S °3 
Resolution of Algebraic ’ Equations. 
preceding, by developing every part of the functions that enter 
into the resolution, so as to be able to compose it at once, or 
make a complete reduction, of the: equation, without the inter- 
vention of any other steps. 
33 . Let (n) be taken sta (5), or any higher number. Here, the 
number of differences is increased to twenty ; and, the higher 
we go the more they increase, so that a direct simple resolution is 
out of the question. Nor are we yet acquainted with any pecu- 
liarity attending five, or any higher number of quantities, upon 
which we can ground a relation to effect a reduction of any 
sort ; wherefore, no method is known for equations of this and 
the higher orders. Whether any may ever be discovered, it is 
not easy to pronounce : if the reasoning from Art. 8 to Art. 15, 
of this Paper, be correct, there can be no chance, until some pe- 
culiar property of quantities of this class can be hit upon. It is 
perhaps a discouraging presumption against the existence of any 
such property, that no art nor labour has hitherto afforded the 
least clue to lead to one ; but, on the other hand, it is impossible 
to tell what general properties of quantity may remain to be dis- 
covered ; and, from the great distance the peculiarities of the 
degrees we have treated of lie from the surface, and their total 
want of order or connection with each other, it may be justly 
expected those of the higher degrees may lie still more de- 
tached and remote, beyond any efforts that have yet been made 
upon the subject. The proper method to proceed seems there- 
fore to be, abandoning all projects for the general resolution of 
equations , to investigate regularly the abstract properties of each 
separate order or number of quantities , turning them into all shapes , 
sifting all their combinations, and constructing and examining the 
equations of different complex functions of them, in order to see 
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