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

64 Chapter 5. Towards unsolvable equations
ALEXANDRE-THÉOPHILE VANDERMONDE. Another very important component of
the theory of equations in the early nineteenth century was the turn towards focusing
on the expressive powers of algebraic expressions. This approach can be traced back
to VANDERMONDE who in 1770 presented the Académie des Sciences in Paris with a trea-
tise entitled Mémoire sur la résolution des équations. 23 There, he described the purpose
of his investigations:
“One seeks the most simple general values which can conjointly satisfy an
equation of a certain degree.” 24
As H. WUSSING has remarked, this weakly formulated program only gained importance
through VANDERMONDE’S use of examples from low degree equations. 25
VANDERMONDE’S aim was to build algebraic functions from the elementary symmet-
ric ones which could assume the value of any root of the given equation. His approach
was very direct, constructive, and computationally based. For example the elementary
symmetric functions in the case of the general second degree equation
(x − x1) (x − x2) = x 2 − (x1 + x2) x + x1x2 = 0
are x1x2 and x1 + x2. The well known solution of the quadratic is
� �
1
x1 + x2 + (x1 + x2)
2
2 �
− 4x1x2 ,
which gives the two roots x1 and x2 when the square root is considered to be a two-
valued function. Similarly, although with greater computational difficulties, VANDER-
MONDE treated equations of degree 3 or 4. In those cases, he also constructed algebraic
expressions having the desired properties. When he attacked equations of degree 5,
however, he ended up with having to solve a resolvent equation of degree 6. Simi-
larly, his approach led from a sixth degree equation to resolvent equations of degrees
10 and 15. Having seen the apparent unfruitfulness of the approach, VANDERMONDE
abandoned it. Later, the idea of studying the algebraic expressions formed from the
elementary symmetric functions became central to ABEL’S research.
Both EULER’S and VANDERMONDE’S approaches are, in spite of their apparently
unsuccessful outcome, interesting in interpreting ABEL’S work on the theory of equa-
tions. Firstly, ABEL’S proof of the impossibility of solving the general quintic by
radicals (see chapter 6) is a fusion of ideas advanced by LAGRANGE and VANDER-
MONDE, although there is no evidence that ABEL was familiar with VANDERMONDE’S
work. Secondly, ABEL’S attempted general theory of algebraic solubility (see chapter
8) bears resemblances to paths followed by EULER, VANDERMONDE and LAGRANGE.
23 (Vandermonde, 1771). This paragraph on VANDERMONDE is largely based on (Wussing, 1969, 52–
53).
24 “On demande les valeurs générales les plus simples qui puissent satisfaire concurremment à une
Équation [sic] d’un degré déterminé.” (Vandermonde, 1771, 366).
25 (Wussing, 1969, 53)