James Bernoulli published a ‘solution’ to the most interesting special case in 1694. Bernoulli choose the case where the rod underwent 90° of turning which corresponds to our case where parameter T = 2. (T is proportional to the weight.) Our picture is rotated 90° from Bernoulli’s conventions.

Huygens soon replies to Bernoulli, generalizes, and draws more of our pictures. The reply is also described in Levien’s modern paper.

Neither Bernoulli nor Huygens had the calculus, or numerical methods, let alone a computer with which to draw such figures as ours. Bernoulli’s theorems do, however, mathematically characterize the solutions we approximate here. His methods cover the more general cases we explore here. T ≃ 3.30373 gives the (rotated) curve where the weight is sufficient to bring the beam’s end down to ground level. Bernoulli would have understood the mathematical parts of my program without introduction to calculus.

Huygens’ pictures stop about where the family of curves begin to represent unstable equilibria which we include.

The implicit goal of identifying the resulting curve among those already known was not achieved, then or now.

Euler comes next and describes our family of curves with parameter λ which is the reciprocal of our T. Euler computes the critical value more accurately as 3.30373 than I had. Euler draws all of our pictures fairly accurately, even without a computer. Levien’s paper includes modern computer drawn versions, like ours.

I believe that the critical value that produces the figure 8 is indeed not a lemniscate. According to some sources the curvature of a lemniscate is propositional to the distance from the origin, not from the axis which is the case for the bent beam. I have not yet verified this but I try for a definitive answer here. It is not done yet.