184 A SHORT HISTORY OF SCIENCE 



This method was considered simpler than our modern long 

 division, and remained in use till the seventeenth century. 



The signs +, , -T-, and the use of decimal fractions belong to 

 a somewhat later period. 



The characteristics of the algoristic arithmetic are: (1) the use 

 of the Hindu- Arabic system of notation ; (2) the system of local value ; 

 (3) the use of the zero ; (4) the entire discarding of the abacus ; (5) 

 the combined use of symbols and numbers (in reality a combination 

 of algebra and arithmetic, as these terms are understood to-day) ; 

 and (6) the introduction into Western Europe of a vast amount of 

 arithmetical material from the East by means of Latin translations 

 from Arabian sources. While the general tendency of this period 

 was to approach the study of arithmetic from its practical and 

 scientific sides, the mystical aspects of the subject so popular 

 in the earlier periods are by no means neglected. The fantas- 

 tic treatment of the properties of numbers is still common in this 

 age. . . . 



Thus the beginning of the thirteenth century marks the introduc- 

 tion of the Arabian system of notation and its adoption in place of 

 both the Roman notation and the abacus. This fundamental revolu- 

 tion was brought about only gradually, and that of the algorism can 

 be traced in the translated literature of the Hindu-Arabian arith- 

 metic. Abelson. 



MATHEMATICS IN THE MEDIEVAL UNIVERSITIES. The state of 

 mathematics in the universities toward the close of the fourteenth 

 century may be inferred from the requirements for the master's 

 degree at Prague (1384) and Vienna (1389). The former included 

 Sacrobosco's Sphere, Euclid Books I-VI, optics, hydrostatics, 

 theory of the lever, and astronomy. Lectures were given on 

 arithmetic, finger-reckoning, almanacs, and Ptolemy's Almagest. 

 At Vienna, Euclid I-V, perspective, proportional parts, mensu- 

 ration, and a recent version of Ptolemy were required. In Leipsic, 

 however, in 1437 and 1438 mathematical ( ?) lectures were confined 

 to astrology, and conditions seem to have been much the same at 

 the Italian universities, while Oxford and Paris probably occupied 

 an intermediate level. 



