198 Mr. Babbage on the analogy between the 
the means by which he attains it, that it not unfrequently 
happens, that while we admire the happiness of a discovery, 
we are totally at a loss to conceive the steps by which its 
author ascended to it. 
From these considerations, I think it will appear, that any 
successful attempt to embody into language those fleeting 
laws by which the genius of the inventor is insensibly guided 
in the exercise of the most splendid privilege of intellect, 
would contribute more to the future progress of mathema- 
tical science than any thing which has hitherto been accom- 
plished. Amidst the total absence of all attempts of this 
kind, the following illustrations of one of the most obvious 
assistants of the inventive faculty, will not, I hope, be consi- 
dered useless, even though it should have no other effect than 
that of directing the attention of those who are engaged in 
mathematical enquiries, to this most interesting and impor- 
tant subject. 
At our first entrance into algebra, one of the most 
remarkable circumstances which presents itself is, that 
some fractions which in certain cases apparently vanish, 
have in fact a finite value. Such is the fraction a -^-~ which 
when x — o becomes -1— ^ = — , and yet its real value is well 
known to be log. -j. 
Here then by assigning a certain value to a variable quan- 
tity which is capable of all degrees of magnitude , the expres- 
sion apparently becomes illusory: let us now examine a 
parallel case in the Calculus of Functions. Take the expres- 
4'^— 4-r 
sion — -gj— , and let us suppose *1/ to be an arbitrary charac- 
