Carl Friedrich Gauss

Johann Carl Friedrich Gauss was born on April 30, 1777, in Brunswick, then the capital of the Duchy of Brunswick-Wolfenbüttel in northern Germany ^[1]. His father, Gebhard Dietrich Gauss, worked variously as a gardener, canal laborer, and bricklayer's foreman; his mother, Dorothea Benze, was the daughter of a stonemason and could barely read. Neither parent recorded the boy's exact birth date. Gauss later worked it out himself from his mother's memory that he had been born on a Wednesday, eight days before the feast of the Ascension, a small calculation problem that reportedly helped push him toward a general method for dating Easter ^[2].

Stories of his precocity are among the most repeated of all Carl Friedrich Gauss facts. He is said to have corrected an error in his father's wage calculations at the age of three, and a famous anecdote holds that as a young schoolboy he summed the integers from 1 to 100 almost instantly by pairing the numbers from opposite ends of the sequence ^[1]. Whatever the precise truth of these tales, his teachers in Brunswick, including the schoolmaster J. G. Büttner and his assistant Martin Bartels, recognized an extraordinary talent and brought the boy to wider attention.

That attention proved decisive. In 1791 Gauss was presented to Carl Wilhelm Ferdinand, Duke of Brunswick, who granted him a stipend that freed the family from worry about his education ^[2]. The duke's support carried Gauss through the Collegium Carolinum in Brunswick from 1792 to 1795 and then to the University of Göttingen, where he enrolled in 1795 still undecided between mathematics and classical philology.

The decision came on March 30, 1796, when the eighteen-year-old Gauss proved that a regular polygon with seventeen sides can be constructed with straightedge and compass alone ^[3]. Greek geometers had known constructions for polygons with 3, 4, 5, and 15 sides and their doublings, and nothing had been added in more than two thousand years. Gauss showed that constructibility depends on the arithmetic of what are now called Fermat primes, a result so striking that it settled his choice of career. He began keeping a mathematical diary that same year; its 146 terse entries, discovered only in 1898, record a torrent of discoveries, many never published ^[3].

His doctoral dissertation, accepted by the University of Helmstedt in 1799, gave a proof of the fundamental theorem of algebra, the statement that every polynomial equation with complex coefficients has a complex root ^[1]. Gauss criticized earlier attempts by d'Alembert and others as incomplete, and he returned to the theorem throughout his life, publishing three further proofs, the last in 1849.

In 1801 he brought out the work that made his European reputation: the Disquisitiones Arithmeticae, written in Latin and financed in part by his patron the duke. The book introduced the congruence notation still used today, gave the first complete proof of the law of quadratic reciprocity, and developed the theory of binary quadratic forms into a systematic discipline ^[4]. Later number theorists, including Dirichlet, Kummer, and Dedekind, treated it as the foundation of their subject. That same year Gauss achieved a very different kind of fame. The asteroid Ceres, discovered by Giuseppe Piazzi in January 1801, had been lost in the sun's glare after only a few weeks of observation. Gauss computed its orbit from the scanty data using new methods of his own, and in December astronomers recovered the object almost exactly where he predicted ^[2]. The feat made his name known far beyond mathematical circles.

Any account of Carl Friedrich Gauss achievements has to range across half a dozen fields, because he treated pure mathematics, astronomy, and physics as parts of a single enterprise. In 1807 he became director of the Göttingen Observatory and professor of astronomy, positions he held until his death ^[1]. His treatise Theoria Motus Corporum Coelestium (1809) set out his orbit determination methods and described the technique of least squares for extracting the best estimate from imperfect observations. Adrien-Marie Legendre had published least squares first, in 1805, and a long priority dispute followed, but Gauss connected the method to the theory of errors and to the bell-shaped curve now called the gaussian or normal distribution ^[5].

From 1818 Gauss directed the geodetic survey of the Kingdom of Hanover, a task that occupied him for years of fieldwork and millions of calculations. Practical triangulation fed directly into deep theory. He invented the heliotrope, an instrument that used reflected sunlight to make distant survey stations visible, and in 1827 he published the Disquisitiones generales circa superficies curvas, which founded the differential geometry of surfaces ^[3]. Its central result, which Gauss named the Theorema Egregium, shows that the curvature of a surface is intrinsic: it can be measured by a being confined to the surface, without any reference to the surrounding space. This idea prepared the ground for Riemann's geometry and, eventually, for general relativity.

In the 1830s he turned to physics in collaboration with the young physicist Wilhelm Weber. Together they investigated terrestrial magnetism, organized the Magnetischer Verein, a network of observatories making simultaneous magnetic measurements across Europe, and in 1833 constructed an electromagnetic telegraph that carried signals about a kilometer across Göttingen, years before commercial telegraphy ^[6]. Gauss also gave magnetism an absolute system of units based on length, mass, and time, and his 1839 paper on the general theory of the earth's magnetic field applied spherical harmonic analysis to locate the magnetic poles. The CGS unit of magnetic flux density was later named the gauss in his honor ^[5].

Much of his most far-reaching work never appeared in print during his lifetime. His diary and letters show that he had developed the essentials of non-Euclidean geometry decades before János Bolyai and Nikolai Lobachevsky published, but he withheld the results, telling a correspondent that he feared the outcry of the Boeotians, his term for uncomprehending critics ^[4]. He worked to a personal motto, pauca sed matura (few, but ripe), and released nothing he did not consider polished and complete.

Gauss married Johanna Osthoff, the daughter of a Brunswick tanner, in 1805, and by his own account the marriage was deeply happy. It ended abruptly: Johanna died in October 1809, shortly after the birth of their third child, and the infant son Louis died the following spring ^[2]. Within a year Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children. Minna suffered long illness, probably tuberculosis, and died in 1831, after which Gauss's youngest daughter Therese ran his household for the rest of his life ^[1].

His relations with his children were uneven. He steered his sons away from mathematics, reportedly unwilling to see the family name attached to lesser work, and quarreled with Eugen, who emigrated to the United States in 1830; a second son, Wilhelm, followed later. Both eventually prospered in America ^[2]. Gauss himself almost never traveled, attending only a single scientific conference, in Berlin in 1828, and leaving Göttingen rarely in his last decades.

In person he was conservative, frugal, and intensely private. He read widely in European literature, followed politics with a skeptical eye, kept careful accounts, and built a modest fortune through shrewd investments ^[6]. Colleagues found him reserved and sometimes forbidding, yet his correspondence, notably with the French number theorist Sophie Germain, whom he praised warmly after learning that his correspondent Monsieur Le Blanc was a woman, shows genuine generosity toward talent ^[4].

Gauss remained scientifically active well past sixty. After Weber was dismissed from Göttingen in 1837 as one of the Göttingen Seven, professors who protested the King of Hanover's revocation of the constitution, the great period of magnetic research wound down, though Gauss himself avoided open political conflict ^[6]. He turned to problems in optics, producing the theory of lens systems still taught as gaussian optics, and to studies in capillarity and mechanics.

In his final decade he took on new subjects with undiminished appetite. He learned Russian in his sixties, well enough to read literature and scientific papers in the original, and he supervised a small number of doctoral students, among them Richard Dedekind and, for his habilitation, Bernhard Riemann, whose 1854 lecture on the foundations of geometry Gauss personally selected and reportedly praised with rare warmth ^[3]. He also worked on the practical mathematics of pensions and annuities for the widows' fund of the university.

His health declined from the early 1850s with an enlarged heart and worsening dropsy. Carl Friedrich Gauss died in his sleep in Göttingen on February 23, 1855, at the age of 77, and was buried in the Albani cemetery there ^[1]. His brain was preserved for study at the university, one of the odder footnotes to a very orderly life.

Few scientists have a comparable claim on posterity. The normal distribution that anchors modern statistics is called gaussian; so are integers of the form a + bi in number theory, the elimination method every linear algebra student learns, a fundamental theorem of vector calculus, and the physical unit of magnetic flux density ^[5]. The German ten mark banknote issued in 1991 carried his portrait together with the bell curve, and a crater on the moon and an expedition ship of the first German Antarctic expedition were named for him.

Within mathematics his influence ran through his students and readers rather than through any school he organized. The Disquisitiones Arithmeticae set the agenda for nineteenth century number theory, his surface theory led through Riemann to the geometry underlying Einstein's physics, and his insistence on rigorous proof helped move mathematics away from the looser standards of the eighteenth century ^[4]. The posthumous publication of his diary and collected works revealed how much he had anticipated: elliptic functions, non-Euclidean geometry, and results in analysis that others rediscovered decades later.

For readers of any Carl Friedrich Gauss biography, the abiding puzzle is the combination of range and reticence. He calculated the orbit of a lost asteroid, mapped a kingdom, measured the earth's magnetism, built a telegraph, and quietly rewrote geometry, while publishing only what met his own severe standard. Laplace is said to have called him the greatest mathematician in Europe, and the title Princeps Mathematicorum, the Prince of Mathematicians, was struck on a commemorative medal ordered by the King of Hanover after his death ^[2].