EARLY METHODS IN ARITHMETIC. 209 



De Burgo, the most noted medieval writer on arithmetic, thinks this 

 last process — our long division — much less pleasant than the following 

 method. Surely tempora mutcmtur, et )ios mutamur in illis. 

 Example 9. Divide 97535399 by 98T6. 



1^ 9 8 7 6 



9 



In this work the divisor is placed next below the dividend, and 

 removed one place to the right since it is not contained in the first four 

 figures of the dividend. The process with the first figure of the quo- 

 tient, placed as usual at j^resent, is as follows : The first number of 

 the divisor, 9, is contained in 97 nine times with a remainder 16. The 

 first figure of the divisor having been used is canceled ; as are also 

 the first two figures of the dividend. (The " scratches " or canceling- 

 marks are omitted in the illustration.) The remainder, being of the 

 same denomination as the first two figures of the dividend, is put 

 directly above them. The next number to be used is 165. Multiply- 

 ing the second figure of the divisor, 8, by 9, and subtracting from 165, 

 93 remains ; 165 and 8 are now canceled, having been used. The 

 remainder 93 is placed above in the proper orders, the 6th and 7th 

 places. So it continues, leaving, after completing the work with the 

 first figure of the quotient, the remainder 8651399. The divisor is 

 now set down again, taking one place to the right as it should to cor- 

 respond to the highest order now in the dividend ; the last figure 

 being raised to the line above, probably for symmetry. The process 

 is continued as before. 



All writers upon arithmetic appear to have agreed in commenda- 

 tion of this method as late as the end of the seventeenth century. It 

 was, in fact, the only method thought necessary to notice. The Eng- 

 lish arithmeticians, from evident cause, called it the " scratch way " 

 of division. Our present method was known specifically as Italian 

 division, and was not introduced until the beginning of the last cen- 

 tury. 



One writer on arithmetic, a pious monk, furnishes a good illustra- 

 tion of medieval logic. He is embarrassed by the usage and meaning 

 of the term "multiplication" in the case of fractions in which the 

 product is less than the multiplicand, and he proposes the question, 



VOL. XTI. 14 



