EARLY METHODS IN ARITHMETIC. 205 



according to Theological, Arithmetical, Geometrical, and Harmonical 

 Computation. Pleasing to read, profitable to under stande, opening 

 themselves to the capacities of both learned and unlearned y being no 

 other than a Keg to lead Man to any Doctrinal Knoicledge ichatsoever.^^ 



But, in addition, there was difficulty and complexity in the science 

 as practiced then that made it no boy's play. Even making allowance 

 for the great advantage of " being used to a thing," the middle-age 

 processes in the fundamental rules were often much more intricate 

 than those practiced nowadays. In his incomparable history of the 

 science of arithmetic, in the " Encyclopedia Metropolitana," Dr. Pea- 

 cocke gives many interesting illustrations, some of which will doubt- 

 less strike the reader as novel. Some of their steps are easily ex- 

 plained, but others are by no means so simple. It might prove of 

 interest and advantage to test the higher grades in some modern 

 schools in regard to their actual comprehension of the first four rules 

 by requiring them to explain the philosophy, not the process merely, 

 of a few of these medieval "sums." Explanations further than a 

 description of the process are purjDosely omitted. 



In subtraction they usually began at the left hand instead of the 

 right. Inconvenient as it is, the method was continued as late as the 

 end of the sixteenth century. The difference was placed above the 

 numbers instead of below. 



Example 1. Subtract 35843 from 54612. 

 When the digits in the subtrahend are greater 

 than those in the minuend, units are placed be- 

 neath them as in the example ; 3 being increased 

 by the unit in the next place to the right, and 

 similarly for 5, 8, and 4. 



Process. Example 2. Subtract 23245 from 30024. Of 



06779 remainder. course with such an arrangement it is of no con- 

 2991 sequence whether the operation proceeds from right 



30024 minuend. to left or from left to right. It will be easily seen 

 23245 subtrahend, how the substituted minuend is obtained, with the 

 exception of the one ten. Suppose the figure 4 in 

 the subtrahend had been 1 ; then to what device would the boys and 

 girls of the time of Luther and of Queen Elizabeth have had to resort 

 to save their credit ? 



There is reason for thinking that the modern method of subtrac- 

 tion was the invention of an English mathematician of the first part 

 of the seventeenth century, by the name of Gath. 



In multiplication there were some ten or twelve different processes 

 in practical use ; but, strange to say, our present mode is not found 

 among them. A few of the subjoined examples are easily intelligible. 

 A little study will make the others plain : 



Example 1. Multiply 135 by 12. 



Process. 

 18769 remainder. 

 54612 minuend. 

 35843 subtrahend. 

 1111 



