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

196 Chapter 10. Toward rigorization of analysis
theorem from Taylor theorem recurred throughout the century and even into the nineteenth
century. 9
EULER’S second proof of the binomial theorem based on functional equations. In
his second proof of the binomial theorem, published in 1775, 10 EULER devised his
proof following an outline which would recur in most subsequent “rigorous” proofs.
EULER introduced the notation
[m] = 1 + m
1
m (m − 1)
x + x
1 · 2
2 + . . .
to denote the binomial series associated with the exponent m. Thus, proving the bino-
mial theorem thus amounted to proving the equality [m] = (1 + x) m . The central step
in the proof was the realization that the brackets satisfied a functional equation 11
[m + n] = [m] · [n] .
EULER’S proof of the functional equation was based on formally multiplying the corre-
sponding infinite series. Once EULER had obtained the above functional equation and
the binomial formula secured the equality [m] = (1 + x) m for integral m, he extended
the domain for m by the computation
[1] =
�
m · 1
� � �m 1
=
m m
⇒
m,n∈N
�
n
�
= [n]
m
1 m = (1 + x) n m .
Thus, EULER proved the binomial theorem for all fractional exponents and claimed —
without giving any proof — that it extended to all real exponents by way of continuity
(see below). In summary, the central steps of EULER’S second proof of the binomial
theorem are:
1. The binomial formula, [m] = (1 + x) m for m ∈ N.
2. The functional equation [m + n] = [m] · [n] proved by manipulating the associ-
ated power series.
3. An extension to rational exponents.
4. A further extension to real exponents by continuity arguments.
In complete correspondence with his views on formal equality, EULER did not ven-
ture into considerations of the convergence of the infinite expression contained in the
binomial theorem. To him, the theorem simply stated a formal equivalence of two
different representations of the same function (expression).
9 On the proof by WALLACE, see (Craik, 1999, 252–253).
10 (L. Euler, 1775).
11 For the history of functional equations, mainly with CAUCHY, see (J. Dhombres, 1992).