He considered his discovery, which had eluded scientists and mathematicians for thousands of years, of such significance that he requested a heptadecagon be inscribed on his headstone when he died.
The task seemed too daunting to stonemasons at the time, however. Also in , he advanced number theory through his development of modular arithmetic and the formulation of the law of quadratic reciprocity, demonstrated a theorem of prime numbers, and found that all integers can be represented as a sum of no more than three triangular numbers.
At the request of his patron, Gauss submitted thesis work for a doctorate to the University of Helmstedt, which granted him a degree for his initial proof of the fundamental theorem of algebra he would improve on this proof throughout his life.
With his doctorate in hand, Gauss was assured of the continued patronage of the duke, allowing him to pursue his mathematical and scientific interests with little concern for money. In , he published two notable works. One was Disquisitiones arithmeticae , a treatise he had begun several years before and which was a comprehensive treatment of number theory. In it Gauss recounted the work of his mathematical predecessors, but also corrected all errors and shortcomings he found and included his own new contributions to the subject.
The treatise served as a foundational text in number theory throughout most of the 19th century. In , the asteroid Ceres had been discovered by Giuseppe Piazzi, an astronomer from Italy. The discovery sparked the interest of the scientific community, but Ceres moved behind the sun before anyone was able to calculate its orbit very accurately.
As a result, no one knew where to look for the asteroid when it reemerged, though numerous scientists tried. Gauss was the first to succeed in the task, which required his use of the least squares method of approximation and an improved estimate of orbit shape. When Gauss published his finding, he gained wide recognition and became sought after for his skills in astronomy. He turned down several offers to direct foreign observatories due to his loyalty to his German patron.
His research there led to the writing of a number of other works relating to astronomy. Its significance lies not in the result but in the proof, which rested on a profound analysis of the factorization of polynomial equations and opened the door to later ideas of Galois theory.
His doctoral thesis of gave a proof of the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has as many roots solutions as its degree the highest power of the variable.
Gauss later gave three more proofs of this major result, the last on the 50th anniversary of the first, which shows the importance he attached to the topic. Foremost was his publication of the first systematic textbook on algebraic number theory, Disquisitiones Arithmeticae. This book begins with the first account of modular arithmetic, gives a thorough account of the solutions of quadratic polynomials in two variables in integers, and ends with the theory of factorization mentioned above.
The second publication was his rediscovery of the asteroid Ceres. Its original discovery, by the Italian astronomer Giuseppe Piazzi in , had caused a sensation, but it vanished behind the Sun before enough observations could be taken to calculate its orbit with sufficient accuracy to know where it would reappear.
Many astronomers competed for the honor of finding it again, but Gauss won. His success rested on a novel method for dealing with errors in observations, today called the method of least squares. Thereafter Gauss worked for many years as an astronomer and published a major work on the computation of orbits—the numerical side of such work was much less onerous for him than for most people.
Gauss began corresponding with Bessel , whom he did not meet until , and with Sophie Germain. Gauss married Johanna Ostoff on 9 October, Despite having a happy personal life for the first time, his benefactor, the Duke of Brunswick, was killed fighting for the Prussian army.
In his father died, and a year later Gauss's wife Johanna died after giving birth to their second son, who was to die soon after her. Gauss was shattered and wrote to Olbers asking him to give him a home for a few weeks, to gather new strength in the arms of your friendship - strength for a life which is only valuable because it belongs to my three small children.
Gauss was married for a second time the next year, to Minna the best friend of Johanna, and although they had three children, this marriage seemed to be one of convenience for Gauss.
Gauss's work never seemed to suffer from his personal tragedy. In the first volume he discussed differential equations , conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit.
Gauss's contributions to theoretical astronomy stopped after , although he went on making observations until the age of Much of Gauss's time was spent on a new observatory, completed in , but he still found the time to work on other subjects. The latter work was inspired by geodesic problems and was principally concerned with potential theory. In fact, Gauss found himself more and more interested in geodesy in the s.
Gauss had been asked in to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. He regularly wrote to Schumacher, Olbers and Bessel , reporting on his progress and discussing problems. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope.
However, inaccurate base lines were used for the survey and an unsatisfactory network of triangles. Gauss often wondered if he would have been better advised to have pursued some other occupation but he published over 70 papers between and From the early s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry.
He discussed this topic at length with Farkas Bolyai and in his correspondence with Gerling and Schumacher. In a book review in he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague.
Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry. Gauss replied to praise it would mean to praise myself.
Again, a decade later, when he was informed of Lobachevsky 's work on the subject, he praised its "genuinely geometric" character, while in a letter to Schumacher in , states that he had the same convictions for 54 years indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age this seems unlikely. Gauss had a major interest in differential geometry , and published many papers on the subject. In fact, this paper rose from his geodesic interests, but it contained such geometrical ideas as Gaussian curvature.
The period - was a particularly distressing time for Gauss. He took in his sick mother in , who stayed until her death in , while he was arguing with his wife and her family about whether they should go to Berlin. He had been offered a position at Berlin University and Minna and her family were keen to move there.
In Gauss's second wife died after a long illness. Gauss had known Weber since and supported his appointment. These papers were based on Gauss's potential theory, which proved of great importance in his work on physics. He later came to believe his potential theory and his method of least squares provided vital links between science and nature.
In , Gauss and Weber began investigating the theory of terrestrial magnetism after Alexander von Humboldt attempted to obtain Gauss's assistance in making a grid of magnetic observation points around the Earth. These papers all dealt with the current theories on terrestrial magnetism, including Poisson 's ideas, absolute measure for magnetic force and an empirical definition of terrestrial magnetism.
Dirichlet 's principle was mentioned without proof.
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