148 A SHORT HISTORY OF SCIENCE 



metic, music, and astronomy, of great and lasting educational 

 influence. His classification of these "seven liberal arts" main- 

 tained itself throughout the Middle Ages and is not yet wholly 

 extinct. Gregory of Tours for example says: "If thou wilt 

 be a priest of God, then let our Martianus instruct thee first in 

 the seven sciences." 



BOETHIUS (480-524) born at Rome on the eve of its fall in 476 

 is the author not only of the famous Consolations of Philosophy 

 but also of works on Music and on Arithmetic which long served 

 to represent Greek mathematics to the medieval world. In the 

 course of his public-spirited career, Boethius interested himself 

 in the reform of the coinage and in the introduction of water- 

 clocks and sun-dials. His geometry consists merely of some of 

 the simpler propositions of Euclid, with proofs of the first three 

 only, and with applications to mensuration. Yet the intellectual 

 poverty of the age was such that this remained long the standard 

 for mathematical teaching. Boethius' Arithmetic begins : 



By all men of old reputation who following Pythagoras' reputation 

 have distinguished themselves by pure intellect it has always been 

 considered settled that no one can reach the highest perfection of 

 philosophical doctrines, who does not seek the height of learning at 

 a certain crossway the quadrivium. 



For him the things of the world are either discrete (multitudes), 

 or continuous (magnitudes). Multitudes are represented by 

 numbers, or in their ratios by music; magnitudes at rest are 

 treated by geometry, those in motion by astronomy. These four 

 of the seven liberal arts form the quadrivium; grammar, dialec- 

 tics and rhetoric, the trivium. A Christian in faith, a pagan in 

 culture, Boethius has been called the " bridge from antiquity to 

 modern times." (See page 50.) 



The scholars of the time were almost without exception men 

 whose first interests were theological. Mathematics, having no 

 direct moral significance, seemed to them in itself unworthy of 

 attention. On the other hand, they attached exaggerated im- 

 portance to all sorts of mystical attributes of numbers and to the 



