go 8 Extenjion of Cardan’s Rule to the 
greater of the two parts be called a , and the leffer b* 
Then will ab be = L, and confequently 3 ab will be = q y 
and %abx [ a+b will b e-qxa+b. 
Now, iince a+b is equal to x, we fhall have al+^aab 
+ 3 abb + b l — x 3 , and qx a + b- qx. Therefore x 3 — qtz 
will be =a l + $aab+ ^abb+b 3 - qx a + b-aP + ^abx a+b + 
b 3 -qx a+b ; that is (becaufe 3 abx a+b is —qx a + b) x 3 — 
qx will be - a? + b\ Thei'efore r (which is -x 3 - qx) 
will be = a 3 + b 3 . 
But, lince %ab is ~q y we fliall have b ~~- y and b 3 — 
Therefore a l +b 3 is —a 3 +~. and r (which is-a 3j r 
2ja* 2ja* 7 v 
3 3 
b 3 )is=a 3 +~. Therefore ra z is = <? 6 +— and ral-al’ is 
' 27a* 2; 7 
-i*. 
2 7 * 
But ra 3 — a^ is the product of the multiplication of 
r-a 3 into a 3 , which are together equal to r. Therefore 
(by El. II. 5.) rcP—cf muft be lefs than the Iquare of half 
r, that is, than and conlequently may be fubtra&ed 
from it. Let it, and its equal be fo fubtradled. And 
■* 
we fliall have -- ra 3 4 - — . Therefore the fquare- 
4 4 *7 
root of ~~ ra 3 + a'" will be equal to But the 
fquare-root of + al is the difference of -j and a\ 
that 
