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

15.2. The lemniscate 291
Figure 15.2: LEONHARD EULER (1707–1783)
In the letter, EULER applied the theorem to demonstrate how the difference be-
tween two segments of the arc of an ellipse could be rectified. There is no proof of the
theorem in the letter but EULER published a proof in 1756/56. 10 The proof progressed
by direct differentiation of the purported integral
to obtain
y 2 + x 2 = c 2 �
+ 2xy 1 − c4 − c 2 x 2 y 2
dx
√ 1 − x 4 =
dy
� 1 − y 4 .
(15.1)
The theorem founded a particular branch of the theory of elliptic integrals as it con-
tained the so-called addition theorem for lemniscate integrals. If x and y were related
by (15.1), the equation
� dx
(1−x4 ) =
dy
�
(1−y4 )
� x
0
das integrale:
dt
√
1 − t4 =
� y
0
dt
√ + C
1 − t4 �
yy + xx = cc + 2xy (1 − c4 ) − ccxxyy.”
(Euler→Goldbach, 1752. Euler and Goldbach, 1965, 347–348); also (Fuss, 1968, I, 567).
10 (L. Euler, 1756/57).