Squaring the Pendulum’s Arc: Motion Along Broken Chords

  • Jochen Büttner
Part of the Boston Studies in the Philosophy and History of Science book series (BSPS, volume 335)


The chapter deals with Galileo’s considerations regarding motion along polygonal paths inscribed into an arc that have been preserved in the Notes on Motion. Galileo’s approach was motivated by the assumption that fall of a heavy body along a concave surface supported from below was kinematically equivalent to the swinging of a pendulum bob along the same arc supported, however, from above. As Galileo was not able to make an inference regarding motion of fall along the arc directly, he instead investigated motion along a series of adjoined inclined planes inscribed as chords into the arc. If the number of inclined planes is increased above all limits, the polygonal path exhausts the arc and in this way Galileo hoped to eventually be able to construe an argument regarding falling, and thus swinging, along the arc. Galileo’s approach yielded a number of tangible results, among them a proof for the statement that motion along a broken chord takes less time than motion along a single chord spanning the same upright arc subtending an angle of 90 degrees or less, a proof he mentioned in a letter to Guidobaldo del Monte in 1602. Based on manuscript evidence, Galileo’s extensive work on the problem is indeed dated to 1602. Despite considerable partial accomplishments Galileo’s approach was not crowned with success. By yielding indications that motion along the arc could not be isochronous, it indeed rather pointed to a problem in the underpinning conceptualization.


  1. Clavelin, M. (1983). Conceptual and technical aspects of the Galilean geometrization of the motion of heavy bodies (pp. 23–50). Dordrecht: Springer.Google Scholar
  2. Drake, S. (1970). Galileo studies: Personality, tradition, and revolution. Ann Arbor: The University of Michigan Press.Google Scholar
  3. Drake, S. (1978). Galileo at work: His scientific biography. Chicago: University of Chicago Press.Google Scholar
  4. Erlichson, H. (1998). Galileo’s work on swiftest descent from a circle and how he almost proved the circle itself was the minimum time path. The American Mathematical Monthly, 105(4), 338–347.CrossRefGoogle Scholar
  5. Favaro, A. (1886). La libreria di Galileo Galilei. Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, 19, 249–293.Google Scholar
  6. Galilei, G. (1914). Dialogues concerning two new sciences. New York: Macmillan.Google Scholar
  7. Gilbert, W. (1958). On the magnet: [Magnetick bodies also, and on the great magnet the earth; a new physiology, demonstrated by many arguments & experiments] (Nachdr. der Ausg. London 1900 edn.). New York: Basic Books.Google Scholar
  8. Heilbron, J.L. (2010). Galileo. Oxford: Oxford University Press.Google Scholar
  9. Wisan, W.L. (1974). The new science of motion: A study of Galileo’s De motu locali. Archive for History of Exact Sciences, 13, 103–306.CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  • Jochen Büttner
    • 1
  1. 1.Max Planck Institute for the History of ScienceBerlinGermany

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