Resolution of Algebraic Equations. 295 
Example 3d Let ( — 1, — 4 ,4-5), the roots of the equation 
(jr 3 — 2ix — 20 = o), which are the differences used in the 
last example, be next taken ; 
l + 4 = + 3 }— 5 + 3 v /- J ---5 + i a /-3 
— ^6 j’+4+^v'— J — 4+2/ — 3^ 22—10-/ 
— <; + 1/ 
4+5 = + * ). 
4_ 5 -_ 9 / 
-9/— 1—3/— 3 
5 + 1 / — 3 
4 + z /-3 
1— 3 v "“ 3 
— 5 + 1 v' — 3 
4 + 3 
1— 3 \/ — 3 
25-10/— 3 
16+16/— 3 
1- 6/— 3 
- 3 
— 12 
-27 
22— 10/— 3 
4 + *6/ — 3 
—26— 6/— 3 
- 5 + 1 v' — 3 
4+ 2/ 3 
1- 3^-3 
-110+50/-3 
16 + 64/ — 3 
—26— 6/— 3 
30+22/— 3 
-96+ 8/ — 3 
— 54+78A/ — 3 
80+72/ — 3 —80 + 72/ — 3 —80+72/ — 3 
which last cube, if divided by (3), becomes (— - 3 - -f- 2 4\/~3)» 
or exactly the reverse of the first ; the reason of which will be 
shewn in the next section of this Article. 
The cube ( 24 — - 3 - v/ — +) , when its equation (x 3 = 2 4 — - 3 - \/ — y) 
is made rational, gives the quadratic-formed equation of the 6th 
degree (x 6 — 48 x 3 — -{- 57 6 = — L t? ., ? .) ; or, transposing all the 
terms to one side, and dividing it by (2), to reduce it, as before, 
(x 6 — 6x 3 - f 3 2 4 7 3 = o) ; the same equation that results from the 
common methods. 
2dly. The differences of the three differences (a — b, a -f 26, 
— 2 a — b) are (3 a, 3 b, 3 x a -j- 6), or merely three times the 
original quantities. Had, therefore, the differences themselves 
been taken as original quantities, and binomials been formed 
from them, according to the directions before observed, those 
binomials , and the ultimately resulting cubes, would differ from 
the former, in nothing essential but the place of the surd. The 
differences which were affected with it before, would now be 
Qq 2 
