoretical astronomical work, but he continued positional observing, calculating, and reporting his results until his final illness.⁠[2][2][2][2]

He had explored the consequences of denying the parallel postulate as early as his student days, and his conception of non-Euclidean geometry gradually matured during his early years in Göttingen, but he hinted at his views publicly only once, in a book review of 1816, and privately feared ridicule. In 1831 Farkas Bolyai sent him the work of his son, János Bolyai, on the subject, and Gauss replied that to praise it would be to praise himself; ten years later he learned of Lobachevsky’s work and arranged for Lobachevsky to be elected a corresponding member of the Göttingen Academy, but he never publicly supported these new ideas.⁠[2][2][1][1][2]

In 1818, Gauss was asked to carry out a geodetic survey of the Kingdom of Hanover to link up with the existing Danish grid. The project was not officially approved until 1820, but from 1818 he did the fieldwork himself every summer and reduced the data in winter, persisting for eight years despite difficulties with transport, weather, funding, and manpower; after 1825 he confined himself to supervision and calculation, and the triangulation of Hanover was completed in 1847, by which time he had handled more than a million numbers on his own.⁠[1][2][2][2]

The actual accuracy of the survey fell short of his expectations: the base lines were laid out carelessly and the network of triangles was unsatisfactory, so the results were good only for rough geographic and military maps. The work did, however, drive his research on conformal mapping, from which German geodesists later developed the Gauss–Krüger projection (1912), which became the basis for topographic grids that take into account the spheroidal shape of the earth.⁠[1][2][2]

To observe distant survey points by day, Gauss hit on the idea of using reflected sunlight; after working out the optical principles, he designed the heliotrope, and the first one was built in 1821. Combining a mirror with a small telescope, it had the brightness of a first-magnitude star at a distance of 15 miles, and it became standard equipment for large-scale triangulation until it was superseded by improved models after 1840 and by aerial surveying in the 20th century.⁠[2][2][2][2]

On February 15, 1821, Gauss presented to the Royal Academy of Sciences in Göttingen the first part of Theoria combinationis observationum erroribus minimis obnoxiae, giving a new formulation of the method of least squares that does not depend on any particular error distribution: as long as the variances of the observations are finite, the least squares estimator has the smallest mean squared error among linear unbiased estimators. The second part in 1823 and a supplement in 1828 completed the theory, and the related result later became known as the Gauss–Markov theorem.⁠[2][2][1][2]

In 1822, in response to a prize competition of the Copenhagen Academy, Gauss wrote a paper on mapping one surface onto another so that the two are similar in their smallest parts (that is, conformal mapping); it won first prize and was published in 1825.⁠[2][1]

In 1828, Gauss published Disquisitiones generales circa superficies curvas, which grew out of three decades of geodetic reflection; it introduced concepts such as Gaussian curvature and proved the theorema egregium (“remarkable theorem”): when a surface is deformed isometrically, the Gaussian curvature at corresponding points is unchanged. The paper opened more than a century of research in differential geometry.⁠[1][1][2]

In the same year, Alexander von Humboldt persuaded him to attend the Naturforscherversammlung in Berlin, the only scientific convention of his life. He stayed for three weeks in Humboldt’s house and met the young experimental physicist Wilhelm Weber; the visit became the occasion for his turn to research on geomagnetism. In 1829, he published the principle of least constraint.⁠[2][2][2][2]

On September 13, 1831, Gauss’s second wife died after a long illness; two days later, Wilhelm Weber arrived in Göttingen to take up the professorship of physics, an appointment Gauss had supported. The two began a close collaboration, with Gauss leading on the theoretical side and Weber on the experimental side.⁠[2][1][2]

On December 15, 1832, Gauss presented Intensitas vis magneticae terrestris ad mensuram absolutam revocata to the Royal Academy of Sciences in Göttingen, making the first systematic use of “absolute units” based on the three fundamental quantities of length, mass, and time to measure a nonmechanical quantity, the intensity of the earth’s magnetism. The paper acknowledged Weber’s help but did not list him as a coauthor.⁠[2][2][2]

Stimulated by Faraday’s discovery of induced current in 1831, Gauss and Weber investigated electrical phenomena and in 1833 arrived at what were later called Kirchhoff’s laws. Weber strung a double wire about a mile long over houses and two towers, connecting the observatory with the physics laboratory in town; in 1833 the two began sending telegraph messages over the line, first single words and later complete sentences.⁠[2][2][4][2][2]

Gauss recognized the military and economic value of the telegraph and tried to promote its large-scale adoption by government and industry; in 1835–1836 plans were also drawn up for its use on the Leipzig–Dresden railroad, but they were dropped when the railroad required the wires to be laid underground. Others, such as Steinheil (1837) and Morse (1838), later independently developed more practical methods, and the two men’s pioneering work was gradually forgotten; by the time lightning destroyed the line in 1845, it was no longer in use.⁠[2][2][2][2]

In the same year, at Gauss’s suggestion, the University of Göttingen built a magnetic observatory containing no magnetic metal.⁠[2]

Gauss and Weber organized the Magnetischer Verein (Göttingen Magnetic Association), which brought together observatories in various places for simultaneous geomagnetic observations; its Resultate aus den Beobachtungen des magnetischen Vereins appeared in six volumes from 1836 to 1841, containing 15 papers by Gauss, 23 papers by Weber, and the two men’s joint Atlas des Erdmagnetismus (1840). In 1837, the two jointly invented the bifilar magnetometer.⁠[2][2][2]

His Allgemeine Theorie des Erdmagnetismus of 1839, based on observational data, expressed the magnetic potential at any point on the earth’s surface as an infinite series of spherical functions and used data from the worldwide observation network to compute the first 24 coefficients; his paper of 1840 was the first systematic treatment of potential theory as a mathematical subject. After this he ended his research on magnetism.⁠[2][2][2][2]

In November 1837, Ernst August, the new King of Hanover, abrogated the constitution of 1833 and required public servants to swear a personal oath of allegiance to him. Seven Göttingen professors, including Weber and Gauss’s son-in-law, the orientalist G. H. A. von Ewald, jointly protested and were promptly dismissed, an episode known as the “Göttingen Seven.” Gauss took no public action, and his private efforts had no effect. Weber later took a post in Leipzig, ending the two men’s collaboration; when Weber regained his position in 1848, Gauss was no longer doing research in physics.⁠[2][2][2][2][2][2]

In 1838, Gauss received the Copley Medal from the Royal Society. Over his lifetime he was elected to academies of sciences in countries including Germany, Russia, Britain, Spain, and France and received about 75 foreign honors; because he disliked traveling far, physical items such as medals were mostly brought to Göttingen by colleagues or students.⁠[2][3][3]

In 1841, Gauss published the optical work Dioptrische Untersuchungen, which analyzed the path of light through a system of lenses and showed that any lens system is equivalent to a properly chosen single lens. This was his last significant scientific contribution; thereafter the intensity of his research gradually declined.⁠[2][2][2]

In 1842, Gauss was awarded the Pour le mérite, the highest order of the Kingdom of Prussia.⁠[2]

From 1845 to 1851, Gauss recalculated the actuarial tables for the University of Göttingen’s widows’ fund, putting it on a sound actuarial basis. The work gave him practical financial experience; he later invested in bonds issued by private companies and left an estate equal to about 200 times his annual salary. In his later years he also learned to read and speak Russian fluently, and he served several times as dean of the faculty at the University of Göttingen.⁠[1][2][1][2][2][2]

On July 16, 1849, exactly 50 years after he received his doctorate, Göttingen held a celebration for Gauss, and he was made an honorary citizen of both Brunswick and Göttingen. From the mathematical community only Jacobi and Dirichlet attended; at the celebration Gauss presented his fourth proof of the fundamental theorem of algebra, a variation of the proof in his 1799 doctoral dissertation.⁠[2][2][1][2]

In 1851, at the age of 74, Gauss made his last astronomical observation, and in the same year he approved Bernhard Riemann’s doctoral thesis on the foundations of complex analysis. In 1854 he was still studying a modified Foucault pendulum; in June of that year, he heard Riemann’s probationary lecture on “the foundations of geometry,” a topic Gauss had chosen. A few days later, he left Göttingen for the last time to see the construction of the railway from Kassel.⁠[2][1][2][2][2]

Gauss died in his sleep in Göttingen early on the morning of February 23, 1855, at the age of almost 78. While seriously ill and confined to bed, he kept up his reading, correspondence, and trading in securities.⁠[1][3][2]

Gauss’s income came mainly from his astronomy post; by one count, of the courses he gave at Göttingen, 70% were on astronomy, 15% on mathematics, 9% on geodesy, and 6% on physics. From 1835 until his death he taught a course on the method of least squares every year. His students included astronomers such as Schumacher, Encke, and Möbius, and mathematicians such as Riemann, Dedekind, and Moritz Cantor.⁠[3][2][3]

In his lifetime he published only about half of his recorded innovative ideas, in a terse style that found few readers; the rest of his results were scattered in notes, correspondence, and reports to official bodies, and became public only many years later. More than 7,000 letters to and from him are known to survive. His contemporaries called him the “prince of mathematicians” (princeps), and he is often ranked alongside Archimedes and Newton.⁠[2][2][2][2]

His method of least squares had already become a basic tool of astronomy and geodesy worldwide during his lifetime; when Karl Pearson and G. Udny Yule developed the theory of correlation in the 1890s, they found that the mathematical tools he had devised for least squares could be applied directly to correlation analysis.⁠[2][2]

On May 17, 1962, 14 people founded the Gauss-Gesellschaft (Gauss Society) in Göttingen to preserve the memory of Gauss and study his life and works, and since 1964 it has published an annual, Mitteilungen der Gauss-Gesellschaft. In 1977 Brunswick and Göttingen held commemorations of the 200th anniversary of his birth; 2005 was designated the “Gauss Year”; and in 2009 a bust of him was installed at the Walhalla memorial near Regensburg. The old Göttingen observatory where he worked for many years is now called the “Historic Observatory” and is no longer used for astronomical observation.⁠[3][3][3][3][3][3][3]