[515] Adrien Marie Legendre (1752-1833) was one of the great mathematicians of the opening of the nineteenth century. His _Elements de geometrie_ (1794) had great influence on the geometry of the United States. His _Essai sur la theorie des nombres_ (1798) is one of the classics upon the subject. The work to which Kircher refers is the _Nouvelle theorie des paralleles_ (1803), in which the attempt is made to avoid using Euclid's postulate of parallels, the result being merely the substitution of another assumption that was even more unsatisfactory. The best presentations of the general theory are W. B. Frankland's _Theories of Parallelism_, Cambridge, 1910, and Engel and Staeckel's _Die Theorie der Parallellinien von Euclid bis auf Gauss_, Leipsic, 1895. Legendre published a second work on the theory the year of his death, _Reflexions sur ... la theorie des paralleles_ (1833). His other works include the _Nouvelles methodes pour la determination des orbites des cometes_ (1805), in which he uses the method of least squares; the _Traite des fonctions elliptiques et des integrales_ (1827-1832), and the _Exercises de calcul integral_ (1811, 1816, 1817). [516] Johann Joseph Ignatz von Hoffmann (1777-1866), professor of mathematics at Aschaffenburg, published his _Theorie der Parallellinien_ in 1801. He supplemented this by his _Kritik der Parallelen-Theorie_ in 1807, and his _Das eilfte Axiom der Elemente des Euclidis neu bewiesen_ in 1859. He wrote other works on mathematics, but none of his contributions was of any importance. [517] Johann Karl Friedrich Hauff (1766-1846) was successively professor of mathematics at Marburg, director of the polytechnic school at Augsburg, professor at the Gymnasium at Cologne, and professor of mathematics and physics at Ghent. The work to which Kircher refers is his memoirs on the Euclidean _Theorie der Parallelen_ in Hindenburg's _Archiv_, vol. III (1799), an article of no merit in the general theory. [518] Wenceslaus Johann Gustav Karsten (1732-1787) was professor of logic at Rostock (1758) and Butzow (1760), and later became professor of mathematics and physics at Halle. His work on parallels is the _Versuch einer voellig berichtigten Theorie der Parallellinien_ (1779). He also wrote a work entitled _Anfangsgruende der mathematischen Wissenschaften_ (1780), but neither of these works was more than mediocre. [519] Johann Christoph Schwab (not Schwal) was born in 1743 and died in 1821. He was professor at the Karlsschule at Stuttgart. De Morgan's wish was met, for the catalogues give "c. fig. 8," so that it evidently had eight illustrations instead of eight volumes. He wrote several other works on the principles of geometry, none of any importance.