PROGRESS OF SCIENCE TO 1450 A.D. 



183 



fingers closed; and the number nine by the hand with four fingers 

 closed. To multiply one number by another let the multiplier be 

 represented by one hand, and the number multiplied by the other, 

 according to the above convention. Then the required answer is 

 the product of the number of fingers (counting the thumb as a finger) 

 open in the one hand by the number of fingers open in the other to- 

 gether with ten times the total number of fingers closed. 1 



Long division naturally required the skill of a mathematical 

 expert. For example, if it is necessary to divide 1330 by 84 

 (Ball, p. 191) the Arabic or Persian method may be represented 

 as follows, the right hand figure summing up the whole process: 



. The galley or " scratch " method generally employed in Italy 

 would take for the same problem, the successive forms (Ball, p. 

 192) : 



5 

 #30(1 



4 

 09 



4 

 09 

 #00(15 



UJ 

 IrH 



8 



& 



m 



9 



;#o(i6 



1 In modern notation : If x is the number of fingers closed in one hand, y the 

 number closed in the other, then 



(5 + x) (5 + y) = (5 - x) (5 - y) + 10 ( + y). 



