The result just mentioned was bound to be questioned. Indeed, Fourier
series supplied a host of examples of series of continuous functions with
discontinuous sums. Of course, Cauchy was aware of this and did not
respond to objections such as Abel's. {\it Cauchy's theorem is true if we
require convergence for all x+\alpha, x real and \alpha infinitesimal, and
false if all we are given is convergence for all real x in the interval!}
Since Cauchy did not bother to issue a clarification and the use of
infinitely small quantities seemed to be questionable, they were gradually
eliminated.
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He added:
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I think that there is a gap between the sentence "Thus ... uniformly" in
the paragraph due to Judith Grabiner and the passage due to Detlef
Laugwitz....
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This "gap" occurs because Laugwitz is mistaken. I do not mean,
of course, that you can't invent a way of assigning meanings
to Cauchy's words that would give that result; but I don't think
such invention has much historical importance. The important thing
is that Laugwitz's assertion
"Of course, Cauchy was aware of this and did not respond to
objections such as Abel's....Since Cauchy did not bother to
issue a clarification..."
is demonstrably false.
The relevant paper is in the _Oeuvres_ I.12 (p.30ff) and comes from
the Comptes Rendus for 1853. Its opening words are (in quick
translation):
In establishing the general rules for convergence of series,
in my _Analyse alge'brique_, I stated in addition the following
theorem:
When the different terms of the series
(1) u_0, u_1, u_2, ... u_n, u_{n+1}, ...
are functions of the same variable x, continuous with respect
to this variable in the neighborhood of a particular value for
which the series is convergent, the sum s of the series is
also a continuous function of x in the neighborhood of this
particular value.
As has been pointed out by MM. Bouquet and Briot, this theorem
is true for series arranged according to the increasing powers
of a variable. But for other series, it cannot be accepted
without restriction ["il ne saurait e^tre admis sans
restriction"].
Cauchy proceeds to give the example with terms [sin(kx)]/k, says
that "it is easy to see" how to modify the statement of the
theorem, states a uniform version of the Cauchy criterion, and
makes the distinction clear by computing to show explicitly how the
counterexample fails to satisfy this modified criterion.
William C. Waterhouse
Penn State