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

2.2. “Study the masters” 21
HOLMBOE’S list of the masters whom they studied together sheds interesting light
on the mathematical literature of the early nineteenth century as seen from the pe-
riphery. 8 In the eyes of the two Norwegians, the two most influential writers were
L. EULER (1707–1783) and J. L. LAGRANGE (1736–1813); but the list also included
S. F. LACROIX (1765–1843), L. B. FRANCOEUR (1773–1849), S.-D. POISSON (1781–
1840), C. F. GAUSS (1777–1855), and J. G. GARNIER (1766–1840). In the following
paragraphs, the influences from these authors are outlined.
EULER. EULER and HOLMBOE studied algebra and calculus from EULER’S works;
ABEL’S first independent adventures into creative mathematics were greatly inspired
by the great Swiss mathematician. Although the sources are not very explicit, it is be-
yond doubts that ABEL studied EULER’S Introductio in analysin infinitorum [Introduc-
tion to the infinite analysis] of 1748. 9 To what extent ABEL also knew of EULER’S other
publications including his papers in the transactions of the St. Petersburg Academy
is left for speculation; we shall return to the question when we see examples of EU-
LER’S — possibly indirect — influence on ABEL in parts II and IV.
The big four. LAGRANGE, LACROIX, POISSON, and GAUSS all belong to the heavy-
weight division of mathematics in the late eighteenth century with massive and im-
portant works on the calculus and algebra. Although a writer of very influential text-
books on the calculus, 10 LAGRANGE mainly inspired ABEL through his work on the
theory of equations which redefined the viewpoint from which this theory was to be
attacked. 11 LACROIX’ effort laid more in organization and presentation than in cre-
ative research; his three volume textbook on the calculus, Traité de calcul différentiel et
intégral [Treatise on differential and integral calculus], ran multiple editions beginning
in 1797–1800. 12 In the Traité, LACROIX presented an survey of the calculus based on
the research of his contemporaries and picking up a variety of approaches and foun-
dations from different authors.
The lesser souls. The two authors in HOLMBOE’S list who today are lesser known,
FRANCOEUR and GARNIER, both wrote textbooks on mathematics which found wide
circulation toward the end of the eighteenth century. ABEL almost certainly studied
7 “Si l’on veut savoir comment on doit faire pour parvenir à un resultat plus conforme à la nature il
faut consulter l’ouvrage du celebre Laplace où cette theorie est exposée avec la plus grande clarté
et dans une extension convenable à l’importance de la matière. Il est en outre aisé de voir que une
theorie ecrite par M. Laplace doit être bien superieure à toute autre donnée des geometres d’une
claire inferieure. Au reste il me parait que si l’on veut faire des progres dans les mathématiques il
faut étudier les maitres et non pas les écoliers.” (Abel, MS:351:A, 79, marginal note).
8 (Holmboe, 1829, 335).
9 (L. Euler, 1748).
10 E.g. (Lagrange, 1813).
11 (Lagrange, 1770–1771).
12 (Lacroix, 1797; Lacroix, 1798; Lacroix, 1800).