ARITHMETIC. 319 



simple numbers ; except in the item of carrying from a lower to a 

 higher denomination. To add them, we commence with those of 

 the lowest denomination, and find how many units these will make 

 of the next higher ; carrying therefor ; and setting down the excess 

 or surplus as a part of the sum ; thus proceeding through all the 

 denominations, to the highest, in which we set down the total sum. 

 To subtract denominate numbers, we proceed as in simple numbers: 

 only, when the upper number is the smallest, we add to it as many 

 units as are required of this denomination to make one of the next 

 higher; in return for which, we add one to the lower number of the 

 next denomination, before subtracting it from that above. 



Multiplication of a Denominate number by a simple one, is per- 

 formed as in simple numbers ; only carrying by the proper ratios 

 in passing from one denomination to the next higher. We cannot 

 properly multiply one denominate number, by another, without con- 

 sidering one of the two abstractly, as composed of certain units and 

 fractional parts ; as is sometimes done in the Rule of Three. Divi- 

 sion of Denominate numbers, by a simple number, is performed as 

 in simple division : only, when we have a remainder of a higher 

 denomination, we reduce it to the next lower, by multiplying by the 

 proper ratio, and to the product we add the number of the same 

 denomination in the dividend, before dividing, to find the number of 

 that denomination in the quotient. 



3. Fractions, are broken numbers, or parts of entire numbers ; 

 the common kinds of which are Vulgar, and Decimal. A Vulgar 

 Fraction, is expressed by two numbers, written one above the other, 

 with a line drawn between them. The lower number, called the 

 denominator, shows into how many equal parts a unit is supposed 

 to be divided ; and the upper number, called the numerator, shows 

 how many of these parts the fraction expresses. By increasing the 

 denominator, we diminish the value of the fraction ; because while 

 the number of parts remain the same, the value of each of these parts 

 is diminished, as more of them are required to make one unit. To 

 add, or subtract, vulgar fractions, we must first reduce them to a 

 common denominator ; in order that they may express like parts of 

 unity. This may be done by multiplying both the numerator and 

 denominator of each fraction by the product of all the other denomi- 

 nators, as the value of the fractions will not be changed thereby. We 

 have then only to add or subtract the numerators, and write the sum 

 or difference over the common denominator, for the result required. 



To multiply, or divide, a vulgar fraction by a whole number, we 

 have only to multiply or divide the numerator; preserving the deno- 

 minator unchanged. Or instead of dividing the numerator, we may 

 multiply the denominator, to perform the division. To multiply two 

 fractions together, we have only to write the product of the nume- 

 rators over that of the denominators : but to divide one fraction by 

 another, we invert the terms of the divisor, that is, make its numera- 

 tor and denominator change places, and then multiply the fractions 

 together. A mixed number, consisting of a whole number and a 

 fraction, is reduced to a fractional form, by multiplying the whole 

 number by the denominator, adding the product to the numerator, 



