This Petro Ciruelo was born in Arragon, and died in 1560 at Salamanca. He studied mathematics and philosophy at Paris, and took the doctor's degree there. He taught at the University of Alcalà and became canon of the Cathedral at Salamanca. Besides his editions of Bradwardine he wrote several works, among them the Liber arithmeticae practicae qui dicitur algorithmus (Paris, 1495) and the Cursus quatuor mathematicarum artium liberalium (Alcalà, 1516).
[511] Star polygons, a subject of considerable study in the later Middle Ages. See note [35] on page [44].
[512] "A new theory that adds lustre to the fourteenth century."
[513] There is nothing in the edition of 1495 that leads to this conclusion.
[514] The full title is: Nouvelle théorie des parallèles, avec un appendice contenant la manière de perfectionner la théorie des parallèles de A. M. Legendre. The author had no standing as a scientist.
[515] Adrien Marie Legendre (1752-1833) was one of the great mathematicians of the opening of the nineteenth century. His Eléments de géométrie (1794) had great influence on the geometry of the United States. His Essai sur la théorie des nombres (1798) is one of the classics upon the subject. The work to which Kircher refers is the Nouvelle théorie des parallèles (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 Stäckel'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, Réflexions sur ... la théorie des parallèles (1833). His other works include the Nouvelles méthodes pour la détermination des orbites des comètes (1805), in which he uses the method of least squares; the Traité des fonctions elliptiques et des intégrales (1827-1832), and the Exercises de calcul intégral (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 völlig berichtigten Theorie der Parallellinien (1779). He also wrote a work entitled Anfangsgründe 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.