Skip to main content
SpringerLink
Log in

History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher

  • Chapter
  • First Online:
Excursions in the History of Mathematics

Abstract

The infinitely small and the infinitely large – in one form or another – are essential in calculus. In fact, they are among the distinguishing features of calculus compared to some other branches of mathematics, for example algebra. They have appeared throughout the history of calculus in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the seventeenth through the twentieth centuries. This will, in fact, entail discussing central issues in the development of calculus.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. K. Andersen, Cavalieri’s method of indivisibles, Arch. Hist. Exact Sciences 31 (1985) 291–367.

    MATH  Google Scholar 

  2. M. Baron, The Origins of the Infinitesimal Calculus, Dover, 1987.

    Google Scholar 

  3. E. T. Bell, The Development of Mathematics, McGraw-Hill, 1945.

    Google Scholar 

  4. J. L. Bell, Infinitesimals, Synthese 75 (1988) 285–315.

    Article  MathSciNet  Google Scholar 

  5. H. J. M. Bos, Differentials, higher-order differentials and the derivative in the Leibnizian calculus, Arch. Hist. Exact Sciences 14 (1974) 1–90.

    Article  MATH  MathSciNet  Google Scholar 

  6. U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer, 1986.

    Google Scholar 

  7. C. B. Boyer, The History of the Calculus and its Conceptual Development, Dover, 1959.

    Google Scholar 

  8. C. B. Boyer, Cavalieri, limits and discarded infinitesimals, Scripta Mathematica 8 (1941) 79–91.

    MathSciNet  Google Scholar 

  9. D. Bressoud, A Radical Approach to Lebesgue’s Theory of Integration, Math. Assoc. of Amer., 2008.

    Google Scholar 

  10. D. Bressoud, A Radical Approach to Real Analysis, Math. Assoc. of Amer., 1994.

    Google Scholar 

  11. F. Cajori, Indivisibles and “ghosts of departed quantities” in the history of mathematics, Scientia 37 (1925) 301–306.

    Google Scholar 

  12. F. Cajori, Grafting of the theory of limits on the calculus of Leibniz, Amer. Math.Monthly 30 (1923) 223–234.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Cajori, Discussion of fluxions: from Berkeley to Woodhouse, Amer. Math. Monthly 24 (1917) 145–154.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Calinger (ed.), Vita Mathematics: Historical Research and Integration with Teaching, Math. Assoc. of Amer., 1996.

    Google Scholar 

  15. A. – L. Cauchy, Cours d’Analyse, translated into English, with annotations, by R. Bradley and E. Sandifer, Springer, 2009 (orig. 1821).

    Google Scholar 

  16. J. W. Dauben, Abraham Robinson and nonstandard analysis: history, philosophy, and foundations of mathematics. In W. Aspray and P. Kitcher (eds.), History and Philosophy of Modern Mathematics, Univ. of Minnesota Press, 1988, pp. 177–200.

    Google Scholar 

  17. M. Davis and R. Hersh, Nonstandard analysis, Scient. Amer. 226 (1972) 78–86.

    Article  MathSciNet  Google Scholar 

  18. R. Dedekind, Essays on the Theory of Numbers, Dover, 1963 (orig. 1872 & 1888).

    Google Scholar 

  19. U. Dudley (ed.), Readings for Calculus, Math. Assoc. of Amer., 1993.

    Google Scholar 

  20. W. Dunham, The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton Univ. Press, 2005.

    MATH  Google Scholar 

  21. W. Dunham, Euler: The Master of Us All, Math. Assoc. of Amer., 1999.

    Google Scholar 

  22. W. Dunham, Journey Through Genius: The Great Theorems of Mathematics, Wiley, 1990.

    Google Scholar 

  23. C. H. Edwards, The Historical Development of the Calculus, Springer, 1979.

    Google Scholar 

  24. L. Euler, Inrtoductio in Analysin Infinitorum, vols. 1 and 2, English tr. by J. Blanton, Springer, 1989 (orig. 1748).

    Google Scholar 

  25. H. Eves, Great Moments in Mathematics, 2 vols., Math. Assoc. of Amer., 1983.

    Google Scholar 

  26. J. Fauvel (ed.), The use of history in teaching mathematics, Special Issue of For the Learning of Mathematics, vol. 11, no. 2, 1991.

    Google Scholar 

  27. J. Fauvel (ed.), History in the Mathematics Classroom: The IREM Papers, vol. 1, The Mathematical Association (London), 1990.

    Google Scholar 

  28. J. Fauvel and J. van Maanen (eds.), History in Mathematics Education, Kluwer, 2000.

    Google Scholar 

  29. C. Fraser, The calculus as algebraic analysis: some observations on mathematical analysis in the 18th century, Arch. Hist. Exact Sciences 39 (1989) 317–335.

    MATH  MathSciNet  Google Scholar 

  30. A. Gardiner, Infinite Processes: Background to Analysis, Springer, 1979.

    Google Scholar 

  31. J. W. Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, Amer. Math. Monthly 90 (1983) 185–194.

    Article  MATH  MathSciNet  Google Scholar 

  32. J. W. Grabiner, The changing concept of change: the derivative from Fermat to Weierstrass, Math. Mag. 56 (1983) 195–206.

    Article  MATH  MathSciNet  Google Scholar 

  33. J. W. Grabiner, The Origins of Cauchy’s Rigorous Calculus, MIT Press, 1981.

    Google Scholar 

  34. I. Grattan-Guinness (ed.), From the Calculus to Set Theory, 1630–1910, Princeton Univ. Press, 2000.

    Google Scholar 

  35. I. Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, MIT Press, 1970.

    Google Scholar 

  36. E. Hairer and G. Wanner, Analysis by its History, Springer, 1996.

    Google Scholar 

  37. V. Harnik, Infinitesimals from Leibniz to Robinson: time to bring them back to school, Math. Intelligencer 8 (2) (1986) 41–47, 63.

    Google Scholar 

  38. K. Kalman, Six ways to sum a series, College Math. Jour. 24 (1993) 402–421.

    Article  Google Scholar 

  39. V. J. Katz, A History of Mathematics, 3rd ed., Addison-Wesley, 2009.

    Google Scholar 

  40. V. J. Katz (ed.), Using History to Teach Mathematics: An International Perspective, Math. Assoc. of America, 2000.

    Google Scholar 

  41. J. Keisler, Elementary Calculus: An Infinitesimal Approach, 2nd ed., Prindle, Weber & Schmidt, 1986.

    Google Scholar 

  42. J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, 1976.

    Google Scholar 

  43. P. Kitcher, The Nature of Mathematical Knowledge, Oxford Univ. Press, 1983.

    MATH  Google Scholar 

  44. P. Kitcher, Fluxions, limits, and infinite littlenesse: a study of Newton’s presentation of the calculus, Isis 64 (1973) 33–49.

    Article  MATH  MathSciNet  Google Scholar 

  45. M. Kline, Euler and infinite series, Math. Mag. 56 (1983) 307–314.

    Article  MATH  MathSciNet  Google Scholar 

  46. M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford Univ. Press, 1972.

    MATH  Google Scholar 

  47. I. Lakatos, Cauchy and the continuum: the significance of non-standard analysis for the history and philosophy of mathematics, Math. Intelligencer 1 (1978) 151–161.

    Article  MATH  MathSciNet  Google Scholar 

  48. R. E. Langer, Fourier series: the genesis and evolution of a theory, Amer. Math. Monthly 54 (S) (1947) 1–86.

    Google Scholar 

  49. R. Laubenbacher and D. Pengelley, Mathematical Expeditions: Chronicles by the Explorers, Springer, 1999.

    Google Scholar 

  50. D. Laugwitz, On the historical development of infinitesimal mathematics, I, II, Amer. Math. Monthly 104 (1997) 447–455, 660–669.

    Google Scholar 

  51. N. MacKinnon (ed.), Use of the history of mathematics in the teaching of the subject, Special Issue of Mathematical Gazette, Vol. 76, no. 475, 1992.

    Google Scholar 

  52. P. Mancosu, The metaphysics of the calculus: a foundational debate in the Paris Academy of Sciences, 1700–1706, Hist. Math. 16 (1989) 224–248.

    Article  MATH  MathSciNet  Google Scholar 

  53. K. O. May, History in the mathematics curriculum, Amer. Math. Monthly 81 (1974) 899–901.

    Article  MathSciNet  Google Scholar 

  54. NCTM, Historical Topics for the Mathematics Classroom, 2nd ed., National Council of Teachers of Mathematics, 1989.

    Google Scholar 

  55. D. Pimm, Why the history and philosophy of mathematics should not be rated X, For the Learning of Mathematics 3 (1982) 12–15.

    Google Scholar 

  56. A. Robinson, Non-Standard Analysis, North-Holland, 1966.

    Google Scholar 

  57. R. Roy, The discovery of the series formula for π by Leibniz, Gregory, and Nilakantha, Math. Mag. 63 (1990) 291–306.

    Article  MATH  MathSciNet  Google Scholar 

  58. A. Shell-Gellasch (ed.), Hands on History: A Resource for Teaching Mathematics, Math. Assoc. of Amer., 2007.

    Google Scholar 

  59. A. Shell-Gellasch and R. Jardine (eds.), From Calculus to Computers: Using the Last 2000 Years of Mathematical History in the Classroom, Math. Assoc. of Amer., 2005.

    Google Scholar 

  60. G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, McGraw-Hill, 1992.

    Google Scholar 

  61. G. F. Simmons, Differential Equations, McGraw-Hill, 1972.

    Google Scholar 

  62. S. Stahl, Real Analysis: A Historical Approach, Wiley, 1999.

    Google Scholar 

  63. L. A. Steen, New models of the real-number line, Scient. Amer. 225 (1971) 92–99.

    Article  MathSciNet  Google Scholar 

  64. J. Stillwell, Mathematics and its History, 2nd ed., Springer-Verlag, 2002.

    Google Scholar 

  65. D. J. Struik, A Concise History of Mathematics, 4th ed., Dover, 1987.

    Google Scholar 

  66. F. Swetz et al (eds.), Learn from the Masters!, Math. Assoc. of Amer., 1995.

    Google Scholar 

  67. O. Toeplitz, The Calculus: A Genetic Approach, The Univ. of Chicago Press, 1963.

    Google Scholar 

  68. N. Y. Vilenkin, In Search of Infinity, transl. from the Russian by A. Shenitzer, Birkhäuser, 1995.

    Google Scholar 

  69. D. T. Whiteside, The Mathematical Papers of Isaac Newton, 8 vols., Cambridge Univ. Press, 1967–1981.

    Google Scholar 

  70. R. L. Wilder, History in the mathematics curriculum: its status, quality and function, Amer. Math. Monthly 79 (1972) 479–495.

    Article  MathSciNet  Google Scholar 

  71. R. M. Young, Excursions in Calculus, Math. Assoc. of Amer., 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Israel Kleiner

Authors
  1. Israel Kleiner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Israel Kleiner .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Kleiner, I. (2012). History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher. In: Excursions in the History of Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8268-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation