The Brachistochrone Problem: The Curve of Fastest Descent

Johann Bernoulli was twenty-eight, newly appointed professor at Groningen, brilliant and quarrelsome. He was also one of the very few people who had mastered the calculus Leibniz had published twelve years earlier. In the June 1696 issue of Acta Eruditorum, Europe's leading scientific journal, he printed a short notice with a grand title: a new problem that mathematicians are invited to solve.

Given two points A and B in a vertical plane, find the path AMB along which a moving body M, descending under its own weight, travels from A to B in the shortest possible time.

Johann Bernoulli, Acta Eruditorum, 1696 (our translation)

He gave the field six months. When the deadline came, only Leibniz had sent an answer, privately, and he asked Bernoulli to extend the contest to Easter 1697 so that mathematicians abroad would have a fair chance. Bernoulli agreed. In January 1697 he reprinted the problem as a broadside, addressed to “the sharpest mathematicians in the whole world”, and mailed copies across Europe. He hinted that he had a solution of his own, and he warned readers that the answer was not the obvious one.

He also gave the curve a name. Brachistochrone comes from the Greek brachistos, shortest, and chronos, time. The name is about time, not length. The shortest path is obviously the straight line, and the whole point of the problem is that the shortest path is not the fastest.

Johann's solution is one of the most elegant arguments in all of mathematics. He did not attack the falling bead at all. He solved a problem about light instead.

Light is lazy in a precise sense: going from one point to another, it takes the path of least time, not least distance. That is Fermat's principle, and it is why light bends when it passes from air into water, the same bending that makes rainbows possible. At every boundary the ray obeys Snell's law: the sine of its angle divided by its speed stays the same on both sides.

So Bernoulli imagined the space between A and B as a stack of thin glass sheets, each one letting light travel a little faster than the sheet above, with the speed in each sheet matched to the speed a falling bead would have at that depth. A ray of light finds the fastest route through the stack automatically. Make the sheets thinner and more numerous, and the ray's zigzag smooths into a curve. That curve is the fastest path for light, and so it must be the fastest path for the bead.

Slide the layers up to sixty and the yellow ray lies almost on the green dashed curve. Bernoulli recognised the equation the limit satisfies: it is the equation of a cycloid. The argument takes half a page, and it borrows its whole idea from optics to settle a question in mechanics. That link between light and motion grew, over the next century, into the least-action principles that run through physics, from planetary orbits to the equations that changed the world.

The winning curve was already famous. Mark a point on the rim of a wheel and roll the wheel along a straight line: the point traces a row of arches called a cycloid. Roll the wheel along the underside of a ceiling instead, and each arch hangs down, the same shape as the slide that wins the race.

Galileo named the curve around 1599 and tried to find the area under one arch by cutting the shape out of metal and weighing it. The scales said roughly three times the area of the wheel, but he suspected the true ratio was not a whole number. It is exactly three: Gilles de Roberval proved it in the 1630s. In 1658 Christopher Wren, before he became an architect, showed that one arch is exactly eight times the wheel's radius long. That same year Blaise Pascal ran a prize contest on the cycloid that ended in bitter disputes over who had solved what.

The curve caused so many priority fights that it was nicknamed the Helen of Geometers, after Helen of Troy, the beauty who started a war. The brachistochrone contest was one more fight over it, and it would not be the last.

Johann's light trick was beautiful, but it worked only because this one problem happened to look like optics. His brother Jakob's solution was clumsier and far more important. Jakob treated the curve as a whole and asked how the travel time changes when you nudge it slightly anywhere along its length. On the fastest curve, every small nudge makes things worse. That idea works for any problem of the form “find the curve that makes this quantity as small as possible”.

The brothers were rivals more than collaborators. Jakob answered the challenge with a harder problem of his own and an open offer of a prize to Johann. Their quarrel over the solutions lasted until Jakob's death in 1705. The method outlived both of them. Leonhard Euler, Johann's student, made it systematic in 1744. In 1755, a nineteen-year-old Joseph-Louis Lagrange wrote to Euler with a cleaner approach. Euler was impressed, adopted it, and later gave the new field its name: the calculus of variations.

That equation is now everywhere. Physicists build mechanics out of it, engineers use it to shape wings and bridges, and the shortest paths on curved surfaces in general relativity come from it. It belongs on the same shelf as the other great tools of analysis, like the Fourier transform and the Gaussian integral. The journal pages where the 1697 answers appeared belong next to the original papers that shaped modern mathematics. And the contest itself was a forerunner of puzzles like the overhanging branch problem, in which a simple-sounding question turns out to have a deep answer.