Abstract
The Principle of Continuity was a very broad law, used widely and importantly – though often not explicitly formulated – throughout the seventeenth, eighteenth, and nineteenth centuries. In general terms, the Principle of Continuity says that what holds in a given case continues to hold in what appear to be like cases. Specifically, it maintains that (1) What is true for positive numbers is true for negative numbers. (2) What is true for real numbers is true for complex numbers. (3) What is true up to the limit is true at the limit. (4) What is true for finite quantities is true for infinitely small and infinitely large quantities. (5) What is true for polynomials is true for power series. (6) What is true for a given figure is true for a figure obtained from it by continuous motion. (7) What is true for ordinary integers is true for (say) Gaussian integers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. F. Atiyah, Bakerian lecture, 1975: global geometry, Amer. Math. Monthly 111 (2004) 716–723.
I. G. Bashmakova, Diophantus and Diophantine Equations, Math. Assoc. of Amer., 1997.
G. Birkhoff and M. K. Bennett, Felix Klein and his “Erlanger Programm.” In History and Philosophy of Modern Mathematics, ed. by W. Aspray and P. Kitcher, Univ. of Minnesota Press, 1988, pp. 145–176.
H. J. M. Bos, Differentials, higher-order differentials and the derivative in the Leibnizian calculus, Arch. Hist. Exact Sc. 14 (1974) 1–90.
H. J. M. Bos, C. Kers, F. Oort, and D. W. Raven, Poncelet’s closure theorem, Expositiones Math. 5 (1987) 289–364.
U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag,1986.
N. Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1991.
E. Brieskorn and H. Knörrer, Plane Algebraic Curves, Birkhäuser, 1986.
D. Corfield, Towards a Philosophy of Real Mathematics, Cambridge Univ. Press, 2003.
J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard Univ. Press, 1979.
C. H. Edwards, The Historical Development of the Calculus, Springer-Verlag, 1979.
C. G. Fraser, The calculus as algebraic analysis: some observations on mathematical analysis in the 18th century, Arch. Hist. Exact Sci. 39 (1989) 317–335.
H. Freudenthal, The Didactic Phenomenology of Mathematical Structures, Reidel, 1983.
J. V. Grabiner, The Origins of Cauchy’s Rigorous Calculus, MIT Press, 1981.
H. Grant, Leibniz and the spell of the continuous, College Math. Jour. 25 (1994) 291–294.
J. Gray, The Worlds out of Nothing: A Course in the History of Geometry in the19 th Century, Springer, 2007.
T. L. Hankins, Sir William Rowan Hamilton, Johns Hopkins Univ. Press, 1980.
V. J. Katz, A History of Mathematics: An Introduction, 3rd ed., Addison-Wesley, 2009.
J. Keisler, Elementary Calculus: An Infinitesimal Approach, 2nd ed., Prindle, Weber & Schmidt, 1986.
F. Klein, A comparative review of recent researches in geometry, New York Math. Soc. Bull. 2 (1893) 215–249.
I. Kleiner, A History of Abstract Algebra, Birkhäuser, 2007.
E. Knobloch, Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums, Synthese 133 (2002) 59–73.
E. Knobloch, Analogy and the growth of mathematical knowledge. In The Growth of Mathematical Knowledge, ed. by E. Grosholz and H. Breger, Kluwer Acad. Publ., Amsterdam, 2000, pp. 295–314.
E. Koppelman, The calculus of operations and the rise of abstract algebra, Arch. Hist. Exact Sc. 8 (1971/72) 155–242.
M. H. Krieger, Some of what mathematicians do, Notices of the Amer. Math. Soc. 51 (2004) 1226–1230.
D. Laugwitz, Bernhard Riemann, 1826–1866, Birkhäuser, 1999.
D. Laugwitz, On the historical development of infinitesimal mathematics I, II, Amer. Math. Monthly 104 (1997) 447–455, 660–669.
P. J. Nahin, An Imaginary Tale: The Story of \(\sqrt{-1}\), Princeton Univ. Press, 1998.
G. Polya, Mathematics and Plausible Reasoning, 2 Vols., Princeton Univ. Press, 1954.
H. M. Pycior, George Peacock and the British origins of symbolical algebra, Hist. Math. 8 (1981) 23–45.
A. Robinson, Non-Standard Analysis, North-Holland, 1966.
B. A. Rosenfeld, The analytic principle of continuity, Amer. Math. Monthly 112 (2005) 743–748.
A. Weil, Number Theory: An Approach through History, Birkhäuser, 1984.
A. Weil, De la métaphysique aux mathématiques, Collected Papers, Vol. 2, Springer-Verlag, 1980, pp.408–412. (Translation into English in J. Gray, Open Univ. Course in History of Math., Unit 12, p. 30.)
A. N. Whitehead, An Introduction to Mathematics, Oxford Univ. Press, 1948.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Kleiner, I. (2012). Principle of Continuity: Sixteenth–Nineteenth Centuries. In: Excursions in the History of Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8268-2_9
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8268-2_9
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-8267-5
Online ISBN: 978-0-8176-8268-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)