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Tautochrone Moby Dick

A fascinating glimpse of deep mathematics rises to the surface and spouts geometrically in Herman Melville's Moby-Dick, which mentions the "tautochrone" — a curve along which all sliding objects falling in a uniform gravitational field reach the bottom at the same time. (Starting higher up a body has farther to go, but it also can accelerate to a higher speed, so by making its path steeper at the start like a ski jump it can catch up with another body that starts lower down.)

From Chapter 96, "The Try-Works", describing the kettles where whale blubber is melted:

... Removing this hatch we expose the great try-pots, two in number, and each of several barrels' capacity. When not in use, they are kept remarkably clean. Sometimes they are polished with soapstone and sand, till they shine within like silver punch-bowls. During the night-watches some cynical old sailors will crawl into them and coil themselves away there for a nap. While employed in polishing them — one man in each pot, side by side — many confidential communications are carried on, over the iron lips. It is a place also for profound mathematical meditation. It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time. ...

Moby-Dick was published in 1851; the tautochrone curve was proved to be a cycloid in 1659 by Christiaan Huygens. The cycloid also is the shape along which a falling object reaches the finish line as quickly as possible, the "brachistochrone", as proved in the late 1600's by one of the Bernoulli brothers who noted:

... Before I end I must voice once more the admiration I feel for the unexpected identity of Huygens' tautochrone and my brachistochrone. I consider it especially remarkable that this coincidence can take place only under the hypothesis of Galileo, so that we even obtain from this a proof of its correctness. Nature always tends to act in the simplest way, and so it here lets one curve serve two different functions, while under any other hypothesis we should need two curves ...

(cf. Richard Feynman on Alternative Paths (2017-01-15), ...) - ^z - 2017-02-18