2o8 THE POPULAR SCIENCE MONTHLY. 



Processes. 

 Ex. 6. Ex. 7. 



2 3 4 

 2 3 4 



4 6 116 



6 8 2 

 1 2 

 9 

 8 



5 4 7 5 6 



2 2 110 6 



Though the multiplication-table was in use by the Arabians and 

 Italians at an early date, no notice Avas taken of it during the middle 

 ages in the rest of Europe. It may give us more charity for the boys 

 and girls who are now wrestling with it — although nowadays it 

 does not seem to require the struggle that it used to — to know that 

 grown men, and wise men probably, sought for devices by which the 

 labor might be avoided which we go through in childhood. Outside 

 of Italy, many writers considered it necessary to relieve the memory 

 from retaining the products of digits above five. The principal rule 

 — known as the " sluggard's rule " — given for this purpose during the 

 last half of the sixteenth century, the half century after the time of 

 Luther, Melanchthon, and Erasmus, was this : Subtract each digit from 

 ten, and write down the differences ; rtiultiply the differences together 

 and add as many tens to their product as the first digit exceeds the 

 second difference, or the second digit the first difference. 



Examples. 7 X 8 == (3 X 2) + (7 — 2 = 5) tens = 56. 

 6X9= (4X1) + 5 tens = 54. 



The method which we call short division was largely used in the 

 middle ages, as was also the method of dividing by using the factors 

 of the divisor. The process by long division was known, but was not 

 so commonly used as others. It was called the process " by giving," 

 since after subtraction we give or add (bring down) one or more fig- 

 ures to the remainder. Here is an example set down after the fashion 

 of those times : 



Example 8. Divide 97335376 by 9876. 



Divisior. Proveniens. 



9 8 7 6 9 8 7 6 



9 7 3 3 5 3 7 6 



8 8 8 8 4 



8 6 5 13 



7 9 8 



7 5 5 7 



6 9 13 2 



5 9 2 5 6 

 5 9 2 5 6 



