Skip to main content
SpringerLink
Log in

Identifying Adequate Models in Physico-Mathematics: Descartes’ Analysis of the Rainbow

  • Conference paper
  • First Online:
Book cover Model-Based Reasoning in Science and Technology

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 8))

Abstract

The physico-mathematics that emerged at the beginning of the seventeenth century entailed the quantitative analysis of the physical nature with optics, meteorology and hydrostatics as its main subjects. Rather than considering physico-mathematics as the mathematization of natural philosophy, it can be characterized it as the physicalization of mathematics, in particular the subordinate mixed mathematics. Such transformation of mixed mathematics was a process in which physico-mathematics became liberated from Aristotelian constraints. This new approach to natural philosophy was strongly influenced by Jesuit writings and experimental practices. In this paper we will look at the strategies in which models were selected from the mixed sciences, engineering and technology adequate for an analysis of the specific phenomena under investigation. We will discuss Descartes’ analysis of the rainbow in the eight discourse of his Meteorology as an example of carefully selected models for physico-mathematical reasoning. We will further demonstrate that these models were readily available from Jesuit education and literature.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    [6], I, 244: “Dicit tamen se nunquam neminem reperisse, praeter me, qui hoc modo, quo ego gaudeo, studendi utatur accuratèque cum Mathematicâ, Physicam jungat. Neque etiam ego, praeter illum, nemini locutus sum hujusmodi studij”.

  2. 2.

    From a letter from Descartes to Vatier, February 1638, [8], I, pp. 559–660: “Nor could I show the use of that method in the three treatises that I included, since it prescribes an order of investigation which is different enough from the order I believed I must use in order to explain them. However, I have given a sample of it in describing the rainbow, and if you take the trouble to reread it, I hope it will satisfy you better that it did the first time”. Translation from [10], p. 86.

  3. 3.

    For a forthcoming critical edition of the English translation of this book see [15].

  4. 4.

    SP, Geometry, prop. XII, p. 8: “Potest unum & idem corpus solidum ac durum, per tria foramina transmitti, quorum unum sit rotundum, secundum quadratum, tertium ovale, ita ut singula in transitu compleat; potest aliud eandem conditione transmitti per rotundum, ovale & quadr angulare quantaelibet longitudinis: aliud, per rotundum & quandrangulare; alius per rotundum & triangulare duntaxat”.

  5. 5.

    SP, Optics, prop. XX, p. 35: “Refractio et reflexio, varia coelestibus ac sublunaribus pingit phasmata, halones, parelia, paraselenias, irides in nubibus, in crystallo trigonam, in ampulis vitreus, in polyoptris, in aquam fontium arcuaram et irroratam, in lucernic ardentibus. Ubi certum videtur, colores hic apparentes, causam habere congenerem, puta per mixtionem refractae lucis cum refrigente subiecto; figures autem diversas, incidentium, refractorum simul et reflexorum radiorum legibus gubernari. Probabile est, geminari posse lunarem iridem, sicut et solarem, sic tamen, ut secundaria iris non sit imago alterius ex reflexione producta”.

  6. 6.

    Compare Descartes, [8], VI, p. 343: “qui pourroient causer grande admiration a ceux qui en ignoreroient les raisons”, with one of several similar formulations in RM: “avec grand estonnement des assistans particulierement s’ils en ignorent la cause” (1630, p. 73).

  7. 7.

    From the English edition: 1633, p. 68: “But to have one more stable and permanent in his colours. Take a glass full of water, and expose it to the sun, so that the rays that pass through strike upon a shadowed place, you will have pleasure to see the fine form of a rainbow by this reflection”.

  8. 8.

    SP, Geometry, XV: “Mathematicae iucunditatis est, in sphaerulis eburneis, aut buxeis, et similibus, ita ludere, ut reflecionem legibus, una in planum, aut duo vel plura plana impingens, ad destinatum locum emittatur: vel ita una sphaerula, caeteras impellens, ad quamcunque volveris partem dirigat, assignatis etiam, si lubet, variorum motuum lineis, variorum contactuum et allisionum punctis, sive in plano, sive per ambitu[m] circuli”.

  9. 9.

    From the English edition which does not mention billiard but tennis and trap-ball. This reference to the game of trap-ball predates the earliest entry in the Oxford English Dictionary (1658). The figure is taken from the author’s copy of the 1672 Lyon edition. Johannes Marcus expands on these rules in a later treatise of 1636 (see Fig. 5). The earliest depiction of tennis is shown in Fig. 6.

  10. 10.

    Philosophical Transactions of the Royal Society, No. 80 (19 Feb. 1671/2), pp. 3075–3087.

  11. 11.

    Philosophical Transactions of the Royal Society, No. 80 (19 Feb. 1671/2), pp. 3075–3087.

  12. 12.

    [6], III, 237: “Iris oculi est humor corneae concavo adhaerens a parte sui pupillae, limbum (ubi lux dissolvens humores, est debilior quam in medio) inficiens, id est tegens. Hinc sequitur iridem circa candelam, aut quodvis lumen pupilla majus, visam, eo videri majorem quo lumen id est ab oculo remotius”.

References

  1. Armogathe, J.-R.: The rainbow: a privileged epistemological model. In: Gaukroger, S., Schuster, J., Sutton, J. (eds.) Descartes’ Natural Philosophy, pp. 249–257. London, Routledge (2000)

    Google Scholar 

  2. Boyer, C.: The Rainbow. From Myth to Mathematics. Sagamore Press, New York (1959)

    Google Scholar 

  3. Buchwald, J.: Descartes’s experimental journey past the prism and through the invisible world to the rainbow. Ann. Sci. 65(1), 1–46

    Google Scholar 

  4. Cusanus.: De ludo globi, 1463, (English translation by Jasper Hopkins), Metaphysical Speculations, vol. 2. The Arthur J. Banning Press, Minneapolis (2000)

    Google Scholar 

  5. Dear, P.: Discipline and Experience: The Mathematical Way in the Scientific Revolution. Chicago University Press, Chicago (1995)

    Book  Google Scholar 

  6. de Waard, C., (ed.): Journal tenu par Isaac Beeckman de 1604 à 1634. Tome 1: 1604–1619, Martinus Nijhoff, Den Haag (1939)

    Google Scholar 

  7. Mersenne, M. Correspondance du P. Marin Mersenne, Religieux Minime, publiée et annotée par Cornelis de Waard; avec la collaboration de Armand Beaulieu; édition entreprise sur l’initiative de madame Paul Tannery et continuée par le C.N.R.S., Editions du Centre national de la recherche scientifique (1932–1988), 18 vols

    Google Scholar 

  8. Descartes, R.: Oeuvres de Descartes (11 vols). In: Adam, C., Tannery, P. (eds.) Librairie Philosophique, Paris. J. Vrin (1983)

    Google Scholar 

  9. Descartes, R.: Discourse on Method, Optics, Geometry and Meteorology, P. Olscamp (tr.), Hackett, Indianapolis (2001)

    Google Scholar 

  10. Descartes, R.: Philosophical Essays and Correspondence. Ariew R. (tr.), Hacket, Indianapolis (2000)

    Google Scholar 

  11. Descartes R.: The Philosophical Writings of Descartes, 3 vols., Cottingham, J., R. Stoothoff, D. Murdoch, (tr.) Cambridge University Press, Cambridge (1991)

    Google Scholar 

  12. Galison, P.: Descartes’s comparisons: from the invisible to the visible. Isis 75, 311–326 (1984)

    Article  Google Scholar 

  13. Giere, R.: Explaining Science: A Cognitive Approach. University of Chicago Press, Chicago (1988)

    Book  Google Scholar 

  14. Heeffer, A.: The logic of disguise: descartes’ discovery of the Sine law, Historia scientiarum. Int. J. Hist. Sci. Soc. Jpn 16(2), 144–165 (2006). November

    Google Scholar 

  15. Heeffer, A.: Wonder to those that are ignorant in the cause, A critical English edition of Récréations Mathématiques (1624), with a glossary and commentaries. Springer, Heidelberg (2012)

    Google Scholar 

  16. Hesse, M.: Models and Analogies in Science. Sheet and Ward, London (1963)

    Google Scholar 

  17. Leurechon, J.: Selectae propositiones, Sébastien Carmoisy, Pont-à-Mousson (1622)

    Google Scholar 

  18. Mac Lean, J.: De historische ontwikkeling der stootwetten van Aristoteles tot Huyghens. Ph.D. dissertation, Free University of Amsterdam, Van Sijn en zonen, Rotterdam (1959)

    Google Scholar 

  19. Magnani, L., Nersessian, N.J., Thagard, P. (eds.): Model-Based Reasoning in Scientific Discovery. Kluwer, New York (1999)

    Google Scholar 

  20. Lorenzo Magnani, L., Nersessian, N.J. (eds.): Model Based Reasoning, Science, Technology, Values. Springer, New York (2002)

    Google Scholar 

  21. Meheus, J., Nickles, T. (eds.): Models of Discovery and Creativity. Springer, New York (2009)

    Google Scholar 

  22. Sabra, A.I.: Theories of Light from Descartes to Newton. Cambridge University Press, Cambridge (1981)

    Google Scholar 

  23. Scaino, A.: Trattato del giuoco della palla, Venice: Gabriele Giolito de’Ferrari and brothers, 1555, (Modern editions: a cura di Giorgio Nonni, Urbino Quattroventi 2000. Scaino on tennis, Translated into english by W.W. Kershaw at the instance of C.B. Gabriel, Honorary Secretary and Treasurer of the Royal Tennis Court 1932–1947; with acknowledgements, corrigenda and historical notes by P.A. Negretti. [London]: Strangeways Press, 1951)

    Google Scholar 

  24. Shea, W.R.: The Magic of Numbers and Motion. The Scientific Career of René Descartes, Canton (1991)

    Google Scholar 

  25. Schuster, J.R.: Physico-mathematics and the Search for Causes in Descartes’ Optics—1619–1637. Synthese 185(3), 467–499 (2012)

    Article  Google Scholar 

  26. Simpson, G.C.: Ocular haloes and coronas. Br. J. Ophtalmol. 37, 450–486 (1953)

    Article  Google Scholar 

  27. Westfall, R.S.: The development of Newton’s theory of color. Isis 53, 339–580 (1962)

    Article  Google Scholar 

  28. Vaccari, A.: Legitimating the machine: the epistemological foundation of technological metaphor in the natural philosophy of René Descartes. In: Claus, Z., Romano, N. (eds.) Philosophies of Technology: Francis Bacon and His Contemporaries (2 vols.). Bril, Leiden, pp. 287–336

    Google Scholar 

  29. Van Dyck, M.: Argumentandi modus huius scientiae maximè proprius’. Guidobaldo’s mechanics and the question of mathematical principles. In: Gamba, E., Becchi, A., Bertoloni Meli, D. (eds.) Mathematiche e tecnica da Urbino all’Europa. Max Planck Research Library for the History and Development of Knowledge, Berlin (2013)

    Google Scholar 

  30. van Etten, H.: Recreation Mathematique. Jean Appier Hanzelet, Pont-à-Mousson (1624)

    Google Scholar 

Download references

Acknowledgments

The author is a research fellow of the Research Foundation Flanders (FWO Vlaanderen) and member of the Centre for History of Science at Ghent University, Belgium. The paper benefited from discussions with Delphine Bellis, Madelina Giurgea and Maarten Van Dyck.

Author information

Authors and Affiliations

  1. Research Foundation Flanders (FWO Vlaanderen), Ghent University, Gent, Belgium

    Albrecht Heeffer

Authors
  1. Albrecht Heeffer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albrecht Heeffer .

Editor information

Editors and Affiliations

  1. Department of Humanities Philosophy Section, University of Pavia, Pavia, Italy

    Lorenzo Magnani

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Heeffer, A. (2014). Identifying Adequate Models in Physico-Mathematics: Descartes’ Analysis of the Rainbow. In: Magnani, L. (eds) Model-Based Reasoning in Science and Technology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37428-9_23

Download citation

Publish with us

Policies and ethics

Search

Navigation