John Bernoulli, in 1696, published a mathematical challenge called the brachistochrone problem. The problem is to determine the shape of a curve that has the property of the fastest decent.
Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half. On 29 January 1697 the challenge was received by Isaac Newton, who found it in his mail, in a letter from Johann Bernoulli, when he arrived home from the Royal Mint at 4 p.m., and stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he recognizes a lion from his claw mark. This story gives some idea of Newton’s power, since Johann Bernoulli took two weeks to solve it. Newton also wrote, I do not love to be dunned [pestered] and teased by foreigners about mathematical things
Ishmael is cleaning the Try-Pots , used for rendering oil from blubber, on the Pequod and this task brings to mind the Cycloid.
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
From Wikipedia: A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
Also mentioned in the Wikipedia article is the Cycloidal pendulum which was discovered while searching for a more accurate chronometer.