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XIV. Observations on the Analogy which subsists between the 
Calculus of Functions and other branches of Analysis. By 
Charles Babbage, Esq. M. A. F. R. S. 
Read April 17, 1817. 
It is my intention in the following Paper to offer to the 
Royal Society some remarks on the utility of analogical rea- 
soning in mathematical subjects, and to illustrate them by 
some striking facts which have occurred to me, when com- 
paring the calculus of functions with other modes of calcula- 
tion with which mathematicians have been long acquainted. 
The employment of such an instrument may, perhaps, create 
surprise in those who have been accustomed to view this sci- 
ence as one which is founded on the most perfect demonstra- 
tion, and it may be imagined that the vagueness and errors 
which analogy , when unskilfully employed, has sometimes 
introduced into other sciences, would be transferred to this. 
It is, however, only as a guide to point out the road to 
discovery, that analogy should be used, and for this purpose 
it is admirably adapted. 
It is usually more difficult to discover than to demonstrate 
any proposition ; for the latter process we may have rules, 
but for the former we have none. The traces of those ideas 
which, in the mind of the discoverer of any new truth, con- 
nect the unknown with the known, are so faint, and his atten- 
tion is so much more intensely directed to the object, than to 
