9 1 o 'Extenfion of Cardan’s Rule to the 
x^-qx-r will appear by fubftituting it inftead of at in 
the compound quantity x r -qx, which will thereby be 
■ 
feen to be equal to r, as it ought to be. 
This may be done in the manner following. 
Since x is=^ 3 17 
+3+ 
3 ^ 
or 
3 i r 
+ J 
3 + 
+ s 
we fhall have x 3 - — + s + 3 x I— +3) 1 
3x1— +3 P 
3 * 
+ 3 x| t +3! 
K 
3fJ. 
27 X — + J’ 
= -j+s + qx 
— + 
z 1 
3 x 7+ J 
1 
2 7 r 
-f+ 2 J.S 
; and qx~qx 
— + 3F + 
Z 
qi 
3 x !— l 
^ 1 2 1 
and confe- 
•quently x‘ s -qx=—+ 3 + 
27^ 
+ 27 s 
= — + ,f+— 
2 27^ 
— =r — 
5 V 2 
a* r 
? = — +S + 
n* 
■* 7 * 1 + 54 * 
27 'r+ 54 J 
Now j’i, or ' — — , is 
2 7 
7 ;T - 4 r 
108 
zqrr-A q 
36x3 
; or, if we put 
2*00= iqrr-\q\ we fhall have 33, or 5 - U, and 
3 = 
0/ 
6 v ^3 
1 2 -v/3 x <7 3 
Therefore 
2^- 
2 7 r + S 4 i 
2V3 X-? 3 
. 2^ 3 
-.IS ==7^ 1 
2 T 
27 r -f 54 x nt 6X2)'X V3 X r+ 5402 
6^3 6 V3 
— t — — — . Therefore 3+ — ~~~is=-fr+ 
*7 v 3 X r+ 5430 2 7 v 3 .x r -h 9 »; ; 27^4- 54* oi/3 
a / /> V/ ^ *7 i / '■y V 
2 V$x > i 
27 v' 3 X m + 9^02 -f- 367 s 3 a/ 3 x m 4- 4- 47 s _ 
27 v^3 x r 4-902 6 x 27 x 3^4-6 x 9’x V 3 X 54r-{-6 ^3 x 20 
gv^ 3 Xm 4 27rr-r 47 3 4 - 47 s 3 ^3 X^4 27^ - __ V^X m + 9 r r v n 
54^ 4- 6 V 3 X ^ ** £4*^15 ~~ i%r + 2 */ i 
■ J* 
