Carl Friedrich Gauss

Biography

Youth and education

Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel (now in the German state of Lower Saxony). His family was of relatively low social status. His father Gebhard Dietrich Gauss (1744–1808) worked variously as a butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate. He had one elder brother from his father's first marriage.

Gauss was a child prodigy in mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum, which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the University of Göttingen until 1798. His professor in mathematics was Abraham Gotthelf Kästner, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams. Astronomy was taught by Karl Felix Seyffer, with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. On the other hand, he thought highly of Georg Christoph Lichtenberg, his teacher of physics, and of Christian Gottlob Heyne, whose lectures in classics Gauss attended with pleasure. Fellow students of this time were Johann Friedrich Benzenberg, Farkas Bolyai, and Heinrich Wilhelm Brandes.

He was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved a geometrical problem that had occupied mathematicians since the Ancient Greeks when he determined in 1796 which regular polygons can be constructed by compass and straightedge. This discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss's mathematical diary, a collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time.

An apocryphal story recounts that as an elementary student, Gauss and his class were tasked by their teacher, J.G. Büttner, to sum the numbers from 1 to 100. Much to Büttner's surprise, Gauss replied with the correct answer of 5050 in a vastly faster time than expected. Gauss had presumably realised that the sum could be rearranged as 50 pairs of 101 (1 + 100 = 101, 2 + 99= 101, etc.). Thus, he simply multiplied 50 by 101.

Private scholar

Gauss graduated as a Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated, but from the University of Helmstedt, the only state university of the duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the Russian Academy of Sciences in St. Peterburg and Ludwig-Maximilians-Universität in Landshut. Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans: the Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped.

When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical communities of Bremen and Lilienthal, especially Wilhelm Olbers, Karl Ludwig Harding, and Friedrich Wilhelm Bessel, forming part of the informal group of astronomers known as the Celestial police. One of their aims was the discovery of further planets. They assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium in sectionibus conicis solem ambientium ("Theory of the motion of celestial bodies that orbit the sun in conic sections") (1809).

Professor in Göttingen

In November 1807, Gauss was hired by the University of Göttingen, then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory, and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt, later discovered to be Prince-primate Dalberg, paid the sum.

Gauss took on the directorship of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach, and a heliometer from Fraunhofer.

The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade.

Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about the burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics, and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor, Dedekind, Dirksen, Encke, Gould, Heine, Klinkerfues, Kupffer, Listing, Möbius, Nicolai, Riemann, Ritter, Schering, Scherk, Schumacher, von Staudt, Stern, Ursin; as geoscientists Sartorius von Waltershausen, and Wappäus.

Gauss did not write any textbook and disliked the popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German. He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians.

Gauss's brain

The day after Gauss's death his brain was removed, preserved, and studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams (3.29 lb). Wagner's son Hermann, a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after the first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of either person. Thus, all investigations of Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs.

Family

Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after the birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids.

On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he 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 being seriously ill for more than a decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald. Gauss's mother Dorothea lived in his house from 1817 until she died in 1839.

The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in the summer of 1821. After a short time at university, in 1824 Joseph joined the Royal Hanoverian Army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network into the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways. In 1836 he studied the railroad system in the US for some months.

Eugen left Göttingen in September 1830 and emigrated to the United States, where he spent five years with the army. He then worked for the American Fur Company in the Midwest. He later moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel; he then moved to Missouri, started as a farmer and became wealthy in the shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children.

Personality

In the first two decades of the 19th century, Gauss was the only important mathematician in Germany comparable to the leading French mathematicians. His Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into the French language.

Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. In contrast to previous mathematicians like Leonhard Euler, who let their readers take part in their reasoning, including certain erroneous deviations from the correct path, Gauss introduced a new style of direct and complete exposition that did not attempt to show the reader the author's train of thought.

Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments.

But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai 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.

His posthumous papers, his scientific diary, and short glosses in his own textbooks show that he empirically worked to a great extent. He was a lifelong busy and enthusiastic calculator, working extraordinarily quickly and checking his results through estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used numerous mathematical tables, examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination. Gauss's calculations and the tables he prepared were often more precise than practically necessary. Very likely, this method gave him additional material for his theoretical work.

Gauss was only willing to publish work when he considered it complete and above criticism. This perfectionism was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself by claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left a considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", 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.

Mathematics

Algebra and number theory

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's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts led to considerable clarification of the concept of complex numbers.

In the preface to the Disquisitiones, Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered. The Disquisitiones Arithmeticae, written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces the triple bar symbol (≡) for congruence and uses it for a clean presentation of modular arithmetic. It deals with the unique factorization theorem and primitive roots modulo n. In the main sections, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms.

The Disquisitiones include the Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares, he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards the end of the fifth section, it appears that Gauss already knew the class number formula in 1801.

In the last section, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. This was the first progress in regular polygon construction in over 2000 years. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or the product of a power of 2 and any number of distinct Fermat primes. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields, which amounts to counting integral points on an elliptic curve. An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.

Analysis

One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through Landen's transformation, and a diary entry recorded the discovery of the connection of Gauss's constant to lemniscatic elliptic functions, a result that Gauss stated "will surely open an entirely new field of analysis". He also made early inroads into the more formal issues of the foundations of complex analysis, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – Cauchy's integral theorem – and understood the notion of complex residues when integrating around poles.

Euler's pentagonal numbers theorem, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on Jacobi theta functions, culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case. His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.

Several mathematical fragments in his Nachlass indicate that he knew parts of the modern theory of modular forms. In his work on the multivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM and its two "simplest values". In his unpublished writings he recognized and made a sketch of the key concept of fundamental domain for the modular group. One of Gauss's sketches of this kind was a drawing of a tessellation of the unit disk by "equilateral" hyperbolic triangles with all angles equal to

π

/

4

{\displaystyle \pi /4}

.

An example of Gauss's insight in analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the lemniscate curve, which inspired Abel's theorem on lemniscate division. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of quadratic Gauss sums, in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem from his work on elliptic function theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic function theory, and only the later work of mathematicians such as Jacobi and Hermite has exposed the crux of his argument.

Geometry

The geodetic survey of Hanover fuelled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led him in 1828 to the publication of a work that marks the birth of modern differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as cartesian graphs of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, regardless of the embedding of the surface in three-dimensional or two-dimensional space.

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended manifolds; it clarifies the distinction between the intrinsic properties of the manifold (the metric) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing projections for geographical maps. A portion of this essay is dedicated to a profound study of geodesics. In particular, Gauss proves the local Gauss–Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior.

Gauss's memoir from 1828 lacks the conception of geodesic curvature. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its invariance under isometric transformations, a result that was later obtained by Ferdinand Minding and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by Pierre Ossian Bonnet in 1848 and known as the Gauss–Bonnet theorem.

During Gauss's lifetime, the parallel postulate of Euclidean geometry was heavily discussed. Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it. Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s that a non-Euclidean geometry without the parallel postulate could solve the problem. In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry", but he strongly forbade Taurinus to make any use of it. Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.

Minor mathematical accomplishments

Gauss applied the concept of complex numbers to solve well-known problems in a new concise way. For example, in a short note from 1836 on geometric aspects of the ternary forms and their application to crystallography, he stated the fundamental theorem of axonometry, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers. He described rotations of this sphere as the action of certain linear fractional transformations on the extended complex plane, and gave a proof for the geometric theorem that the altitudes of a triangle always meet in a single orthocenter.

Gauss was concerned with John Napier's "Pentagramma mirificum" – a certain spherical pentagram – for several decades; he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane.

Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given quadrilateral, and discovered a surprising result about the computation of area of pentagons.

Sciences

Astronomy

On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered a new celestial object, presumed it to be the long searched planet between Mars and Jupiter according to the so-called Titius–Bode law, and named it Ceres. He could track it only for a short time until it disappeared behind the glare of the Sun. The mathematical tools of the time were not sufficient to predict the location of its reappearance from the few data available. Gauss tackled the problem and predicted a position for possible rediscovery in December 1801. This turned out to be accurate within a half-degree when Franz Xaver von Zach on 7 and 31 December at Gotha, and independently Heinrich Olbers on 1 and 2 January in Bremen, identified the object near the predicted position.

Gauss's method 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.

The discovery of Ceres led Gauss to the theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum. It introduced the Gaussian gravitational constant.

Since the new asteroids had been discovered, Gauss occupied himself with the perturbations of their orbital elements. Firstly he examined Ceres with analytical methods similar to those of Laplace, but his favorite object was Pallas, because of its great eccentricity and orbital inclination, whereby Laplace's method did not work. Gauss used his own tools: the arithmetic–geometric mean, the hypergeometric function, and his method of interpolation. He found an orbital resonance with Jupiter in proportion 18:7 in 1812; Gauss gave this result as cipher, and gave the explicit meaning only in letters to Olbers and Bessel. After long years of work, he finished it in 1816 without a result that seemed sufficient to him. This marked the end of his activities in theoretical astronomy.

One fruit of Gauss's research on Pallas perturbations was the Determinatio Attractionis... (1818) on a method of theoretical astronomy that later became known as the "elliptic ring method". It introduced an averaging conception in which a planet in orbit is replaced by a fictitious ring with mass density proportional to the time the planet takes to follow the corresponding orbital arcs. Gauss presents the method of evaluating the gravitational attraction of such an elliptic ring, which includes several steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral.

Chronology

Gauss's first publication following his doctoral thesis dealt with the determination of the date of Easter (1800), an elementary mathematical topic. Gauss aimed to present a convenient algorithm for people without any knowledge of ecclesiastical or even astronomical chronology, and thus avoided the usual terms of golden number, epact, solar cycle, domenical letter, and any religious connotations. This choice of topic likely had historical grounds. The replacement of the Julian calendar by the Gregorian calendar had caused confusion in the Holy Roman Empire since the 16th century and was not finished in Germany until 1700, when the difference of eleven days was deleted. Even after this, Easter fell on different dates in Protestant and Catholic territories, until this difference was abolished by agreement in 1776. In the Protestant states, such as the Duchy of Brunswick, the Easter of 1777, five weeks before Gauss's birth, was the first one calculated in the new manner.

Error theory

Gauss likely used the method of least squares to minimize the impact of measurement error when calculating the orbit of Ceres. The method was published first by Adrien-Marie Legendre in 1805, but Gauss claimed in Theoria motus (1809) that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". Gauss proved that the method has the lowest sampling variance within the class of linear unbiased estimators under the assumption of normally distributed errors (Gauss–Markov theorem), in the two-part paper Theoria combinationis observationum erroribus minimis obnoxiae (1823).

In the first paper he proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of the Gauss-Winckler inequality). He derived lower and upper bounds for the variance of the sample variance. In the second paper, Gauss described recursive least squares methods, which he discovered. Gauss's work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert to the Gauss-Helmert model.

Gauss also contributed to problems in probability theory that are not directly concerned with the theory of errors. One example appears as a diary note where he tried to describe the asymptotic distribution of entries in the continued fraction expansion of a random number uniformly distributed in (0,1). He derived this distribution, now known as the Gauss-Kuzmin distribution, as a by-product of the discovery of the ergodicity of the Gauss map for continued fractions. Gauss's solution is the first-ever result in the metrical theory of continued fractions.

Geodesy

Gauss was busy with geodetic problems since 1799 when he helped Karl Ludwig von Lecoq with calculations during his survey in Westphalia. Beginning in 1804, he taught himself some practical geodesy in Brunswick and Göttingen.

Since 1816, Gauss's former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg as head of an observatory, carried out a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south. This project was the basis for map production but also aimed at determining the geodetic arc between the terminal sites. Data from geodetic arcs were used to determine the dimensions of the earth geoid, and long arc distances brought more precise results. Schumacher asked Gauss to continue this work further to the south in the Kingdom of Hanover; Gauss agreed after a short time of hesitation. Finally, in May 1820, King George IV gave the order to Gauss.

An arc measurement needs a precise astronomical determination of at least two points in the network. Gauss and Schumacher used the coincidence that both observatories in Göttingen and Altona, in the garden of Schumacher's house, lay nearly in the same longitude. The latitude was measured with both their instruments and a zenith sector of Ramsden that was transported to both observatories.

Gauss and Schumacher had already determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818. During the summers of 1821 until 1825 Gauss directed the triangulation work personally, from Thuringia in the south to the river Elbe in the north. The triangle between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum size of 107 km (66.5 miles). In the thinly populated Lüneburg Heath without significant natural summits or artificial buildings, he had difficulties finding suitable triangulation points; sometimes cutting lanes through the vegetation was necessary.

For pointing signals, Gauss invented a new instrument with movable mirrors and a small telescope that reflects the sunbeams to the triangulation points, and named it heliotrope. Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope. Gauss was assisted by soldiers of the Hanoverian army, among them his eldest son Joseph. Gauss took part in the baseline measurement (Braak Base Line) of Schumacher in the village of Braak near Hamburg in 1820, and used the result for the evaluation of the Hanoverian triangulation.

Magnetism and telegraphy

Gauss had been interested in magnetism since 1803. After Alexander von Humboldt visited Göttingen in 1826, both scientists began intensive research on geomagnetism, partly independently, partly in productive cooperation. In 1828, Gauss was Humboldt's guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquainted with the physicist Wilhelm Weber.

When Weber got the chair for physics in Göttingen as successor of Johann Tobias Mayer by Gauss's recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of magnetism with a representation for the unit of magnetism in terms of mass, charge, and time. They founded the Magnetic Association (German: Magnetischer Verein), an international working group of several observatories, which carried out measurements of Earth's magnetic field in many regions of the world using equivalent methods at arranged dates in the years 1836 to 1841.

In 1836, Humboldt suggested the establishment of a worldwide net of geomagnetic stations in the British dominions with a letter to the Duke of Sussex, then president of the Royal Society; he proposed that magnetic measures should be taken under standardized conditions using his methods. Together with other instigators, this led to a global program known as "Magnetical crusade" under the direction of Edward Sabine. The dates, times, and intervals of observations were determined in advance, the Göttingen mean time was used as the standard. 61 stations on all five continents participated in this global program. Gauss and Weber founded a series for publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to Leipzig in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Association activity.

Following Humboldt's example, Gauss ordered a magnetic observatory to be built in the garden of the observatory, but the scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thought to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity and differentiated, unlike Humboldt, between "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of the intensity of the magnetic field and constructed a suitable magnetometer to measure absolute values of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus. The precision of the magnetometer was about ten times higher than that of previous instruments. With this work, Gauss was the first to derive a non-mechanical quantity by basic mechanical quantities. He devised spherical harmonic analysis as a technique to describe potential fields, and used it to show most of Earth's magnetic field was from internal sources.

Potential theory

Since Isaac Newton had shown theoretically that the Earth and rotating stars assume non-spherical shapes, the problem of attraction of ellipsoids gained importance in mathematical astronomy. In his first publication on potential theory, the "Theoria attractionis..." (1813), Gauss provided a closed-form expression to the gravitational attraction of a homogeneous triaxial ellipsoid at every point in space. In contrast to previous research of Maclaurin, Laplace and Lagrange, Gauss's new solution treated the attraction more directly in the form of an elliptic integral. In the process, he also proved and applied some special cases of the so-called Gauss's theorem in vector analysis.

In the General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances (1840) Gauss gave a basic theory of magnetic potential, based on Lagrange, Laplace, and Poisson; it seems rather unlikely that he knew the previous works of George Green on this subject. However, Gauss could never give any reasons for magnetism, nor a theory of magnetism similar to Newton's work on gravitation, that enabled scientists to predict geomagnetic effects in the future.

Optics

Gauss's calculations enabled instrument maker Johann Georg Repsold in Hamburg to construct a new achromatic lens system in 1810. A main problem, among other difficulties, was that the refractive index and dispersion of the glass used were not precisely known. In a short article from 1817 Gauss dealt with the problem of removal of chromatic aberration in double lenses, and computed adjustments of the shape and coefficients of refraction required to minimize it. His work was noted by the optician Carl August von Steinheil, who in 1860 introduced the achromatic Steinheil doublet, partly based on Gauss's calculations. Many results in geometrical optics are scattered in Gauss's correspondences and hand notes.

In the Dioptrical Investigations (1840), Gauss gave the first systematic analysis of the formation of images under a paraxial approximation (Gaussian optics). He characterized optical systems under a paraxial approximation only by its cardinal points, and he derived the Gaussian lens formula, applicable without restrictions in respect to the thickness of the lenses.

Mechanics

Gauss's first work in mechanics concerned the earth's rotation. When his university friend Benzenberg carried out experiments to determine the deviation of falling masses from the perpendicular in 1802, what today is known as the Coriolis force, he asked Gauss for a theory-based calculation of the values for comparison with the experimental ones. Gauss elaborated a system of fundamental equations for the motion, and the results corresponded sufficiently with Benzenberg's data, who added Gauss's considerations as an appendix to his book on falling experiments.

After Foucault had demonstrated the earth's rotation by his pendulum experiment in public in 1851, Gerling questioned Gauss for further explanations. This instigated Gauss to design a new apparatus for demonstration with a much shorter length of pendulum than Foucault's one. The oscillations were observed with a reading telescope, with a vertical scale and a mirror fastened at the pendulum. It is described in the Gauss–Gerling correspondence and Weber made some experiments with this apparatus in 1853, but no data were published.

Gauss's principle of least constraint of 1829 was established as a general concept to overcome the division of mechanics into statics and dynamics, combining D'Alembert's principle with Lagrange's principle of virtual work, and showing analogies to the method of least squares.

Metrology

In 1828, Gauss was appointed as head of the board for weights and measures of the Kingdom of Hanover. He created standards for length and measure. Gauss himself took care of the time-consuming measures and gave detailed orders for the mechanical construction. In the correspondence with Schumacher, who was also working on this matter, he described new ideas for high-precision scales. He submitted the final reports on the Hanoverian foot and pound to the government in 1841. This work achieved international importance due to an 1836 law that connected the Hanoverian measures with the English ones.

Recognition

Gauss is frequently regarded as one of the three greatest mathematicians of all time alongside Isaac Newton and Archimedes. He has been called the "Prince of Mathematics".

According to mathematician Felix Klein, when asked who was the greatest mathematician in Germany, French polymath Pierre-Simon Laplace reportedly replied, "Pfaff." When his questioner expressed surprise that he had not named Gauss, Laplace responded, "Oh, Gauss is the greatest mathematician in Europe."

Mathematician and astronomer Henry John Stephen Smith ranked Gauss second only to Newton, writing that:

If we except the great name of Newton (and the exception is one that the great Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute vigorousness in demonstration, which the ancient Greeks themselves might have envied.

Honours and awards

Gauss first became member of a scientific society, the Russian Academy of Sciences, in 1802. Further memberships (corresponding, foreign or full) were awarded by the Academy of Sciences in Göttingen (1802/ 1807), the French Academy of Sciences (1804/ 1820), the Royal Society of London (1804), the Royal Prussian Academy in Berlin (1810), the National Academy of Science in Verona (1810), the Royal Society of Edinburgh (1820), the Bavarian Academy of Sciences of Munich (1820), the Royal Danish Academy in Copenhagen (1821), the Royal Astronomical Society in London (1821), the Royal Swedish Academy of Sciences (1821), the American Academy of Arts and Sciences in Boston (1822), the Royal Bohemian Society of Sciences in Prague (1833), the Royal Academy of Science, Letters and Fine Arts of Belgium (1841/1845), the Royal Society of Sciences in Uppsala (1843), the Royal Irish Academy in Dublin (1843), the Royal Institute of the Netherlands (1845/ 1851), the Spanish Royal Academy of Sciences in Madrid (1850), the Russian Geographical Society (1851), the Imperial Academy of Sciences in Vienna (1848), the American Philosophical Society (1853), the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem.

Both the University of Kazan and the Philosophy Faculty of the University of Prague appointed him honorary member in 1848.

Gauss received the Lalande Prize from the French Academy of Science in 1809 for the theory of planets and the means of determining their orbits from only three observations, the Danish Academy of Science prize in 1823 for his memoir on conformal projection, and the Copley Medal from the Royal Society in 1838 for "his inventions and mathematical researches in magnetism".

Gauss was appointed Knight of the French Legion of Honour in 1837, and became one of the first members of the Prussian Order Pour le Merite (Civil class) when it was established in 1842. He received the Order of the Crown of Westphalia (1810), the Danish Order of the Dannebrog (1817), the Hanoverian Royal Guelphic Order (1815), the Swedish Order of the Polar Star (1844), the Order of Henry the Lion (1849), and the Bavarian Maximilian Order for Science and Art (1853).

The Kings of Hanover appointed him the honorary titles "Hofrath" (1816) and "Geheimer Hofrath" (1845). In 1949, on the occasion of his golden doctor degree jubilee, he received honorary citizenship of both Brunswick and Göttingen. Soon after his death a medal was issued by order of King George V of Hanover with the back inscription dedicated "to the Prince of Mathematicians".

Names and commemorations

List of things named after Carl Friedrich Gauss

Selected writings

Mathematics and astronomy

1799: 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 of the first or second degree]. Helmstedt: C. G. Fleckeisen. (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) Original book

1816: "Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 107–134. Original

1816: "Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 135–142. Original

1850: "Beiträge zur Theorie der algebraischen Gleichungen". Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 4: 34–35. Original (Lecture from 1849)

Die vier Gauss'schen Beweise für die Zerlegung ganzer algebraischer Funktionen in reelle Faktoren ersten und zweiten Grades. (1799–1849) [The four Gaussian proofs of the fundamental theorem of algebra]. Translated by Netto. Leipzig: Wilhelm Engelmann. 1890. (German)

1800: "Berechnung des Osterfestes" [Calculation of Easter]. Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde (in German). 2: 121–130. Original

1801: Disquisitiones Arithmeticae. Leipzig: Gerh. Fleischer jun.

Gauss, Carl Friedrich (1986). Disquisitiones Arithmeticae & other papers on number theory. Translated by Clarke, Arthur A. (2nd, corrected ed.). New York: Springer. doi:10.1007/978-1-4939-7560-0. ISBN 978-0-387-96254-2. (translated from the second German edition, Göttingen 1860)

Physics

1804: Fundamentalgleichungen für die Bewegung schwerer Körper auf der Erde ( in original book: Benzenberg, Johann Friedrich. Versuche über das Gesetz des Falls, über den Widerstand der Luft und über die Umdrehung der Erde [Experiments on the Law of falling Bodies, on the Resistance of Air, and of the Rotation of the Earth]. Dortmund: Gebrüder Mallinckrodt. pp. 363–371. Original)

1813: "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 2: 1–24. Original (contains Gauss's theorem of vector analysis)

1817: "Ueber die achromatischen Doppelobjective besonders in Rücksicht der vollkommnern Aufhebung der Farbenzerstreuung" [On achromatic double lenses with special regard to a more complete dispersion of colours]. Zeitschrift für Astronomie und verwandte Wissenschaften (in German). IV: 345–351.

1829: "Über ein neues allgemeines Grundgesetz der Mechanik" [On a new General Fundamental Law of Mechanics]. Journal für die reine und angewandte Mathematik. 1829 (4): 232–235. 1829. doi:10.1515/crll.1829.4.232. S2CID 199545985.

1830: "Principia generalia theoriae figurae fluidorum in statu aequilibrii". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 7: 39–88. Original (from 1829)

1841: "Intensitas vis magneticae terrestris ad mensuram absolutam revocata". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 8: 3–44. Original (from 1832)

The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement. Translated by Susan P. Johnson.

1836: Erdmagnetismus und Magnetometer (Original book: H.C. Schumacher (ed.). Jahrbuch für 1836 (in German). Vol. 1836. Tübingen: J.G.Cotta'sche Buchhandlung. pp. 1–47.)

1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte (Original book: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte [General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances] (in German). Leipzig: Weidmannsche Buchhandlung. 1840.

Collected works

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

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)

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)

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)

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 978-3945941379. (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)