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§§ 276, 277] observations of its position, which was published in his, Theoria Motus (1809). As we have seen (chapter, § 236), the complete determination of a planet's orbit depends on six independent elements: any complete observation of the planet's position in the sky, at any time, gives two quantities, e.g. the right ascension and declination (chapter , § 33); hence three complete observations give six equations and are theoretically adequate to determine the elements of the orbit; but it had not hitherto been found necessary to deal with the problem in this form. The orbits of all the planets but Uranus had been worked out gradually by the use of a series of observations extending over centuries; and it was feasible to use observations taken at particular times so chosen that certain elements could be determined without any accurate knowledge of the others; even Uranus had been under observation for a considerable time before its path was determined with anything like accuracy; and in the case of comets not only was a considerable series of observations generally available, but the problem was simplified by the fact that the orbit could be taken to be nearly or quite a parabola instead of an ellipse (chapter, § 190). The discovery of the new planet Ceres on January 1st, 1801 (§ 294), and its loss when it had only been observed for a few weeks, presented virtually a new problem in the calculation of an orbit. Gauss applied his new methods—including that of least squares—to the observations available, and with complete success, the planet being rediscovered at the end of the year nearly in the position indicated by his calculations.

277. The theory of the "reduction" of observations (chapter, § 218) was first systematised and very much improved by Friedrich Wilhelm Bessel (1784-1846), who was for more than thirty years the director of the new Prussian observatory at Königsberg. His first great work was the reduction and publication of Bradley's Greenwich observations (chapter, § 218). This undertaking involved an elaborate study of such disturbing causes as precession, aberration, and refraction, as well as of the errors of Bradley's instruments. Allowance was made for these on a uniform and systematic plan, and the result was the publication in 1818,