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 the theory of biquadratic residues (1825 and 1831), the second of which contains a theorem of biquadratic reciprocity.

Gauss was led to astronomy by the discovery of the planet Ceres at Palermo in 1801. His determination of the elements of its orbit with sufficient accuracy to enable Olbers to rediscover it, made the name of Gauss generally known. In 1809 he published the Theoria motus corporum coelestium, which contains a discussion of the problems arising in the determination of the movements of planets and comets from observations made on them under any circumstances. In it are found four formulæ in spherical trigonometry, now usually called "Gauss' Analogies," but which were published somewhat earlier by Karl Brandon Mollweide of Leipzig (1774–1825), and earlier still by Jean Baptiste Joseph Delambre (1749–1822).[44] Many years of hard work were spent in the astronomical and magnetic observatory. He founded the German Magnetic Union, with the object of securing continuous observations at fixed times. He took part in geodetic observations, and in 1843 and 1846 wrote two memoirs, Ueber Gegenstände der höheren Geodesie. He wrote on the attraction of homogeneous ellipsoids, 1813. In a memoir on capillary attraction, 1833, he solves a problem in the calculus of variations involving the variation of a certain double integral, the limits of integration being also variable; it is the earliest example of the solution of such a problem. He discussed the problem of rays of light passing through a system of lenses.

Among Gauss' pupils were Christian Heinrich Schumacher, Christian Gerling, Friedrich Nicolai, August Ferdinand Möbius, Georg Wilhelm Struve, Johann Frantz Encke.

Gauss' researches on the theory of numbers were the starting-point for a school of writers, among the earliest of whom was Jacobi. The latter contributed to Crelle's Journal an article on cubic residues, giving theorems without proofs. After the