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

98 Chapter 6. Algebraic insolubility of the quintic
an application to a specific numerical example.
“As for the talented Mr. Abel, I will be happy to present his treatise to the
Royal Academy of Science. It shows, even if the goal has not been reached, an extraordinary
head and extraordinary insights, especially for someone his age. Nevertheless,
I excuse myself to require the condition that Mr. A. sends an elaborated
deduction of his result together with a numerical example, taken from, for instance,
an equation such as x 5 − 2x 4 + 3x 2 − 4x + 5 = 0. I believe that this will be a rather
necessary lapis lydius [Lydian stone] for him, as I recall what happened to Meier
Hirsche 3 and his ενρηκα [Eureka]; item [furthermore] I would, since the latter part
of the communicated manuscript would not be easily readable to the majority of
the members of the Academy, ask for another copy of it.” 4
We have no indication that ABEL ever produced an elaborated deduction; appar-
ently the numerical examples worked their part — as the probes of truth — as DEGEN
had suggested and led ABEL to a radically new insight. In 1824, he published, at his
own expense, a short work in French entitled Mémoire sur les équations algébriques ou
l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré. 5 It
demonstrated the impossibility of solving the general equation of the fifth degree by
algebraic means — ABEL had left the last essential requirement out of the title. ABEL
intended the memoir to be his best introduction on his planned tour of the Continent.
Since he had to pay for the publication himself, he compressed the proof to cover
only six pages and his style of presentation suffered accordingly. In numerous points
he was unclear or left advanced arguments out. When ABEL came into contact with
A. L. CRELLE (1780–1855) in Berlin, he found himself in a position to make his dis-
covery available to a broader public. He rewrote the argument elaborating the ideas
of the 1824 proof, and had CRELLE translate it into German for publication in the very
first issue of Journal für die reine und angewandte Mathematik which appeared in 1826. 6
Through this paper — and the French report of it, 7 which ABEL wrote for the Bulletin
des sciences mathématiques, astronomiques, physiques et chimiques edited by BARON DE
FERRUSAC (1776–1836) 8 — the world gradually came to know that a young Norwe-
gian had settled the question of solubility of the general quintic in the negative.
3 M. HIRSCHE (1765–1851) was a teacher of mathematics in Berlin who in 1809 published a collection
of exercises. There, he thought he had given the general solution to all equations. He quickly discovered
his error, perhaps by a Lydian probe as DEGEN recommends. (N. H. Abel, 1902e, Oplysninger
til Brevene, p. 125)
4 “Hvad den talentfulde Hr. Abel angaar, da vil jeg med Fornøielse fremlægge hans Afhandling for
det Kgl. V. S. Den viser, om end ikke Maalet skulde være opnaaet, et ualmindeligt Hoved og ualmindelige
Indsigter, især i hans Alder. Dog maatte jeg som Bøn tilføie den Betingelse: At Hr. A. sender
en udførligere Deduction af sit Resultat og tillige et numerisk Exempel, tagen f. Ex. af en Ligning
som denne: x 5 − 2x 4 + 3x 2 − 4x + 5 = 0. Dette vil efter min Overbevisning være en saare nødvendig
lapis lydius for ham Selv, da man veed, hvorledes det gik Meier Hirsche med hans ενρηκα; item
maatte jeg, da den sidste Deel af den mig communicerede Afh. ikke vilde være ret læselig for de fleeste
af S.’s Medlemmer, udbede mig en anden Afskrift af samme.” (Degen→Hansteen, Kjøbenhavn,
1821/05/21. N. H. Abel, 1902b, 93).
5 (N. H. Abel, 1824b).
6 (N. H. Abel, 1826a).
7 (N. H. Abel, 1826c).
8 Dates from (Stubhaug, 1996, 580).