The science

Three frictionless ramps join the same two points: a straight line, a circular arc ending at its lowest point (the curve Galileo guessed was quickest, in 1638) and a cycloid, the curve traced by a point on a rolling wheel. The cycloid is the longest, 11% longer than the straight line, and even dips below the finish, yet its bead arrives first. With gravity scaled so the race lasts a few seconds the times are 2.60 s, 2.82 s and 3.46 s; the ratios (1 : 1.083 : 1.330) are fixed by the geometry. Every position is computed from energy conservation, v = √(2gh).

Johann Bernoulli posed the problem in Acta Eruditorum in June 1696. Newton received Bernoulli's printed challenge on 29 January 1697 (Old Style), and his solution, in a letter dated the next day, was printed in the Philosophical Transactions without his name. His niece Catherine Conduitt recalled that he came home from the Mint at four, very tired, and did not sleep until he had solved it, by four in the morning. Bernoulli wrote that several circumstances told him the author was Newton, and that the answer alone would have shown it, "comme ex ungue leonem": like the lion by its claw.

The same curve has a second trick, published by Huygens in 1673: let beads go from any height on it and they reach the bottom at the same moment.