334 Mr. BaBBaGE on the Influence of Signs 
accomplish his object, but the results soon became so complicated 
that little expectation could be formed of succeeding by that 
means. In this difficulty Mr. Spence introduced into the equation 
to be solved an arbitrary quantity a, which is merely employed 
as a letter by whose powers the resulting series may be arranged : 
if the attempt is now made by continually eliminating, a series 
arises proceeding according to the powers of a, and equations are 
found for determining their coefficients: finally, the arbitrary 
quantity a having performed its office is made equal to unity, 
and the result is the solution of the equation. The success of 
this plan depends entirely on breaking into a number of separate 
parts a very complicated expression, each of these portions being 
separately reducible to known laws. 
On resolving into their separate parts a vast variety of 
questions which have occurred, it has been found that the number 
of individual difficulties is by no means so large as had been 
originally supposed; many of very different kinds have been 
found to depend perhaps on the same integral, or on the solution 
of the same equation. In proportion to the number of questions 
which are reduced to these new difticulties, they themselves assume 
importance, and the celebrity which always attaches to those who 
remove obstacles regarded as insuperable by their predecessors, in- 
duces many to attempt the solution of these purely abstract ques- 
tions. Perhaps these ultimate points of reference may not from their 
nature admit of a comparison with, or reduction to, existing tran- 
scendents: the labour and ingenuity employed in the attempt are 
not however thrown away; relations are discovered by which, 
from a certain number of particular cases numerically given, all 
others may be readily calculated, approximations are discovered 
for determining the cases which are required as data, and, finally, 
they are arranged in tables and accompanied by rules for their 
employment, by which, as far as results in pure numbers are 
