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SECTION XI. 
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THE APPLICATION OF ALGEBRA TO THE RESOLUTION 
OF NUMERICAL PROBLEMS. 
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THEN a Problem is proposed to be solved alge- 
f braically, its true design and signification ought, 
in the first place, to be perfectly understood, so that (if 
heedful) it may be abstracted from all ambiguous and 
unnecessary phrases, and the conditions thereof exhibit 
ed in the clearest light possible. This being done, and 
the several quantities therein concerned being denoted 
by proper symbols, let the true sense and meaning of 
the question be translated from the verbal, to a symboli 
cal form of expression; and the conditions thus express 
ed in algebraic terms, will, if it be properly limited, give 
as many equations as are necessary to its solution. But, 
if such equations cannot be derived without some previ 
ous operations ( which frequently happens to be the case), 
then let the Learner observe this rule, viz. let him con 
sider what method or process he would use to prove, or 
satisfy himself in, the truth of the solution, were the 
numbers that answer the conditions of the question to 
be given, or affirmed to be so and so; and then, by fol 
lowing the very same steps, only using unknown 
symbols instead of known numbers, the question will 
be brought to an equation. 
Thus, if the question were to find a number, which 
being multiplied by 5, and 8 subtracted from the pro 
duct, the square of the remainder shall be 144; then, 
having put a — 5, b — 8, and c — 144, suppose the 
number sought 
to be —. — — — 4} (or) a? 
then 5 ? or or times that num- ¡> 
ber will be — — — i 
from which 8,or b being sub-} 
tracted, there remains — 5 
which, squared, is 1441 a‘x~ 
SO! 
12 
144 
ax 
ax—b 
zaxb b 1 
Therefore a x x 2 — 2axb + b 1 is - c (or 144) accord 
ing to the conditions of the question. In the same man- 
.