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

296 Chapter 15. Elliptic integrals and functions: Chronology and topics
In order to facilitate computation of numerical values, LEGENDRE developed a the-
ory by which a sequence of moduli could be constructed which allowed the numerical
approximation of elliptic integrals to be reduced.
15.4 Left in the drawer: GAUSS on elliptic functions
When C. F. GAUSS (1777–1855) was informed of ABEL’S first publication on elliptic
functions, the Recherches, his answer was extraordinary. 16 GAUSS was impressed with
ABEL’S work and was happy to see that the young Norwegian had relieved him of the
obligation to publish a third of his own knowledge concerning these elliptic functions.
Furthermore, GAUSS was surprised to see that ABEL had followed almost exactly the
same route as he, himself, had taken to the point where their symbols were the same.
As GAUSS never published any of the monographs on elliptic functions which he had
intended, historians have had to look in his diary and in some of his manuscripts for
hints concerning his results and methods.
From 1797, GAUSS’ diary documents an increasing interest in the lemniscate inte-
gral. 17 Despite a lasting interest and many connections to other parts of his research,
GAUSS never published on the theory of lemniscate integrals. Thus, GAUSS’ ideas
only indirectly influenced the development of the theory of elliptic functions in the
1820s. In order to illustrate how GAUSS arrived at some of his insights and to under-
stand his remarks on ABEL’S Recherches, a brief discussion of important points in his
manuscripts and in his mathematical diary is given. Emphasis is here put on the in-
version of the lemniscate integral into GAUSS’ lemniscate function, the periods of the
lemniscate function, and GAUSS’ representation of the lemniscate function by various
infinite expressions.
Whereas GAUSS’ diary obviously provides a strict chronological frame, his manu-
scripts are less clearly ordered. The manuscripts contained in the Werke have been
compiled and put into an order which fit the editor. Therefore, and because GAUSS’
ideas had no direct impact on his immediate successors, I have taken the liberty to
treat his production in its thematic contexts.
The role of GAUSS’ knowledge. As mentioned, GAUSS deferred publication on the
subject of lemniscate functions. According to SCHLESINGER, GAUSS had hoped to
publish on his research on higher transcendentals in a form which would combine his
three greatest interests in the field: the lemniscate function, the arithmetic-geometric
means and the hypergeometric series. 18 GAUSS did publish on the hypergeometric
series, 19 and there is a brief description of arithmetic-geometric means in his work De-
16 (Gauss→Bessel, 1828.03.30. In Gauss and Bessel, 1880).
17 (C. F. Gauss, 1981; J. J. Gray, 1984).
18 (Schlesinger, 1922–1933, 27).
19 (C. F. Gauss, 1813).