Skip to main content
SpringerLink
Log in

Leibniz's Rigorous Foundation Of Infinitesimal Geometry By Means Of Riemannian Sums

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  • Knobloch, Eberhard: 1990, ‘L'infini dans les mathématiques de Leibniz’, in L'infinito in Leibniz, Problemi e terminologia, Simposio Internazionale Roma, 6–8 Novembre 1986, Roma, pp. 33–51.

  • Knobloch, Eberhard: 1993, ‘Les courbes analytiques simples chez Leibniz’, in Sciences et techniques en perspective, Vol. 26, pp. 74–96.

    Google Scholar 

  • Knobloch, Eberhard: 1994, ‘The Infinite in Leibniz's Mathematics — The Historiographical Method of Comprehension in Context’, in Kostas Gavroglu, Jean Christianidis and Efthymios Nicolaïdis (eds), Trends in the Historiography of Science, Dordrecht [Boston Studies in the Philosophy of Science 151], pp. 265–278.

  • Knobloch, Eberhard: 1999, ‘Galileo and Leibniz: Different Approaches to Infinity’, Archive for the History of Exact Sciences 54 (1999), 87–99.

    Google Scholar 

  • Leibniz, Gottfried Wilhelm: 1993, De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis, kritisch herausgegeben und kommentiert von Eberhard Knobloch, Göttingen [Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-physikalische Klasse 3; 43].

Download references

Author information

Authors and Affiliations

  1. Institut für Philosophie, Wissenschaftstheorie, Wissenschafts- und Technikgeschichte der Technischen Universität Berlin, Ernst–Reuter–Platz 7, 10587, Berlin, Germany

    Eberhard Knobloch

Authors
  1. Eberhard Knobloch
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knobloch, E. Leibniz's Rigorous Foundation Of Infinitesimal Geometry By Means Of Riemannian Sums. Synthese 133, 59–73 (2002). https://doi.org/10.1023/A:1020859101830

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020859101830

Keywords

Search

Navigation