With an auxiliary cubic 
( 
3 ^2 
2 
nh\ cn^ ac ^ 
“^ T >--8 + 2-8 = ‘^' 
,( 23 ) 
Newton and Maclaurin pass over this auxiliary cubic in 
silence, that is, their minds were evidently occupied with 
illustrating the method of surd divisors to such an extent 
that they did not obtain this cubic at all. 
Simpson, following somewhat in the v/ake of Newton 
and Maclaurin, does indeed slightly refer to it as being an 
equation which “rises to the sixth dimension, and would 
perhaps require more trouble to reduce it than even the 
original propounded, little advantage would be reaped 
therefrom.” (Simpson’s Tracts, page 108.) 
This view is surprising from Simpson, by far the greatest 
mathematician of his age. Had he used instead of 
/3, V, in all probability he would have found the auxiliary 
cubic referred to. 
Emerson, writing twenty years after Simpson, in his 
solution of biquadratic equations, makes no use of N ewton’s 
method of surd divisors, a system which is never likely to 
be resuscitated in the presence of Horner’s method of 
solving numerical equations of all dimensions. 
5. The form (D) is entirely due to Euler alone, and, like 
many of the mathematical investigations of this wonderful 
man, it is characterised by great clearness, simplicity, and 
analytical beauty. 
As there is no restriction on the constants a, /3, v, the 
values v/a, \//3, will be used instead. 
The following operations are readily performed from (C) ; 
a + /3 + V + 2{ \/ a/3 + V 
[a + ft + vY + 4(a + ft + v){\/ aft + \/ av + \/ ftv) + aft + av + \J ftvY 
(a + ft-\-vY + 2(a + ft + v)\^0G^ — (a + /3 + j/)| + 4(a/3 + av + ftv) + 8 v/ aftv.X 
- 2(a + /3 + v)x^ - S^/ aftv.X +{a+ft + v)^- 4(a/3 + av + ftv) = 0...(24) 
The coefficient of x in (24) is the only term affected by 
changing the signs of V Therefore, when any 
