algebra. 
291 
becomes a-\-x: b-\-x, which is greater or lefs than the for¬ 
mer, according as is greater or lefs than or by re- 
0 ,b+x 0 b 
. ab-\-bx . 
ducing them to a common denominator, as — . is 
;i b . bA-x 
greater or lefs than — that is, as b is greater or lets 
b. b-\-x 
tlian a. 
Hence a ratio of greater inequality is increafed, and of 
lefs inequality diminiihed, by taking from the terms a 
quantity lefs than either of them. 
If the antecedents of any ratios be multiplied together, 
and alfo the conlequents, a new ratio refults which is faid 
to be compounded of the former. Thus ac : bd is faid to be 
compounded of the two, a\b and c:d. It is alio fome- 
times called the fum 'of the ratios; and, when the ratio 
a : b is compounded with itfelf, the refulting ratio a 1 : is 
called the double of the ratio of U:b-, and, if three of 
thefe ratios be compounded together, the refult, a? : b 3 , is 
called the triple of the firft, &c. Alfo the ratio of a \ b 
JL JL 
is faid to be one third of the ratio of a 3 :b 3 y and a m :b"‘ is. 
faid to be an jath part of the ratio of a : b. 
If the confequent of the preceding ratio be the antece¬ 
dent of the fucceeding one, and any number of fuch ra¬ 
tios be taken, the ratio which arifes from their compofi- 
tion is that of the firft antecedent to the laft confequent. 
Let a: b, b: c, c: d, See. be the ratios, the compound ratio 
is axbX.c-bx.cxd', or, dividing by by.c, a:d. 
A ratio of greater inequality compounded with another 
increafes, and a ratio of lefs inequality diminiflies, it. Let 
the ratio of x :y be compounded with the ratio of a: b, the 
refulting ratio ax: by is greater or lefs than the ratio a: b, 
according as ~ is greater or lefs than i. e. according 
by , b 
as x is greater or lefs than_y. 
On Proportion. 
Four quantities are faid to be proportionals , when the 
firft is the fame multiple, part, or parts, of the fecond, 
CL C 
that the third is of the fourth. That is, when-=:-, the 
b d 
four quantities, a, b, c, d, are called proportionals. This 
is ufually exprefled by faying, a is to b as c to d\ and thus 
rep relented, a:b::c:d. The terms a and d are called the 
extremes, and b and c the means. 
When four quantities are proportionals, the product of 
the extremes is equal to the product of the means. Let 
a , b, c, d, be the four quantities ; then, fince they are pro- 
a c 
portionals, anc h hy multiplying both Tides by bd, 
ad=bc. 
Cor. 1. If the firft be to the fecond as the fecond to 
the third, the produdt of the extremes is equal to the 
fquare of the mean. 
Cor. 2. Any three terms in a proportion being given, 
the fourth may be determined from the equation ad—be. 
If the product of two quantities be equal to the product 
of two others, the four are proportionals, making the 
terms of one prodinft the means, and the terms of the 
other the extremes. Let xy—ab ; then, dividing by ay, 
x b , 
or x: a:: b:y. 
a y 
If a: b::c: d, and c:d::e:f, then will a: b:: e :f; be- 
caufe and therefore ~=t, or a : b :: e : f. 
b d d J b f J 
If four quantities be proportionals, they are alfo pro¬ 
portionals when taken inverfely. If a : b:: c : d, then b : a 
:\d:c, F01 't—% and dividing unit by each of thefe 
b a 
equal quantities, or taking their reciprocals,; i. e. 
. a c 
b : a : : d: c. 
If four quantities be proportionals, they are alfo pro¬ 
portionals when taken alternately. It a:b::c:d, then 
a:c:\b:d. Becaufe the quantities are proportionals, 
a c . . .... b a b 
; and, multiplying by -, -; or a: c ::b:d. Un- 
0 d c c d 
lefs the four quantities are of the fame kind, the alterna¬ 
tion cannot take place, becaufe this fuppofes tire firft to be 
fome multiple, part, or parts, of the third. One line may 
have to another line the fame ratio that one weight has to 
another; but a line has no relation, in refpeft of magni¬ 
tude, to a weight. In cafes of this kind, if the four 
quantities be reprefented by numbers, or other quantities 
which are fimilar, the alternation may take place, ai d th 2 
conclufions drawn from it will be juft. 
When four quantities are proportionals, the firft toge¬ 
ther with the fecond is to the fecond as the third together 
with the-fourth is to the fourth. 
Let a:b::e:d, then 
componendoj a\b:b\: c-\-d: d. 
Becaufe by adding one to both Tides, — 
b d b 
, +I, that is, 
a-\-b c-\-d 
, or a-\-b :b:: c-\-d: d. 
Alfo dividendo, a—b : b :: c — d \ d. 
Cl c 
Becaufe by fubtradting one from ’ both Tides, 
c - . a —, 
—-i, that is —— 
a b 
d 
or a — b :b\: c — d : d. 
-b :: r. : c — d. 
By the laftarticle 3 
b b c.—d d 
-X-=—r-X—, or 
c 
Again convertendo, a: a 
a—b c—d bd a— 
"———■—; and ———, therefore—— 
0 d a c b 
a—b c—d 
————-—; that is, a—b :a\: c — d : c, and, inverfely, 
a : a—b :: c : c — d. 
When any number of quantities are proportionals, as 
one antecedent is to its confequent, fo are all the antece¬ 
dents taken together to all the confequents taken together. 
Let a : b :: c: d :: e:f See. 
Then a: b:: a-\-c-\-e : b-\-d-\-f. 
a c . 
Becaufe dd"z. 
b d 
-.be ; in the fame manner, af-=ebe, alfo 
ab—ba\ hence ab-\-ad-{-a/—ba-\-bc-\-be, or a.b-\-d-^f— 
b.a-\-c-\-e, and a : b\: a-\-c-\-e : b-\-d~\-f. 
When four quantities are proportionals, if the firft aud 
fecond be multiplied, or divided, by any quantity, as alfo 
the third and fourth, the refulting quantities will be pro¬ 
portionals. 
c d 
Let a: b ::c: d, then will ma: mb 
_ a c ’ 
For ; 
b d ’ 
therefore, 
mb 
x 
-. a 
c d 
or ma : mb 
7i n 
. If the firft and third be multiplied, or divided, by any 
quantity, and alfo the fecond and fourth, the refulting 
quantities will be proportionals. 
a c ma me 
For therefore — ——e, 
b d b d 
ma me b 
and--, or ma . - 
-.b-~.d ' U 
n n 
d 
;: me : —. 
n 
Cor. Hence in any proportion, if, infteadof the fecond 
and fourth terms, quantities proportional to them be fub- 
ftituted, we have (till a proportion. For -and - are in the 
n n 
fame proportion with b and d. 
In 
