9 1 6 Extenfton of Cardan’s Rule to the 
thence arifing, will be equal to r. This may be done 
in the following rnan ner. 
If x is - \j 1 7 + -f + sj 1 -j- s > or j ~ + s 1 J + 1t _ we 
£hallliavea: 3 = r - +r+ 3 xU-kt]- 
* fr-J T + 3 *l- + J| 
f7 ]i fr 
I— + S \ 3 x’ — - 
' z 
S -T 
+ y— J='r+ 3 x 
f r ] 2 j r 1 1 f r 
1 
r ! 
l_ +J ji x j_ rp + 3 l* |~ + s 
J X 
H 
3 * l T + J| J x 
— + JT*x 
2 L 
■ j n +3 
• + J T X 
i fr I r 
■-JF’X 
r+ 3 x 
rr rr <r~) 1 frr rr 
+h i+tx +- * 
4 4 27I 1 4. 4 27' 
V 
+ 3 X'|--wp;x 
r 1 1 
ji=r+ 3 x 
~ ijt 
T + ' J - 
3 Xv— H 
0 1 27 1 
X. J 
1 2 
T~r+ 3 x — +j |3 
2 i 
2 
s j=r- f-3 x,: 
- + J T x 
* 4 + 
q fr 
— X i— — , 
X--+ 5 X-X 
3 • 3 ' U 
•27 
-r + qx 
|~ + "j|i + q x - j 3.. And qx will be = ? x j— + j|* + q x 
2 1 2 2 
jf— jji. Therefore a: 3 - will be = r + qx j-f+ r | ^ + q x 
— j p* — 7 x. 
+ r 
+ jp - ^ x |f — j f = r ; and confequently 
, is the true value 
i+ fF^> or ^\T+s^^\IZs 
of x in the cubick equation x^-qx-r:' 
13. N. B. I do not remember to have feen thefe fub- 
ftitutions, or fynthetical demonftrations of the truth of 
the expreffions given by cardan’s rule, in any book of 
algebra. 
14. An 
