Science & Education

, Volume 18, Issue 8, pp 1057–1082 | Cite as

The History of the Planar Elastica: Insights into Mechanics and Scientific Method

  • Victor Geoffrey Alan Goss


Euler’s formula for the buckling of an elastic column is widely used in engineering design. However, only a handful of engineers will be familiar with Euler’s classic paper De Curvis Elasticis in which the formula is derived. In addition to the Euler Buckling Formula, De Curvis Elasticis classifies all the bent configurations of elastic rod—a landmark in the development of a rational theory of continuum mechanics. As a historical case study, Euler’s work on elastic rods offers an insight into some important concepts which underlie mechanics. It sheds light on the search for unifying principles of mechanics and the role of analysis. The connection between results obtained from theory and those obtained from experiments on rods, highlights two different approaches to scientific discovery, which can be traced back to Bacon, Descartes and Galileo. The bent rod also has an analogy in dynamics, with a pendulum, which highlights the crucial distinctions between initial value and boundary value problems and between linear and nonlinear differential equations. In addition to benefiting from the overview which a historical study provides, the particular problem of the elastica offers students of science and engineering a clear elucidation of the connection between mathematics and real-world engineering, issues which still have relevance today.


Scientific Methodology Flexural Stiffness Natural Philosopher Dynamic Analogy Elastic Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Antman SS (1995) Nonlinear problems of elasticity. Springer-Verlag, New YorkGoogle Scholar
  2. Ariew R (1986) Descartes as critic of Galileo’s scientific methodology. Synthese 67:77–90CrossRefGoogle Scholar
  3. Bacon F (1620/1902) Novum Organum. From Essays Civil and Moral, London. Ward Lock and Co., Ltd, New YorkGoogle Scholar
  4. Balaeff A, Mahadevan L, Schulten K (2006) Modeling DNA loops using the theory of elasticity. Phys Rev E 73:1919–1923CrossRefGoogle Scholar
  5. Bell JF (1973) Mechanics of solids 1. Encyclopedia of physics, vol VIa(1). Springer-Verlag, BerlinGoogle Scholar
  6. Bennett J, Cooper M, Hunter M, Jardine L (2003) London’s Leonardo. The life and work of Robert Hooke. Oxford University Press, OxfordGoogle Scholar
  7. Benvenuto E (1991) An introduction to the history of structural mechanics, part 1: statics and resistance of solids, English edn. Springer-Verlag, New YorkGoogle Scholar
  8. Berteaux HO (1976) Buoy engineering. Wiley, New YorkGoogle Scholar
  9. Burckhardt JJ (1983) Leonhard Euler, 1707–1783. Math Mag 56(5):262–273Google Scholar
  10. Capechhi D, Drago A (2005) On Lagrange’s history of mechanics. Meccanica 40:19–33CrossRefGoogle Scholar
  11. Ceccarelli M (2006) Early TMM in Le Mecaniche by Galileo Galilei in 1593. Mech Mach Theory 41:1401–1406CrossRefGoogle Scholar
  12. Coyne J (1990) Analysis of the formation and elimination of loops in twisted cable. IEEE J Oceanic Eng 15(2):72–83CrossRefGoogle Scholar
  13. Descartes R (1637) La Geometrie. Dover reprint 1954Google Scholar
  14. Dugas R (1955) A history of mechanics. Dover, New YorkGoogle Scholar
  15. Euler L (1744) Methodus inveniendi lineas curvas maximi minimivi propreitate gaudentes. Additamentum I (De curvis elasticas). Leonardi Euleri Opera Omnia IXXIV, 231–297. English translation: Oldfather WA, Ellis CA, Brown DM (1930) Leonhard Euler’s elastic curves. Isis 20:72–160Google Scholar
  16. Euler L (1947) On the strength of columns. Am J Phys 15:315–318CrossRefGoogle Scholar
  17. Fellmann EA (2007) Leonhard Euler. Birkhäuser, BaselGoogle Scholar
  18. Frasier CG (1991) Mathematical technique and physical conception in Euler’s investigation of the elastica. Centaurus 34:211–246CrossRefGoogle Scholar
  19. Galileo G (1623) Il Saggiatore. Opere VI George Polya, trGoogle Scholar
  20. Galileo G (1638/1988) Discorsi E Dimostrazioni Matematiche intorno à due nuoue fcienze. Leyden. (Reprint: Dialogues concerning two new sciences. Dover, New York)Google Scholar
  21. Goss VGA, Van der Heijden GHM, Neuikrich S, Thompson JMT (2005) Experiments on snap buckling, hysteresis and loop formation in twisted rods. Exp Mech 45(2):101–111CrossRefGoogle Scholar
  22. Gower B (1997) Scientific method. RoutledgeGoogle Scholar
  23. Hearle JWS (2000) A critical review of the structural mechanics of wool and hair fibres. Int J Biol Macromol 27:123–138CrossRefGoogle Scholar
  24. Hilbert D (1900) Mathematical problems. Lecture delivered before the International Congress of Mathematicians at Paris in 1900. For full text see web site djoyce/hilbert/problems.html
  25. Hooke R (1665/2006) Micrographica. Hard PressGoogle Scholar
  26. Johnson BG (1983) Column buckling theory: historic highlights. J Struct Eng 109(9):2086–2096CrossRefGoogle Scholar
  27. Kipnis N (2005) Scientific analogies and their use in teaching science. Sci & Educ 14:199–233CrossRefGoogle Scholar
  28. Kirchhoff G (1859) Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastichen Stabes. Journal fü die reine und angewandte Mathematik 56:285–316CrossRefGoogle Scholar
  29. Kline M (1972) Mathematical thought from ancient to modern times. OUP, New YorkGoogle Scholar
  30. Lagrange JL (1770–1773) Sur la figure des colones. Ouevres de Lagrange 2:125–137Google Scholar
  31. Love AEH (1927) The mathematical theory of elasticity, 4th edn. CUP, CambridgeGoogle Scholar
  32. Lyons JS, Brader JS (2004) Using the learning cycle to develop freshmen’s abilities to design and conduct experiments. Int J Mech Eng Educ 32(2):126–134Google Scholar
  33. Metz D, Klassen S, McMillan B, Clough M, Olson J (2007) Building a foundation for the use of historical narratives. Sci & Educ 16:313–334CrossRefGoogle Scholar
  34. Mikhailov GK, Stepanov SYa (2007) Leohnard Euler and his contribution to the development of mechanics. J Appl Math Mech. doi: 10.1016/j.jappmathmech.2007.06.001.
  35. Newburgh R, Newburgh GAM (2000) Finding the equation for a vibrating car antenna. Phys Teach 38:31–34CrossRefGoogle Scholar
  36. Ravi-Chandar K (2005) Murray lecture: the role of experiments in mechanics. Exp Mech 45(6):478–492Google Scholar
  37. Sargant R (1989) Scientific experiment and legal expertise: the way of experience in seventeeth century England. Stud Hist Philos Sci Part A 20(1):19–45CrossRefGoogle Scholar
  38. Schecker P (1992) The paradigmatic change in mechanics: implications of historical processes for physics education. Sci & Educ 1:71–76CrossRefGoogle Scholar
  39. Sönmez Ü (2006) Synthesis methodology of a compliant exact long dwell mechanism using elastica theory. Int J Mech Mater Des 3:73–90CrossRefGoogle Scholar
  40. Speiser D (2003) The importance of concepts of science. Meccanica 38:483–492CrossRefGoogle Scholar
  41. Stillwell J (1989) Mathematics and its history. Springer-Verlag, New YorkGoogle Scholar
  42. Timoshenko SP (1953) History of the strength of materials. Reprint: Dover, New YorkGoogle Scholar
  43. Timoshenko SP, Gere JM (1961) Theory of elastic stability, 2nd edn. McGraw Hill, New YorkGoogle Scholar
  44. Travers AA, Thompson JMT (2004) An introduction to the mechanics of DNA. Phil Trans R Soc Lond A 362:1265–1279CrossRefGoogle Scholar
  45. Truesdell C (1960) L. Euler Opera Omnia II The Rational Mechanics of Flexible or Elastic Bodies 1638–1788. Füssli, ZurichGoogle Scholar
  46. Truesdell C (1968) Essays in the history of mechanics. Springer-Verlag, New YorkGoogle Scholar
  47. Truesdell C (1984) An idiot’s fugitive essays on science. Springer Verlag, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Faculty of Engineering Science and the Built EnvironmentLondon South Bank UniversityLondonUK

Personalised recommendations