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

7.2. Elliptic functions 153
Figure 7.1: ABEL’S drawing of the lemniscate in one of his notebooks. (Stubhaug, 1996,
270)
ABEL’S inspirations for this problem were twofold. The case in which m = 2 and φ
was the lemniscate function useful in measuring the arc length of the lemniscate curve
(see figure 7.1)
φ (x) =
� x
0
dx
√ 1 − x 4
had been settled in the eighteenth century by G. C. FAGNANO DEI TOSCHI (1682–
1766). 21 In his study of the equivalent problem for circular functions GAUSS had
expressed his conviction that his approach would apply equally well to other tran-
scendentals, for instance the lemniscate integral (see the quotation in section 5.3.1, p.
74).
ABEL had learned of FAGNANO DEI TOSCHI’S work and the tradition in research
on elliptic integrals through his studies of the much more advanced works on the
subject by EULER and A.-M. LEGENDRE (1752–1833). 22 Complementary to his gen-
eralization of FAGNANO DEI TOSCHI’S result to the bisection of elliptic functions of
the first kind, ABEL gave a detailed investigation of the division of such functions
into 2n + 1 parts. Reformulated in the light of the addition formulae, which he had
previously developed, ABEL obtained a different version of the problem, summarized
in:
Problem 2 (Division Problem) Given n, solve the equation
φ ((2n + 1) β) = P2n+1 (φ (β))
Q2n+1 (φ (β))
which has degree (2n + 1) 2 . ✷
ABEL’S central insight was that the equation of degree (2n + 1) 2 could be reduced
to lower degree equations which were always solvable if the divisions of the periods
21 (Houzel, 1986, 298).
22 See chapter 15.