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

6.7. ABEL and RUFFINI 123
The notebook has been dated to 1828 by P. L. M. SYLOW (1832–1918) — a date
which implies that ABEL disclosed his knowledge of RUFFINI only after returning to
Christiania. 46 It is most likely that ABEL learned about RUFFINI during his European
tour, and two instances are of particular importance. During his stay in Vienna in
April and May 1826, ABEL became acquainted with the local astronomers K. L. VON
LITTROW (1811–1877) and A. VON BURG (1797–1882). In the first volume of their
journal Zeitschrift für Physik und Mathematik, occurring while ABEL was in town, an
anonymous paper on the theory of equations was published. 47 The author, 48 who was
inspired by ABEL’S proof and praised it highly, reviewed RUFFINI’S proof. Therefore it
is not unlikely that ABEL learned of RUFFINI’S proof from his Viennese connections. 49
Once in Paris, ABEL took on the duty of writing unsigned reviews for FERRUSAC’S
Bulletin des sciences mathématiques, astronomiques, physiques et chimiques of papers pub-
lished in CRELLE’S Journal für die reine und angewandte Mathematik. We know from
one of ABEL’S letters that he, himself, wrote the review of his Beweis der Unmöglichkeit
which gave a short exposition of the flow of the proof. 50 However, appended to the re-
view was a short note by the editor, J. F. SAIGEY (1797–1871), 51 which drew attention
to the works of RUFFINI. 52 SAIGEY mentioned CAUCHY’S favorable review of RUF-
FINI’S treatise and made it clear that CAUCHY’S view was not universally accepted:
“Other geometers have not understood this demonstration and some have
made the justified remark that by proving too much, Ruffini could not prove anything
in a satisfactory manner; to be sure it was not known how an equation of
the fifth degree, e.g., could not have transcendental roots, equivalent to infinite series
of algebraic terms, since one demonstrates that every equation of odd degree
necessarily has some root. By a more profound analysis, M. Abel proves that such
roots cannot exist algebraically; but he has not solved the question of the existence
of transcendental roots in the negative.” 53
Thus, at two instances in 1826, ABEL had been in close contact with journals, in
which his result was linked to that of RUFFINI. A third possible source of information
46 (L. Sylow, 1902, 16).
47 (Anonymous, 1826).
48 Or authors? Unlike the review in FERRUSAC’S Bulletin (see below), ABEL is not likely to be the
author, himself.
49 See (Ore, 1957, 125).
50 (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a, 44). The paper reviewed is, of course, (N.
H. Abel, 1826a).
51 (Stubhaug, 1996, 589).
52 (N. H. Abel, 1826c, 353–354).
53 “D’autres géomètres avouent n’avoir pas compris cette démonstration, et il y en a qui ont fait la
remarque très-juste que Ruffini en prouvant trop pourrait n’avoir rien prouvé d’une manière satisfaisante;
en effet, on ne conçoit pas comment une équation du cinquième degré, par exemple,
n’admettrait pas de racines transcendantes, qui équivalent à des séries infinies de termes algébriques,
puisqu’on démontre que toute équation de degré impair a nécessairement une racine quelconque.
M. Abel, au moyen d’une analyse plus profonde, vient de prouver que de telles racines ne
peuvent exister algébriquement; mais il n’a pas résolu négativement la question de l’existence des
racines transcendantes.” (Saigey in ibid., 354).