Page 18 - Foundations Of Differential Calculus

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On Finite Differences

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1. From what we have said in a previous book about variables and func-
tions, it should be clear enough that as a variable changes, the values of
all functions dependent on that variable also change. Thus if a variable
quantity x changes by an increment ω, instead of x we write x + ω. Then
2
2
3
2

such functions of x as x , x ,(a + x) / x + a , take on new values. For
2
3
2
2
2
3
2
3
instance, x becomes x +2xω + ω ; x becomes x +3x ω +3xω + ω ;
2 2
(a + x) / a + x is transformed into
a + x + ω .
2
2
a + x +2xω + ω 2
This kind of change always occurs unless the function has only the appear-
ance of a function of a variable, while in reality it is a constant, for example,
0
x . In this case the function remains constant no matter how the value of
x changes.
2. Since these things are clear enough, we move now to those results
concerning functions upon which rests the whole of analysis of the infinite.
Let y be any function of the variable x. Successively we substitute for x
the values of an arithmetic progression, that is, x, x + ω, x +2ω, x +3ω,
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x +4ω,... . We call the value of the function y when x + ω is substituted
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for x; likewise, y is the value of the function when x +2ω is substituted
1 L. Euler, Introductio in Analysin Infinitorum. English translation: Introduction to
Analysis of the Infinite, Books I, II, Springer-Verlag, New York, 1988.