eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Research article
Generalized approach to Galileo’s swiftest descent problem on a circle; pp. 77–95

Matti Selg

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.


1. Galilei, G. Two New Sciences, Including Centers of Gravity and Force of Percussion (translated with a new introduction and notes by S. Drake). 2nd ed. Wall & Emerson, Toronto, 2008. 

2. Mach, E. The Science of Mechanics: a Critical and Historical Account of its Development (translated by T. McCormack). 4th ed. The Open Court Publishing Co. Chicago, London, 1919. 

3. Supplementary material of this article:

4. Erlichson, H. Galileo’s work on swiftest descent from a circle and how he almost proved the circle itself was the minimum time path. Amer. Math. Monthly, 1998, 105, 338–347.

5. Nahin, P. J. When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. Princeton University Press, Princeton, Oxford, 2007.

6. Selg, M. Galileo’s Swiftest Descent Problem Revisited: Complete Analytic Solution. Amer. Math. Monthly, 2023, 130, 103–113.

7. Mandl, R., Pühringer, R. T. and Thaler, M. Analytic approach to Galileo’s theorem on the descent time along two-chord paths in a circle. Amer. Math. Monthly, 2012, 119, 468–476.

8. Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series, and Products. 7th ed. Academic Press, New York, 2007.

9. Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. 2nd ed. Dover Publications, Inc., Mineola, New York, 2000.

10. Rolle, M. Démonstration d’une méthode pour résoudre les égalitès de tous les degrés. Paris, 1691.

11. Adams, R. A. and Essex, C. Calculus: Several Variables. 7th ed. Pearson, Toronto, 2010.

12. Nelder, J. A. and Mead, R. A. Simplex method for function minimization. Comput. J., 1965, 7, 308–313.

13. Hannas, B. L. and Goff, J. E. Inclined-plane model of the 2004 Tour de France. Eur. J. Phys., 2005, 26, 251–259.

14. Hanin, L. G. Which Tanks Empty Faster? Amer. Math. Monthly, 1999, 106(10), 943–947.

15. Agmon, D. and Yizhaq, H. From the discrete to the continuous brachistochrone: a tale of two proofs. Eur. J. Phys., 2021, 42

Back to Issue