Abstract
The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history 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 17th through the 20th centuries. We will also present `didactic observations' at relevant places in the historical account.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Andersen, K.: 1985, ‘Cavalieri's method of indivisibles’, Archive for History of the Exact Sciences 31, 291–367.
Baron, M.E.: 1987, The Origins of the Infinitesimal Calculus, Dover, New York.
Bell, E.T.: 1945, The Development of Mathematics, McGraw-Hill, New York.
Bell, J.L.: 1988, ‘Infinitesimals’, Synthese 75, 285–315.
Bos, H.J.M.: 1974, ‘Differentials, higher-order differentials and the derivative in the Leibnizian calculus’, Archive for History of the Exact Sciences 14, 1–90.
Bottazzini, U.: 1986, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, New York.
Boyer, C.B.: 1959, The History of the Calculus and its Conceptual Development, Dover, New York. Boyer, C.B.: 1941, ‘Cavalieri, limits and discarded infinitesimals’, Scripta Mathematica 8, 79–91.
Bressoud, D.: 1994, A Radical Approach to Real Analysis, Mathematical Association of America, Washington, DC.
Cajori, F.: 1925, ‘Indivisibles and ‘ghosts of departed quantities’ in the history of mathematics’, Scientia 37, 301–306.
Cajori, F.: 1923, ‘Grafting of the theory of limits on the calculus of Leibniz’, American Mathematical Monthly 30, 223–234.
Cajori, F.: 1917, ‘Discussion of fluxions: from Berkeley to Woodhouse’, American Mathematical Monthly 24, 145–154.
Calinger, R. (ed.): 1996, Vita Mathematics: Historical Research and Integration with Teaching, Mathematical Association of America, Washington, DC.
Dauben, J.W.: 1988, ‘Abraham Robinson and nonstandard analysis: history, philosophy, and foundations of mathematics’, in W. Aspray and P. Kitcher (eds.), History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, pp. 177–200.
Davis, M. and Hersh, R.: 1972, ‘Nonstandard analysis’, Scientific American 226, 78–86.
Dedekind, R.: 1963, Essays on the Theory of Numbers, Dover, New York.
Dudley, U. (ed.): 1993, Readings for Calculus, Mathematical Association of America, Washington, DC.
Dunham, W.: 1990, Journey Through Genius: The Great Theorems of Mathematics, John Wiley & Sons, New York.
Edwards, C.H.: 1979, The Historical Development of the Calculus, Springer-Verlag, New York.
Eves, H.: 1983, Great Moments in Mathematics, 2 vols., Mathematical Association of America, Washington, DC.
Fauvel, J. (ed.): 1991, The use of history in teaching mathematics, Special Issue of For the Learning of Mathematics 11(2).
Fauvel, J. (ed.): 1990, History in the Mathematics Classroom: The IREM Papers, vol. 1, The Mathematical Association, London.
Fauvel, J. and van Maanen J. (eds.): 2000, History in Mathematics Education: The ICMI Study, Kluwer Academic Publishers, Dordrecht.
Fraser, C.: 1989, ‘The calculus as algebraic analysis: some observations on mathematical analysis in the 18th century’, Archive for History of the Exact Sciences 39, 317–335.
Grabiner, J.W.: 1983a, ‘Who gave you the epsilon? Cauchy and the origins of rigorous calculus’, American Mathematical Monthly 90, 185–194.
Grabiner, J.W.: 1983b, ‘The changing concept of change: the derivative from Fermat to Weierstrass’, Mathematics Magazine 56, 195–206.
Grabiner, J.W.: 1981, The Origins of Cauchy's Rigorous Calculus, MIT Press, Cambridge, Mass.
Grattan-Guinness, I. (ed.): 1980, From the Calculus to Set Theory, 1630- 1910, Duckworth, London.
Grattan-Guinness, I.: 1970, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, MIT Press, Cambridge, Mass.
Hairer, E. and Wanner, G.: 1996, Analysis by its History, Springer-Verlag, New York.
Harnik, V.: 1986, ‘Infinitesimals from Leibniz to Robinson: time to bring them back to school’, Mathematical Intelligencer 8(2), 41–47, 63.
Kalman, K.: 1993, ‘Six ways to sum a series’, College Mathematics Journal 24, 402–421.
Katz, V.J. (ed.): 2000, Using History to Teach Mathematics: An International Perspective, Mathematical Association of America, Washington, DC.
Katz, V.J.: 1998, A History of Mathematics, 2nd ed., Addison-Wesley, New York.
Keisler, J.: 1986, Elementary Calculus: An Infinitesimal Approach, 2nd ed., Prindle,Weber & Schmidt, Boston.
Keisler, J.: 1976, Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston.
Kitcher, P.: 1983, The Nature of Mathematical Knowledge, Oxford University Press, New York.
Kitcher, P.: 1973, ‘Fluxions, limits, and infinite littlenesse: a study of Newton's presentation of the calculus’, Isis 64, 33–49.
Kline, M.: 1983, ‘Euler and infinite series’, Mathematics Magazine 56, 307–314.
Kline, M.: 1972, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York.
Lakatos, I.: 1978, ‘Cauchy and the continuum: the significance of non-standard analysis for the history and philosophy of mathematics’,Mathematical Intelligencer 1, 151–161.
Langer, R.E.: 1947, ‘Fourier series: the genesis and evolution of a theory’, American Mathematical Monthly 54(S), 1–86.
Laubenbacher, R. and Pengelley, D.: 1999, Mathematical Expeditions: Chronicles by the Explorers, Springer-Verlag, New York.
Laugwitz, D.: 1997, ‘On the historical development of infinitesimal mathematics’, I, II, American Mathematical Monthly 104, 447–455, 660- 669
MacKinnon, N. (ed.): 1992, Use of the history of mathematics in the teaching of the subject, Special Issue of Mathematical Gazette 76(475).
May. K. O.: 1974, ‘History in the mathematics curriculum’, American Mathematical Monthly 81, 899–901.
NCTM: 1989, Historical Topics for the Mathematics Classroom, 2nd ed., National Council of Teachers of Mathematics, Reston, Virginia.
Pimm, D.: 1982, ‘Why the history and philosophy of mathematics should not be rated X’, For the Learning of Mathematics 3, 12–15.
Polya, G.: 1962, Mathematical Discovery, combined ed., John Wiley & Sons, New York.
Robinson, A.: 1966, Non-Standard Analysis, North-Holland, Amsterdam.
Roy, R.: 1990, ‘The discovery of the series formula for π by Leibniz, Gregory, and Nilakantha’, Mathematics Magazine 63, 291–306.
Simmons, G.F.: 1992, Calculus Gems: Brief Lives and Memorable Mathematics, McGraw-Hill, New York.
Simmons, G.F.: 1972, Differential Equations, McGraw-Hill, New York.
Stahl, S.: 1999, Real Analysis: A Historical Approach, John Wiley & Sons, New York.
Steen, L.A.: 1971, ‘New models of the real-number line’, Scientific American 225, 92–99.
Stillwell, J.: 1989, Mathematics and its History, Springer-Verlag, New York.
Struik, D.J.: 1987, A Concise History of Mathematics, 4th ed., Dover, New York.
Swetz, F. et al. (eds.): 1995, Learn from the Masters!, Mathematical Association of America, Washington, DC.
Toeplitz, O.: 1963, The Calculus: A Genetic Approach, The University of Chicago Press, Chicago.
Vilenkin, N.Y.; 1995, In Search of Infinity (translated from the Russian by A. Shenitzer), Birkhäuser, Boston.
Wilder, R.L.: 1972, ‘History in the mathematics curriculum: its status, quality and function’, American Mathematical Monthly 79, 479–495.
Young, R.M.: 1992, Excursions in Calculus, Mathematical Association of America, Washington, DC.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kleiner, I. History of the Infinitely Small and the Infinitely Large in Calculus. Educational Studies in Mathematics 48, 137–174 (2001). https://doi.org/10.1023/A:1016090528065
Issue Date:
DOI: https://doi.org/10.1023/A:1016090528065