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§39] the Almagest of. These were afterwards elaborated by the Arabs, and in the middle of the fifteenth century by, for use in Astronomy.

In modern times the study of Spherical Trigonometry received a fresh impetus from the writings of, who published several memoirs on the subject. The first appeared in the Mémoires de l'Académie Royale de Berlin in 1753, and was followed some years later by a series of papers in the Acta Petropolitana; of these the most important are those entitled "De Mensura Angulorum Solidorum" (1778, p. 31), and "Trigonometria Sphaerica Universa ex primis principiis derivata" (1779, p. 79). gave an investigation of the formulae of the spherical triangle a few years later in the Journal de l'École Polytechnique (1799, Cahier 6, p. 270).

The chief contributors to the science of Spherical Geometry, in addition to those already named, are (1595), (1614),  (1626),  (1629),  (1782),  (1787),  (1831),  (1833),  (1835), and  (1847).

The extension of the standard formulae to triangles whose sides and angles are not necessarily less than $180^\circ$ is generally ascribed to (“Entwickelungen der Grundformeln der sphärischen Trigonometrie in grösstmöglicher Allgemeinheit,” Verhandlungen der kön. säch, Gesellschaft der Wissenschaften zu Leipzig, 1860, p. 51). But from a remark of ’s in §54 of his Theoria Motus Corporum Coelestium (1809), it is plain that he had thought of this generalisation, and had worked it out, though he did not publish it. Professor, in the preface to his work on Astronomy, points out that in his own Treatise on Trigonometry, published in 1850 (a work at present out of print and difficult to procure), the standard formulae are proved for the general triangle; however, ’s first memoir on the subject appeared in 1846 (cf. §302). We shall discuss the generalisation of the triangle in Chapter.