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Nearly four centuries ago, Galileo Galilei proved that descent along any two-chord path in the lower quadrant of a circle is faster than descent along the straight one-chord path from the same starting point to the same endpoint, if the initial speed is zero, and the paths end in the lowest point of the circle. Moreover, Galileo posed two conjectures: (1) the main conclusion remains the same if the body initially at rest falls to a starting point and thus the speed there is not zero; (2) descent along the circular arc itself is faster than along any broken line of chords. In this article, analytical proofs are given that confirm the validity of both of these conjectures.

Galileo’s Swiftest Descent Problem Revisited: Complete Analytic Solution