148 » 
And I now find that I have to pay for 11 tons, 
1 cwt. 2 qr. 19 lb. 
Subtraction. 
Subtraction is the operation by which we take 
a less number from a greater, and find their dif- 
ference. The greater is called the minuend, and 
the less the subtrahend. 
If any figure of the subtrahend be greater 
than the corresponding figure of the minuend, 
and having found and marked the difference, we 
add one to the next place of the .subtrahend. 
This is called borrowing ten, because adding 
one to the subtrahend produces the same effect 
as taking one from the minuend. 
Rule. Subtract units from units, tens from 
tens, and so on. If any figure of the subtrahend 
be greater than the corresponding one of the 
minuend, borrow ten. 
Example. Minuend 5173694 47386414 
Subtrahend 2421453 23792352 
Remainder 2752241 ^ 23594062 
To prove subtraction, add the remainder and 
subtrahend together ; if their sum be equal to 
the minuend, the sum is right. 
Or subtract the remainder from the minuend. 
If the difference be equal to the subtrahend, the 
sum is right. 
Rule for Compound Subtraction. “ Place 
like denominations under like, and borrow, 
when necessary, according to the value of the 
higher place.” 
lb. oz. dnvts. grs. £,• i. d- 
45 8 14 15 95 7 6^ 
29 8 17 17 59 16 9| 
15 11 16 22 .35 10 8i 
The reason for borrowing is the same as in 
simple subtraction. Thus in subtracting pence, 
we add 12 pence when necessary to the minu- 
end, and at the next step, we add one shilling 
to the subtrahend. 
The learner should acquire the habit, when 
two numbers are marked down, of placing such 
a number under the less, that, when added to- 
gether, the sum may be equal to the greater. 
The operation is the same as subtraction, though 
conceived in a different manner, and is useful 
in balancing accounts, and on other occasions 
in the concerns of life. 
Multiplication. 
Multiplication is a compendious mode of ad- 
dition, and teacheth to find the amount of any 
given number by repeating it any proposed 
number of times. Thus, 8 multiplied by 5, or 
5 times 8, is 40. The given numbers (8 and 5) 
are called factors ; the' first (8) the multiplicand, 
the second (5) the multiplier ; and the amount 
(20) the product. 
Ex. 76859 mult, by 4, or 76859 added 4 times. 
4 7685-9 
76859 
807436 76859 
307436 
If the multiplier be 10, we annex a cypher to 
the multiplicand. If the multiplier be 100, we 
annex two cyphers ; and so on. d he reason is 
obvious, from the use of cyphers in notation. 
RuLp. Place the multiplier under the multi- 
plicand, and multiply the latter successively by 
the significant figures of the former; placing the 
right-hand figure of each product under the 
figure of the multiplier from which it arises; 
then add the product. 
Ex.< a) 7329 37846 93956 
365 235 8704 
ARITHMETIC. 
multiplication of two others is called a prime 
number ; as 3, 5, 7,11. 
A number which may be produced by the 
multiplication of two or more smaller ones, is 
called a composite number. For example, 27, 
which arises from the multiplication of 9 by 3; 
and these numbers (9 and 3) are called the com- 
ponent parts of 27. 
If the multiplier be a composite number, we 
may multiply successively by the component 
parts. 
Ex.] 7638 by 45, or 5 times 9 7638 
45 9 
38190 
30552 
68742 
5 
343710 . 343710 
Because the second product is equal to. five 
times the first, and the first is equal to nine times 
the multiplicand, it is obvious that the second 
product must be five times nine, or forty-five 
times, as great as the multiplicand. 
If the multiplier be 5, which is the half of 10, 
we may annex a cypher and divide by 2. If it 
be 25, which is the fourth part of 100, we may 
annex two cyphers, and divide by 4. Other con- 
tractions of the like kind will readily occur to 
the learner. 
To multiply by 9, which is one less than 10, 
we may annex a cypher ; and subtract the mul- 
tiplicand from the number it composes. To 
multiply by 99,999, or any number of 9’s, an- 
nex as many cyphers, and subtract the multipli- 
cand. The reason is obvious; and a like rule 
may be found, though the unit place be different 
from 9. 
Multiplication is proved by repeating the 
operation, using the multiplier for the multipli- 
cand, and the multiplicand for the multiplier. 
Or it may be done by casting out the nines ; 
that is, cast out the nines of the multiplier and 
multiplicand, and set down the remainders. 
Multiply the remainders together, and if the 
excess of nines in their product be equal to the 
excess of 9’s in the total product, the work may 
be deemed right: thus, in Ex. (a) above, the 
excess of nines in the multiplicand is 3, and in 
the multiplier it is 5, and 3x5= 15, or 6 
above nine, which I find is the excess of nines 
in the total product. The best method of prov- 
ing Multiplication is by division ; and if that 
be adopted, the two rules must be learnt at the 
samexime. Then the proof of each example in 
Multiplication becomes a sum in Division, and 
nice versa. 
Compound Multiplication. 
Compound Multiplication teacheth to find 
the amount of any given number of different 
denominations, by repeating it any proposed 
number of times. 
Rule. (1) Place the multiplier under the 
lowest denomination of the multiplicand. (2) 
Multiply the number of the lowest denomina- 
tion by the multiplier, and find how many ones 
of the next higher denomination are contained 
in the product. (3) Write down the excess, and 
carry the ones to the product of the next higher 
denomination, with which proceed as before ; 
and so on, through all the denominations to the 
highest ; whose product, together with the se- 
veral excesses, taken as one number, will be the 
whole amount required. 
Examples of Money. 
9 lb. of tobacco, at 4s. 8-§d. per lb. 
4 8| 
9 
36645 
43974 
21987 
189230 375824 
113538 657692 
75692 751648 
2675085 8893810 817793024 
Jknuimber which cannot be produced by the 
£. 2 2 4§ the answer. 
The product of a number consisting of seve- 
ral parts, or denominations, by any simple num- 
ber whatever, will evidently be expressed by 
taking the product of that simple number and 
each part by itself as so many distinct questions; 
thus, 2 51. 12s. 6d. multiplied by 9, will be 22 54 
108s. 54 d. = (by taking the shillings from the 
pence, and the pounds from the shillings, and 
placing them in the shillings and pounds re- 
spectively) 230/. 12s.6a'. which is the same as the 
rule ; and this will be true when the multipli- 
cand is any compound number whatever. 
Case I. If the multiplier exceed ] 2, multiply 
successively by its component parts, instead of 
the whole number at once, as in simple multi- 
plication. 
i'.v.l 16 cwt. of cheese, at 11. 18s. 8 d. per cwU 
1/. 18s. 8 d. 
4 
30/. 18 s. 8 d. the answer. 
Case II. If the multiplier cannot be produced 
by the multiplication of small numbers, find the 
nearest to it, either greater or less, which can 
be so produced ; then, multiply by the compo- 
nent parts as before, and for the odd parts, add 
or subtract according as is required. 
Ex.] 17 ells of holland, at 7s. 8 \d. per ell. 
7s. 8 kd. 
4 
1 10 
10 
4 
6 3 
4 
7 
61. 11s. 
0 \d. the answer. 
Division. 
Division teacheth to find how often one num- 
ber is contained in another of the same denomi- 
nation, and thereby performs the work of many 
subtractions. 
The number to be divided is called the divi- 
dend. 
The number you divide by is called the divisor- 
The number of times the dividend contains 
the divisor is called the quotient. 
If the dividend contains the divisor any num- 
ber of times, and some part or parts over, those 
parts are called the remainder. 
Rule. (1) On the right and left of the divi- 
dend draw a curved line, and write the divisor 
on the left hand, and the quotient as it arises on 
the right. (2) Find how many times the divi- 
sor may be had in as many figures of the divi- 
dend as are just necessary, and write the number 
in the quotient. (3) Multiply the divisor by 
the quotient figure, and set the product under 
that part of the dividend used. (4). Subtract 
the last-found product from that part of the di- 
vidend under which it stands, and to the right 
hand of the remainder bring down the next 
figure of the dividend ; which number divide as- 
before ; and so on, till the whole is finished. 
When the divisor does not exceed 12, the 
whole computation may be performed without 
setting down any figures except the quotient. 
Ex.] 7)35868(5124 or 7)85868 
5124 
When the divisor is a composite number, and 
one of the component parts also measures the 
dividend, we may divide successively by the 
component parts. 
EXAMPLE I. EXAMPLE II. 
30114 by 63 975 by 105 = 5 X 7x3 
9)30114 5)975 
7)3346 3)195 
Quotient 478 7)65 
Quotient 92. 
This method might he also used, although the 
component parts of the divisor do not measure 
the dividend ; but the learner will not under- 
