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advantage/' remarks Murphy, “ in all branches of analysis 
to bind together isolated methods of solving problems, 
which are at bottom of the same nature; the advance- 
ment of analysis has been principally thus achieved; classi- 
fication to the chemist and the naturalist generate science; 
comprehensive processes are the perfection of analysis.” 
Methods of solving equations of the first four degrees in 
finite terms have long been known; but all attempts to solve 
the higher equations have, except in particular cases, failed. 
Many distinguished analysts, both in the last and present 
century, have attacked the problem, but although their 
labours have resulted in throwing much light on the theory 
of equations, and in advancing the knowledge of algebraical 
science in general, yet not one of them has succeeded in ob- 
taining the coveted solution. Towards the close of the last 
century both Lagrange and Vandermonde attacked it with 
all the advantages of an improved analysis, but they found 
the theoretical difficulties encircling it altogether insuperable. 
Since then this celebrated problem has been generally re- 
garded as incapable of solution; and indeed, analysts of the 
highest eminence have put forward what most mathemati- 
cians accept as demonstrations of its absolute impossibility. 
