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XXIII. An essay towards the calculus of functions. By C. 
Babbage, Esq. Communicated by IV. H. Wollaston, M. D. 
Sec. R. S. 
Read June 15, 1815. 
The term function has long been introduced into analysis 
with great advantage, for the purpose of designating the 
result of every operation that can be performed on quantity. 
This extent of signification has rendered it of essential use, 
but the various applications of which it admits, and the ques- 
tions to which it gives rise, do not appear to have met with 
sufficient attention. 
I propose in the following paper to present an outline of a 
new calculus, which naturally results from it. It compre- 
hends questions of the greatest generality and difficulty, and 
will probably require the invention of new methods for its 
improvement. 
Many of the calculations with which we are familiar, con- 
sist of two parts, a direct, and an inverse ; thus, when we 
consider an exponent of a quantity : to raise any number to a 
given power, is the direct operation : to extract a given root 
of any number, is the inverse method. The differential cal- 
culus, which is a direct method, naturally gave rise to the 
integral, which is its inverse : the same remark is applicable 
to finite differences. In all these cases the inverse method is 
by far the most difficult, and it might perhaps be added, the 
most useful. 
