over smooth curves connecting two given points \(A\) and \(B\); here \(F(y) > 0\) is a given function and \(ds\) is an element of arc length. Bernoulli based his beautiful solution on the equivalence between Fermat’s principle and Snell’s law.

The following solution, in addition to being shorter, substitutes a mechanical analogy for Bernoulli’s optical one – and thus could have been given by Archimedes.

can be interpreted mechanically as the potential energy of the system of rings and springs shown in Figure 1. Each of the \(N\) rings slides without friction on its own line; the neighboring rings are coupled by constant-tension springs whose tensions are given by the discretized values of \(F_k = F(y_k)\). If \(P_N\) is minimal, each ring is in equilibrium, implying the balance of horizontal forces on the ring: