Abstract
We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
Abel N.H. (1826), Untersuchungen über die Reihe, u.s.w, Journal für die reine und angewandte Mathematik. 1: 311–339
F. Arndt (1852), Bemerkungen zur Convergenz der unendlichen Reihen, Grünert’s Archiv, Th. XX.
Björling E.G. (1846), Om en anmärkningsvärd klass af infinita serier, Kongl. Vetens. Akad. Förh. Stockholm 13: 17–35
E.G. Björling (1846/47), Doctrinae serierum infinitarum exercitationes, Nova Acta Soc. Sci. Upsala 13, 61–66, 143–187.
Björling E.G. (1852a), Sur une classe remarquable des séries infinies. J. math. pures appl., 17(1): 454–472
E.G. Björling (1852b), Om det Cauchyska kriteriet på de fall, då functioner af en variabel låta utveckla sig i serie, fortgående efter de stigande digniteterna af variabeln, Kongl. Vetens. Akad. Förh. Stockholm, 167–228.
Björling E.G. (1853), Om oändliga serier, hvilkas termer äro continuerliga functioner af en reel variabel mellan ett par gränser, mellan hvilka serierna äro convergerande. öfvers. Kongl. Vetens. Akad. Förh. Stockholm 10: 147–160
A.L. Cauchy (1821), Cours d’analyse de l‘École royale polytechnique (Analyse Algébrique).
Cauchy A.L. (1853), Note sur les séries convergentes dont les divers termes sont des fonctions continues d’une variable réelle ou imaginaire, entre des limites données. Oeuvres Complètes I 12: 30–36
Domar Y. (1987), E.G. Björling och seriesummans kontinuitet. Normat 35: 50–56
Giusti E. (1984), Gli “errori” di Cauchy e i fondamenti dell’analisi. Bollettino di Storia delle Scienze Matematiche 4: 24–54
Grattan-Guinness I. (1986), The Cauchy–Stokes–Seidel story on uniform convergence: was there a fourth man?. Bull. Soc. Math. Belg. 38: 225–235
I. Grattan-Guinness (2000), The search for mathematical roots 1870–1940, Princeton Paperbacks.
Grattan-Guinness I. (2004), The mathematics of the past: distinguishing its history from our heritage. Historia Mathematica. 31: 163–185
L. Gårding (1998), Mathematics and mathematicians: Mathematics in Sweden before 1950, Lund University Press.
Laugwitz D. (1980), Infinitely small quantities in Cauchy’s textbooks. Historia Mathematica 14: 258–274
E. Palmgren and A. Öberg (2006+), Ickestandardanalys, infinitesimaler och Cauchys behandling av kontinuitet och konvergens, about to appear in Normat.
Pringsheim A. (1897), Ueber zwei Abelsche Sätze, die Stetigkeit von Reihensummen betreffend. Sitzungsberichte Königl. Akad. Wiss. München 27: 343–358
Schmieden C., Laugwitz D. (1958), Eine Erweiterung der Infinitesimalrechnung. Mathematisches Zeitschrift 69: 1–39
Seidel P.L. (1848), Note über eine Eigenschaft der Reihen, welche Discontinuerliche Functionen darstellen, Abh. Akad. Wiss. München 5: 381–393
Spalt D. (2002), Cauchys Kontinuum – eine historiografische Annäherung via Cauchys Summensatz. Arch. Hist. Exact Sci. 56: 285–338
S. Stenlund (2004), Kommentarer med anledning av Kajsa Bråtings lic-avhandling, manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Lützen.
Rights and permissions
About this article
Cite this article
Bråting, K. A new look at E.G. Björling and the Cauchy sum theorem. Arch. Hist. Exact Sci. 61, 519–535 (2007). https://doi.org/10.1007/s00407-007-0005-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00407-007-0005-7