Carl Friedrich Gauss

• 1. Introduction

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  Johann Carl Friedrich Gauss was nineteen years old when he solved a problem the Ancient Greeks had left open for more than two thousand years. He proved that a regular seventeen-sided polygon, the heptadecagon, could be drawn with nothing but a compass and a straightedge. It was the first progress in regular polygon construction in over 2000 years. The discovery convinced him to choose mathematics over philology as a career. Gauss was born on the 30th of April 1777 in Brunswick and died on the 23rd of February 1855. In between, this German mathematician, astronomer, geodesist, and physicist reached into number theory, algebra, analysis, geometry, statistics, and probability. More than 100 mathematical and scientific concepts now carry his name. Yet he published only what he judged complete and above criticism, and that habit hid much of what he knew. What drove a butcher's son from a family of low social status toward the queen of sciences? Why did he refuse to release work that would have made him famous sooner? And how did one man come to predict the path of a lost planet, measure the Earth's magnetic field, and quietly invent ideas that others would rediscover decades later?

• 2. The Prodigy's Origins

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  Gebhard Dietrich Gauss worked as a butcher, a bricklayer, a gardener, and treasurer of a death-benefit fund. His son characterized him as honourable and respected, but rough and dominating at home. Carl Friedrich's mother, Dorothea, was nearly illiterate. The family held a relatively low social status in the Duchy of Brunswick-Wolfenbüttel. An apocryphal story captures the boy's gift. As an elementary student, Gauss and his class were told by their teacher, J.G. Buttner, to sum the numbers from 1 to 100. Gauss answered 5050 far faster than expected. He had realised the sum could be rearranged as fifty pairs that each added to 101, then simply multiplied 50 by 101. The same rule appears centuries earlier in the 12th-century Tosafot commentary on the Babylonian Talmud. When elementary teachers noticed his abilities, they brought him to the Duke of Brunswick, who sent him to the local Collegium Carolinum from 1792 to 1795. The Duke then funded studies at the University of Gottingen until 1798. There his mathematics professor was Abraham Gotthelf Kastner, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians." Gauss graduated as a Doctor of Philosophy in 1799, not at Gottingen but from the University of Helmstedt, receiving the degree in absentia without an oral examination.

• 3. Few But Ripe

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  "Pauca sed Matura," meaning "Few, but Ripe," was the motto on Gauss's personal seal. He was willing to publish only when he considered a work complete and above criticism. This perfectionism delayed the dissemination of many discoveries, and he left several works to be edited posthumously. Many colleagues encouraged him to publicize new ideas and rebuked him when he hesitated too long. Gauss defended himself by saying the initial discovery of ideas was easy, but preparing a presentable elaboration was demanding, for either lack of time or "serenity of mind." His ideal lay in the work itself, not its reward. In a letter to Farkas Bolyai he wrote that it is the act of learning and the act of getting there, not knowledge or possession, that grants the greatest enjoyment. When he had clarified and exhausted a subject, he turned away from it to go into darkness again. He called mathematics "the queen of sciences" and arithmetic "the queen of mathematics." Gauss broke from earlier mathematicians like Leonhard Euler, who let readers follow their reasoning, including wrong turns. Instead he introduced a direct and complete style that did not reveal the author's train of thought. His concept of priority was "the first to discover, not the first to publish." That stance, and his negligent citations, set him apart from his contemporaries.

• 4. The Lost Planet

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  On the 1st of January 1801, Italian astronomer Giuseppe Piazzi discovered a new celestial object and named it Ceres. He could track it only briefly before it vanished behind the glare of the Sun. The mathematical tools of the time could not predict where it would reappear from so few observations. Gauss took up the problem and predicted a position for rediscovery in December 1801. He proved accurate within half a degree when Franz Xaver von Zach at Gotha, and independently Heinrich Olbers in Bremen, found the object near the predicted spot. The method led to an equation of the eighth degree. Gauss's success drew him into the theory of the motion of planetoids disturbed by large planets. He published it in 1809 as Theoria motus corporum coelestium, which introduced the Gaussian gravitational constant. His favorite object was the asteroid Pallas, prized for its great eccentricity and orbital inclination. There Laplace's method failed, so Gauss used his own tools: the arithmetic-geometric mean, the hypergeometric function, and his method of interpolation. In 1812 he found an orbital resonance with Jupiter in proportion 18:7, giving the result first as cipher and revealing its meaning only in letters to Olbers and Bessel. He finished the work in 1816 without a result that satisfied him, ending his activities in theoretical astronomy. His last observation was the solar eclipse of the 28th of July 1851.

• 5. Surveying A Kingdom

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  In May 1820, King George IV ordered Gauss to extend a triangulation southward into the Kingdom of Hanover. The arc measurement project ran from 1820 to 1844. During the summers of 1821 to 1825, Gauss directed the triangulation personally, from Thuringia in the south to the river Elbe in the north. The largest triangle he ever measured ran between Hoher Hagen, the Grosser Inselsberg in the Thuringian Forest, and the Brocken in the Harz mountains. In the thinly populated Luneburg Heath, with no significant summits or buildings, he struggled to find triangulation points and sometimes cut lanes through vegetation. To aim signals across these distances, Gauss invented a new instrument in 1821 with movable mirrors and a small telescope that reflected sunbeams to the triangulation points. He named it the heliotrope. He was assisted by soldiers of the Hanoverian army, among them his eldest son Joseph. The survey fed his interest in differential geometry. In 1828 he published a work marking the birth of modern differential geometry of surfaces, treating a surface from the inner viewpoint of a two-dimensional being constrained to move on it. From this came the Theorema Egregium, the remarkable theorem. It holds that the curvature of a surface can be found by measuring angles and distances on the surface alone, regardless of how the surface sits in space. A consequence is that a sphere cannot be flattened to a plane without distortion, the fundamental problem of map projections. In 1828, studying differences in latitude, Gauss defined the figure of the Earth as the surface everywhere perpendicular to gravity, which his student Johann Benedict Listing later called the geoid.

• 6. Measuring The Earth's Magnetism

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  After Alexander von Humboldt visited Gottingen in 1826, both scientists began intensive research on geomagnetism. The decisive partnership came with the physicist Wilhelm Weber, who took the chair for physics in Gottingen on Gauss's recommendation in 1831. Gauss had been interested in magnetism since 1803. With Weber he developed methods to measure the components of the field and built a magnetometer for absolute values of the Earth's magnetic field strength, not the relative values earlier instruments gave. Its precision was about ten times higher than previous instruments. Gauss provided the first absolute measurement of Earth's magnetic field in 1832. He devised spherical harmonic analysis to describe potential fields and used it to show that most of Earth's magnetic field comes from internal sources. Gauss and Weber founded the Magnetic Association, an international group of observatories that measured the field at arranged dates from 1836 to 1841. Sixty-one stations on all five continents joined a global program, using Gottingen mean time as the standard. The discoveries of Hans Christian Orsted and Michael Faraday turned Gauss toward electromagnetism. He and Weber found rules for branched electric circuits, later rediscovered and published by Gustav Kirchhoff as Kirchhoff's circuit laws. In 1833 they built the first electromechanical telegraph, and Weber connected the observatory with the institute for physics in the town centre. They made no commercial use of it. Weber's departure to Leipzig in 1843 marked the end of the Magnetic Association's activity.

• 7. Ideas Kept In The Dark

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  In a letter to Franz Taurinus in 1824, Gauss outlined what he named a "non-Euclidean geometry," then strongly forbade Taurinus from making any use of it. He had thought about the basics of geometry since the 1790s but only realized in the 1810s that a geometry without the parallel postulate could resolve the long debate. Gauss is credited as the first to discover and study non-Euclidean geometry, and with coining the term. He never published his ideas, avoiding influence on the scientific discussion. The first publications instead came from Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832. Only the publication of his Nachlass in 1900 showed Gauss's own thoughts on the matter. The same silence shadowed his other inventions. Gauss invented an algorithm for what is now called the discrete Fourier transform while calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey found their similar algorithm. The paper appeared only posthumously in 1876. He likely used the method of least squares to minimize error when calculating the orbit of Ceres. Adrien-Marie Legendre published it first in 1805, but Gauss claimed in Theoria motus that he had used it since 1794 or 1795. The disagreement is called the "priority dispute over the discovery of the method of least squares." His seal had warned what to expect: few works, but ripe ones, with the rest left waiting in the dark.

• 8. A Man Of Difficult Character

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  Gauss married Johanna Osthoff on the 9th of October 1805 in St. Catherine's church in Brunswick. She died on the 11th of October 1809, one month after the birth of their son Louis, who died a few months later. Gauss revealed his grief in a last letter to his dead wife in the style of an ancient threnody, the most personal of his surviving documents. He named his children after Giuseppe Piazzi, Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids. On the 4th of August 1810 he married Wilhelmine Waldeck, a friend of his first wife. She died on the 12th of September 1831 after being seriously ill for more than a decade. His close contemporaries agreed that Gauss was a man of difficult character. He often refused compliments, and visitors were sometimes irritated by his grumpy behaviour before his mood shifted to that of a charming host. Apart from his closer circle, others saw him as reserved and unapproachable, "like an Olympian sitting enthroned on the summit of science." His family life was overshadowed by severe problems. His second wife and two daughters suffered from tuberculosis. In a letter to Bessel in December 1831 he called himself "the victim of the worst domestic sufferings." His son Eugen wanted to study philology while Gauss wanted him to become a lawyer; after debts and a public scandal, Eugen left Gottingen in September 1830 and emigrated to the United States. At the age of 62, Gauss began teaching himself Russian, very likely to read Lobachevsky on non-Euclidean geometry. Only his youngest daughter Therese accompanied him in his last years.