and Infinite Series. 
403 
determining the value of n from the equation ^ 
hy an equation under the 3d degree ; but by re- 
ducing this equation out of radicals, there refults ano- 
ther cubic equation of no lefs difficulty to refolve than 
the original one. We muft therefore fearcn for other 
methods of determining the roots ; and firft it will be 
proper to treat of the rule which is called cardan’s. 
41. Let a? 3 + px- q be the general equation where 
p and q denote any given numbers with their figns, pofi- 
tive or negative. And let z +/ denote one of the roots 
of this equation, that is, let the root be divided into any 
two parts z and/. Hence then x = z +/; which value 
of x being fubftituted for it in the original equation 
x* + px - q> that equation will become z* + 3 z*y + 3 zfi 
+/ 3 +p.z +y = £,or # 3 +/ 3 + 3 zy ,z +y+p.z+y=q. 
Now on introducing the two unknown quantities z and/, 
we fuppofed only one condition or equation, namely, 
z +/ = x; we are therefore yet at liberty to affume any 
other poffible condition we pleafe : but this other condi- 
tion ought to be fuch as will make the equation reduci* 
ble to a limple one, or to a quadratic, in order to obtain 
from it the value of z or/: and for this purpofe there 
does not feem to be any other proper condition beiide 
Vol. LXX 
H h h 
that 
