290 
ALGEBRA. 
ftituted for x in the original equation, anfwers the condi¬ 
tion, that is, makes the whole equal to nothing'. 
It ftiould be particularly obferved, that, fince d-A-x+y 
is equal to —in the multiplication and involution 
of quantities, new values are always introduced, which, 
if not again excluded by the nature of the queftion, will 
at length appear. 
Every equation, where the unknown quantity is found 
in two terms, and its index in one is twice as great as in 
the other, may be refolvea as follows : 
Ex. 3. Let z-\-i t z 2 — n 
2+4^+4=21+42=25 
z ^-f 2=±5 
JL 
z 2 =±s— 22=3, or —7. 
Therefore, z—y, or 49. 
When therehire more equations and unknown quantities 
than one, a tingle equation, involving only one of the un¬ 
known quantities, may fometimes be obtained by the rules 
laid down for the folution of iimple equations ; and, one 
of the unknown quantities being difeovered, the others 
may be obtained by fubftituting its value in the preceding- 
equations. 
r *-^=4 ] 
Ex. 4, Let < x+3y f To find x and y. 
X -\-2 J 
From the firft equation, 2x — 
x-\-y— 8 
And .*=28— y 
From the 2d equation, xy-\-2y —a-— 3_y=A-+2 
Or xy —2a-— -y—i 
By fubftitution, 8— yXy —2X8— -y — y— 
Sjy— -y 2 —16+24*— -y—2 
9 y I 6 —1—2.22= I 8 
f— 9>——18 
, 81 81 
y — 97+ —=- 1S— 
4 4 
y 9 — 3 
y - 
2 
9 — 3 
16, or 3 
y— 2, or 5. 
rendered more Ample by 
a~8- 
The folution will often be 
particular artifices, the proper application of which can 
only be learned by experience. 
Problems producing Quadratic Equations, 
'Prob. 1. A perfon bought a certain number of oxen 
for eighty guineas, and, if he had bought four more for 
the fame fum, they would have cofi a guinea a-piece lefs ; 
required the number of oxen and price of each. 
Let x be the number 
Then — is the price of each 
a: 
And 
the price of each on the fecond fuppofition 
So 
•*+4 
8° 80 
——=-1, by the queftion, 
W+4 A' 
8oa'+3 20 
80=--- x —4 
X 
8 o.v= 8 Oa: +3 2 o—,v E —4-V 
x"-\-4.x~3 20 
A?“+4^+4=324 
■*+22= rh 18 
x -1-18—2—16, or—20. 
80 So . , . c , 
—=2—-=25 guineas, the price 01 each. 
In this and in many other cafes, efpecially in the folu¬ 
tion of philofophical queftions, we deduce, from tire alge¬ 
braical procefs, anfwers which do not correfpond with the 
conditions. The reafon feems to be, that the algebraical 
expreilion is more general than the-common language, and 
the equation which is a proper reprefentation of the con¬ 
ditions will alfo exprefs other conditions and anfwer other 
fuppofitions.' In the foregoing infiance, fince x may either 
reprefent a pofitive or a negative quantity, the equation 
——-1, when x is negative or reprefentsthe diminu- 
A—p 4 X 
tion of flock, will be a proper exprefilon for the folution 
of the following problem. A perfon fells a certain num¬ 
ber of oxen for 80 guineas, had lie fold four lefs for the 
fame fum lie would have received a guinea a-piece more for 
them ; required the number fold. 
Prob. 2. To divide a line of 20 inches into two parts, 
fuch, that the redlangle under the whole and one part may 
be equal to the fquare of the other. 
Let x be the greater part; then will 20—.v be the lefs. 
And a c —20—.y.20~4oo— 20A-, by the queftion, 
w 3 +20a-—400 
A"-j- 20 A'+I 00^400 +1 O 0 ~ 50 O 
w+iom± \J 500 
at— + \J 5°°—10 or —4/5°°— I0 - 
The obfervation contained in the preceding article may' 
be applied here; and it is to, be remarked, that the nega¬ 
tive values thus deduced are not infignificant or ufelefs. 
Here tlie negative value (hews, that, if the line be pro¬ 
duced 4/500+10 inches, the fquare of the part produced 
is equal to the redtangle under the line given, and the line- 
made up of the whole and part produced. 
On Ratios. 
Ratio is the relation w hich one quantity bears to ano¬ 
ther in refpeif of magnitude, the companion being made 
by confidering how often one contains, or is contained by, 
the other. 
Thus, in comparing 6 with 3, we obferve that it has a 
certain magnitude with refpeCt to it, as it contains it twice; 
again, in comparing it with 2, we fee that it has a different 
relative magnitude, for it contains 2 three times, or it is 
greater when compared with 2 than it is when compared 
with 3. The ratio of a to b is ufualiy expreifed by two 
points placed between them, thus, a : b ; and the former, is 
called the antecedent of the ratio, the latter the confequent. 
Cor. i. When one antecedent is the fame multiple, 
part, or parts, of its confequent, that another antecedent is 
of its confequent, the ratios are equal. Thus the ratio ot 
4 : 6 is equal to the ratio of 2:3, i. e. 4 has the fame mag¬ 
nitude when compared with 6, that 2 has when compared 
alfo the ratio of a : b is equal to 
with 3, 
the ratio of c 
fince -=- 
6 3 
becaufe 
b a 
and 
- reprefent 
a 
the multiple, part, or parts, that a is of b, and c of d. 
Cor. 2. If the terms of a ratio be multiplied or di¬ 
vided by the fame quantity, the ratio is not altered. For 
a ma 
b mb 
Cor. 3. That ratio is greater than another, whofe an¬ 
tecedent is the greater multiple, part, or parts, ol its con¬ 
fequent. Thus the ratio of 7 : 4 is greater than the ratio 
8 %% Thefe 
of 8 : 5, becaufe-, or —, is greater than -, or 
32 
4 20 5 20 
conclufions follow immediately from our idea of ratio. 
A ratio is called a ratio of greater inequality, of lefs ine¬ 
quality , or of equality, according as the antecedent is great¬ 
er, lefs than, or equal to, the confequent. 
A ratio of greater inequality is diminiftied, and of lefs 
inequality increafed, by adding any quantity to both its 
terms. If to theterms of the ratio 7 : 4, 1 be added, it 
becomes the ratio of 8 85, which is lefs than the former. 
And, in general, let x be added to the ratio a:b, and it 
- - becomes 
