RATIO. 
2 : 3, j. «. 4 lias the same magnitude when 
compared with 6, tliat 2 has when com- 
4 2 
pared with 3, since - = - ; the ratio of 
o o 
a c 
a', bis equal to tlie ratio of c : d, if ^ 
fl c 
because^ and represent the multiple, 
part, or p.arts, that a is of b, and c of d. 
If the terms of a ratio be multiplied or 
divided by the same quantity, the ratio is 
. , _ a m a 
not altered. For -r = — 
b mb 
That ratio is greater than another, whose 
antecedent is the greater multiple, part, 
or parts, of its consequent. Thus, the 
ratio of 7 : 4 is greater than the ratio 
7 35 
of 8 : 5 ; because - , or -j, is greater than 
4 zO 
8 3^^ 
or — . These conclusions follow imme- 
5’ 20 
diately from our idea of ratio. 
“ A ratio is called a ratio of greater in- 
equality, of less inequality, or of equality, 
according as the antecedent is greater, 
less than, or equal to the consequent.” 
“ A ratio of greater inequality is dimi- 
nished, and of less inequality increased, by 
adding any quantity to both its terms. If 
to the terms of the ratio 7 : 4, 1 be added, 
it becomes the ratio of 8 : 5, which is less 
than the former. And in general, let x be 
added to the terms of the ratio a : b, and 
it becomes a -}- a: : 6 a:, which is greater 
/ ^ - I - ^ 
or less than the former, according as 
is greater or less than 2 ; or by reducing 
. . ^ ab-\-bx 
them to a common denominator, as- —'- — = 
&.6 + .'c 
, , ab-\-a X . , ^ . 
IS greater or less than ; that is, as 
b.b-\-x 
b is grfeater or less than a. Hence, a ratio 
of greater inequality is increased, and of 
less inequality diminished, by taking from 
the terms a quantity less than either of 
them. 
If the antecedents of any ratios be mul- 
tiplied together, and also the consequents, 
a new ratio results, which is said to be com- 
pounded of the former. Thus, ac : b d is 
said to be compounded of the two a : b 
and c : d. It is also sometimes called the 
sum of the ratios ; and when the ratio a : b 
is compounded with itself, the resulting 
ratio, : 6^, is called the double of the 
ratio of a : 6, and if three of these ratios be 
compounded together, the result, n’ ; 
is called the triple of the first, &c. Also, 
the ratio of a -.bis said to be one third of 
X i 
the ratio of a? ■. V •, and a™ : b is said to 
be an part of the ratio of a : 
Let the first ratio be a ; 1 ; then : 1, 
: 1, ....a" : 1, are twice, three times, ....n 
times the first ratio ; where n, the index of 
a, shows what multiple, or part, of the 
ratio aP- : 1, the first ratio, a : 1, is. On 
this account, the indices, 1, 2, 3, ...n, are 
called measures of the ratios : 1, a? : 1, 
: 1, a" : 1. 
“ If the consequent of the preceding ratio 
be the antecedent of the succeeding one, 
and any number of such ratios be taken, 
the ratio, which arises from their compo- 
sition, is that of the first antecedent to the 
last consequent.” Let a : b, b ; c, c : d, 
&c. be the ratios, the compound ratio is 
a X b X c : b X c X d; or dividing by 
b X c, a : d. 
“ A ratio of ^'reater inequality, com- 
pounded with another, increases it ; and a 
ratio of less inequality diminishes it.” Let 
the ratio of ar : y be compounded with the 
ratio of a : 6, and the resulting ratio ax : 
is greater or less than the ratio a : b, accord- 
ing as is greater or less than ” 
b’' 
cording as x is greater or less than y. 
“ If the difference between the antece- 
dent and consequent of a ratio be small 
when compared with either of them, the 
double of the ratio, or the ratio of their 
squares, is nearly obtained by doubling this 
difference.” 
Let a -|- X : a be the proposed ratio, 
where x is small when compared with a ; 
then «“-}- 2 ax is the ratio of the 
squares of the antecedent and consequent; 
but since x is small when compared with «, 
X- or X X ^ is small when compared with 
2a X X, and much smaller than a X a; 
therefore, aP -{- 2 ax ; a^, or a -|- 2x ; a 
will nearly express the ratio of a^-j- 2 
ax -j- : aP. 
Thus the ratio of the square of f OOl to 
tlie square of 1000 is nearly 1002 : 1000 ; the 
real ratio is 1002.001 : 1000, in which the 
antecedent differs from its approximate va- 
lue, only by one thousandth part of an 
unit. 
Hence, the ratio of the square root of 
a-\-2xto the square root of a is the ratio 
a X X : a, nearly ; that is, if the difference 
of two quantities be small with respect to 
either of theili, the ratio of their square 
