A bead slides from point A to a lower point B under gravity. What curve gets it there fastest? Not a straight line. Not a circular arc. The answer — discovered by Johann Bernoulli in 1696 and famously solved by Newton overnight when challenged — is a cycloid: the curve traced by a point on a rolling wheel.
Bernoulli's challenge and Newton's overnight reply
In June 1696, Johann Bernoulli posed a challenge to "the cleverest mathematicians in the world": find the curve between two points such that a bead sliding under gravity (without friction) reaches the lower point in minimum time. The straight line, the obvious guess, takes longer than several other curves.
Five mathematicians solved it within months: Leibniz, l'Hôpital, Jakob Bernoulli, von Tschirnhaus, and an anonymous response. Bernoulli recognized Newton's hand immediately — "ex ungue leonem" (the lion by his claw). Newton, by his own report, received the problem at 4pm and solved it before 4am the next morning.
The answer is a cycloid — the curve traced by a point on the rim of a wheel rolling along a line. The cycloid's parametric equations: x = R(θ − sin θ), y = R(1 − cos θ).
Why a cycloid? Bernoulli's brilliant proof used Snell's law: light traveling through layers of decreasing index of refraction (~ inverse speed) refracts to minimize travel time. A bead descending under gravity acquires speed ∝ √h. Treating each infinitesimal horizontal strip as a Snell layer, the path satisfies Snell's law, and that path is the cycloid.
Bonus: the cycloid is also the tautochrone — release a bead from anywhere along the cycloid, and it takes exactly the same time to reach the bottom. Christiaan Huygens used this property in 1673 to design pendulum clocks with cycloidal arcs that swung in equal time regardless of amplitude. This is one of the deepest dualities in classical mechanics: the curve of fastest descent is also the curve of time-independent oscillation.