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

19.3. Additional, tentative remarks on ABEL’s tools 373
century. I believe that the problem of elimination can be considered sufficient inspira-
tion for ABEL to treat Lagrange interpolation but the very central role which it together
with the operator Π played in the Paris memoir (see above) may also suggest that ABEL
had actually picked up his tools from CAUCHY’S new theory of residues although he
did not adapt the full theory.
The degree-operator. ABEL’S arguments about the number of independent integrals
γ relied extensively on ways of determining the number of independent (or free) coef-
ficients. In turn, their determination was based on an extended degree operator which
could apply to the implicitly defined algebraic functions with which he dealt. ABEL
introduced the fractional degree operator hR as the highest exponent in a develop-
ment of R according to decreasing powers. As was the case with the residue-operator,
no explicit considerations as to the validity of this definition are presented and it ap-
pears to be a well known, formal trick. ABEL presented the basic rules for the de-
gree operator and when he wanted to apply it to differences between to expressions
among y1, . . . , yn, he insisted that these expressions be ordered according to their de-
gree. However, as observed on page 362, particular cases could still arise in which the
identity
h (ym − y k) = max {hym, hy k} (19.34)
did not hold. Such cases were apparently peculiar situations of little interest, and
ABEL dismissed them by claiming that the equation (19.34) would hold “in general”.
In his comments, 41 SYLOW made a considerable effort in clarifying ABEL’S arguments
and in particular in revising his deduction of the properties of γ by making explicit
some of the assumptions which ABEL had not made when he simply argued “in gen-
eral”. The notion of equations being “generally valid” will be addressed further in
chapter 21 where it was be interpreted and explained based on the notion of formula
based mathematics.
The genus. The number which ABEL denoted γ expressed the number of indepen-
dent integrals related to a particular algebraic differential. As we have seen, ABEL’S
deduction of the invariance of the number γ was cumbersome and hampered by cer-
tain points where it was not completely clear and rigorous. Furthermore, although his
arguments were highly explicit they did not immediately produce a way of generally
computing the number γ. In the subsequent decades, it became an extremely promi-
nent mathematical problem to rigorously establish the basis for ABEL’S theorems and
to investigate the number γ further. Eventually, RIEMANN presented an approach
based on multi-sheeted surfaces and introduced the name genus and the symbol p for
ABEL’S γ. 42 The further description of RIEMANN’S theory is, unfortunately, way be-
41 (N. H. Abel, 1881, II).
42 (B. Riemann, 1857).