Cognate Linkages the Roberts – Chebyshev Theorem

Conference paper
Part of the History of Mechanism and Machine Science book series (HMMS, volume 15)

Abstract

This article deals with investigations by Pafnuty Chebyshev and Samuel Roberts in the late 1800s, which led them independently to the conclusion that for each curve that can be drawn by four bar linkages, there are always three linkages describing the same curve. These different linkages resulting in the same curve can be called cognate linkages.

We will see that Chebyshev and Roberts had very different interests and motivations for studying four bar linkages. Despite of these differences, they came to a similar result concerning cognate linkages.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Thurston, T.: A History of the Growth of the Steam-engine. Appleton and Company, New York (1878)Google Scholar
  2. 2.
    Muirhead, J.P.: The Origin and Progress of the Mechanical Inventions of James Watt, vol. III. John Murray, London (1854)Google Scholar
  3. 3.
    Muirhead, J.P.: The Origin and Progress of the Mechanical Inventions of James Watt, vol. II. John Murray, London (1854)Google Scholar
  4. 4.
    Muirhead, J.P.: The Life of James Watt. John Murray, London (1858)Google Scholar
  5. 5.
    de Prony, G.: Nouvelles Architecture Hydraulique, vol. I. Firmin Didot, Paris (1790)Google Scholar
  6. 6.
    de Prony, G.: Nouvelles Architecture Hydraulique, vol. II. Firmin Didot, Paris (1796)Google Scholar
  7. 7.
    Vincent, A.J.H.: Mémoires de la Société royal des Sciences, de l’Agriculture et des Arts, de Lille (1836, 1837, 1838)Google Scholar
  8. 8.
    Markoff, A., Sonin, N.: Oeuvres de P.L. Tchebychef, Tome II, l’académie impériale des sciences. St. Petersburg (1907)Google Scholar
  9. 9.
    Markoff, A., Sonin, N.: Oeuvres de P.L. Tchebychef, Tome I, l’académie impériale des sciences. St. Petersburg (1899)Google Scholar
  10. 10.
    Roberts, S.: On the Mechanical Description of some Species of Circular Curves of the third and fourth Degrees. Proceedings London Mathematical Society, 125–136 (1869)Google Scholar
  11. 11.
    Roberts, S.: On three bar motion in plane space. Proceedings London Mathematical Society, 14–23 (1875)Google Scholar
  12. 12.
    Schor, J.B.: On the theorem of Roberts – Tchebyshev. Journal of Applied Mathematics and Mechanics 5, 323–324 (1941)Google Scholar
  13. 13.
    Richard de Jonge, A.E.: The Correlation of Hinged Straight-Line Four Bar Devices by means of Roberts Theorem and a New Proof of the Latter. Annals of the New York Academy of Sciences 84, 77–145Google Scholar
  14. 14.
    Nolle, H.: Linkage Coupler Curve Synthesis: A Historical Review -I and II. Mechanism and Machine Theory 9, 147–168 (1974)CrossRefGoogle Scholar
  15. 15.
    Ceccarelli, M., Vinciguerra, A.: Approximate Four-Bar Circle-Tracing Mechanisms: Classical and New Synthesis. Mechanism and Machine Theory 35, 1579–1599 (2000)MATHCrossRefGoogle Scholar
  16. 16.
    Koetsier, T.: A contribution to the history of kinematics, Part I and Part II. Mechanism and Machine Theory 18, 37–42, 43–48 (1983)CrossRefGoogle Scholar

Copyright information

© Springer Netherlands 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceVrije UniversiteitAmsterdamThe Netherlands

Personalised recommendations