Of the elastica (i.e., the shape assumed by a flexible inextensible line subjected to lateral forces) studied by Euler, a unique classification—class V—is the lemniscate, lemnoid, or "figure eight." This is the shape assumed if the two ends of the line are brought together, or alternately (if doubled) the shape assumed by twisting one side of a circular loop by 180°:
Of the general elastica solution
$$y^\prime=\frac{a^2-c^2+x^2}{\sqrt{(c^2-x^2)(2a^2-c^2+x^2)}},$$
Euler reported (p. 97) that the lemniscate arises for the special case of $\frac{c^2}{a^2}\approx 1.651868$:
Let there be sought first, by methods familiar to everyone, or
else by mere trial, the limits between which the true value of $\left[\frac{c^2}{2a^2}\right]$
is contained, and these limits will be found to be $\left[\frac{c^2}{2a^2}\right] = 0.824$,
and $\left[\frac{c^2}{2a^2}\right]=0.828$. But if now both of these values be substituted
in the equation, from the two errors which are certain to arise,
it will finally be concluded that $\left[\frac{c^2}{2a^2}\right]=0.825934$, whence $\frac{c^2}{a^2}=1.651868$.
The series Euler used was (p.97 again)
$$1=\frac{1\cdot 3}{2\cdot 2}\left(\frac{c^2}{2a^2}\right)+\frac{1\cdot 1\cdot 3\cdot 5}{2\cdot 2\cdot 4\cdot 4}\left(\frac{c^2}{2a^2}\right)^2+\frac{1\cdot 1\cdot 3\cdot 3\cdot 5\cdot 7}{2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6}\left(\frac{c^2}{2a^2}\right)^3+\cdots,$$
(I’ve corrected an apparent factor-of-two inconsistency in the first term) which can be rewritten (Eq. 175 of Truesdell's The Rational Mechanics, where the two ends have been brought together such that distance $\text{AD}=b=0$) as
$$1=\sum_{n=1}^\infty\left[\frac{2n+1}{2n-1}\left(\frac{(2n-1)!!}{(2n)!!}\right)^2\left(\frac{c^2}{2a^2}\right)^n\right].$$
This particular value of 1.651868 is repeated across the literature (Hayman's Elements of the Theory of Structures, p. 58, Kot's A First Course
in the Calculus of Variations, p.128, Levien's From Spiral to Spline: Optimal Techniques in Interactive Curve Design, p.107, Levien's "The elastica: a mathematical history" p. 11).
So what happens if we evaluate this series using, for instance, Mathematica? We certainly see convergence to a value near 1.
Notably, however, the convergence is much closer for, say, 1.652229532.
Thus, there seem to be two possibilities:
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The value of 1.651868 is often repeated as Euler's initial estimate, but the true value is different. I find this hard to believe, as the sources report the constant not just in the context of historical development but as the current best-known value. Researchers must have checked this aspect previously.
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The Mathematica/Wolfram Alpha solution is incorrect.
Therefore, as a check on both Euler's value and the Mathematica value, and for completeness, I'm wondering if there's a different way to calculate $\frac{c^2}{a^2}$ for the lemniscate, perhaps involving the modern machinery of elliptical integrals and functions. I'm somewhat surprised that no source provides this link, but perhaps I just haven't found it yet. I suspect that work such as Yoshizawa's "The critical points of the elastic energy among curves pinned at endpoints" could be useful here, but I'm unable to decipher a connection to $\frac{c^2}{a^2}$.
Best Answer
In the comments, @TravisWillse provides a very useful link to Djondjorov et al.'s "Explicit parameterization of Euler’s elastica." The authors express the limits of the lemniscate as $\left(\pm\sqrt{\frac{1-\mu}{2}}\right)$ (left and right) and $\left(\pm\sqrt{-\frac{\mu}{2}}\right)$ (up and down), where $\mu=1-2k^2$, where $k$ solves $$2E(k)-K(k)=0,$$ where $K(k)$ and $E(k)$ are the complete elliptic integrals of the first and second kind, respectively.
Thus, $k= 0.9089085575\dots$, and $\mu=-0.6522295320\dots$. Because these extremes correspond to points $x^2=c^2$ and $x^2=c^2-a^2$ as defined above (different from $a$ and $c$ used in Djondjorov et al.'s paper), the number we are looking for is indeed $\frac{c^2}{a^2}=2k^2=1.6522295320\dots$.
Books, theses, and articles reporting that the characteristic value for the lemniscate is 1.651868 are thus repeating Euler's initial imprecise estimate and have not checked for a more accurate value.