318 MATHEMATICS. 



Subtraction, is the taking of a smaller number, or subtrahend, 

 from a larger number, or minuend, and finding what remains. To 

 do this, we write the subtrahend underneath the minuend, units under 

 units, and so on : then beginning with the units, we take each lower 

 figure from the one above it, and write the difference below, to form 

 the remainder sought. If the upper figure happen to be the smallest, 

 we add ten to it, and subtract as before ; and to compensate for this, 

 we add one to the next lower figure, before subtracting it; which 

 increases the lower line as much as the upper, and thus preserves 

 their difference unchanged. 



Multiplication, is the repeating of a given number, called the 

 multiplicand, as many times as are denoted by another given num- 

 ber, or multiplier : the two numbers thus employed being called 

 factors ; and the sum obtained being the product. The operation 

 might be performed, by writing down the multiplicand as many times 

 as the multiplier denotes, and adding the whole together : but this 

 would be tedious. Hence, we write the multiplicand only once, and 

 the multiplier underneath ; then multiply the upper line by the unit 

 figure of the multiplier ; carrying as in addition, and writing the 

 result. If the multiplier have a second figure, we multiply the 

 upper line by this also, setting the first figure of its product, which 

 expresses tens, under the tens of the first product ; and so proceed- 

 ing to the left. If the multiplier contain hundreds, the first figure of 

 their product must come under the place of hundreds ; and so to the 

 end. Then, adding all these partial products together, the sum will 

 be the total product required. 



Division, is the process of finding how many times one number, 

 called the divisor, may be contained in, or taken from, another, called 

 the dividend ; and also whether a surplus number remains. This 

 last, if there be any, is called the remainder : and the number which 

 expresses how many times the divisor is contained in the dividend, 

 is termed the quotient. To find it, we take as many figures on the 

 left of the dividend, as are sufficient to contain the divisor ; and the 

 number of times they contain it, will be the left hand figure of the 

 quotient. We multiply the whole divisor by this figure, and subtract 

 the product from that part of the dividend used. To the right of 

 the remainder, if any, we bring down the next figure of the dividend, 

 and divide again to obtain the next figure of the quotient ; or if the 

 remainder thus increased be too small, we place a cypher in the 

 quotient, and bring down another figure to the remainder, with which 

 we obtain another quotient figure, as in the first instance. When all 

 the figures of the dividend are brought down, and all those of the 

 quotient found, the last subtraction will give the final remainder. 

 The reasons for this rule, we have no room here to present. 



2. By Denominate Numbers, called also Compound, or Com- 

 plex Numbers, we mean those that refer to certain kinds of quantity, 

 having different denominations; as pounds, shillings, and pence; miles, 

 rods, feet, and inches ; days, hours, minutes, and seconds ; and other 

 like series. The different tables, expressing the ratios of these de- 

 nominations, we have no room to insert. Denominate numbers of 

 the same kind, can be added or subtracted in the same manner as 



