MATHEMATICAL SCIENCES. 



Ancient Systems of the Planetary World. Ptolemy and Aristarchus of Samos. Boethius, 

 Pappus, and Gerbert. Schools of Bagdad. Mathematical School in Spain, Italy, England, 

 and France. Astronomical Researches of the Arabs. Roger Bacon and Master Pierre. 

 Alhertus Magnus and St. Thomas Aquinas. Progress of Mathematics. Popes and Kings 

 protectors of the Exact Sciences. The King of Hungary, Matthias Corvinus. Principal 

 Works composed in the Fifteenth Century. Pic Mirandola. Peter Ramus. Tycho Brahe 

 and Copernicus. 



S a proof of the forward state of the exact 

 sciences in the Middle Ages, it would be 

 sufficient to instance a Roman basilica or a 

 Gothic cathedral. What immensity and 

 depth of mathematical calculations ; what 

 knowledge of geometry, statics, and optics ; 

 what experience and skill in execution must 

 have been possessed by the architects and 

 builders in hewing, carving, and fitting the 

 stones, in raising them to great heights, in 



constructing enormous towers and gigantic belfries, in forming the 

 many arches, some heavy and massive, others light and airy, in com- 

 bining and neutralising the thrust of these arches which interlace and 

 hide each other up to the very summit of the edifice all as if 

 the most complicated science had humbly made herself the servant 

 of art, placing no obstacle in the way of its free development ! 



From the commencement of the Middle Ages and henceforward, 

 mat hematics were not so much the object of special and public teaching as 

 of individual and solitary study, either in the shade of the cloisters or amidst 

 associations of artisans who zealously preserved the traditions of their 

 predecessors. 



In the University centres, as in the Arab and Jewish schools which had 



