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Superposition: on Cavalieri’s practice of mathematics

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Abstract

Bonaventura Cavalieri has been the subject of numerous scholarly publications. Recent students of Cavalieri have placed his geometry of indivisibles in the context of early modern mathematics, emphasizing the role of new geometrical objects, such as, for example, linear and plane indivisibles. In this paper, I will complement this recent trend by focusing on how Cavalieri manipulates geometrical objects. In particular, I will investigate one fundamental activity, namely, superposition of geometrical objects. In Cavalieri’s practice, superposition is a means of both manipulating geometrical objects and drawing inferences. Finally, I will suggest that an integrated approach, namely, one which strives to understand both objects and activities, can illuminate the history of mathematics.

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References

  • Agazzi E. (1967) La ricomparsa di Bonaventura Cavalieri. Periodico di Matematiche 45: 1–11

    MATH  Google Scholar 

  • Agostini Amedeo (1940) I baricentri di gravi non omogenei e la formola generale per il loro calcolo determinati da Bonaventura Cavalieri. Bollettino dell’ Unione Matematica Italiana, 2(d series 2): 147–171

    MATH  MathSciNet  Google Scholar 

  • Andersen K. (1985) Cavalieri’s method of indivisibles. Archive for the History of Exact Sciences 31: 291–367

    MATH  Google Scholar 

  • (1544) Archimedis Syracusani Philosophi ac Geometrae Excellentissimi Opera. Ioannes Hervagius, Basel

    Google Scholar 

  • (1558) Opera nonnulla (Federico Commandino, Ed). Paulus Manutius, Venice

    Google Scholar 

  • (1615) Opera quae extant(David Rivault, Ed). Apud Claudium Morellum, Paris

    Google Scholar 

  • Archimedes, 1880. Opera omnia, Vol. I., (I. L. Heiberg, Ed.). Teubner, Leipzig.

  • Archimedes, 1974. Opere di Archimede (Attilio Frajese, Ed.). UTET, Turin.

  • Arrighi G. (1973) La ‘Geometria indivisibilibus continuorum’ di Bonaventura Cavalieri nella ritrovata stesura del 1627. Physis 15: 133–147

    MathSciNet  Google Scholar 

  • Beeley P. (1996) Kontinuität und Mechanismus. Zur Philosophie des jungen Leibniz in ihrem ideengeschichtlichen Kontext. Stuttgart, Steiner

    MATH  Google Scholar 

  • Cavalieri Bonaventura (1635) Geometria indivisibilibus continuorum nova quadam ratione promota. Clemente Ferroni, Bologna

    Google Scholar 

  • Cavalieri Bonaventura (1647) Exercitationes geometricae sex. Giacomo Monti, Bologna

    Google Scholar 

  • Cavalieri, Bonaventura, 1966. Geometria degli indivisibili (L. Lombardo-Radice, Ed. and Tr.). UTET, Turin.

  • Cellini G. (1966) Le dimostrazioni di Cavalieri del suo principio. Periodico di Matematiche 44: 85–105

    MATH  MathSciNet  Google Scholar 

  • (1966b) Gli indivisibili nel pensiero matematico e filosofico di Bonaventura Cavalieri. Periodico di Matematiche 44: 1–21

    MATH  MathSciNet  Google Scholar 

  • Clavius, Christoph, 1999. Commentaria in Euclidis Elementa Geometrica. Olms-Weidmann, Hildesheim [facsimile edition of part of the first volume of Christophori Clauii Bambergensis e Societate Iesu Opera mathematica V tomis distributa (1611–1612). Sumptibus Antonij Hierat excudebat Reinhard Eltz, Mainz].

  • De Gandt, F. 1992. Cavalieri’s indivisibles and Euclid’s canons. In: Revolution and continuity: Essays in the history and philosophy of early modern science (P. Barker, R. Arew, Eds.), Washington, DC., The Catholic University of America Press, pp. 157–182.

  • Euclid, 1557. In Euclidis elementa geometrica demonstrationum libri sex (Jacques Peletier, Ed.). Apud Tornaesium and Gazeium, Lyon.

  • Euclid, 1575. De gli Elementi d’ Euclide libri quindici (Federico Commandino, Ed.). Frisolino, Urbino.

  • Euclid, 1883. Opera omnia, vol. I, (I. L. Heiberg, H. Menge, Eds.). Teubner, Leipzig.

  • Euclid, 1884. Opera omnia, vol. II, (I. L. Heiberg, H. Menge, Eds.). Teubner, Leipzig.

  • Euclid, 1956. The Thirteen Books of the Elements.Translated with introduction and commentary by Sir Thomas Heath. 3 vols. 2nd edition. New York: Dover Publications.

  • Favaro A. (1885) Bonaventura Cavalieri nello Studio di Bologna. Fava e Garagnani, Bologna

    Google Scholar 

  • Festa E. (1992) La notion d’ ‘aggrégat d’ indivisibles’ dans la constitution de la cinématique galiléenne: Cavalieri, Galilée, Torricelli. Revues d’ histoire des sciences 45: 307–336

    MATH  MathSciNet  Google Scholar 

  • Giuntini, S., 1985, [Ed.]. De spatijs helicis (Il trattatello delle spirali). Bollettino di storia delle scienze matematiche 5, 7–36.

  • Giusti E. (1980) Bonaventura Cavalieri and the theory of indivisibles. Edizioni Cremonese, Rome

    MATH  Google Scholar 

  • Guldin, Paulus, 1635. De centro gravitatis, Vol. I. Formis G. Gelbhaar, Vienna.

  • Guldin, Paulus, 1640–2641. De centro gravitatis, Vols. II-IV. Formis Matthaei Cosmerovii, Vienna.

  • Koyré, A., 1973. Bonaventura Cavalieri et la géométrie des continus. In: Koyré, A., Études d’ histoire de la pensée scientifique. Gallimard, Paris [1st edition, Presses Universitaires de France, Paris], pp. 334–361.

  • Malet, A., 1996. From indivisibles to infinitesimals. Universitat Autònoma de Barcelona, Bellaterra (Barcelona).

  • Mancosu Paolo (1996) Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford University Press, New York and Oxford

    MATH  Google Scholar 

  • Mueller, Ian, 2006. Philosophy off mathematics and deductive structure in Euclid’s Elements. New York: Dover Publications (1st ed. Cambridge, MA: MIT Press).

  • Muntersbjorn, M., 2000. The qauadrature of parabolic segments 1635–2658: A response to Herbert Breger. In: The growth of mathematical knowledge (E. Grosholz, H. Breger, Eds.). Kluwer Academic Press, Dordrecht/ Boston/ London, pp. 229–256.

  • Piola Gabrio (1844) Elogio di Bonaventura Cavalieri. Bernardoni Di Giovanni, Milan

    Google Scholar 

  • Proclus, 1560. In primum Euclidis elementorum librum (Francesco Barozzi, Ed.). Perchacinus, Padua.

  • Sestini, G., 1996/97. La ‘Geometria indivisibilibus’ di Bonaventura Cavalieri nelle edizioni manoscritte del 1627.Annali della facoltà di Lettere e Filosofia. 4. Studi Filosofici. Vol. 34, n. 20. Università degli Studi di Perugia, Perugia, pp. 85–119.

  • Struik D.J. (1969) A source book in mathematics, 1200-1800. Harvard University Press, Cambridge, MA

    MATH  Google Scholar 

  • Terregino G. (1980) Sul principio di Cavalieri. Archimede 32: 59–65

    MathSciNet  Google Scholar 

  • Terregino G. (1987) La seconda ‘ratio’ di Bonaventura Cavalieri. Archimede 39: 209–213

    MathSciNet  Google Scholar 

  • Torricelli, Evangelista, (1919–2944). Opere (G. Loria, G. Vassura, Eds.), 4 vols. G. Montanari, Faenza.

  • Vita V. (1972) Gli indivisibili curvi in Bonaventura Cavalieri. Archimede 24: 16–24

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of History and Philosophy of Science, HPS Department, University of Pittsburgh, 1017 Cathedral of Learning, Pittsburgh, PA, 15260, USA

    Paolo Palmieri

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  1. Paolo Palmieri
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Correspondence to Paolo Palmieri.

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Communicated by U. Bottazzini.

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Palmieri, P. Superposition: on Cavalieri’s practice of mathematics. Arch. Hist. Exact Sci. 63, 471–495 (2009). https://doi.org/10.1007/s00407-008-0032-z

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