Instead of considering the surface as part of a three-dimensional displaying depth space, Gauss set up a network of coordinates on the surface itself, showing that the geometry of the surface can be described completely in terms of measurements in this network. Quadratic residues, which led to the law of quadratic reciprocity that Gauss had discovered before he was 18, and indeed power residues in general, are treated extensively. At the age of three, Gauss is said to have corrected an arithmetical error for his father. In June 1802 Gauss visited Olbers who had discovered Pallas in March of that year and Gauss investigated its orbit. Gauss and Weber achieved much in their six years together. Another topic on which Gauss largely concealed his ideas from his contemporaries was. He showed that the series, called the hypergeometric series, can be used to define many familiar and many new functions.
The paper Theoria combinationis observationum erroribus minimis obnoxiae 1823 , with its supplement 1828 , was devoted to mathematical statistics, in particular to the least squares method. It was appropriately a variation on his dissertation of 1799. He wrote many articles, and a biography: Carl Frederick Gauss: Titan of Science. Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. He also worked on a new method for determining the orbits of new asteroids.
From that day Gauss was popular in the whole school. As adviser on geodesy to the Hanoverian government, Gauss had to consider the problem of surveying hilly country. These investigations, an extension of his earlier work on quadratic residues, involved the use of complex numbers. At age 19, Gauss demonstrated a method for constructing a using only a and which had eluded the Greeks. As a student at Göttingen, he began to the a priori truth of and suspected that its truth might be. It is clear from Gauss's notebooks that he recognized the double periodicity of the elliptic functions; however, the work was unpublished, and discovery of the property is credited to N. Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick now Germany Died: 23 Feb 1855 in Göttingen, Hanover now Germany At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately.
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. 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. Gauss solved the general problem of making a of one surface onto another. He was more interested in the task of establishing a world-wide net of magnetic observation points. There were seven sections, all but the last section, referred to above, being devoted to number theory. Its original discovery, by the Italian astronomer in 1800, had caused a sensation, but it vanished behind the Sun before enough observations could be taken to calculate its with sufficient accuracy to know where it would reappear.
Gauss proved the , which states that every has a root of the form a+ bi. New York: Exposition Press, 1955. He refused to publish theories that were not finished and perfect. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen. Defining a straight line as the shortest distance between two points, measured along the surface, the geometry of a curved surface can be regarded as a two-dimensional non-Euclidean geometry. His father labored as a gardner and brick-layer and was regarded as an upright, honest man. Apart from his books Gauss published a number of memoirs, mainly in the journal of the Royal Society of Göttingen.
Gauss gave three proofs for this: the first of these, given in his thesis, assumes that a continuous function which takes positive and negative values is necessarily zero for some value of the variable. Impressed by this ability and by his gift for languages, his teachers and his devoted mother recommended him to the in 1791, who granted him financial assistance to continue his education locally and then to study at the from 1795 to 1798. Gauss was the first to prove the fundamental theorem of algebra; in fact, he produced four entirely different proofs for this theorem over his lifetime, clarifying the concept of complex number considerably along the way. If he proceeded from an explanation of principles to the development of mathematical formulas, then he got up, and in a stately very upright posture he wrote on a blackboard beside him in his peculiarly beautiful handwriting: he always succeeded through economy and deliberate arrangement in making do with a rather small space. This groundbreaking discovery made while he was still a student, the construction of a regular heptadecagon a 17-sided polygon , encouraged him to choose a career in mathematics.
He completed Disquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though it was not published until 1801. In 1832, 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. Gauss had known Weber since 1828 and supported his appointment. In 1818, Gauss started a geodesic survey of the state of Hanover, work which later lead to the development of the normal distribution for describing measurement errors and an interest in differential geometry and his theorema egregrium establishing an important property of the notion of curvature. In 1733 an attempt to prove the postulate was made by Girolamo Saccheri, who, in fact, invented two non-Euclidean geometries only to reject them for unsound reasons. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss returned to Brunswick where he received a degree in 1799.
Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70. Among other times, Gauss visited Munich in 1816. Gauss's method involved determining a conic section in space, given one focus the Sun and the conic's intersection with three given lines lines of sight from the Earth, which is itself moving on an ellipse, to the planet and given the time it takes the planet to traverse the arcs determined by these lines from which the lengths of the arcs can be calculated by Kepler's Second Law. This led him to develop the idea that the measurements of a curved surface could be developed in terms of Gaussian coordinates points. New ones were ordered in Munich. The second publication was his rediscovery of the. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method.