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

15.2. The lemniscate 289
in which the semi-axes were related by
a 2 = (n + 1) b 2 .
EULER then expanded the square root by use of the binomial theorem
�
(b 2 + t 2 ) + nt 2 = �
b 2 + t 2 +
∞
∑
µ=1
(−1) µ−1 ∏ µ−1
k=1 (2k − 1)
∏ µ
k=1 (2k)
n µ t 2µ
(b2 + t2 ) 2µ−1
2
Thus, EULER obtained the differential in the form (A0, A1, . . . were specified constants)
∞
2
ds = b ∑
µ=0
Aµn µ t 2µ dt
(b2 + t2 .
µ+1
)
which he next integrated term-wise from 0 to ∞ to obtain the rectification of a quarter
of the ellipse in the form
�AMB = π
2
15.2 The lemniscate
∞
∑
µ=0
∏ µ−1
k=0 (2k + 1)2
∏ µ
k=1 (2k)2 (2µ − 1) n µ .
Another curve which received the attention of mathematicians starting with the broth-
ers JAKOB I BERNOULLI (1654–1705) and JOHANN I BERNOULLI (1667–1748) was the
so-called lemniscate. 6 The curve was defined by the Cartesian equation
�
x 2 + y 2� 2 �
2 = a x 2 − y 2�
,
and both brothers recognized that the arc length of the curve depended on an integral
of the form �
(see box 7).
dz
√ 1 − z 4
In Italy, the autodidact nobleman FAGNANO DEI TOSCHI took up the study of the
lemniscate. 7 By a set of theorems, FAGNANO DEI TOSCHI was able to prove that the
division of the quadrant of the lemniscate into k parts could be constructed by ruler
and compass if k was of one of the forms 2 × 2 m , 3 × 2 m , or 5 × 2 m . By elimination of
the intermediate variable x in the substitutions
x =
�
1 − √ 1 − z 4
FAGNANO DEI TOSCHI obtained that
dz
√ 1 − z 4
z
and x =
du
= 2√
1 − u4 √ 2u
√ 1 − u 4 ,
and he had obtained the duplication of any segment of the lemniscate arc.
6 See (H. J. M. Bos, 1974). Mostly, discovery of the curve is attributed to BERNOULLI alone as he holds
priority of publication and gave the curve its name.
7 Unfortunately, I not have had access to FAGNANO DEI TOSCHI’S original works. Instead, the short
outline is based on (R. Ayoub, 1984; Siegel, 1959).
.