Skip to main content
SpringerLink
Log in

Using Invariances in Geometrical Diagrams: Della Porta, Kepler and Descartes on Refraction

  • Chapter
  • First Online:
The Optics of Giambattista Della Porta (ca. 1535–1615): A Reassessment

Part of the book series: Archimedes ((ARIM,volume 44))

Abstract

In this paper, I will demonstrate how geometrical diagrams on refraction were instrumental in the discovery of the sine law of refraction. In particular, I will show how a specific diagram in the Paralipomena assisted Kepler in looking for invariances of proportions under different angles of incidence. Eventually, Kepler failed in finding a quantitative law of refraction, but it will be shown that his basic hypothesis and methodology can lead to the discovery of a quantitative law and that probably this was Descartes’ path to the discovery of the sine law. Both Kepler and Descartes could build on a tradition of geometrical reasoning which accounted for co-exact properties in geometrical diagrams. Della Porta was the first to recognize such properties in diagrams dealing with refraction.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
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.

    For a recent survey on diagrams in the philosophy of mathematical practice see Giardino (2017).

  2. 2.

    Kirsh and Maglio (1994).

  3. 3.

    Mumma (2010).

  4. 4.

    Manders (2008).

  5. 5.

    Smith (1996).

  6. 6.

    Mach (1926: 33).

  7. 7.

    Heeffer (2006).

  8. 8.

    AT (VI, 76, p. 61).

  9. 9.

    From a letter from Descartes to Vatier, February 1638 (AT, I, 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 than it did the first time”. Translation from Ariew (2000, 86).

  10. 10.

    Kramer (1882).

  11. 11.

    Milhaud (1921), Buchdahl (1969), Malet (1990).

  12. 12.

    Sabra (1967).

  13. 13.

    Sabra (1967: 113).

  14. 14.

    Shea (1991).

  15. 15.

    Smith (1996: 46).

  16. 16.

    Schuster (1978).

  17. 17.

    Schuster (2013).

  18. 18.

    Schuster (2013: 160).

  19. 19.

    Kepler (1604).

  20. 20.

    Mersenne (1636).

  21. 21.

    Hérigone (1634–1644).

  22. 22.

    De Waard (1939, III: 97).

  23. 23.

    CM (I, 404).

  24. 24.

    Schuster (2013: 187–8).

  25. 25.

    Mydorge (1631), more particularly book I, prop. 49, 51, concerning proposition II, and “now all this is done, it will be easy to describe the hyperbole,” in proposition III, as he explained in book II, prop. 20, 26. Schuster does not mention the book in this context.

  26. 26.

    The text by Mydorge is contained in a rough draft by Mersenne of an unpublished treatise on optics, kept as manuscript Fr. 5176 at the BNF in Paris. The manuscript is 30 pages long and is an unorganized and unnumbered collection of notes, bearing no resemblance to a published work by Mersenne. Lenoble (1957, 239) considers it a sketch to an abandoned continuation of the treatise on sound in the Quaestiones in Genesim (BNF. Lat. 17,261 and 17,262) although some of its propositions also appear in Mersenne’s treatise on Optics and Catoptrics included in the second edition of Niceron’s book on Perspective published in 1651.

  27. 27.

    Mydorge (1631: 157–8).

  28. 28.

    Schuster (2013: 186).

  29. 29.

    CM (I, 404).

  30. 30.

    Schuster (2013, 188).

  31. 31.

    Mydorge (1631).

  32. 32.

    Mersenne (1636, prop. 29: 65–6).

  33. 33.

    De Waard, (1939, III: 97).

  34. 34.

    This was pointed out to me by Yaakov Zik.

  35. 35.

    Kepler 1604, Ch. 4, § 2, Frish, 1859 Bd II, p. 182: “Cum ergo densitas plane sit in causa refractionum, et refractio ipsa compressio quaedam videtur lucis, utpote ad perpendicularem, subiit animum inquirere, an quae proportio mediorum causa densitatis eadem sit proportio fundi spatiorum, quae lux primum in vacuum vas, dein aqua superfusa, introgressa feriat.” Translation by Donahue 2000, p. 102.

  36. 36.

    Heeffer (2003).

  37. 37.

    De Waard (1939, III: 97, translation by Malet (1990), emphasis mine. Also compare with Fig. 7.7.

  38. 38.

    Malet (1990).

  39. 39.

    Schuster (2013: 200–203).

  40. 40.

    Kepler (1604, prop. 20: 17–21).

  41. 41.

    Kepler (1604: 18), translation by Donahue (2000).

  42. 42.

    Schuster (2013: 186).

  43. 43.

    The manuscript BNF NAF 5176, from which De Waard fabricated the “letter” from Mydorge to Mersenne contains at least two additional propositions which should have been included in his transcription. An interesting proposition (unnumbered, f. 28r) also concerns an application of the sine law to lens theory, in particular the focusing of parallel rays into a single point by use of a plano-convex (be it spherical or hyperbolic) lens. Descartes uses such a lens in his seventh discourse (AT, VI, 161).

  44. 44.

    Della Porta (1593, Bk. 1, prop. 8).

  45. 45.

    Netz (1999) and Manders (2008).

References

  • Ariew, R. (tr.) 2000. René Descartes. Philosophical essays and correspondence, Indianapolis: Hacket Publishing.

    Google Scholar 

  • Buchdahl, G. 1969. Metaphysics and the Philosophy of Science. Cambridge MA: MIT Press.

    Google Scholar 

  • Della Porta, G. 1593. De refractione optices parte: libri novem. Neapoli: Ex officina Horatii Salviani, apud Jo. Jacobum Carlinum, & Antonium Pacem.

    Google Scholar 

  • Descartes, R. 1897-1913. Oeuvres De Descartes, 12 vols., eds. by Charles Adam and Paul Tannery, Paris: Librairie Philosophique J. Vrin, 1983 [AT].

    Google Scholar 

  • Donahue, W. 2000. Kepler’s Optics. Santa Fe: Green Lion Press.

    Google Scholar 

  • Frish, C. 1858. Joannis Kepleri Astronomi opera omnia. Francofurti: A.M., Heyder & Zimmer.

    Google Scholar 

  • Giardino, V. 2017. Diagrammatic reasoning in Mathematics. In Springer Handbook of Model Based Science, eds. Lorenzo Magnani & Tommaso Bertolotti, New York: Springer (forthcoming).

    Google Scholar 

  • Heeffer, A. 2003. Kepler’s near discovery of the sine law: A qualitative computational model”, In Computer Modeling of Scientific Reasoning, eds. Claudio Delrieux & Javier Legris, Bahia Blanca, Argentinia: Universidad Nacional Del Sur: EDIUNS, 93–102.

    Google Scholar 

  • ———. 2006. The Logic of Disguise: Descartes’ Discovery of the Sine Law. Historia Scientiarum 16(2): 144–165.

    Google Scholar 

  • Hérigone, P. 1634. Cursus mathematicus: nova, brevi, et clara methodo demonstratus ... Cours mathematique, demonstré d’une nouvelle, briefve, et claire methode. Paris: Chez Henri Le Gras.

    Google Scholar 

  • Kepler, J. 1604. Ad Vitellionem paralipomena quibus Astronomiæ pars optica traditvr: potissimum de artificiosa observatione et æstimatione diametrorvm deliquioruḿ[que] solis et lunæ, cvm exemplis insignivm eclipsivm : Habes hoc libro, lector, inter alia multa noua, tractatum luculentum de modo visionis, & humorum oculi vsu, contra opticos & anatomicos. Francofvrti: Apud Claudium Marnium & Hæredes Ioannis Aubrii.

    Google Scholar 

  • Kirsh, D., and P. Maglio. 1994. On Distinguishing Epistemic from Pragmatic Action. Cognitive Science 18: 513–549.

    Article  Google Scholar 

  • Kramer, P. 1882. Descartes und das Brechungsgesetz des Lichtes. Abhandlungen zur Geschichte der Mathematik 4: 233–278.

    Google Scholar 

  • Mach, E. 1926. The principles of physical optics: an historical and philosophical treatment. New York: Dover.

    Google Scholar 

  • Malet, A. 1990. Gregorie, Descartes, Kepler, and the law of refraction. Archives Internationales D’histoire Des Sciences. 40: 278–304.

    Google Scholar 

  • Manders, K. 2008. The Euclidean diagram. In Philosophy of Mathematical Practice, ed. Paolo Mancosu, 112–183. Oxford: Clarendon Press.

    Google Scholar 

  • Mersenne, M. 1636. Harmonie universelle contenant la théorie et la pratique de la musique: où il est traité de la nature des sons, & des mouuemens, des consonances, des dissonances, des genres, des modes, de la composition, de la voix, des chants, & de toutes sortes d’instrumens harmoniques. A Paris: Chez Sebastien Cramoisy.

    Google Scholar 

  • Milhaud, G. 1921. Descartes savant. Paris: Felix Alcan.

    Google Scholar 

  • Mumma, J. 2010. Proofs, Pictures and Euclid. Synthese 175(2): 255–287.

    Article  Google Scholar 

  • Mydorge, C. 1630. Examen du livre des Recreations mathematiques et de ses problemes en geometrie, mechanique, optique, & catoptrique. Où sont aussi discutées & restablies plusieurs experiences physiques y proposées. Par Claude Mydorge, escuyer, sieur de La Maillarde, conseiller du Roy, & tresorier general de France en Picardie. A Paris: chez Rolet Boutonné, au Palais, en la premiere boutique de la gallerie des prisonniers.

    Google Scholar 

  • Mydorge, C. 1631. Clavdii Mydorgii patricii Parisini Prodromi catoptricorvm et dioptricorvm: sive, Conicorvm operis ad abdita radii reflexi et refracti mysteria praeuij & facem praeferentis: libri primvs et secvndvs : D.A.L.G. Parisiis: Ex Typographia I. Dedin.

    Google Scholar 

  • Netz, R. 1999. The shaping of deduction in Greek mathematics: a study of cognitive history. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Sabra, A.I. 1967. Theories of light from Descartes to Newton. London: Oldbourne.

    Google Scholar 

  • Schuster, J.A. 1978. Descartes and the Scientific Revolution 1618–1634: An Interpretalion. PhD thesis Princeton University. Ann Arbor. Mi: University Microfilms International.

    Google Scholar 

  • ———. 2013. Descartes-Agonistes: Physico-Mathematics, Method and Corpuscular-Mechanism. Studies in History and Philosophy of Science. New York: Springer.

    Book  Google Scholar 

  • Shea, W.R. 1991. The magic of numbers and motion: the scientific career of René Descartes. Canton, MA: Science History Publications, U.S.A.

    Google Scholar 

  • Smith, A.M. 1996. Ptolemy’s theory of visual perception: an English translation of the Optics. Philadelphia: American Philosophical Society.

    Google Scholar 

  • Waard, C. De. 1939. Journal tenu par Isaac Beeckman de 1604 à 1634. La Haye: M. Nijhoff.

    Google Scholar 

  • ———, 1986. Correspondance du P. Marin Mersenne, religieux minime. Paris: Éditions du Centre national de la recherche scientifique, 16 vols [CM].

    Google Scholar 

Download references

Acknowledgments

This research paper resulted from the research project 3G002713 funded by the Flemish Research Foundation (FWO Vlaanderen), titled A Study on Material Models from Engineering and Technology used in Discovery, Explanation and Negotiation in Early-Modern Philosophical Debates. The paper benefitted from critical comments and suggestions by Maarten Van Dyck, Boris Demarest, Arianna Borrelli, A. Mark Smith, Yaakov Zik, John Schuster and proofreading by Saskia Willaert.

Author information

Authors and Affiliations

  1. Ghent University, Ghent, 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. History of Science, Technische Universität Berlin, Berlin, Germany

    Arianna Borrelli

  2. Department of Philosophy, University of Haifa, Haifa, Israel

    Giora Hon

  3. Department of Philosophy, University of Haifa, Haifa, Israel

    Yaakov Zik

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Heeffer, A. (2017). Using Invariances in Geometrical Diagrams: Della Porta, Kepler and Descartes on Refraction. In: Borrelli, A., Hon, G., Zik, Y. (eds) The Optics of Giambattista Della Porta (ca. 1535–1615): A Reassessment. Archimedes, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-50215-1_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50215-1_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50214-4

  • Online ISBN: 978-3-319-50215-1

  • eBook Packages: HistoryHistory (R0)

Publish with us

Policies and ethics

Search

Navigation