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PLANETARY ORBITS AND ELLIPTIC FUNCTION THEORY. First edition, very rare separately-paginated offprint, preceding the journal appearance in Commentationes Societatis Regiae Scientiarum Gottingensis (vol. 4, pp. 21-48) by two years."In his first years at Göttingen, Gauss experienced a second upsurge of ideas and publications in various fields of mathematics. Among the latter were several notable papers inspired by his work on the tiny planet Pallas, perturbed by Jupiter . [including] Determinatio attractionis quam in punctum quodvis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita (1818)" (DSB). In this paper, Gauss's last important contribution to theoretical astronomy, he showed "that the secular variations which the elements of the orbit of a planet would experience from another planet which disturbs it are the same as if the mass of the disturbing planet were distributed in an elliptic ring coincident with its orbit, in such a manner that equal masses of the ring would correspond to portions of the orbit described in equal times" (Dunnington, p. 111). (The 'secular variations' are those which occur over a period of time that is long compared to the orbital period of the planet; the 'elements' of the orbit are the parameters which determine it, such as its eccentricity - the extent to which it differs from a circle - and the inclination of the orbit to the plane of the ecliptic.) Gauss showed thatthe attraction caused by such a 'Gaussian ring' could be expressed in terms of elliptic integrals, and he related the evaluation of these integrals to the 'arithmetic-geometric mean' (see below). "In a letter, dated April 16, 1816, to a friend, H. C. Schumacher, Gauss confided that he discovered the arithmetic-geometric mean in 1791 at the age of 14. At about the age of 22 or 23, Gauss wrote a long paper describing his many discoveries on the arithmetic-geometric mean. However, this work, like many others by Gauss, was not published until after his death . Gauss obviously attached considerable importance to his findings on the arithmetic-geometric mean, for several of the entries in his diary, in particular, from the years 1799 to 1800, pertain to the arithmetic-geometric mean" (Almkvist & Berndt, p. 586). Determinatio attractionis was the only work Gauss published on elliptic integrals, although much more remained in manuscript and was published after his death (this showed that he had anticipated some of the later work of Abel and Jacobi in this field). It perfectly illustrates the great breadth of Gauss's interests and expertise, combining as it does studies in both pure and applied mathematics: Gauss did not recognise the barrier between these two disciplines that exists today. "In 1813 on a single sheet appear notes relating to parallel lines, declinations of stars, number theory, imaginaries, the theory of colors, and prisms" (DSB). No copy of the offprint listed on ABPC/RBH. Already well known for contributions to algebra and number theory, Gauss (1777-1855) made his dramatic entry into the astronomical world at the age of 24. Having graduated from the University ofGöttingensome three years earlier, he had returned to his native city of Brunswick to continue the intense mathematical studies for which the Duke of Brunswick had been supporting the precocious youth since he had been in his mid-teens. "In January 1801 Giuseppe Piazzi had briefly observed and lost a new planet [the asteroid Ceres]. During the rest of that year the astronomers vainly tried to relocate it. In September, as his Disquisitiones arithmeticaewas coming off the press, Gauss decided to take up the challenge. To it he applied both a more accurate orbit theory (based on the ellipse rather than the usual circular approximation) and improved numerical methods (based on least squares). By December the task was done, and Ceres was soon found in the predicated position. Seller Inventory # 5312
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