island how to manage the remainder till he be 
acquainted with the doctrine of vulgar fractions. 
When there are cyphers annexed to the di- 
visor, cut them off, and cut off an equal number 
of figures from the dividend ; annex these figures 
to the remainder. 
Ex.] To divide 378643 by 5200. 
52!00)3786l43(72f|4§-. 
364 
146 
104 
4243 
The reason will appear by performing the 
Operation at large, and comparing the steps. 
Compound Division. 
Compound Division teacheth to find how 
often one given number is contained in another 
of different denominations. 
Rule. (1) Place the numbers as in simple di- 
vision. (2) Begin at the left-hand, and divide 
each denomination by the divisor, setting the 
quotients under their respective dividends. (3) 
But if there be a remainder, after dividing any 
of the denominations. except the least, find how 
many of the next lower denomination it is 
equal to, and add it to the number, if any, 
which was in this denomination before ; then 
divide the sum as usual, and so on till the whole 
is finished. 
The method of proof 19 the same as in simple 
division. 
Examples of Money. 
Divide 225/. 2s. 47. by 2.. 
2)225/. 2s. 47. 
112/. ID. 2d. the quotient. 
Case I. If the divisor exceed 12, divide con- 
tinually by its component parts, as in simple 
division. 
Ex.] What is cheese per cwt. if 16 cwt. cost 
30/. 18/. 8 d. 
4)30/. 18/. 8 d. 
4) 7/. 14 s. 87. 
1/. 18s. 87. the answer. 
Case II. If the divisor cannot be produced 
by the multiplication of small numbers, divide 
by it after the manner of long division. 
Ex.] Divide 74/. 13s. 6d. by 17. 
17)74/. 13s. 67.(4/. 7s. 107. the quotient. 
68 
' 6~ 
20 
lslT 
119 
“ 
12 
174 
170 
4 
Reduction. 
Reduction teaches to bring numbers from 
©ne name or denomination to another, without 
changing their value. 
Rule. All great names are brought into 
smaller ones by multiplying with so many of the 
next less as make one of the greater, adding to 
the product the parts of the less name, if the 
number to be reduced be a compound one ; — 
and all small names are brought into greater by 
dividing by as many of the less as make one of 
the greater. 
EXAMPLES. 
1. Reduce 2551. 6s. 97. into pence* 
20 
5106 
12 
ARITHMETIC. 
Here it is evident that 61281 pence is equal to 
2551. 6s. 97. 
2. How many pounds are there in 122562 
pence ? 
12)122562 
20)10213 — 6 
Answer £. 510 13 6 
The Rule of Three Direct. 
The Rule of Three Direct teacheth, by having 
three numbers given, to find a fourth, that shall 
have the same proportion to the third as the 
second has to the first. 
Rule. (1) State the question: that is, place 
the numbers so, that the first and third may be 
the terms of supposition and demand, and the 
second of the same kind with the answer re- 
quired. (2) Bring the first and third numbers 
into the same denomination, and the second 
into the lowest name mentioned. (3) Multiply 
the second and third numbers together, and di- 
vide the product by the first, and the quotient 
will be the answer to the question, in the same 
denomination you left the second number in ; 
which may be brought into any other denomi- 
nation required. 
The method of proof is by inverting the 
question. 
Ex. If 12//. of cheese cost 9/. 67. what wall 
4 cheeses cost, each weighing Iqr. 616.} 
If 12//. : 9 j. 67. Iqr. 51b. X 4. 
12 28 
114" ~~33~ 
4 
132 
144 
528 
1452 
12)15048 
12)1254 pence 
20)104 6 
£-5 4 ' 6 
Note. This rule, on account of its great and 
extensive usefulness, is oftentimes called The 
Golden Rule of Proportion; for, on a proper ap- 
plication of it, and the preceding rules, the whole 
business of arithmetic, as well as every mathe- 
matical enquiry, depends. The rule itself is 
founded on this obvious principle, that the mag- 
nitude or quantity of any effect varies constantly 
in proportion to the varying part of the cause : 
thus, the quantity of goods bought is in propor- 
tion to the money laid out; the space gone over 
by an uniform motion is in proportion to the 
time, &c. — As the idea annexed to the term pro- 
portion is easily conceived, it would be more per- 
plexing than instructive to explain, in this place, 
what is meant by it, in a strict geometrical sense. 
It may be sufficient, therefore, to observe, that, 
independent of the precise meaning of that word, 
and its deducible properties, the truth of the 
rule, as applied to ordinary enquiries, may be 
made very evident, by attending only to princi- 
ple? already explained. — It is shewn in multipli- 
cation of money, that the price of one multiplied 
by the quantity is the price of the whole : and in 
division, that the price of the whole divided by 
the quantity is the price of one. Now, in all 
cases of valuing goods, &c. where one is the first 
term of the proportion, it is plain that the an- 
swer found by this rule will be the same as that 
found by multiplication of money ; and where 
one is the last term of the proportion, it will be 
the same as that found by division of money. 
In like manner, if the first term be any number 
whatever, it is plain that the product of the se- 
cond and third terms will be greater than the 
true answer required; by as much as the price in 
1 4g‘ 
the second term exceeds the price of one, or as 
the first term exceeds an unit. Consequently 
this product divided by thefirst term will givethe 
true answer required, and is the rule. 
The Rule of Three Inverse. 
The Rule of Three Inverse teacheth by having 
three numbers given, to find a fourth, that shall 
have the same proportion to the second as the 
first has to the third. 
If more require more, or less require less, the 
question belongs to the rule of three direct. 
But if more require less, or less require more, 
it belongs to the rule of three inverse. 
Rule. (1) State and reduce the terms as in 
the rule of three direct. (2) Multiply the first 
and second terms together, and divide their pro- 
duct by the third, and the quotient is the answer 
to the question, in the same denomination you 
left the second number in. 
The method of proof is by inverting the 
question. 
Ex.] What quantity of shalloon, that is S 
quarters of a yard wide, will line l-\ yards oi 
cloth, that is yard wide ? 
1 yd. 2 qrs. * 7 yds. 2 qrs. ** 3 qrs. 
4 4 
If SO 
6 
3)180 
4)60 
15 yards, the answer. 
More requiring more, is when the third term 
is greater than the first, and requires the fourth 
term to be greater than the second. 
And less requiring less, is when the third term 
is less than the first, and requires the fourth term 
to be less than the second. 
In like manner, more requiring less, is when 
the third term is greater than the first, and re- 
quires the fourth term to be less than the second. 
And less requiring moire, is when the third 
term is less than the first, and requires the fourth 
term to be greater than the second. 
The reason of this rule may be explained from 
the principles of compound multiplication and 
division, in the same manner as the direct rule. 
For example : If 6 men can (jlo a piece of work 
in 10 days, in how many days will 12 men do 
it ? 
, 6X10 
As 6 men * 10 days * * 12 men * — - — = 5 days, 
the answer. And here the product of the first 
and second terms, i. e. 6 times 10, or 60, is evi- 
dently the time in which one man would per- 
form the work: therefore 12 men will do it in 
one twelfth part of that time,or 5 days; and this 
reasoning is applicable to any instance whatever- 
Compound Proportion. 
Compound Proportion teacheth to resolve 
such questions as require two or more statings 
by simple proportion; and that whether they 
are direct or inverse. 
Rule. (1) Let that term be put in the second 
place which is of the same denomination with 
the term sought. (2) Place the terms of sup- 
position, one above another, in the first place ; 
and the terms of demand, one above another, in 
the third place. (3) The first and third term 
of every row will be of one name, and must be 
reduced to the same denomination.. (4) Ex- 
amine every row separately: by saying, if the 
first term give the second, does the third require 
more or less? If it require more, mark the less 
extreme with a cross; but if less, mark the greater 
extreme. (5) Multiply all those numbers to- 
gether which are marked for a divisor, and those 
which are not marked for a dividend, and the 
quotient will be the answer sought. 
Note. When the same numbers are found in 
the divisor as in the dividend, they may be- 
6128 1 
