914 
Extenfion of Cardan’s Rule to the 
u 
9 X 
27 *| 
~~—s + q x 
2 x 2 
-R*+ 
m 
2 7 x\ t ~S 
=, and qx = q x 
3 * T~ s 
I 
* + 
m 
*7x\--s 
quently x z -qx=—-s+ 
3 * 1 - 
and confe- 
-s * 
27 X 
< 7 3 r r 
i == = S + = s 
r 2 27r 2 
— s — — 27 s 
2 2 1 
— = — - s+ 
2 7 r 54 * 2 
Now ss t or - — — is— — 
2 7 r ""54* 7 4 27 108 
37 6x T herefore if we put mm-iqrr-\q 1 ^ we fhall 
36x3 
, mm j 
have w = -7 — , and j = 
36 x 3^ 
Therefore 
2 r 
2 f 
2 r 
(>*/$' 
— 12^/3 xg 3 
IS 
2 ^r— 54J 
2 V ^3 X q 3 
^ 54 ^ 6 X27X 3 X^ — 54 ”* 6X27X 3 X r- 54*2 27X1/ — 
• 6 \/ 3 6 \/ 3 
Therefore — - + 
2 2 
- IS 
7 # 
M/3 X? 3 __ r 
2;r — 54J 2 6^/3 27 x >/3 X r — 
— 27 X ^3 X ^ + 97®*® + 3^ 3 r •— 3^3 X rm + t®®* 4 - — r 
6x 27 x 3^— 34 ^3 X ®z 2 54 r “ 6^/3 X ?n ~ ~~ T 
—3 \/ 3X^4- 2 7?t- 4|7 3 + 4^ 3 >* — 3 \/ 3X^7® 4-2777- 34rr-6i/ 3Xm-6y / 3Xm+ 34^ 
54r— bV$Xm ~~ 2 347*— b */ $Xm io8r~ 12 V ^3 X m 
_io^Wjxw — f. But it has been before fhewn that 
io8r — 12 V 3 Xm 
2 r 
x z -qxis=—s + — - — . Therefore x % - qx is = r ; and 
1 2 2 jr — $ 4 s x 7 
4(F 
s + 
*IW* 
equation x'-qx-r. Q. E. D, 
is the true value of a; in the cubick 
Invejligatiom 
