The tautochrone problem is to find the curve down which an object can slide from any point to the bottom (accelerated by gravity and ignoring friction), always in the same length of time. "Tautochrone" comes from the Greek tauto for "the same" (which also gives us "tautology") and chronos for "time." The solution, first found by Christiaan Huygens and published in his Horologium oscillatorium (1673), is a cycloid. Thus, if you were to upturn a cycloid, in the manner of an inverted arch, and then release a marble from any point on it, it would take exactly the same time to reach the bottom, no matter where on the curve you started from. Huygens used his discovery to design a more accurate pendulum – one with curved jaws from the point of support that forced the string to follow the right curve no matter how large or small the swing.
The cycloid's unique property is mentioned in the following passage from Herman Melville's Moby Dick: "[The try-pot] is also a place 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 a cycloid, my soapstone, for example, will descend from any point in precisely the same time."
The cycloid is also the curve that answers the brachistochrone problem.