298 
Mr. Wilson's Essay on the 
3<Jly. Let [x 2 — 7 x -f 6) be given, whose roots are ( 1, 2 — 3). 
The binomials derived from these have been before given, in 
the 2d Example to the first Section of this Article; and the 
cube they produce shewn to be (24 — - 3 - — ^), the cube 
root of which cannot be extracted ; it being from the quadratic 
surd, it involves, in truth, not a cube, but a truncate sixth power 
in a cubic shape : and when, to remove its equivocal state, it 
is made rational, shews itself to be properly the sixth power 
equation ( x 6 — 6 x 2 L^J- = o), as before demonstrated. 
26. This is the common reduction of a cubic equation, to one 
of the sixth degree but in form a quadratic, obtained, by clear- 
ing of its quadratic surd, the pure cubic formed by either of the 
two sets of binomials before described ; and this is the only 
reduction of it yet discovered. Perhaps the method called Car- 
dan's rule, is the shortest mode of effecting this reduction; but I 
am not aware, that the real principle upon which it is founded 
has been any where fully analysed and explained, except in 
the foregoing investigation of it. The ordinary expositions of it 
certainly disclose nothing of the principle, and are even in 
many respects faulty; for they treat it as the effect of a suppo- 
position or lucky conjecture, when, in fact, there is no supposition 
or conjecture made ; a regular clue, furnished by certain demon- 
strable peculiarities in some functions of this order of quantities, 
being pursued, till such a relationship amongst the roots may 
be inferred, as may be converted into equality at some known 
period. They also fail to account for the most striking part 
of the result; the irreducibility happening uniformly in cases 
where it has been supposed least to be expected, i. e. when 
the roots are real ; which they refer to a particular limitation 
