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

238 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
Comparison of ABEL’S and DIRICHLET’S proofs. It is interesting to note how the
attitude toward infinitesimals and limit arguments evolved over the first 40 years after
CAUCHY’S Cours d’analyse and we can get an indication of this by comparing ABEL’S
and DIRICHLET’S proofs. Contrary to ABEL’S proof, DIRICHLET completely avoided
the use of infinitesimals in his proof. Instead, he argued completely within the process
based interpretation of limits when he reduced the series to finitely many terms, ma-
nipulated the polynomials and applied the limit process n → ∞. However, DIRICH-
LET’S notation still hid the order in which limit processes are to be sequenced.
In the ultimate step of DIRICHLET’S proof, two limit processes were involved; both
expressions (12.12) and (12.13) involved both ε and n which were intended to converge
toward zero and infinity, respectively. A modern reconstruction of the limit processes
of DIRICHLET’S argument could proceed along the lines suggested by the proof in box
3. In the box, it is illustrated how the limit processes can be straightened by first fixing
a value of n such that |sm − A| is sufficiently small for all m ≥ n and then specifying
the ε that will make the power series differ from A by as little as had been required.
The order of DIRICHLET’S proof does not reflect the order in which the limit processes
are to be carried out, and neither does his notation. Therefore, we are still faced with a
line of argument in which limit processes are not as clearly identified and sequentially
ordered as it is required today.
12.5 ABEL’s “exception”
The “exception” in the binomial paper. In proofs of the binomial theorem which
follow EULER’S method of extending the binomial formula through the use of the
functional equation, some argument based on continuity has to be applied to get from
rational to real exponents. To meet this demand in his proof, CAUCHY deduced and
stated the so-called Cauchy’s Theorem (see section 11.5). At the corresponding point of
his binomial paper, ABEL discarded CAUCHY’S version of the theorem because he had
discovered that it “suffered exceptions”: 34
“Remark. In the above-mentioned work of Mr. Cauchy (on page 131) the following
theorem can be found:
»Whenever the different terms of the series
u0 + u1 + u2 + u3 + . . . etc.
are functions of one and the same variable quantity and moreover continuous
functions with regard to this variable in the vicinity of a particular value for which
the series is convergent, then the sum s of the series will also be a continuous
function of x in the vicinity of that particular value.«
34 For CAUCHY’S original formulation, which is authentically translated in ABEL’S paper, see page
217.