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

16.2. Inversion in the Recherches 307
which is nowhere found in the published version in the Recherches. 20 Later in the same
notebook, ABEL wrote
and deduced the differential equations
� �
dp
=
dx
�
φ x + y √ �
−1 = p + q √ −1
� �
dq
dy
and
� �
dp
= −
dy
� �
dq
dx
which are the important Cauchy-Riemann equations. 21 Thus, as SYLOW concludes, 22
there is good reason to believe that ABEL had studied CAUCHY’S works on integra-
tion between imaginary limits. However, there is still no direct indication that ABEL
allowed any of these studies or considerations to have an impact on the way he pre-
sented his inversion of elliptic integrals.
Complex integration or formal substitution in the Recherches? As the evidence
seems to be inconclusive, the interpretation of ABEL’S inversion must be left to the his-
torian and depends on the temper of the interpretor. I believe that ABEL’S inversion
was formal in the sense that he employed a formal, imaginary substitution to obtain
the extension to imaginary arguments. Whether or not, he found any reassurance of
his method in CAUCHY’S theory of integration remains an undecidable question.
16.2.4 GAUSS’ unpublished results on lemniscate functions
The idea of inverting elliptic integrals into elliptic functions did not belong uniquely
to ABEL. Actually, contrary to beliefs expressed throughout the secondary literature,
the idea had occurred to LEGENDRE. 23 What LEGENDRE did not fully realize, though,
was that the inverted functions should most naturally be considered as functions of a
complex variable. This idea is most frequently attributed to GAUSS in whose drawer it
remained, however. We may learn a bit more of the idea of inverting elliptic integrals
by considering extracts from GAUSS’ unpublished works and by comparing with the
approach taken by JACOBI after ABEL’S inversion had been published.
In what appears to be GAUSS’ first manuscript on the lemniscate function, we get
an impression of his approach. GAUSS wrote:
“We designate the value of the integral from x = 0 to x = 1 by 1
2 ¯ω. We denote
the variable x of the respective integral by the sign sin lemn and its complementary
integral to 1
2 ¯ω by cos lemn. Thus,
�
� � �
dx
1 dx
sin lemn √ = x, cos lemn ¯ω − √ = x.”
1 − x4 2 1 − x4 24
20 (Abel, MS:351:A, 64).
21 (ibid., 100).
22 (N. H. Abel, 1881, II, 284).
23 See (Krazer, 1909, 55) and (J. J. Gray, 1984, 103).