168 ON THE EARLY 



proving the operation by throwing out the nines is described under each. 

 of these rules. The author gives the following remarkable definitions of 

 multiplication and division, viz. "■ Multiplication is finding a numbe? 

 se such that the ratio which one of the factors bears to it shall be the 

 " same as that which unity bears to the other factor," and " division is 

 " finding a number which has the same ratio to unity as the dividend has 

 " to the divisor/' 



For the multiplication of even tens, hundreds, &c. into one another, 

 the author delivers the following rule, which is remarkable in this res- 

 pect, that it exhibits an application of something resembling the indexes 

 of logarithms. 



" Take the numbers as if they were units, and multiply them together 

 " and write down the product. Then add the numbers of the ranks to— 

 " gether, (the place of units being one, that of tens, two, &c.)' substract 

 u one from the sum and call the remainder the number of the rank of 

 " the product. For example, in multiplying 30 into 4,0, reckon 12 of the 

 " rank of hundreds ; for the sum of the numbers of their ranks is 4, and 

 u three is the number of the rank of hundreds, multiplying 40 into 500, 

 " reckon 20 of the rank of thousands, for the sum of the numbers of the 

 ." ranks is 5." 



The following contrivances have sufficient singularity to merit par- 

 ticular mention. 



I. To multiply numbers between 5 and 10. Call one of the factors 

 tens, and from the result, subtract the product of that factor by the- dif- 

 ference of the other factor from ten. For example, to multiply 8 into 9. 

 Subtract from 90 the product of 9 by 2, there remains 72. Or add the 



