ARITHMETIC. 
Example. Divide 48764812 by 9. 
9)48764312 
Ans. 5418256—8 remainder. 
9 
Proof 48764312 
In this example we say the 9’s in 48, 5 
times and 3 over ; put down 5 and carry 3, 
and say 9’s in 37, 4 times and 1 over ; put 
down 4 and carry 1 ; 9’s in 16, 1 and 7 over; 
and so on to the end ; there is 8 over as a 
remainder. The proof is obtained by mul- 
tiplying the quotient by the divisor, and 
taking in the remainder: this is called 
« Short Division,’’ of which we give for 
practice the following examples. 
1. Divide 4732157 by 2 
2 342351742 by 3 
3 435234174 by 4 
4 49491244 by 5 
5 94942484 by 6 
6 4434983 by 7 
7 994357971 by 8 
8. .-. 449246812 by 9 
9 557779991 by 11 
10 665594765 by 12 
The second part of this rule is called 
Long Division,” for the practice of which 
we give these directions. 
Count the same number of figures on the 
left of the dividend as the divisor has in it ; 
try whether the divisor be contained in this 
number, if not contained therein, take ano- 
ther dividend figure and then try how many 
times the divisor is contained in it. 
To find more easily how many times the 
divisor is contained in any number; cast 
away in your mind all the figures in the di- 
visor except the left hand one, and cast away 
the same number from the dividend figures 
as you did from the divisor: the two num- 
bers, being thus made small, will be easily 
estimated. 
If the product of the divisor with the quo- 
tient figure be greater than the number from 
which it should be taken, the number 
thought of was too great, the multiplying 
roust be rubbed out, and a less quotient fi- 
gure used. 
When after the multiplying and subtract- 
ing, the remainder is more than the divisor, 
the quotient figure was too small, the work 
must be rubbed out, and a larger number 
supplied. j 
Example. 
Divide 87654213, by 987. 
987) 87654213 (88808 Quotient. 
7896 
.8694 
7896 
.7982 
7896 
.8613 
.7896 
.717 remainder. 
88808 
987 
621663 
710465 
799279 
87654213 proof 
Ans. 88808 |rz. 
To prove the truth of the sum, I multi- 
ply the quotient by the divisor, and take in 
the remainder, which gives the original divi- 
dend. 
Examples for Practice. 
1. Divide 721354 by 21 
2 57214372 by 42 
3 67215731 by 63 
4 ....802594321 by 84 
5. ......... 965314162 by 89 
6 43219875 by 674 
7 57397296 by 714 
8 496521 by 2798 
9 49446327 by 796 
10 47324967 by 699 
11 275472734 by 497 
12 43927483 by 586 
13. 96543245 by 763 
14 25769782 by 469 
A number that divides another without a 
remainder is said to measure it, and the se- 
veral numbers that measure another, are 
called its aliquot parts. Thus 3, 6, 9, 12, 18, 
are the aliquot parts of 36. As it is fre- 
quently necessary to discover numbers which 
measure others, it may be observed, 1. That 
every number ending with an even number, 
that is, with 2, 4, 6, 8, or 0, is measured by 
2. 2. Every number ending with 5, or 0, 
is measured by 5. 3. Every number, whose 
figures, when added, amount to an even 
number of 3’s or 9’s, is measured by 3 or 9 
respectively. 
