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One exhibit in particular caught my eye: a marble run with two paths of descent, the first a straight decline and the other a longer, lazy bend called the brachistochrone curve that goes down and then up again.

Galileo discovered it was the brachistochrone curve, which, despite being longer, delivers the ball first.

In 1696, the Swiss mathematician Johann Bernoulli challenged his colleagues to solve an unresolved issue called the brachistochrone problem, specifying the curve connecting two points displaced from each other laterally, along which a body, acted upon only by gravity, would fall in the shortest time.

Before leaving for work the next morning, he had invented an entire new branch of mathematics called the calculus of variations, used it to solve the brachistochrone problem and sent off the solution, which was published, at Newton’s request, anonymously.

To understand the true relation of these theories in that part of the field where they seem equally applicable we must look at them in the light which Hamilton has thrown upon them by his discovery that to every brachistochrone problem there corresponds a problem of free motion, involving different velocities and times, but resulting in the same geometrical path.