Synthese

, Volume 186, Issue 1, pp 191–229 | Cite as

The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Article
  • 293 Downloads

Abstract

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.

Keywords

Newton’s Principia Measurement Geometry 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aiton E. (1989) Polygons and parabolas. Centaurus 31: 207–221CrossRefGoogle Scholar
  2. Andersen K., Bos H. (2006) Pure mathematics. In: Porter R., Park K., Daston L. (eds) The Cambridge history of science. Cambridge University Press, Cambridge, pp 696–726Google Scholar
  3. Ariotti, P. (1972). Celestial reductionism of time. Studi internazionali di filosofia, Autunno.Google Scholar
  4. Arthur R. T. W. (1995) Newton’s fluxions and equably flowing time. Studies in History and Philosophy of Science 26: 323–351CrossRefGoogle Scholar
  5. Arthur R. T. W. (2008) Leery bedfellows: Newton and Leibniz on the status of infinitesimals. In: Goldenbaum U., Jesseph D. (eds) Infinitesimal differences. de Gruyter, Berlin, pp 7–30Google Scholar
  6. Arthur, R. T. W. (2011, forthcoming). On Newton’s fluxional proof of the vector addition of motive forces. In C. Fraser & W. Harper (Eds.) Infinitesimals.Google Scholar
  7. Barrow, I. (1734). The usefulness of mathematical learning explained and demonstrated: Being mathematical lectures read in the public schools of Cambridge (J. Kirkby, Trans.). London: Stephen Austen.Google Scholar
  8. Barrow, I. (1860). Geometrical lectures. In W. Whewell (Ed.) The mathematical works of Isaac Barrow (E. Stone, Trans., London: Stephen Austen, 1735). Cambridge: Cambridge University Press.Google Scholar
  9. Bos H. J. M. (2001) Redefining geometrical exactness. Springer, New YorkCrossRefGoogle Scholar
  10. Brackenridge B. (1995) The key to Newton’s dynamics. University of California Press, BerkeleyGoogle Scholar
  11. Brackenridge Bruce (2000) Newton’s dynamics: The diagram as a diagnostic device. In: Dalitz R. H., Nauenberg M. (eds) The foundations of Newtonian scholarship. World Scientific, Singapore, pp 71–102CrossRefGoogle Scholar
  12. Brackenridge B., Nauenberg M. (2002) Curvature in Newton’s dynamics. In: Cohen I.B., Smith G.E. (eds) Cambridge companion to Newton.. Cambridge University Press, Cambridge, pp 85–137CrossRefGoogle Scholar
  13. Čapek M. (1976) Concepts of space and time. D. Reidel, DordrechtGoogle Scholar
  14. Cohen I. B. (1970) Newton’s second law and the concept of force in the Principia. In: Palter R. (eds) The Annus Mirabilis of Sir Isaac Newton 1666–1966. MIT Press, Cambridge MA, pp 143–185Google Scholar
  15. Cohen, I. B. (1999). A guide to Newton’s Principia. In: I. Newton The Principia: mathematical principles of natural philosophy (I. Bernard Cohen & A. Whitman, Trans.) (pp. 1–370). Berkeley and Los Angeles: University of California Press.Google Scholar
  16. Cohen, I. B., Smith, G. E. (eds) (2002) The Cambridge companion to Newton. Cambridge University Press, CambridgeGoogle Scholar
  17. De Gandt F. (1995) Force and geometry in Newton’s principia (C. Wilson, Trans). Princeton University Press, PrincetonGoogle Scholar
  18. Densmore D. (1995) Newton’s Principia: The central argument. Green Lion Press, Santa FeGoogle Scholar
  19. DiSalle R. (2002) Newton’s philosophical analysis of space and time. In: Cohen I. B., Smith G. E. (eds) Cambridge companion to Newton.. Cambridge University Press, Cambridge, pp 85–137Google Scholar
  20. Domski M. (2003) The constructible and the intelligible in Newton’s philosophy of geometry. Philosophy of Science 70: 1114–1124CrossRefGoogle Scholar
  21. Dreyer J. L. E. (1953) A history of astronomy from Thales to Kepler. Dover, New YorkGoogle Scholar
  22. Dunlop K. (2011) What geometry postulates. In: Janiak A., Schliesser E. (eds) Interpreting Newton. Cambridge University Press, CambridgeGoogle Scholar
  23. Edwards M. (2008) Time and perception in late Renaissance Aristotelianism. In: Knuuttila S., Kärkkäinen P. (eds) Theories of perception in medieval and early modern philosophy. Springer, Dordrecht, pp 225–244CrossRefGoogle Scholar
  24. Elzinga A. (1972) Huygens’ theory of research and Descartes’ theory of knowledge II. Zeitschrift für Allgemeine Wissenschaftstheorie/Journal for General Philosophy of Science 3: 9–27CrossRefGoogle Scholar
  25. Erlichson H. (1992) Newton’s polygon model and the second order fallacy. Centaurus 35: 243–258CrossRefGoogle Scholar
  26. Erlichson H. (2003) Passage to the limit in Proposition 1, Book I of Newton’s Principia. Historia Mathematica 30: 432–440CrossRefGoogle Scholar
  27. Feingold M. (1993) Newton, Leibniz, and Barrow too: An attempt at a reinterpretation. Isis 84: 310–338CrossRefGoogle Scholar
  28. Gabbey A. (1992) Newton’s Mathematical principles of natural philosophy: A treatise on ‘Mechanics’. In: Harman P., Shapiro A. (eds) The investigation of difficult things. Cambridge University Press, Cambridge, pp 305–322Google Scholar
  29. Galileo, G. (1638). Discorsi e dimonstrazioni mathematichè intorno à due nuove scienze. In Opere (1890–1909) (Vol. viii). Edizione Nazionale. Florence: Barbèra.Google Scholar
  30. Grant E. (1977) Physical science in the middle ages. Cambridge University Press, CambridgeGoogle Scholar
  31. Grosholz E. (1987) Some uses of proportion in Newton’s Principia, Book I. Studies in History and Philosophy of Science 18: 209–220CrossRefGoogle Scholar
  32. Guicciardini N. (1998) Reading the Principia. Cambridge University Press, CambridgeGoogle Scholar
  33. Guicciardini N. (2003) Geometry and mechanics in the preface to Newton’s Principia. Graduate Faculty Philosophy Journal 25: 119–159Google Scholar
  34. Guicciardini N. (2009) Isaac Newton on mathematical certainty and method. MIT Press, Cambridge MAGoogle Scholar
  35. Heath T. E. (Ed.). (1956). Euclid’s elements, 3 vols. New York: Dover.Google Scholar
  36. Janiak A. (2008) Newton as philosopher. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  37. Jesseph D. (1999) Squaring the circle. University of Chicago Press, ChicagoGoogle Scholar
  38. Klein, J. (1992). Greek mathematical thought and the origin of algebra (E. Brann, Trans.). Cambridge: MIT Press (Reprint, New York: Dover).Google Scholar
  39. Knorr, W. (1986). The ancient tradition of geometric problems Boston: Birkhäuser (Reprint, New York: Dover, 1993).Google Scholar
  40. Macbeth D. (2004) Viète, Descartes, and the emergence of modern mathematics. Graduate Faculty Philosophy Journal 25: 87–117Google Scholar
  41. Mahoney M.S. (1980) The beginnings of algebraic thought in the seventeenth century. In: Stephen G. (eds) Descartes: Philosophy, mathematics, and physics.. Totowa, Barnes and Noble Books, pp 141–155Google Scholar
  42. Mahoney M. S. (1990) Barrow’s mathematics: Between ancients and moderns. In: Feingold M. (eds) The life and times of Isaac Barrow. Cambridge University Press, Cambridge, pp 179–249CrossRefGoogle Scholar
  43. Mahoney M. S. (1993) Algebraic vs. geometric techniques in Newton’s determination of planetary orbits. In: Theerman P., Seeff A. (eds) Action and reaction. University of Delaware Press, Newark, pp 183–205Google Scholar
  44. Mancosu P. (1996) Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford University Press, OxfordGoogle Scholar
  45. Manders K. (2008) The Euclidean diagram. In: Mancosu P. (eds) The philosophy of mathematical practice.. Oxford University Press, Oxford, pp 80–133 (First appeared in 1995.)CrossRefGoogle Scholar
  46. Mueller I. (1981) Philosophy of mathematics and deductive structure in Euclid’s elements. MIT Press, Cambridge MAGoogle Scholar
  47. Nauenberg M. (1998) The mathematical principles underlying the Principia revisited. Journal for the History of Astronomy 29: 286–300Google Scholar
  48. Nauenberg M. (2003) Kepler’s area law in the Principia. Historia Mathematica 30: 441–456CrossRefGoogle Scholar
  49. Newton, I. (1959–1977). In H. W. Turnbull, J. F. Scott, A. Rupert Hall & L. Tilling (Eds.), The correspondence of Isaac Newton (7 vols). Cambridge: Cambridge University Press.Google Scholar
  50. Newton, I. (1964–1967). In D. T. Whiteside (Ed.), The mathematical writings of Isaac Newton (2 vols). New York: Johnson Reprint Corp.Google Scholar
  51. Newton, I. (1967–1981). In D. T. Whiteside (Ed.), The mathematical papers of Isaac Newton (8 vols). Cambridge: Cambridge University Press.Google Scholar
  52. Newton, I. (1984). In A. E. Shapiro (Ed.), The optical papers of Isaac Newton (Vol. 1). Cambridge: Cambridge University Press.Google Scholar
  53. Newton, I. (1999). The Principia: mathematical principles of natural philosophy (I. Bernard Cohen & A. Whitman, Trans.). Berkeley and Los Angeles: University of California Press.Google Scholar
  54. Panza M. (2011) From velocities to fluxions. In: Janiak A., Schliesser E. (eds) Interpreting Newton. Cambridge University Press, CambridgeGoogle Scholar
  55. Pourciau B. (2003) Newton’s argument for Proposition I of the Principia. Archive for History of the Exact Sciences 57: 267–311CrossRefGoogle Scholar
  56. Pourciau B. (2004) The importance of being equivalent: Newton’s two models of one-body motion. Archive for History of the Exact Sciences 58: 283–321CrossRefGoogle Scholar
  57. Roche J. (1998) The mathematics of measurement. London: Athlone Press, Springer, New YorkGoogle Scholar
  58. Rynasiewicz R. (1995) By their properties, causes, and effects: Newton’s Scholium on time, space, place, and motion. Studies in History and Philosophy of Science 26: 133–153CrossRefGoogle Scholar
  59. Sasaki C. (1985) The acceptance of the theory of proportion in the sixteenth and seventeenth centuries. Historia Scientiarum 29: 83–116Google Scholar
  60. Schemmel, M. (Forthcoming). Medieval representations of change and their early modern application. In A. Heeffer & M. van Dyck (Eds.), Proceedings of philosophical aspects of symbolic reasoning in the seventeenth century.Google Scholar
  61. Stein H. (1970) Newtonian space-time. In: Palter R. (eds) The Annus Mirabilis of Sir Isaac Newton, 1666–1966. MIT Press, Cambridge, pp 258–278Google Scholar
  62. Strong E. W. (1951) Newton’s mathematical way. Journal of the History of Ideas 12: 90–110CrossRefGoogle Scholar
  63. Sylla E. (1984) Compounding ratios. In: Mendelsohn E. (eds) Transformation and tradition in the sciences. Cambridge University Press, Cambridge, pp 11–43Google Scholar
  64. Westfall R. (1980) Never at rest. Cambridge University Press, CambridgeGoogle Scholar
  65. Westfall R. (1996) Technical Newton. Isis 87: 701–707CrossRefGoogle Scholar
  66. Whiteside D.T. (1966) Newtonian dynamics Review of The background to Newton’s Principia, by John Herivel. History of Science 5: 104–117Google Scholar
  67. Whiteside D. T. (1966) The mathematical principles underlying Newton’s Principia Mathematica. Journal for the History of Astronomy 1: 116–138Google Scholar
  68. Whiteside D. T. (1991) The prehistory of the Principia from 1664 to 1686. Notes and Records of the Royal Society of London 45: 11–61CrossRefGoogle Scholar
  69. Wilson C. (1989) Predictive astronomy in the century after Kepler. In: Taton R., Wilson C. (eds) The general history of astronomy. Cambride University Press, Cambridge, pp 161–206Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Brown UniversityProvidenceUSA

Personalised recommendations