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

292 Chapter 15. Elliptic integrals and functions: Chronology and topics
holds where C is constant (independent of x and y). However, if we let y = 0, we find
x 2 = c 2 , and thus
C =
� c
0
dt
√ 1 − t 4 .
In this form, the addition theorem for lemniscate integrals is apparent,
� x
0
dt
√
1 − t4 +
� y dt
√
0 1 − t4 =
� z
0
dt
√ , if
1 − t4 x 2 + y 2 + z 2 x 2 y 2 = z 2 �
+ 2xy
15.2.2 EULER’s rectification of the lemniscate
1 − z 4 .
At least twice, EULER deduced expressions for the arc length of the lemniscate by
infinite series. In the third part of his trilogy, the Institutiones calculi integralis, 11 EULER
found the relation
� 1
0
x m+1 dx
√ 1 − x 2
�
m 1
=
m + 1 0
xm−1 dx
√ . (15.2)
1 − x2 Subsequently, 12 EULER expressed the length of the first quadrant of the lemniscate
based on the relation
� 1
Using the binomial theorem, he wrote
0
�
1 + x 2� − 1 2
0
dx
√
1 − x4 =
� 1 �
1 + x
0
2� − 1 2
dx
√ 1 − x 2 .
= 1 − 1
2 x2 1 · 3
+
2 · 4 x4 1 · 3 · 5
−
2 · 4 · 6 x6 + . . .
and when the term-wise integration was carried out, EULER found the expression
� �
1 dx π
√ = 1 −
1 − x4 2
12
22 + 1232 2242 − 123252 2242 �
+ . . .
62 using the relation (15.2), above.
The deduction of the result in the Institutiones is less complicated than the similar
result for the ellipse given by EULER in 1732 and described above. However, the basic
tools of the two approaches are the same: expansion by use of the binomial theorem
and term-wise integration of the power series which was thus obtained.
15.3 LEGENDRE’s theory of elliptic integrals
Toward the end of the eighteenth century, LEGENDRE gave the theory of elliptic inte-
grals a new twist with his contributions. In a number of lengthy papers and mono-
11 (L. Euler, 1768, XI, 208).
12 (ibid., XI, 211).