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Original Articles

Did Weierstrass’s differential calculus have a limit-avoiding character? His definition of a limit in ϵδ style

 

Abstract

In the 1820s, Cauchy founded his calculus on his original limit concept and developed his theory by using inequalities, but he did not apply these inequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distinguished by their limit-avoiding character. Dugac’s partial publication of the 1861 lectures makes these differences clear. But in the unpublished portions of the lectures, Weierstrass actually defined his limit in terms of inequalities. Weierstrass’s limit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not provide the basic structure for the modern style analysis. Thus it was Dini’s 1878 textbook that introduced the definition of a limit in terms of inequalities.

Acknowledgements

I would like to thank the staff of the Berlin Humboldt University Library, who kindly supplied a copy of Schwarz’s notes on Weierstrass’s lectures. The paper would not have been possible without Manuel Kraus’s translations. This work was financially supported by the Rikkyo University Special Fund for Research.

Notes

1 Edwards (1979, 310), Laugwitz (1987, 260–261, 271–272), and Fisher (1978, 16–318) point out that Cauchy’s infinitesimals equate to a dependent variable function or that approaches zero as . Cauchy adopted the latter infinitesimals, which can be written in terms of arguments, when he introduced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchy’s is a variable in the parts that the present paper discusses.

2 A forerunner of the Technische Universität Berlin.

3 The present paper also quotes Kurt Bing’s translation included in Calinger’s Classics of mathematics.

4 Dugac (1973, 65) indicated that corresponds to the modern notion of . In addition, corresponds to the function that was introduced as in the former quotation from Weierstrass’s sentences.

5 Weil nun für unendlich kleine Werte von wenig von verschieden ist, so erklärt man den Differentialquotienten häufig als die Grenze, die sich des Verhältis der Veränderung der Funktion zur Aenderung des Arguments [the original description is ‘arguments’] nähert, worin letztere unendlich klein wird.—Ist nämlich die Funktion einer Grösse , die unendlich klein werden kann, so sagt man, es nähern sich für unendlich kleine Werte von der Grenze , worin die Differenz durch Verkleinerung von kleiner als jede noch so kleine Grösse gemacht werden kann.

6 Man sagt auch, (…), dass die Grenze von ist für x = a rechts oder links (…), wenn man nach Gefallen eine von Null verschiedene aber beliebig kleine und positive Zahl nimmt und alsdann eine im ersten Fall positive, im zweiten negative Zahl finden kann, der Art, dass für alle Werthe von , die als zwischen und ( ausgeschlossen) liegend angesehen werden können, die Differenz numerisch immer kleiner als ist.

7 (Mit andern Worten:) ist im Punkt , wo sie den Werth hat, continuirlich, wenn die Grenze ihrer Werthe rechts und links von dieselbe und gleich ist (…)

8 (…) nennt man die Derivirte dieser Function in diesem Punkt den Grenzwerth des Verhältnisseswenn der Null sowohl für positive als für negative Werthe zustrebt und unter der Voraussetzung, dass dieser Grenzwerth bestimmt, endlich und unabhängig von dem Vorzeichen von ist.

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