History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher

  • Israel Kleiner


The infinitely small and the infinitely large – in one form or another – are essential in calculus. In fact, they are among the distinguishing features of calculus compared to some other branches of mathematics, for example algebra. They have appeared throughout the history of calculus 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 seventeenth through the twentieth centuries. This will, in fact, entail discussing central issues in the development of calculus.


Eighteenth Century Seventeenth Century Differential Calculus Early Nineteenth Century Nonstandard Analysis 
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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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