RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

194 Chapter 10. Toward rigorization of analysis
jects of analysis. EULER’S definitions and use of functions have attracted the interest
of historians of mathematics. 5 In the present context, the two most important aspects
of EULER’S approach are:
1. EULER’S variable quantities were universal in the sense that they would “com-
prise all determinate values” including positive and negative, rational and irra-
tional, and real and imaginary values.
2. EULER defined a function of a variable quantity to be an “analytic expression
composed in any way from the variable quantity and numbers or constant quan-
tities”. The operations allowed to form analytic expressions were algebraic opera-
tions, both finite and infinite.
Together, these two aspects entail an important interpretation of the concept of
equality between functions. To EULER, two analytic expressions were considered
equal if one could be transformed into the other by a sequence of (formal) manip-
ulations. For instance, in developing methods for expanding rational functions into
power series, EULER described — in the Introductio — a method by which the two ex-
pressions
1
1 − x and
∞
∑ x
n=0
n
should be considered equal because the latter could be obtained by (formally) carrying
out the division. 6 Of course, EULER was aware that peculiar results would emerge if
certain numerical values were inserted for x and the equality was believed to apply to
this numerical case as well. The proper interpretation of the sum
1 − 1 + 1 − 1 + . . .
had been a controversial subject throughout the first half of the eighteenth century.
To EULER, its sum would be 1 2 by the formal equality above. Generally, EULER chose
to focus on the formal aspect of functional equalities ignoring the “paradoxes” which
might occur if numerical values were inserted.
To a modern reader, EULER’S disregard for numerical convergence may seem odd.
However, it corresponds to a paradigm in analysis — the Euclidean paradigm — which
focused on the fruitful manipulations of finite or infinite expressions; the un-problematic
transition from one such representation to another constituted a cornerstone of EU-
LER’S skillful investigations in analysis.
5 See e.g. (Jahnke, 1999; Lützen, 1978; Youschkevitch, 1976).
6 (L. Euler, 1748, §60–61).