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Francesco Maurolico, Giambattista Della Porta and Their Theories on Refraction

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

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

Abstract

For a reassessment of Della Porta’s optics a broad view is needed. For a better understanding of the similarities and the peculiarities of Della Porta’s approach in comparison with other authors of the same period, we are going to deal with Maurolico as one of the best candidates.

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Notes

  1. 1.

    One of the most influential commentaries on Euclid of the period was (Clavius 1574), but already Campanus maintained that proportion is found not only in quantities (geometrical or numerical), but also in weights, forces and sounds: “Non enim solum in quantitatibus reperitur proportio, sed in ponderibus, potentiis et sonis” (Campanus 1546, p. 103). Quite representative of this spirit was the work by Cardanus Opus novum de proportionibus numerorum, motuum, ponderum, sonorum aliarumque rerum mensurandarum (1570). On the model of specific gravity see (Napolitani 1988). For Galilei, whose model is found in the short text De motu equabili (1609–1612), see (Giusti 1986) and (Giusti 1993). For Maurolico’s statics see (Giusti 2001). Maurolico’s refraction law will be treated in this contribution.

  2. 2.

    “Quantitates quae dicuntur esse secundum proportionem unam, prima ad secundam et tertia ad quartam, sunt quarum primae et tertiae multiplices aequales, multiplicibus secundae et quartae aequalibus fuerint similes, vel additione, vel diminutione, vel aequalitate eodem ordine sumptae”, def. 6, bk. V, (Campanus 1546, p. 105). “Magnitudes are said to be in proportion, the first to the second and the third to the fourth when the equimultiples of the first and the third are, according the to decrease or the increase or the equality, similar to the equimultiples of the second and the fourth taken in the corresponding order”. Clavius wrote that Campanus’ interpretation is absurd because it is simply a petitio principii: “si ita intelligatur definitio, Euclidem idem per idem definire” (Clavius 1574, p. 156).

  3. 3.

    “Nam universa Elementorum Euclidis volumina et eiusdem Optica et Catoptrica …nullo praeeunte praeceptore, per memetipsum intellexi. Alii due [scil. libelli adiicientur] de Photismis unus, alter de Diaphanis. In illo, praeter caetera, patescet cur solaris radius per qualecumque foramen transmissus in circularem redigatur formam; in hoc ratio rotunditatis et colorum iridis aperietur. Quorum utrunque fuit Ioanni vulgatae Perspectivae authori incognitum”.

  4. 4.

    For reference and more information on the complete Maurolician Corpus in optics see (Bellé 2006a).

  5. 5.

    Maurolico’s works on optics are extant in four manuscripts and two printed editions. Only the list of manuscripts and printed editions is give here, see (Bellé 2006a) for detailed information. Paris, Bibliothèque Nationale, Lat. 7249, is an autograph manuscript containing only Diaphana; Lucca, Biblioteca Statale, ms. 2080, is a copy of the seventeenth century containing all the works except Problemata. Problemata are found in Fondo Curia 2052, Archivio della Pontificia Università Gregoriana, a manuscript written by C. Clavius. Finally, Hamburg Staats und Universitätsbibliothek, Cod. Math. 483 is a later copy which includes only the enunciation of theorems. The first printed edition is Photismi de lumine et umbra, Diaphanorum partes, seu libri tres, Problemata ad perspectivam et iridem pertinentia, Napoli, Ex Typographia Tarquinij Longi, 1611. Two years later there was a second edition, with some variations, in Lyon, with title Theoremata de lumine et umbra instead of Photismi.

  6. 6.

    Ronchi translated only the first part (theorems I–III) and the last part (theorems XXX–XXXV) of Photismi, De erroribus speculorum and the third part of Diaphana. In the same book some passages from Della Porta’s Magia naturalis, De refractione and De telescopio are translated.

  7. 7.

    For an electronic version of our edition and further information, see www.maurolico.it.

  8. 8.

    All the reference to Diaphana are given with paragraph number of our edition and with page number of (Maurolico 1611): “ut ait Euclides” (Diaph. § 1, p. 31).

  9. 9.

    “Anguli inclinationum sunt fractionum angulis proportionales” (Diaph. § 33, p. 36). See note 14 for an explaination of the expression fraction angle.

  10. 10.

    The result is incorrect and Maurolico’s analysis is, in this case, unsatisfactory. Della Porta, on the contrary, will find correct results on this topic. See in particular Sect. 8.8.2.

  11. 11.

    This part underwent many changes in the revision of the text dating from 1553–1554. For example, dealing with secondary rainbow, Maurolico changed the value of the angle under which it is seen: 55 degrees in the first version, 56 and one quarter degrees in the second one. For more information, see the Note to the Text of Diaphana in our edition www.maurolico.it/Maurolico/sezione.html?path=10.3.

  12. 12.

    According to Lindberg (1984, p. 141): “Maurolico’s most original contribution to visual theory”.

  13. 13.

    The translation is taken, with some variations, from (Crew 1940, pp. 49–50).

  14. 14.

    As usual for the period, the fraction angle is the angle of deviation, that is the difference between inclination angle and what nowadays is called refraction angle.

  15. 15.

    Angles DAK and FBL are equal to inclination angle, being vertical angles. Angles GAD and HBF are deviation angles, called fraction angles.

  16. 16.

    See Fig. 8.1, where f, k and f′, k′ are fraction angles corresponding to the inclination angles i,h and i′, h′.

  17. 17.

    “Multiplicato angulo inclinationis, angulum quoque fractionis aequaliter multiplicari” (Diaph. § 4, p. 31).

  18. 18.

    “Radios aeque inclinatos aeque frangi, magis vero inclinatum, magis” (Diaph. § 3, p. 31).

  19. 19.

    “Ratio est duarum magnitudinum eiusdem generis aliquatenus ad invicem quaedam habitudo” (Zamberti 1505, def. 3, bk. V).

  20. 20.

    “Pondus quoque ac levitas cum respectivae qualitates sint …ad quantitatem redigi possunt. Qualitas enim talis respectiva est: cum idem corpus alio respectu leve sit, alio autem grave …Similiter vox eadem respectu vocis acutioris gravis est, respectu autem gravioris acuta. Sicut igitur voces inter se proportione, ita et pondera comparantur”.

  21. 21.

    In doing this, Maurolico clearly relied on what previous authors had done about the same subject.

  22. 22.

    In the proportion theory a ratio is a relation between two magnitudes of the same kind. In more complex models, like static momentum, specific gravity or theory of motion, which involve magnitudes of different kind, the alternando rule is not allowed.

  23. 23.

    “Ergo et angulus inclinationis ad angulum suae fractionis semper unam servat rationem. Sicut experimento in crystallina sphaera probavimus. Estque dupla et 2/3 superpartiens” (Diaph. § 4, p. 31).

  24. 24.

    I call here refractive index the fixed ratio between inclination and refraction angle, in analogy with the term used nowadays.

  25. 25.

    In Ptolemy three tables are found: air to water, air to glass and water to glass, see (Smith 1996, pp. 233, 236 and 238). In Witelo also the reciprocal path is found, that is water to air, glass to air and glass to water. The values are erroneous, probably computed by a wrong application of reciprocal law, see (Grant 1974, p. 425).

  26. 26.

    This application justifies, in part, the mentioned lack of interest for media other than glass.

  27. 27.

    “Given three parallel rays incident upon a transparent sphere, one passing through the center, the other two beyond the center; then the one passing through the center will maintain its direction in a straight line; the next ray will be refracted towards the first, but not sufficiently to intersect it; the remaining ray will also be refracted, but will never reach the other refracted ray” (Crew 1940, p. 58). “Transeuntibus per diaphanam sphaeram tribus parallelis radiis uno per centrum reliquis autem praeter centrum; qui per centrum in rectum transibit, at ei propior frangetur, ad ipsum tamen non perveniet. Reliquus quoque frangetur, sed alterum fractum nequaquam continget” (Diaph. § 41, p. 36).

  28. 28.

    For a brief sketch of demonstration see Fig. 8.4. In the demonstration the arcs of the circumference and the corresponding angles are sometimes interchanged.

  29. 29.

    I used G eo G ebra’s tool “trace” to obtain the simulations. For this software, released under Non-commercial License Agreement, see https://www.geogebra.org/ .

  30. 30.

    “Radio intra sphaeram pellucidam diametro sibi parallelo extra congrediente, sphaerae semidiameter ad distantiam congressus maiorem semper sortietur proportionem, quam angulus fractionis ad angulum, qui sub perpendiculari et fracto comprehenditur” (Diaph. § 74, p. 43). See Fig. 8.6: FD:DE>∢BEF:∢BFE.

    Fig. 8.6
    figure 6

    Theorem XIX and its corollary

    Based on the principle of reversibility (explicitly stated in assumption 4 of Photismi), Maurolico refers to the ray inside the sphere as refracted ray (BA) and the ray outside as the incident ray (EB). With this convention:

    ∢EBH=∢FBG is the incidence angle (i);

    ∢ABG=∢BEF is the deviated angle (f, contained by BA and BG);

    ∢FBG−∢ABG=∢FBA=∢BFE is the angle included by the refracted ray (BA) and the perpendicular (FB) (which equals i−f)

    Full size image
  31. 31.

    Maurolico, although drawing a sphere, is dealing with a hemisphere. The ray AB is parallel to the diameter CD and is not refracted at the surface GAC, as entering an hemisphere. From lens-maker’s formula we obtain, for an hemisphere of radius R in air:

    $$ \frac{n}{p}+\frac{1}{q}+\frac{1- n}{R} $$

    with p = +∞ for parallel incident rays. So, in theorem XIX, substituting R = FD = −3 and q = ED = 5 (according to the Cartesian sign convention) n=8:5 is found, the refractive index implied by the corollary to Maurolico’s law. The lens-maker’s formula is obtained in paraxial approximation, in which case a law of proportionality between incidence and refraction angles holds, since θ≈sinθ for θ sufficiently small.

  32. 32.

    “Nam cum horum librorum manuscripta exemplaria in multorum manibus essent, quamquam mendis referta et sine Clavii notis, nec deessent qui Perspectivam suis scriptis illustrare molirentur, dubitandum non erat, quin hi ad suorum librorum campos exornandos e florentibus Maurolyci hortis, suppresso auctoris nomina, omnia transtulissent”. English translation is from (Crew 1940, p. 3).

  33. 33.

    “Vennevi etiando il Clavio giesuita celeberrimo nella professione e strinsero insieme …molta famigliarità; a cui egli assegnò nel prender congedo l’originale de Fotismi e dell’opera delli Diafani, distinto in tre libri, affine di farglili stampar in Roma”.

  34. 34.

    There are more elements on the circulation of Maurolico’s optical works, see (Bellé 2006b) for an analysis of the publication of Maurolico’s optical works by C. Clavius and G. G. Staserius (S.J. 1565–1635), or the Introduction to our edition in volume 10 of Edizione Nazionale, www.maurolico.it/Maurolico/sezione.html?path=10.

  35. 35.

    For example (Gardair 1990; Beaulieu 1990; Battafarano 1990; Eamon,1994; Balbiani 1999,2001).

  36. 36.

    For optics see next section. For Pneumatica refer to the Introduction to the critical edition of the text (Trabucco 2008), published in Edizione nazionale delle Opere di Giovan Battista della Porta, where also the critical edition of De refractione will be published. For further information, http://www.lincei.it/files/convegni/710_invito.pdf.

  37. 37.

    Stigliola to Cesi (10 April 1615): “che visitandolo [Della Porta] due giorni inanzi che si mettesse a letto in quest’ultima sua infermità, mi disse che l’impresa del Telescopio l’haveva ammazzato, essendo, come egli diceva, la piú difficile impresa et la piú ardua di quante mai havesse pigliato”. Letter n. 393 in (Gabrieli 1996).

  38. 38.

    A transcription of the autograph manuscript, Roma, Biblioteca dell’Accademia nazionale dei Lincei e Corsiniana, Archivio Linceo XIV, is found in (Naldoni and Ronchi 1962).

  39. 39.

    Something on this last period, with an analysis of De telescopio can be found in Borrelli (2017).

  40. 40.

    “Geometriae pars …idest perspectiva appellatur, ad oculum pertinens, multa enim huiusmodi demiranda facit experimenta, …rectius speculum inspectando pedes sursum, caput deorsum videntur, et caetera, uti latius in sequentibus perspiciuntur” (Della Porta 1558, p. 141).

  41. 41.

    “Reddere tamen eorum causas, alienum puto, cum disciplina ea affatim satisfaciat, multique eas reddiderint; quas qui praeoptarit, consulat Archimedem Syracusanum, Euclidis Opticam, et Catoptricam, Ptolemaeum, Vitellionem et reliquos, quorum multa desumemus, multaque denuo excogitata adiiciemus, ut hinc quisque possit in infinitum ea propagare; ut in omnibus evenit experientiis” (Della Porta 1558, p. 141).

  42. 42.

    “Quomodo in tenebris ea conspicias, quae foris a sole illustrantur et cum suis coloribus”, in chap. II and “Quomodo res multiplicata videri possit”, in chap. IV. Della Porta, in fact, will deal again with these topics in the 1589 edition, expanding his analysis. Moreover they represent the most important contribution of Della Porta to the history of optics, as testified, for example, by the references found in Kepler’s Paralipomena or by the claimed precedence in the invention of the telescope.

  43. 43.

    “Non quod disgregando dissipet, sed colligendo uniat”, according to Della Porta’s description, (Della Porta 1558, p. 143).

  44. 44.

    “Intromittendi dirimitur quaestio sic agitata … intromittitur enim idolum per pupillam fenestrae instar, vicemque obtinet speculi parva magnae spherae portio, ultimo locata oculi; quod si quis distantiam mensuraverit, centri loco fiet visus” (Della Porta 1558, p. 143).

  45. 45.

    “Inter ludos qui circunferuntur, non parum iucunditatis est specillum; instrumentum vitreum illud, quod oculis apponimus, ut commodius aliquid intueamur” (Della Porta 1558, p. 145).

  46. 46.

    Many descriptions are taken directly from Ptolemy’s De speculis. This work was ascribed to Ptolemy in that period (Ptolemy 1518). W. Schmidt and L. Nix considered it as written by Hero (Hero 1900, pp. 303–315). More recently (Jones 2001) questioned both of these attributions and published it as Pseudo-Ptolemy.

  47. 47.

    “It will be easy to understand all the variations [of the concave mirror], if you find the center of the concave semicircle, being that all the things are regulated and known from it. If you want to see the head upside-down, you have to put your head outside the center of the mirror and you will see the head upside-down and the foot up”, chap. XIV (Della Porta 1558, pp. 150–151): “Centro concavi hemicycli iam invento, diversitates omnes cognoscere facile erit cum ex eo omnia regulentur et noscantur. Si vis ergo inversum spectare caput, extra speculi centrum caput habeto, et inversum statim inspicies, et pedes sursum”. In the 1589 edition, instead, he will define rightly this point as punctum inversionis, which corresponds to our focal point. However Della Porta did not give any quantitative information on its position, but only suggested that it could be found by observing the figure made by solar rays, reflected in the mirror. A mention to this method is already found in the 1558 at the end of chap. XIV: “Splendentis Solis radis speculum opponat, et fomitem vel accendibilem rem centri loco collocet, quod accedendo et removendo invenies lucisque conus ostendit et inducit illico flammam”.

  48. 48.

    For example, a “phiala” filled with water or a round crystal. (Della Porta 1558, pp. 151–152).

  49. 49.

    On this letters in connection with optics see (Bellé 2005, p. 9). All the letters to Cardinal d’Este are found in (Campori 1872).

  50. 50.

    Giacomo Contarini is a well known figure of the period, also in connection with Galilei. See, for example, the letters between Contarini and Galileo in (Galilei 1934, vol. X, pp. 52, 55, 57).

  51. 51.

    “Incontrai il S. Leonardo col quale di nuovo siamo stati a Murano al Ballarino, se havesse fatto le fugacciette di vetro per gli occhiali e la fontana”, 10th December 1580, (Campori 1872, p. 22).

  52. 52.

    “Oppone speculum soli et ubi radios coire videbis, scito ibi esse punctum inversionis”.

  53. 53.

    “Si extra punctum inversionis fueris, inversum carnes caput”. On the discovery of punctum inversionis by Ausonio see (Dupré 2005). Later on in the same chapter IV Della Porta wrongly considered the center of the circumference like the inversion point (as he had done in 1558): “The reason is that rays that go from the center to the circumference after reflection become parallel, since parallel rays, after reflection on the circumference, intersect themselves in the center [our italics]”. “Ratio est, quod radii a centro ad circumferentiam reflectuntur parelleli [sic] quum paralleli ad centrum concurrant” (Della Porta 1589, p. 264). This mistake bears witness to a textual stratification in the evolution of the second edition and to its strong connection with the first one.

  54. 54.

    “Si crystallinam lentem foramini appones, iam iam omnia clariora cernes”, chap. VI (Della Porta 1589, p. 266).

  55. 55.

    “Intromittitur idolum per pupillam, fenestrae foraminis instar, vicemque obtinet tabulae crystallinae spherae portio, in medio oculi locata”, in italics differences with the 1558 version.

  56. 56.

    “Quid enim ingeniosus excogitari potuit, ut imaginariis animi conceptionibus certissima experimenta subsequerentur? Et mathematicarum demonstrationum vera fides etiam ocularibus experimentis comprobaretur?” (Della Porta 1589, p. 259).

  57. 57.

    The statement “in nostris opticis” is found six times in the whole book XVII of the 1589 Magia: chap. VI, on the relation between the eye and the camera obscura: “in nostris opticis fusius declaratum est”, p. 267; chap. X, de crystallinae lentis effectibus, two instances: “sed de his plenius in nostris opticis” and “rationes in opticis reddidimus”, p. 269; chap. XIV, on spherical burning mirror: “latius de ea in opticis”, p. 271; chap. XIX, on burning by refraction: “rationem reddemus in opticis”, p. 277; chap. XX, on a specific kind of mirror: “in opticis diximus”, p. 278.

  58. 58.

    Book III, devoted to the description of the eye and book IV, on the theory of vision, make exception to this “mathematical structure”.

  59. 59.

    “[Optica] ex mathematices et naturalis philosophiae mixtura orta est. …Hinc geometricarum speculationum veritas innotescit, nam quae geometria fingit et speculatur, ipsa explicat et in lucem revocat”. “Causa, quae nos de refractione ad scribendum impulit, fuit, quod videremus ea de re maiores nostros breviter, ac satis oscitanter perscripsisse, ut vestigia potius quaedam, quam artis fundamenta iecisse viderentur” (Della Porta 1593, pp. 5–6).

  60. 60.

    The ultimate goal is to apply what is deduced for the glass sphere to the eye: “Haec enim pila oculi specimen refert, nec vera a nobis loca visionum in oculis decerni possunt”, bk. II, preface, (Della Porta 1593, p. 35). I shall not deal in this paper with this rather problematic issue.

  61. 61.

    Compare for example, the definitions found in the book X of Witelo’s Perspectiva, (Witelo 1572, p. 404); Della Porta’s seven definitions correspond to definition 1, 4, 3, 5, 6, 8, 10 in Witelo’s text, with some differences.

  62. 62.

    As noted before (p. 7, n. 14), also Della Porta calls angulus refractionis the angle of deviation, not “our” refraction angle.

  63. 63.

    “Ingredientes egredientesque semper aequalis in refractione reperiuntur” (Della Porta 1593, p. 10).

  64. 64.

    See Fig. 8.7. “Sit Sol in astrolabio D, in medio quartae, ad gradus 45, non veniet ad E, ubi est etiam 45 gradus, sed accedendo ad HBI in F, erit 35 gradus. At existens Sol in F, [ubi] 35, non veniet ad G, ubi etiam 35 sed ad D, ubi gradus 45. Remanent ergo anguli in medio aequales GBD et EBF ex lineis se decussantibus orti, ad invicem aequales”.

  65. 65.

    Ptolemy and Witelo, in fact, give measures of angles multiples of ten, from ten degrees up to eighty degrees, see (Smith 1996, p. 233) and (Grant 1974, p. 425).

  66. 66.

    There is an error of about three degrees (in water, refractive index 1.33, the angles of refraction is 32 degrees). Three degrees is a quite reasonable value, taking into account the structure used by Della Porta. An error of three degrees, if the alidade was 50 cm long, corresponds to a displacement of about 2.5 cm of the ray under water, compensated by the alidade’s thickness. Della Porta, in this case, seems to consider the angle between refracted ray and perpendicular line, that is the refraction angle.

  67. 67.

    Location of images in Della Porta is analysed in details in (Borelli 2014, pp. 53–57). As said before, cathetus rule is already present in the second edition of Magia (see p. 15).

  68. 68.

    Here is Della Porta enunciation: “Something under water, seen by refraction, will be seen more elevated if it is more distant from the eye”. For a deep examination of this proposition, see (Borelli 2014, p. 50).

  69. 69.

    I shall rely on and follow, in particular, (Dupré 2006, pp. 33–36), (Smith, 2010, xciii-xcv, figures pp. 436–439), (Zik and Hon 2012, pp. 445–456) and (Goulding 2014, pp. 145–147 and 171–174).

  70. 70.

    Hereinafter propositions from De refractione are cited with their number followed by a Roman numeral referring to the book. I call this ray hexagonal, being that BC is equal to the side of inscribed hexagon.

  71. 71.

    Also Kepler in his treatment of law of refraction made a large use of analogy between refraction and reflection. See (Malet 1990, pp. 281–282).

  72. 72.

    I emphasise that we can deduce, but Della Porta did not. A similiar situation is found in (Zik and Hon 2012, p. 455) where, applying proposition 5,II, the authors (not Della Porta) found an estimate of 2.4 times the radius for the focal point of an hemisphere.

  73. 73.

    I add that some authors give a different interpretation of the theorem: in their view there is no need to assume that II,2 deals with double refraction. See for example (Smith 2015, pp. 345–349), (Zik and Hon 2012, pp. 449–450) and Borrelli (2014, p. 52).

  74. 74.

    As will be seen, in De refractione Della Porta introduces inversion point also for the lenses.

  75. 75.

    “Magnitudo extra punctum inversionis posita conversa videbitur” (Della Porta 1593, p. 56).

  76. 76.

    For another analysis of inversion point, with a different point of view, see Zik and Hon (2017).

  77. 77.

    “Punctus inversionis maxime in eiusmodi apparentiis vestigandus est, nam ex eo omnes varietates nascuntur” (Della Porta 1593, p. 56).

  78. 78.

    “Ex D refrangetur ad B per secundam huius et egrediendo veniet ad oculum A per quintam” (Della Porta 1593, p. 57). Proposition 5,II deals with parallel rays exiting from a convex hemisphere (Della Porta 1593, p. 47).

  79. 79.

    The same holds for proposition 5, from which, it is possible to find only where the ray EC will intersect a parallel diameter.

  80. 80.

    Prop. 17: “If the object and the eye are near the sphere, the object will be seen upright, but if the eye goes back, the object is seen inverted”. Prop. 18: “If the object is near the sphere, within inversion point, it is seen upright, whether the eye is far or near to the sphere”.

  81. 81.

    “In specillis convexis inversionis punctum invenire” (Della Porta 1593, p. 175). Proposition 16,II deals with spheres but Della Porta considered lenses as composed by two spherical surfaces.

  82. 82.

    “Puncta concursus refractarum linearum, tam convexi quam concavi corporis, sunt puncta inversionum” (Della Porta 1593, p. 176). There is no attempt to find an estimate of the position of inversion point, although using 5,II it would be possible, see (Zik and Hon 2012) and n. 71 on p. 25.

  83. 83.

    “Nam intra crystalli corpus refranguntur” (Della Porta 1593, p. 176).

  84. 84.

    For the problems in applying cathetus rule in this situation see (Borelli 2014, p. 59).

  85. 85.

    In the text, reference is made to “tertium quinti”, with a misprint in book number. Proposition 3,V deals with parallel lines which appear intersecting (Della Porta 1593, p. 112). The correct book reference is to book II.

  86. 86.

    In the figure point E is wrongly indicated as C.

  87. 87.

    After all, nowadays the study of geometrical optics starts with spherical mirrors and then proceeds to lenses, exploiting the analogy between these two optical artifacts.

  88. 88.

    This trait is peculiar of Della Porta “mathematician”, as happens in Elementa curvilineorum libri III where, according to (Napolitani 1990, p. 156) Della Porta is a researcher fond not of the truth of his results but of the shock they can produce. A more detailed analysis of this aspect can be found in (Gavagna and Leone 2005), the critical edition of the text, published in Edizione nazionale delle Opere di Giovan Battista della Porta.

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  1. Liceo Scientifico “Enrico Fermi”, via E. Fermi 2, 54100, Massa, Italy

    Riccardo Bellé

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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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Bellé, R. (2017). Francesco Maurolico, Giambattista Della Porta and Their Theories 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_8

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