Petro
sanchez ciruelo: operaqz Guidonis mercatoris dilig[=e]tissime impresse
parisi^o in c[=a]po gaillardi. Anno d[=n]i. 1495. die. 20, maij._

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 Alcala 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_ (Alcala, 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 theorie des paralleles, avec un
appendice contenant la maniere de perfectionner la theorie des paralleles
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 _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