RULE. 
igalled the Rule of Three from having three 
numbers given to find a fourth ; but more 
properly, the Rule of Proportion, because 
by it we find a fourth number proportional 
to three given numbers : and because of the 
necessary and extensive use of it, it is called 
the Golden Rule. But to give a definition 
of it, with regard to numbers of particular 
and determinate things, it is the rule by 
which we find a number of any kind of 
things, as money, weight, &c. so propor- 
tional to a given number of the same things, 
as another number of the same or different 
things is to a third number of the last kind 
of thing. For the four numbers that are 
proportional must either be all applied to 
one kind of things : or two of them must 
be of one kind, and the remaining two of 
another ; because there can be no propor- 
tion, and consequently no comparison of 
quantities of different species: as, for ex- 
ample, of three shillings and four days : or 
of six men and four yards. All questions 
that fall under this rule may be distinguished 
into two kinds: the first contains those 
wherein it is simply and directly proposed 
to find a fourth proportional to three given 
numbers taken in a certain order : as if it 
were proposed to find a sum of money so 
proportioned to one hundred pounds as 
sixty-four pounds ten shillings is to eighteen 
pounds six shillings and eight-pence, or as 
forty pounds eight shillings is to six hundred 
weight. The second kind contains all such 
questions wherein we are left to discover, 
from the nature and circumstances of the 
question, that a fourth proportional is 
sought ; and consequently, how the state of 
the proportion, or comparison of the term, 
is to be made ; wliich depends upon a clear 
understanding of the nature of the question 
and proportion. After the given terms are 
duly ordered, what remains to be done is 
to find a fourth proportional. But to re- 
move all difficulties as much as possible, 
the whole solution is reduced to the follow- 
ing general rule, which contains what is 
necessary for solving such questions wherein 
the state of the proportion is given; in order 
to which it is necessary to premise these 
observations. 
j . In all questions that fall under the fol- 
lowing rule there is a supposition and a 
demand : two of the given numbers contain 
a supposition, upon the conditions whereof 
a demand is made, to which the other given 
term belongs y and it is therefore said to 
raise the question; because the number 
smtght has such a connection with it as one 
of these in the supposition has to the other. 
For example : if tlrree yards of cloth cost 
4Z. 10s. (here is the supposition) what are 7 
yards 3 quarters worth? here is the demand 
or question raised upon 7 yards 3 quarters 
and the former supposition. ’ 
2. In the question there will sometimes 
be a superfluous term ; that is, a term which, 
though it makes a circumstance in the ques! 
tion, yet it is not concerned in the proper- 
tion, because it is equally so in both the sup. 
position and demand. Tins superfluous 
term is always known by being twice men- 
tioned either directly, or by some word that 
refers to it. Rxample, if three men spend 
201. in 10 days, how much, at that rate, will 
they spend in 25 days? Here the three men 
is a superfluous term, the proportion being 
among the other three given terms, with 
the number sought; so that any number of 
men may be as well supposed as 3. 
Rule. 1. The superfluous term (if there 
is one) being cast out, state the other thi ee 
terms thus : of the two terms in the suppo- 
sition, one is like the thing sought (that is 
of the same kind of thing the same way ap. 
plied) ; set that one in the second or mid- 
dle place ; the other term of the supposition 
set in the first place, or on the left hand of 
the middle ; and the term that raises the 
question, or with which the answer is con- 
nected, set in the third place, or on the 
right hand ; and thus the extremes are like 
one another, and the middle term like the 
thing sought : also the first and second 
terms cmitain the supposition, and the third 
raises the question ; so that the third and 
fourth have the same dependance or con- 
nection as the first and second. 2. Make 
all the three terms simple numbers of the 
lowest denominations expressed, so that the 
extremes be of one name. Then, 3. Re- 
peat the questions from the numbers thus 
stated and reduced (arguing from the sup- 
position to the demand), and observe whe- 
ther the number sought ought to be greater 
or lesser than the middle term, which the 
nature of the question, rightly conceived, 
will determine ; and, accordingly, multiply 
the middle term by the. greater or lesser ex- 
treme, and divide the product by the other, 
the quote is like the middle term, and is 
the complete answer, if there is no remain- 
der ; but if there is, then, 4. Reduce the 
remainder to tlie denomination next below 
that of the middle term, and divide by the 
same divisor, the quotient is another part 
of the answer in this new denomination. 
And if there is here also a remainder, re- 
