ARITHMETIC. 



In speaking of the contractions and va- 

 riety in division, we have already seen, 

 that when the divisor does not exceed 12, 

 the whole computation may be perform- 

 ed without setting down any figure ex- 

 cept the quotient. 



When the divisor is a composite num- 

 ber, we may divide successively by the 

 component parts : thus, if 678450 is to be 

 divided by 75, we may either perform the 

 operation by long division, or divide by 

 5, 5, and 3, because 5 x 5 x 3=75. 



Where there are cyphers annexed to 

 the divisor, cut them off', and cut off also 

 an equal number of figures from the di- 

 vidend ; annex these figures to the re- 

 mainder. 



Example. 

 Divide 54234564 by 602400. 



54216 

 . . . 18564 



To divide by 10, 100, 1000, &c. Cut off 

 as many figures on the right hand of the 

 dividend as there are cyphers in the di- 

 visor. The figures which remain on the 

 left hand compose the quotient, and those 

 cut off compose the remainder. 



Example. 



Divide 594256 by 100.0. 

 1,000)594,256 



Ans. 



\Vhen the divisor consists of several 

 figures, we may try them separately, by 

 enquiring how often the first figure of the 

 divisor is contained in the first figure of 

 the dividend, and then considering whe- 

 ther the second and following figures of 

 the divisor be contained as often in the 

 corresponding ones of the dividend with 

 the remainder, if any prefixed. If not, 

 we must begin again, and make trial of a 

 lower number. 



We may form a table of the products 

 of the divisor multiplied by the nine di- 

 gits, in order to discover more readily 

 how often it is contained in each part of 

 the dividend. This is always useful, when 

 the dividend is very long, or when it is 

 required to divide frequently by the same 

 divisor. 



VOL. II. 



Example. 



Divide 689543271 by 37. 

 37 X 2= 74 37)689543271(18636304 

 37 



319- 

 226 



3=111 

 4=148 

 5=185 

 6=222 

 7=259 

 8=296 

 9=333 



222 



As multiplication supplies the place of 

 frequent additions, and division of fre- 

 quent subtraction, they are only repeti- 

 tions and contractions of the simple rules, 

 and when compared together, their ten- 

 dency is exactly opposite. As numbers 

 increased by addition are diminished and 

 brought back to their original quantity by 

 subtraction, in the same manner numbers 

 compounded by multiplication are reduc- 

 ed by division to the parts from which they 

 are compounded. The multiplier shews 

 how many additions are necessary to pro- 

 duce the number, and the quotient shows 

 how many subtractions are necessary to 

 exhaust it. Hence it follows, that the pro- 

 duct divided by the multiplicand will give 

 the multiplier; and because either factor 

 may be assumed for the multiplicand,, 

 therefore the product divided by either 

 factor gives the other. It also follow*, 

 that the dividend is equal to the product 

 of the divisor and quotient multiplied to. 

 gether, and of course these operations 

 mutually prove each other. 



To prove Multiplication. Divide the prp- 

 duct by either factor; if the operation be 

 right, the quotient is the other factor, an.d 

 there is no remainder. 



To prove Division. Multiply the divisor 

 and quotient together; to the product 

 add the remainder, if any; and if the 

 operation be right, it makes up the divi- 

 dend. W T e proceed to 



COMPOUND DIVISION. 



For the operation of which the rule is : 

 when the dividend only consists of differ- 

 ent denominations, divide the higher de- 

 nomination, and reduce the remainder to 



C 



