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Programma inaugurale in quo peculiarem differentialia investigandi rationem ex theoria functionum deducit.
Helmstadt: J. H. Kühlin, 1788. An important Gauss manuscript, signed and dated 1800. With Gauss's autograph signature and two geometrical diagrams on front endpaper, and a 12-line mathematical proof in his hand on rear endpaper. Gauss's own personal copy of the inaugural dissertation of Johann Friedrich Pfaff, who supervised Gauss's doctoral thesis and was a close personal friend. This is an important Gauss manuscript, signed and dated 1800 by him, and with a mathematical calculation in his hand relating to orbital mechanics, performed at a time when Gauss was deeply involved in the calculation of the orbit of the minor planet Ceres (see below). "In 1801 the creativity of the previous years was reflected in two extraordinary achievements, the Disquisitiones arithmeticae and the calculation of the orbit of the newly discovered planet Ceres" (DSB). Gauss studied at the Brunswick Collegium Carolinum from 1792 to 1795, and then at the University of Göttingen from 1795 to 1798 (where he formed an intimate friendship with Wolfgang Bolyai), supported throughout by a stipend from the Duke of Brunswick. He returned to his parents in Brunswick on 28 September 1798, where he continued to work alone and without access to a good library. The situation changed with a visit to nearby Helmstedt in October: "I have been in Helmstedt and found a very good reception with Pfaff as well as with the custodian of the library. Pfaff came up to my expectations. He shows the unmistakable sign of the genius, of not leaving a matter until he has dug it out as far as possible" (Gauss to Wolfgang Bolyai, 29 November 1798). Pfaff became the supervisor of Gauss's doctoral thesis, Demonstratio nova theorematis, omnem functionem algebraicam rationalem integram unius vaiabilis in factores reales primi vel secundi gradus resolvi posse, which provided the first rigorous proof of the fundamental theorem of algebra. A doctorate from the University of Helmstedt was conferred upon him on 16 July 1799. "About the middle of December 1799, Gauss returned to Helmstedt again in order to use its library. He was cordially received by the librarian Bruns, as well as by the professor of mathematics Johann Friedrich Pfaff (1765-1825), whose acquaintance Gauss had already made during his stay in Helmstedt in October, 1798. Gauss rented a room in Pfaff's home and furnished it himself, but studied so strenuously and incessantly that the others in the house saw him for only a few hours in the evening. He and Pfaff would then take walks "to the spring and to Harpke." The topics of conversation were usually mathematical in nature" (Dunnington, p. 35). In a letter of 16 December 1799 to Bolyai, Gauss wrote: "I am staying here at Professor Pfaff's whom I esteem as an excellent geometer as well as a good man and my warm friend; a man of an innocent and childlike character, without any of the violent emotions which so dishonor a man and are so widespread among scholars." Gauss returned to Brunswick the following spring, evidently taking with him the present copy of Pfaff's dissertation. The discovery of Ceres on New Year's Day 1801 by Joseph Piazzi was reported in the June issue of Monatliche Correspondenz zur Beföde rderung Erd- und Himmelskunde. The astronomers of Europe raced to observe the new planet for themselves, but without success. Piazzi's detailed observations were finally published in the September issue of Monatliche Correspondenz, which reached Gauss in Brunswick. He immediately began working on the determination of the orbit, temporarily putting aside his mathematical researches. He solved the problem some time in November 1801. The following January, Olbers rediscovered the planet, its position agreeing exactly with Gauss's predicted orbit. "The discovery of the planet Ceres introduced Gauss to the world as a theoretical astronomer of the highest order... It was Piazzi's discovery which gave Gauss the opportunity of revealing, in most impressive form, his remarkable mathematical superiority over all his contemporaries" (Dunnington, pp. 55-6). The mathematical calculations performed by Gauss at the rear of the present volume are difficult to interpret precisely - they were intended only for Gauss himself so naturally detailed explanations were unnecessary. They are titled Determinatur curva per aequationem inter radios vectores...," the determination of curves by equations given in polar coordinates (in modern terminology). Moreover, the calculations bear a close resemblance to some of those eventually published in his Theoria motus corporum coelestium (1809), the mature expression of his work on the calculation of planetary orbits. Together with the dating of the manuscript, the conclusion is inescapable that we have here an early expression of Gauss's thoughts on the determination of orbits, the topic which first brought his genius to the attention of the wider scientific world. As the son of a family serving the government of Württemberg, Pfaff went to the Hohe Karlsschule in Stuttgart at the age of nine, completing his legal studies there in the fall of 1785. On the basis of mathematical knowledge that he acquired by himself, Pfaff soon progressed to reading Euler's Introductio in analysin infinitorum. In the fall of 1785, at the urging of Karl Eugen, the duke of Wurttemberg, he began a journey to increase his scientific knowledge. He remained at the University of Göttingen for about two years, studying mathematics with A. G. Kaestner and physics with G. C. Lichtenberg. In the summer of 1787 he travelled to Berlin, in order to improve his skill in practical astronomy with J. E. Bode. Through the recommendation of Lichtenberg, in 1788 Pfaff was appointed full professor of mathematics at the University of Helmstedt as a replacement for Georg Simon Klügel, who had been called to Halle. New professors at Helmstedt traditionally presented an inaugural essay, printed in the present volume. It deals with the differentials of functions which satisfy functional equations, which provides a new approach to the calculus of logarithmic, exponential and trigonometric functions. Pfaff presented his most important mathematical achievement, the theory of Pfaffian forms, in Methodus generalis, aequationes differentiarum partialium, necnon aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, complete integrandi, which he submitted to the Abhandlungen der Königlichen Akademie der Wissenschaften Berlin (1814-1815, pp. 76-136). Despite receiving an exceedingly favourable review by Gauss, its importance was not appreciated until 1827, when it appeared with a paper by Carl Gustav Jacob Jacobi, Über Pfaff's Methode, eine gewohnliche lineare Differentialgleichung zwischen 2 n Variabeln durch ein System von n Gleichungen zu integrieren (Journal für die reine und angewandte Mathematik, Bd. 2, pp. 347-357). Pfaff's Methodus constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Sophus Lie and Elie Cartan, has developed into the modern theory of exterior differential systems. The present volume is from the personal library of Gauss. It was sold as a duplicate in 1951 by the Göttingen State and University Library, at which time it passed into private hands. VD18 10608788. Pfaff's dissertation is rare: KVK locates copies at National Library of Scotland, Staadtsbibliothek Berlin, Göttingen, Tübingen and Wolfenbüttel only. G. W. Dunnington, J. Gray & F.-E. Dohse, Gauss: Titan of Science, Mathematical Association of America, 2004. 4to (207 x 170 mm), pp [1-3] 4-26, in contemporary plain wrappers, title with ink stamp 'Gauss-Bibliothek'.
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