Carl Friedrich Gauss (1777-1855)
Carl Friedrich Gauss, received a stipend from the Duke of Brunswick- Wolfenbüttel, which enabled him to enter Brunswick Collegium Carolinum in 1792. At the academy Gauss independently discovered Bode's law, the binomail theorem and the arithmetic-geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.
In 1795 Gauss left Brunswick to study at Göttingen University. Gauss left Göttingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses. This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae.
Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. Gauss's dissertation was a discussion of the fundamental theorem of algebra. With his stipend to support him, Gauss did not need to find a job so devoted himself to research. Gauss predicted the orbital positions of Ceres, a new "small planet", which differed greatly from others' predictions. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method. In June 1802 Gauss investigated the orbit of Pallas.
He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a major two volume treatise on the motion of celestial bodies. 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 1817, although he went on making observations until the age of 70.
Much of Gauss's time was spent on a new observatory, completed in 1816. He published Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, which 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 1820s. Gauss had been asked in 1818 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. 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.
From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. In a book review in 1816 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 believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry.
Allgemeine Theorie... showed that there can only be two poles in the globe and went on to prove an important theorem, which concerned the determination of the intensity of the horizontal component of the magnetic force along with the angle of inclination. Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.