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

374 Chapter 19. The Paris memoir
Figure 19.1: GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866)
yond the present scope. 43 However, in the present context, it serves to demonstrate
that ABEL’S ideas were pursued and rigorized over the ensuing decades.
Birth of a new concept: algebraic functions. The final aspect which will be tenta-
tively discussed here concerns the introduction of a new concept of implicitly defined
algebraic functions. In his research on the solubility of equations, ABEL had — for obvi-
ous reasons — only been interested in investigating explicit algebraic functions which
could serve as solutions for equations. However, in the Paris memoir, implicit algebraic
functions were the primary objects of study and ABEL developed a number of tools
for their investigation. Later, implicitly given algebraic functions (or just algebraic
functions) became very important objects of mathematics and — as BRILL remarks —
ABEL may be seen as the initiator of the theory of algebraic functions; in particular, he
proved a very important theorem in the theory. 44 Later, J. LIOUVILLE (1809–1882) also
adopted — following ABEL but departing from his French precursors — the implicitly
given algebraic functions into his theory of integration in finite terms. 45 In summary,
we are faced with an introduction of a new set of objects, and ABEL’S highly formula
based investigations of these objects and developments of tools for their study can be
interpreted as a way of getting to know and express results about this new branch.
43 One rather recent, very brief sketch is given in (Houzel, 1986, 310–313).
44 (Brill and Noether, 1894, 212).
45 (Lützen, 1990, 370).