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

300 Chapter 16. The idea of inverting elliptic integrals
Figure 16.1: Stamp depicting Gauss and the construction of the regular 17-gon.
is undoubtedly the simplest integral of the form (16.1) corresponding to e = c = 1. 4
Thus, there is a suggestion that ABEL’S choice of representation of the integrals is a
direct reflection of one of the main purposes of the Recherches, the adaption of C. F.
GAUSS’ (1777–1855) methods from the Disquisitiones arithmeticae to the division of the
lemniscate. 5
16.2 Inversion in the Recherches
The issue of CRELLE’S Journal which contained ABEL’S inversion of elliptic integrals
into elliptic functions was published on 20 September 1827; 6 the date is of importance
in analyzing the internal relations between ABEL’S and C. G. J. JACOBI’S (1804–1851)
approaches (see below and section 18.1, below).
In the introduction to the Recherches, ABEL described his idea:
“In this memoir, I propose to study the inverse function, i.e. the function φα
determined by the equations
4 See (Glaisher, 1902).
5 See also chapter 7.
6 (N. H. Abel, 1881, II, 305).
�
dθ
α = �
1 − c2 sin 2 θ and
sin θ = φ (α) = x.” 7