936 Extenfion of Cardan’s Rule to the 
But it has been fhewn (in Art. 31.) that, if this fyf- 
tem of terras be multiplied into e% or the cube-root of 
A and the feries thence produced be cubed, and alfo 
multiplied into <7, and from its cube the produdt of its 
multiplication into q be fubtradted, the remainder there- 
by obtained will be (nearly) equal to r. Therefore, if 
the feries 2 + — AA + AA_ _ &: C . (which reprefents, 
or is equal to, the faid fyftem of terms, when — is lefs 
4 * 
than f, and ss is made —j — be multiplied by A, or 
the cube-root of A, and the feries thence produced be 
cubed, and alfo multiplied into q, and from the cube of 
the faid feries the product of its multiplication into q be 
' fubtradted, it will follow that the remainder thereby 
obtained will be (nearly) equal to r; that is, the produdt 
of the multiplication of A, or the cube-root of A, into 
the infinite feries 2 + — - AL + Al_ - Sec. is equal to 
gee 243^ o$oie x 
the value of x in the equation x 3 -qx = r in the fecond 
cafe of it, when - is lefs than A. Q. E. I. 
4 27 
36. This feries 2+— - All + All - See. does not al- 
^ gee 243** 656 1£ 6 
ways converge, but only when ss is lefs than ee, or 
27 4 
is lefs than -, or A is lefs than — or - is greater than 
4 27 4 7 4 O 
half 
I 
