and Infinite Series. 407 
root in an imaginary form when the equation has no 
imaginary roots, but in the form of a real quantity when 
it has imaginary; roots. . _ Y ^ ■ 
47. Jt may, perhaps, feem wonderful that cardan’s 
^ A-ftvv 2 Si £ 4 ' >■ j || ... * v». r ^ 
theorem ihould thus exhibit the root of an equation un- 
der the form of an imaginary or impdffible quantity al- 
ways when the equation has no imaginary roots, but at 
JO M j f ; . }j | hlij ' | £ ?V 
no time elfe; and it may juftly be demanded what can 
be the reafon of fo curious art accid ent. But this Teem- 
ing paradox will be cleared up by - the following eohfi- 
deration. It is plain, that this circumffahee hSMt hkve 
happened either through fome impropriety in thfenhan- 
ner of deducing the values of z and-j/ frohi the4wo 
affumed equations x = z + j f and zy<t: as or Clf^ by 
fome impoffibility in one of thefe two Conditions them- 
felves; butj on examination, the dedu&iohs are found to 
be all fairly drawn, and the operations .rightly. p per- 
formed. The true caufe muft therefore lie concealed in 
one of thefe two conditions x = z + y anti In 
the firft of them it cannot be, becaufe it only fuppofes 
i £ . i v 1 
that a quantity, .v can be divided., into, two part? z and_y, 
which is evidently 3 poffible fuppolitipp^ it can |fyerefore 
no where exift but inthe latter* namely, zy : = *- . Now 
this fupi>oii,tion is this, that the produd of the two parts 
» and_y, into which the conftant quantity x is divided, is 
equal 
> t- 
