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Optical Diagrams as “Paper Tools”: Della Porta’s Analysis of Biconvex Lenses from De refractione to De telescopio

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

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

Abstract

In the last decades, the epistemic relevance of mediation and representation strategies in the construction of scientific knowledge has been demonstrated by a large number of studies. Words, symbols, formulas or diagrams on a page provide an essential and epistemically independent means to connect, reflect and expand instrumental and laboratory experience. Historian of science Ursula Klein has introduced the term "paper tool" to describe this kind of function in the case of early chemical formulas, and in the present contribution I will argue that optical ray-tracing diagrams acted as "paper tools" in Della Porta's optical writings. Thanks to diagrammatic practices, Della Porta was able to extend to lenses the connection between light focusing properties and visually perceivable effects that had been recently established for convex mirrors. At the core of the connection stood the ambiguous concept of "point of inversion", and in the present paper we shall follow Della Porta's innovative attempts of adapting it from reflecting to refracting systems using diagrams as paper tools.

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Notes

  1. 1.

    Dupré 2000, 2003, 2005, Ilardi 2007, Lindberg 1984, Smith 2015, 322–372.

  2. 2.

    On Della Porta’s optical works see: Borrelli 2014, Lindberg 1984, Smith 2015, 344–349, Zik and Hon 2012 and references therein, as well as Zik and Hon’s contribution to the present volume (Chap. 3). On his life, work and methodology see Borrelli 2011, Piccari 2007 and references therein.

  3. 3.

    Borrelli 2011, 2014.

  4. 4.

    Heeffer 2017.

  5. 5.

    See for example: Coopmans et al. 2014, Galison 1997, Gooding 1990, Kaiser 2005, Klein 2003.

  6. 6.

    Klein 2003.

  7. 7.

    I made use of the following editions of these works: Della Porta 1560, Della Porta 1597, Della Porta 1593, Della Porta 1962. The manuscript of DT is now in the Library of the Accademia dei Lincei in Rome and is dated to the last years of Della Porta’s life (Della Porta 1610–1615)

  8. 8.

    Borrelli 2014, Duprè 2006, Smith 2015, 280–391, Zik and Hon 2012.

  9. 9.

    I was not able to find any contemporary manual containing a geometrical-optical diagram illustrating how such images are formed.

  10. 10.

    Bertoloni Meli 2006.

  11. 11.

    Borrelli 2014.

  12. 12.

    Dupré 2006, 33–38. Dupré’s analysis is partly biased by an incorrect reading of Della Porta’s text and drawings, which was in turn largely based on the equally incorrect interpretation of the same passages given by David Lindberg (1964). See Borrelli 2014 for a detailed discussion of this issue.

  13. 13.

    The function of optical diagrams in those two authors are discussed in Heeffer’s (Chap. 7) and Bellé’s (Chap. 8) contributions to the present volume.

  14. 14.

    Klein 2003, 245–46.

  15. 15.

    Klein 2003, 188.

  16. 16.

    Georgescu 2014, Heeffer 2017, Kaiser 2005, Mumma and Panza 2011, Wright 2014.

  17. 17.

    Kaiser 2005.

  18. 18.

    Bertoloni Meli 2006, Georgescu 2014, quotes from Georgescu 2014, 18, 24.

  19. 19.

    Borrelli 2014, Smith 2015, 344–349, Zik and Hon 2012.

  20. 20.

    Zik and Hon 2017.

  21. 21.

    Smith 2015, 59–62.

  22. 22.

    Della Porta’s cathetus line for the glass sphere had the same path as a ray emanating from the object and passing through the centre of the sphere, which therefore remains unrefracted. The refracted ray, as already noted, was drawn according to a qualitatively plausible rule, so that the intersection of ray and cathetus actually provided a good approximation of where an image would be seen, as I have discussed in detail in (Borrelli 2014).

  23. 23.

    “in specillis convexis inversionis punctum invenire” (Della Porta 1593, 175–176).

  24. 24.

    “punctum concursus refractarum linearum” (Della Porta 1593, 175).

  25. 25.

    In the title of the proposition the “centre” is indicated as “centre of the eye” (“centrum oculis”, Della Porta 1593, 181), but in the description it is called “centrum circuli” and the diagram leaves no doubt that the title was a misprint.

  26. 26.

    Della Porta [1610–1615], Della Porta 1962.

  27. 27.

    Naldoni 1962, Ronchi 1962.

  28. 28.

    Naldoni 1946, 1962, Ronchi 1962.

  29. 29.

    Saito 2011, see also Saito 2008.

  30. 30.

    Saito 2011, 219–224.

  31. 31.

    “Ogni copia e’ un tentativo diverso. Ogni volta il Porta ricomincia da capo, modificando completamente, anche addirittura capovolgendo i punti su cui basa la sua teoria. Ed è inutile cercare in tutto questo una linea di progresso: è un brancolare cieco in cerca di un qualche sistema con cui poter convalidare le esperienze, ma senza trovare una via di uscita. Ed è naturale perchè siamo fuori strada” (Naldoni 1946, 67).

  32. 32.

    I have discussed these issue, and particularly Lindberg’s bias in (Borrelli 2014).

  33. 33.

    Galison 1997, Klein 2003, Pickering 1995.

  34. 34.

    “Ex his experientiis ducti indicandum inversionis punctum est superius inferioris circuli per semidiametrum existere” (Della Porta 1962, 113). Naldoni read “superior” instead of superius, but I believe that both the content and the writing in the manuscript support my reading.

  35. 35.

    Borrelli 2015, Steinle 2016.

  36. 36.

    “Magnitudine in specillis utrinque convexis extra inversionis punctum constituta imago minor et inversa, in circumferentia inferioris circuli aequalis et conversa, infra centrum recta et maior” (Della Porta 1962, 122).

  37. 37.

    “Centrum in specillis utrinque convexis investigare” (Della Porta 1962, 128).

  38. 38.

    “Cathetus ex inferioris vel superioris circuli centro ducenda, prius discutendum est quod ratione et esperientia tentandum est” (Della Porta 1962, 128).

  39. 39.

    As usual, the path of the refracted ray was derived in principle according to the refraction rule for the sphere, although here, too, the actual drawings were very approximate.

  40. 40.

    “Nam A in I, C in F conspicabitur, ubi experientia cognoscimus rectam semper videri oculo propinquo, quod est contrarium non igitur superius centrum inferioris portionis attendatur” (Della Porta 1962, 129).

  41. 41.

    Of course Della Porta knew that the lens had two symmetrical points of inversion, but in DT 3C he only used the one towards the eye, because he classified the effects according to whether the eye was above or below the point of inversion. In DT 3B, he had done the opposite, classifying all effects according to the position of the object.

  42. 42.

    Naldoni’s trascription of the title of DT 3C 6 states that the eye is above (“supra”) the point of inversion (Della Porta 1962, 132), but text and image clearly refer to the case in which the eye is below it. A comparison to the manuscript shows that the word Naldoni read as “supra” is in fact “infra” (Della Porta [1610] f. 144r).

  43. 43.

    Despite all efforts, there was here still a contradiction between DT 3C 3 and 6, since it is not clear how the cathetus crossing was avoided in DT 3C 3 when the eye is below the point of inversion. The relevant image is unhelpful and we may regard this as a point on which Della Porta needed to work further: he had now realized how the empirical situation was (i.e. the image is not always inverted if the object is above the point of inversion), and was looking for rules allowing him to make sense also of this result.

  44. 44.

    “unde oculus quae dextra videbit sinistra et quae sinistra dextra et quae infra erunt supra spectabuntur” (Della Porta 1962, 132).

  45. 45.

    “Quibus experientiis specillorum operationes indagandea sint” (Della Porta 1962, 102).

  46. 46.

    “Nunc quoque quomodo indaganda sunt inversionis puncta ex specillorum apparentiis, graphice appositeque inspectorum facies describamus ut ex horum disquisitione discriminum variantium figurarum aucupemur” (Della Porta 1962, 102).

  47. 47.

    “Hanc operam supervacaneam non iudicabis, nec opus reiterare pigeat, non festinando, sed sensim et sedulo, ex hac disquisitione facilitas dependent” (Della Porta 1962, 103).

  48. 48.

    “Oportet igitur manu bilentem capiamus, ac prius eam supra oculum opponamus, ut magnitudinem longe collocatam, ita ut est, recta, clara et paulo maior videatur. Mox specillum paulatim ab oculo recedat et recessum paulo ampliorem magnitudinem repraesentabit, et sensim eundo, ita turgidiour fiet, ut vix specillo complecti possit, post, tabescentie forma, pervertitur et demutatur ut specillum tenebris obducatur, appulsu e refractorum radiorum multitudine in ipso inversionis centro. Paulatim deinde specillum elongando manibus, ut inter visum et rem visam medium interiaceat, evanescunt tenebrae et magnitudo conversa, in angustum orbem collecta et quae ubiubi clarior videbitur. Et antrorsum iterum manus elongando crassescit moles, usque donec iterum totum specillum occuparit. Denuo pellitur imago, et transvolat, et tenebrae glomerantur et conglobantur in puncto inversionis: nam fieri non potest de recto ad inversum transmigratio, nisi per medium. Demum ulterius procedendo pervenit in vicinia magnitudinis, ubi clara, aequalia recta ut erat prope oculum conspicietur” (Della Porta 1962, 103).

  49. 49.

    “His perceptis, age ulterius procedamus, et illustremus exemplo, ut quae verba iam recensita sunt, lineamentorum ductu clariora evadant. Esto visenda magnitudo A e regione oculi quam M litera indicabit: specillum notulis obsignabimus K N, quod supra oculum constitutur. Hinc magnitudo, quam A notula obsignavimus, ut est suo loco recta, aequalis et nitida conspicietur. Mox specillum ulterius, sed non longe removes, videbis magnitudinem incrementum suscipere, et ubi maiorem concepisse magnitudinem comperies, ut specillum maiorem concipere non valeat, tota oculorum acies tenebris offundetur, quod eveniet in ipso inversionis puncto, locum H I notulis decorabimus. Dein ultrius procedendo, moles adhuc expanditur, et cum inter utrumque medium magnitudinis et oculi venerit FG convertitur imago, contrahitur et ubivis clarior cernitur. Abeat longius specillum, et molem laxiorem videbis, usque donec imago specillum tenebris obumbrabit, locum DE notulis configurabimus, quod in puncto inversionis superiori constituitur. Tandem ulterius abeundo angustior fiet magnitudo, et magis inhaerendo, et clara, recta, et ut est conspicabitur.” (Della Porta 1962, 103).

  50. 50.

    “Quia catheti ex centro exeuntes semper lineas formas deferentes transversae decussant, ut dextera pars in sinistram vergat, et sinistra in dexteram, ut magnitudinis imaginem conversam referant” (Della Porta 1962, 106).

  51. 51.

    “In convexis utrinque specillis, oculo sub inversionis puncto statuto magnitudo transversa et pendula in aere conspicabitur” (Della Porta 1962, 108).

  52. 52.

    “in aere pendulam et trasversa extra specillum”, “at si oculus altius subeet in FG suo loco transversa videbitur ut est” (Della Porta 1962, 109).

  53. 53.

    Della Porta 1962, 123 (3B) and 135 (3C).

  54. 54.

    Frangsmyr et al. 1990, Garber 1999, Wise 1997.

  55. 55.

    Lynch 1988.

  56. 56.

    Lynch 1988, 201.

  57. 57.

    Lynch 1988, 211.

  58. 58.

    Lynch 1988, 217–218.

  59. 59.

    “La phase initiale de recherche d’Ampère [...] constitue un example tout différent de processus de mathématisation. Son but n’était pas de mathématiser, mais de formuler les régularités de processus spatiaux. Dans la mesure où il n’y avait pas de concept à disposition qui puisse servire à cette fin, il dut developper lui-même des concepts mixtes, géometrico-physiques, qui furent ensuite généralises de manière ancore plus abstraite en géométrie pure. [...] L’invention du concept de ligne de force per Faraday constitue un cas ultèrieur de type similaire dans lequel la recherche de régularités [...] mena à l’élaboration d’un concept qui devint par la suite un outil mathématique abstrait” (Steinle 2011, 83–84).

  60. 60.

    Heeffer 2017.

  61. 61.

    Steinle 2016.

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Acknowledgements

The author wishes to acknowledge the support of the DFG through the projects BO 4062/2-1 (TU Berlin, “Early particle physics”) and KFOR 1927 (Institude for Advances Studies on Media Cultures of Computer Simulation (MECS), Leuphana Universität Lüneburg).

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

    Arianna Borrelli

  2. Institute for Advanced Study on “Media Cultures of Computer Simulation (MECS)”, Leuphana University, Lüneburg, Germany

    Arianna Borrelli

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  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

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Borrelli, A. (2017). Optical Diagrams as “Paper Tools”: Della Porta’s Analysis of Biconvex Lenses from De refractione to De telescopio . 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_4

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