Nearly 400 years ago Galileo Galilei posed a conjecture about the fastest descent on the lower half of a circle. Namely, he assumed that the descent along the circular arc itself is faster than along any broken line of chords. Galileo studied the case when the initial speed is zero, and the paths end in the lowest point of the circle, but he made a guess that the final conclusion remains the same if the particle initially at rest falls to a starting point where the speed is not zero. Galileo was right, as is now rigorously proved. However, this intuitive idea cannot be extended to a starting point on the upper half of a circle. Indeed, as we demonstrate, the fastest descent would then correspond to a finite (not infinite!) number of connected chords. Moreover, we present an extrapolation method which can be applied to determine the optimal number and the positions of all these chords for any starting point on a circle.
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