Vol 10 (2016) , 183-204

Open Access Open Access  Restricted Access Subscription Access

On History of Epsilontics

Galina Iwanowna Sinkiewicz

Digital Object Identifier (DOI): 10.14708/am.v10i0.805

Abstract

This is a review of genesis of „ε-δ" language in works of mathematicians of the 19th century. It shows that although the symbols ε and δ were initially introduced in 1823 by Cauchy, no functional relationship for δ as a function of ε was ever ever specified by Cauchy. It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass de_nition of a limit. The article gives various interpretations of these issues later provided by mathematicians. This article presents the text [Sinkevich, 2012d] of the same author which is slightly redone and translated into English.


Keywords: History of mathematics, analysis, continuity, Lagrange, Ampére, Cauchy, Bolzano, Heine, Cantor, Weierstrass, Lebesgue, Dini

Subject classification: 01A50, 01A55, 01A60

References

[Alexandrova, 1978] N. Aleksandrova. Matematiqeskie terminy. Spravoqnik
(Mathematical terms. Handbook. In Russian). Moskva: Vysxa xkola, 1978.

[Ampére, 1806] A. Ampére. Recherches sur quelques points de la théorie des fonctions dérivées qui conduisent à une nouvelle démonstration de la série de Taylor et à l'expression nie des termes qu'on néglige lorsqu'on arrete cette sèrie á un terme quelconque. Mémoir par M. Ampére, Répétiteur à l'École Politechnique.
Journal de l'École Politechnique. - 1806. - Cahier 13. - P. 148-181.

[Bashmakova, 1986] I. Baxmakova. O roli interpretacii v istorii
matematiki (On the role of interpretation in the history of mathematics,
in Russian). Istoriko-matematiqeskie issledovani. Moskva: Nauka.
XXX:182-194, 1986.

[Belhoste, 1985] B. Belhoste. Cauchy, un Mathématician Légimiste au XIX siècle.
Paris : Belin, 1985.

[Belhoste, 1997] B . Belhost. Ogsten Koxi (Augustin Cauchy, in Russian). -
Moskva: Nauka, 1997.

[Błaszczyk, Katz, Sherry, 2013] P. Bªaszcyk, M. Katz, D. Sherry. Ten misconceptions
from the history of analysis and their debunking. Foundations of Science.
18:1:43-74, 2013.

[Bolzano, 1817] B. Bolzano. Rein analytischer beweis des Lehrsatzes, dass zwischen
je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine
reelle Wurzel der Gleichung liege. - Prag, 1817. Bernard Bolzano (1781-1848).
Bicentenary. Early mathematical works. Prague. 417-476,1981.

[Bolzano, 1851, 1911] B. Bol~cano. Paradoksy beskoneqnogo.1851. (Paradoxes
of innity, in Russian). Translation edited by I. Sleshinski. Odessa: Mathesis,
1911.

[Cajori, 1919] F. Cajori. A history of the conception of limits and uxions in Great
Britain from Newton to Woodhouse. Chicago, London, 1919.

[Cantor, 1872] G. Cantor. Ueber die Ausdehnung eines Satzes der Theorie der
trigonometrischen Reihen. Mathem. Annalen von Clebsch und Neumann. 5:123
132, 1872.

[Cauchy, 1821] A.-L. Cauchy. Cours d'Analyse de l'École Royale Polytechnique
(1821). Analyse Algébrique. Oeuvres. Ser. 2, t. 3. 1471.

[Cauchy, 1821, 1864] O. Koxi. Algebriqeski$i analiz (Analyse Algébrique, in
Russian), translated by F. val~d, V. Grigor~ev, A. Il~in. - Leipzig :
Druck von Bär & Hermann, 1864.

[Cauchy, 1823] A.-L. Cauchy, 1823. Résumé des leçons données sur le calcul in-
nitésimal. Oeuvres ser. 2, IV:9261.

[Cauchy (1823), 1831] O. Koxi. Kratkoe izloenie urokov o differencial~
nom i integral~nom isqislenii (Résumé des leçons données sur le
calcul innitésimal, in Russian). Translation by Bunkovski$i. St.-Petersburg.
1831.

[Cauchy, 1853] A.-L. Cauchy. Note sur les séries convergentes dont les divers termes
sont des fonctions continues d'une variable réelle ou imaginaire, entre des limites
données. CR t. XXXVI, p.454, Oeuvres complètes série I,12:3036.1853.

[Cauchy, 2009] Cauchy's Cours d'analyse. An annotated English translation by R.E.
Bradley, C.E. Sandifer. Springer, 2009.

[Child, 1920] J. M. Child, (ed.), 1920. The early mathematical manuscripts of Leibniz.
Translated from the Latin texts published by Carl Immanuel Gerhardt with
critical and historical notes by J. M. Child. Chicago-London: The Open Court
Publishing Co. 1920.

[d'Alembert, 1765] J. d'Alembert. Limite. Encyclopédie méthodique ou par ordre
de matières. 1765, T.II Padoue, 311312. 17651789.

[Demidov, 1990] S. Demidov. \Zakon nepreryvnosti"Le$ibnica i pontie
nepreryvnosti funkcii u $ilera (Leibniz' law of continuity and the notion
of continuous function in Euler, in Russian). Istoriko-matematiqeskie
issledovani. XXXIIXXXIII, 3439, 1990.

[Dini, 1878] U. Dini. Fondamenti per la teoria delle funzioni di variabili reali. Pisa,
1878.

[Dorofeeva, 1971] A. Dorofeeva A. Formirovanie ponti nepreryvno$i
funkcii (The formation of the notion of continuous function, in Russian). Istori
i metodologi estestvennyh nauk. Moskva:Moscow State University.
- XI (Mathematics and mechanics). 3750, 1971.

[Dugac, 1972] P. Dugac. Éléments d'analyse de Karl Weierstrass. Paris, 1972.

[Dugac, 1973] P. Dgak Pontie predela i irracional~nogo qisla, koncepcii
Xarl Mere i Karla Ve$ierxtrassa (The notion of a limit and irrational
number, concepts of Charles Méray and Karl Weierstrass, in Russian).
Istoriko-matematiqeskie issledovani. XVIII:176180. 1973.

[Euler, 1748] L. $iler. Vvedenie v analiz beskoneqno malyh (Introductio in
analysin innitorum, in Russian). - Moskva: Fizmatgiz, Vol. II. , 1961.
[Euler, 1755] L. $iler. Nastavlenie po differencia    nomu isqisleni. (Institutiones
calcului dierentialis, in Russian). In : L. $iler. Differencia    -
noe isqislenie (Euler. Dierential calculus, in Russian), 2 vol. Moskva-
Leningrad: Gostehizdat, 1949.

[Fourier, 1822] J.B. Fourier. Théorie analitique de la chaleur. Oeuvres. Paris, v. 1,
1822.

[Gauss, 1800, 1917] K.F. Gauss. Grundbegrie der Lehre von der Reihen. Werke.
Leipzig : B. Bd. X/1: 390394, 1917.

[Grabiner, 1983] J. Grabiner. Who gave you the Epsilon? Cauchy and the Origin of
Rigorous Calculus. The American Mathematical Monthly. March. 90: 3:185194,
1983.

[Grattan-Guinness, 1970] I. Grattan-Guinness. Bolzano, Cauchy and the New
Analysisof the Early Nineteenth Century. Archive for History of Exact Sciences.
Springer Verlag. Berlin-Heidelberg-New York. 6, 35, 372400, 1970.

[Grattan-Guinness 2004] I. Grattan-Guinness. The mathematics of the past: distinguish
its history from our heritage. Historia mathematica 31:163-185, 2004.

[Gray, 2008] J. Gray. Plato's ghost. The modern transformation of mathematics.
NY, Princeton:Princeton University Press, 2008.

[Hankel, 1870/71] H. Hankel. Grenze. Allgemeine Enzyklopädie der Wissenschaften
und Künste. Leipzig. Vol. 90, 185211, 1870/71.

[Heine, 1872] H.E. Heine. Die Elemente der Functionenlehre. J. reine angew. Math.
74, 172-188, 1872.

[L'Huilier, 1786] S.-A.-J. L'Huilier. Exposition Élémentaire des Principles des calcul
supérieurs. 1786.

[Katz, Sherry, 2012] M. Katz, D. Sherry, D. Leibniz's laws of continuity and homogeneity.
Notices of the American Mathematical Society. 59:11,15501558, 2012.

[Koetsier, 2009] T. Koetsier, T. Lakatos, Lako and Nú¬ez: Towards a Satisfactory
Denition of Continuity. In Explanation and Proof in Mathematics. Philosophical
and Educational Perspectives. Edited by G. Hanna, H. Jahnke, and H. Pulte.
Springer, 2009.

[Lacroix, 1797] S.F. Lacroix. Traité du calcul dierentiel et du calcul intégral. Paris,
1797, 1798, 1800.

[Lacroix, 1806] S.F. Lacroix, S. F. Traité élementaire de calcul diérentiel et de
calcul intégral. Paris, 1806, 1828.

[Lagrange, 1797, 1881] J. Lagrange. Théorie des fonctions analytique. Oeuvres de
Lagrange. 9. Paris,1881.

[Lagrange, 1811] J. Lagrange. Mécanique analytique. V.1 Paris. 2-m ed., 1811.

[Leathem, 1905] J.G. Leathem. Volume and Surface Integrals Used in Physics. Cambridge,
1905.

[Lebesgue, 1904, 1934] A. Lebeg. Integrirovanie i otyskanie primitivnyh
funkci$i. (Leçons sur l'intégration et la recherché des fonctions primitives, in
Russian). Moskva-Leningrad, 1934.

[Méray, 1872] Ch. Méray. Nouveau précis d'analyse innitésimale. Par Charles
Méray. Publication : F. Savy. XXIII. Paris, 1872.

[Putnam, 1974] H. Putnam. What is mathematical truth? Proceedings of the American
Academy Workshop on the Evolution of Modern Mathematics. Boston,
Mass., 1974; Historia Mathematica. 2:4:529533, 1975.

[Seidel, 1847] Ph.L. Seidel. Note über eine Eigenschaft der Reihen, welche discontinuirliche
Functionen darstellen. Abhandl. Der Math. Phys. Klasse der Kgl.
Bayerschen Akademie der Wissenschaften V: 381394, München 1847.

[Sinkevich, 2012a] G.I. Sinkeviq. Uliss Dini i pontie nepreryvnosti
(Uliss Dini and the notion of continuity, in Russian). Istori nauki i tehhniki.
Moskva. 10:3-11, 2012.

[Sinkevich, 2012b] G.I. Sinkeviq. Genrih duard Ge$ine. Teori funkci$i.
(Heinrich Eduard Heine. Function theory. In Russian). Matematiqeskoe modelirovanie,
qislennye metody i kompleksy programm. St.-Peterburg.
18:646, 2012.

[Sinkevich, 2012c] G.I. Sinkeviq. Razvitie ponti nepreryvnosti u X.
Mere. (The development of continuity notion in Ch. Méray'). Trudy X Me-
dunarodnyh Kolmogorovskih qteni$i. (Proceeding of X Kolmogorov's reading).
roslavl~. 180185, 2012.

[Sinkevich, 2012d] G.I. Sinkeviq. K istorii psilontiki (To the history of
epsilontics, in Russian). Matematika v vysxem obrazovanii. 10:149166.
2012.

[Stokes, 1849] G. G. Stokes. On the critical values of the sums of periodic series,
Transactions of the Cambridge Philosophical Society, 8 : 533583. (presented:
1847 ; published: 1849).

[Stolz, 1881] O. Stolz. B. Bolzano's Bedeutung in der Geschichte der Innitesimalrechnung.
Mathematische Annalen. 18: 255279, 1881.

[Stolz, 1885] O. Stolz. Vorlesungen über allgemeine Arithmetik: Nach den neueren
Ansichten. Leipzig, Bd. I:156157, 1885.

[Wallis (1656), 2004] L. Wallis. The arithmetic of Innitesimals. Translated by J.A.
Stedall. USA: Springer. 2004.

[Weierstrass, 1886,1989] K. Weierstrass. Ausgewählte Kapitel aus der Funktionenlehre.
Vorlesung gehalten in Berlin 1886 mit der Akademischen Antrittsrede,
Berlin 1857 und drei weiteren Originalarbeiten von K.Weierstrass aus den Jahren
1870 bis 1880/86. Teubner-Archiv zur mathematic. Band 9. Reprint 1989.

[Yushkevich, 1972] A.xkeviq. Istori matematiki (The History of Mathematics,
in Russian). Edited by A.xkeviq. Moskva: Nauka. Vol. 3, 1972.

[Yushkevich, 1973] A.xkeviq. L. Karno i konkurs Berlinsko$i Akademii
Nauk 1786 na temu o matematiqesko$i teorii beskoneqnogo (L.Carnot
and the competition of Berlin Academy of Sciences 1786 on the mathematical
theory of innite, in Russian). Istoriko-matematiqeskie issledovani.
XVIII:132156, 1973.

[Yushkevich, 1977] A. xkeviq. Hrestomati po istorii matematiki.
Matematiqeski$i analiz (Khrestomatiya po istorii matematiki. Matematicheskiy
analizReading book on the history of mathematics. Analysis. In Russian),
edited by A.xkeviq. Moskva: Prosvewenie, 1977.

[Yushkevich, 1986] A. xkeviq. Razvitie ponti predela do K.
Ve$ierxtrassa (The development of the notion of limit till K. Weierstrass.
In Russian). Istoriko-matematiqeskie issledovania. Moskva: Nauka.
XXX:181, 1986.


Pages: 183-204

Full Text: PDF (Polski)


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.


Antiquitates Mathematicae by Polish Mathematical Society is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at http://am.ptm.org.pl/.
Permissions beyond the scope of this license may be available at http://www.ptm.org.pl/.

Print ISSN: 1898-5203, Online ISSN: 2353-8813


The journal is abstracted and indexed in: