of the Machinery for Calculating Tables . i&f 
to you some of the mathematical principles on which it is founded. 
The contrivances of Pascal and others have, as far as I am awate, 
been directed to an entirely different object. Machinery which will 
perform the usual operations of common arithmetic, will never, in 
my opinion, be of that essential utility which must arise from an 
engine that calculates tables ; and although mine is not defec- 
tive in these points, and will extract the roots of numbers, and 
approximate to the roots of equations, and even, as I believe, to 
their impossible roots, yet, had this been its only office, I should 
have esteemed it of comparatively but little value. As far as I 
have inquired, I believe the method of differences has now, for 
the first time, been embodied into machinery ; and in speaking 
of this method, I am far from meaning to confine myself to cal- 
culating tables by constant differences. The same mechanical 
principles which I have already proved, enable me to integrate 
innumerable equations of finite differences, if I may be allowed 
to use the term integrate , in a sense somewhat different from its 
usual acceptation. My meaning is, that the equation of diffe- 
rences being given, I can, by setting an engine, produce, at the 
end of a given time, any distant term which may be required ; 
or, if a succession of terms are sought, commencing at a distant 
point, these shall be produced. Thus, although I do not de- 
termine the analytical law, I can produce the numerical result 
which it is the object of that law r to give. Some kinds of equa- 
tion of differences, can be adapted to machinery with much 
greater facility than others ; and hence it will become an object 
of inquiry, how, when we wish to calculate that of any trans- 
cendant, we may deduce from some approximate equation the 
differences which may be suitable to our purpose. Thus, you 
see, one of the first effects of machinery adapted to numbers, 
has been to lead us to surmount new difficulties in analysis ; and 
should it be carried to perfection, some of the most abstract 
parts of mathematical science will be called into practical utility, 
to facilitate the formation of tables. The more I examine this 
theoretical part, the more I feel convinced that it will belong 
before the novel relations which it presents will be exhausted ; 
and if the absence of all encouragement to proceed with the me- 
chanism I have contrived, shall prove that I have anticipated too 
far the period at which it shall become necessary, I will yet ven- 
