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

290 Chapter 15. Elliptic integrals and functions: Chronology and topics
Rectification of the lemniscate To find the arc length of the lemniscate given by the
polar equation
we compute
i.e.
r 2 = a 2 cos 2θ,
2r dr = −2a 2 sin 2θ dθ,
ds 2 = r 2 dθ 2 + dr 2 = a 2 cos2 2θ + sin2 2θ
dθ
cos 2θ
2 ,
a dθ a dθ
ds = √ = �
cos 2θ
�
s (θ) = a
dθ
�
1 − 2 sin2 θ .
1 − 2 sin 2 θ ,
This is an example of an elliptic integral of the first kind. With the substitution z =
sin θ, we find (see box 6)
�
s = a
1 dz
√ √
1 − 2z2 1 − z2 �
= a
dz
√ 1 − 3z 2 + 2z 4 .
Box 7: Rectification of the lemniscate
15.2.1 Addition of lemniscatic arcs
EULER took his inspiration directly from the works of FAGNANO DEI TOSCHI. In 1750,
after FAGNANO DEI TOSCHI had published his collected works, the author sent a copy
to the Berlin Academy of Sciences of which he was a member. The following year,
on 23 December 1751, the work came into the hands of EULER who was given the
assignment of commenting upon it. 8 C. G. J. JACOBI (1804–1851) has called this date
the birthday of elliptic functions.
In the process of preparing an answer for FAGNANO DEI TOSCHI, EULER became
very interested in the topic of lemniscate integrals. EULER commented on his new
research in a letter to C. GOLDBACH (1690–1764):
“Recently, I have come across a curious integration. Just as the integral of
√ is yy + xx = cc + 2xy
1−yy √ 1 − cc, the integral of the
the equation dx
√ 1−xx = dy
equation dx
√ 1−x 4
= dy
√ 1−x 4 is
�
yy + xx = cc + 2xy 1 − c4 − ccxxyy.” 9
8 (Siegel, 1959).
9 “Neulich bin ich auch auf curieuse Integrationen verfallen. Dann gleich wie von dieser Äquation
√ dx =
(1−xx)
dy
√ (1−yy) das integrale ist yy + xx = cc + 2xy � (1 − cc), also ist von dieser Äquation