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

16.2. Inversion in the Recherches 309
16.2.5 JACOBI’s inversion in the Fundamenta nova
As will be described in section 18.1, a third inversion of elliptic integrals was per-
formed by CARL GUSTAV JACOB JACOBI who in 1829 published the first book entirely
devoted to the study of the new elliptic functions. 28 As will also be illustrated in
section 18.1, JACOBI’S main objective with his research on elliptic integrals and func-
tions was the development of transformation theory. After having devised the first set
of theorems concerning the transformation of elliptic integrals, JACOBI presented his
version of the inversion:
“Letting � φ
0
dφ
√ 1−k 2 sin 2 φ = u, geometers have accustomed themselves to call
the angle φ the amplitude of the function u. In the following, this angle is denoted
by amplu or shorter by
Thus, if
then
φ = amu.
u =
� x
0
dx
� (1 − x 2 ) (1 − k 2 x 2 )
x = sin amu.” 29
JACOBI then introduced the complete integrals already stressed by LEGENDRE,
K =
K ′ =
28 (C. G. J. Jacobi, 1829).
29 “Posito � φ
0
� 1
0
� π 2
0
� π
dx
2
� =
(1 − x2 ) (1 − k2x2 ) 0
�
dφ
1 − k ′ k ′ sin2 φ
where k ′ k ′ + kk = 1.
�
dφ
1 − k2 sin2 φ
and
dφ
√ 1−k 2 sin 2 φ = u, angulum φ amplitudinem functionis u vocare geometrae consueverunt.
Hunc igitur angulum in sequentibus denotabimus per amplu seu brevius per:
Ita, ubi
erit:
(ibid., 81).
φ = amu.
u =
� x
0
x = sin amu.”
dx
� (1 − x 2 ) (1 − k 2 x 2 ) ,