182 A SHORT HISTORY OF SCIENCE 



and the Earth (De Aqua et Terra) which, according to himself, 

 was delivered at Mantua in 1320 as a contribution to the question, 

 then much discussed, "whether on any part of the earth's surface 

 water is higher than the earth." In his cosmology, Dante seems 

 to derive from Aristotle and Pliny, without having attained 

 familiarity with the Ptolemaic system. 



COMPUTATION IN THE MIDDLE AGES. During the fourteenth 

 century there was continued activity in the gradual dissemination 

 of Arabic learning, largely through the medium of almanacs and 

 calendars, so that Arabic computations, Euclidean geometry, and 

 Ptolemaic astronomy became widely known. Some of these 

 calendars emphasized the religious side and gave dates of church 

 festivals for a series of years, others specialized in astrology, 

 medicine, or astronomy. For ecclesiastical purposes Roman 

 numerals were preferred, but at least an explanation of the new 

 Arabic characters and their use was generally given. 



The arithmetic of Boethius, based on Roman numerals, retained 

 its vogue in northern Europe as late as about 1600. Arabic 

 arithmetic, or algorism, based on the Liber Abaci of Leonardo 

 Pisano, employing the decimal scale and including the elements 

 of algebra, came into general use among the Italian merchants 

 in the thirteenth and fourteenth centuries, though not without 

 meeting serious opposition. Outside of Italy, however, accounts 

 were kept in Roman numerals till about 1550, and in the more 

 conservative religious and educational institutions, for a hundred 

 years longer. The Florentines at the same time considerably 

 simplified the classification of arithmetical operations, in accord- 

 ance with our modern list : numeration, addition, subtraction, 

 multiplication, division, involution and evolution. 



Addition and subtraction were begun at the left. The multi- 

 plication table, at first little known, ended with 5x5. For 

 further products up to 10 X 10, a system of finger reckoning was 

 widely used, the rule running : 



Let the number five be represented by the open hand ; the number 

 six by the hand with one finger closed ; the number seven by the hand 

 with two fingers closed; the number eight by the hand with three 



