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Chapter 2 1653 — ‘Tractatus’

The mathematical understanding of telescopes
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Part of the Archimedes book series (ARIM, volume 9)

Keywords

Solar Eclipse Focal Distance Spherical Aberration Spherical Lens Convex Lens 
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References

  1. 1.
    OC 1, 215. “Nunc autem in dioptricis totus sum...”Google Scholar
  2. 2.
    Berkel, “Illusies”, 83–84. In the 1660s Huygens would start to seek patronage abroad, first in Florence and then, successfully in Paris.Google Scholar
  3. 3.
    The most thorough-going account still are the ‘avertissements’ by the editors of the Oeuvres Complètes. Southall, “Some of Huygens’ contributions” reported on Huygens’ dioptrics after the publication of volume 13. Harting, Christiaan Huygens had earlier discussed it briefly. In relationship with his astronomical work and his practical dioptrics, Albert van Helden, “Development” and Anne van Helden/Van Gent, The Huygens collection and “Lens production” discuss some topics. In the context of the history of seventeenth-century geometrical optics — which in its own right has little been studied — Shapiro, “‘Optical Lectures’” mention Huygens’ contributions. They are remarkably absent from the Malet, “Isaac Barrow” and “Kepler and the telescope”. Hashimoto, “Huygens, dioptrics” is the only effort to discuss Huygens’ dioptrics in the context of his broader oeuvre.Google Scholar
  4. 4.
    OC 13, 1–271. The editors of the Oeuvres Complétes have labeled it Dioptrica, Pars I. Tractatus de refractione et telescopiis. Its content stems from the 1650s. The original version of Tractatus does not exist anymore. A copy was made in Paris by Niquet — probably in 1666 or 1667, at the beginning of Huygens’ stay in Paris — on which the text of the Oeuvres Complètes is based. The editors assume Niquet’s copy of Tractatus is largely identical with the original 1653 manuscript; “Avertissement”, XXX.Google Scholar
  5. 6.
    OC 1, 305–305.Google Scholar
  6. 7.
    Descartes, Geometrie, 352 (AT6, 424). “Au reste affin que vous sçachiées que la consideration des lignes courbes icy proposée n’est pas sans usage, & qu’elles ont diverses proprietés, qui ne cedent en rien a celles des sections coniques, ie veux encore adiouster icy lľexplication de certaines Ovales, que vous verrés estres tres utiles pour la Theorie de la Catoptrique, & de la Dioptrique.”Google Scholar
  7. 8.
    Descartes, Geometrie, 358–359 (AT6, 430–431). The left part 2A2 is a mirror that reflects rays intersecting in G so that they (virtually) intersect in F, provided that it diminishes the ‘tendency’ of the rays to a given degree.Google Scholar
  8. 9.
    Descartes, Geometrie, 353–354 (AT6, 424–426). The curve satisfies the equation F2 − FA = n(G2 − GA).Google Scholar
  9. 10.
    OC 1, 305. See note 9: the equation becomes AF = nAG.Google Scholar
  10. 11.
    Reproduced in OC 14, 419.Google Scholar
  11. 12.
    In Tractatus, he merely mentioned that a spherical surface is aplanatic for certain points: OC13, 64–67.Google Scholar
  12. 13.
    OC 1, 190–192.Google Scholar
  13. 14.
    OC 1, 192.Google Scholar
  14. 15.
    OC 1, 201–205.Google Scholar
  15. 16.
    OC 1, 204. “... adeo ut nullius radij concursus cum axe contingat ultra punctum O.”Google Scholar
  16. 17.
    OC 1, 224–226.Google Scholar
  17. 18.
  18. 19.
    OC1, 219–223.Google Scholar
  19. 22.
    OC13, 16–19Google Scholar
  20. 23.
    OC13, 40–79.Google Scholar
  21. 24.
    OC13, 70–73.Google Scholar
  22. 25.
    OC13, 42–47.Google Scholar
  23. 26.
    OC13, 88–89. Equivalent to the modern formula 1/f=(n−1)(1/R1+1/R2)Google Scholar
  24. 27.
    OC13, 98–109.Google Scholar
  25. 28.
    OC13, 122–125.Google Scholar
  26. 29.
    OC13, 118–123. In modern terms, L is the optical center.Google Scholar
  27. 30.
    OC13, 114–119.Google Scholar
  28. 32.
    OC13, 176n1.Google Scholar
  29. 33.
    OC13, 174–179.Google Scholar
  30. 34.
    OC13, 186–197.Google Scholar
  31. 35.
    “This is the great Proposition asserted by most Dioptrick Writers, but hitherto proved by none (for as much as I know)...” Molyneux, Dioptrica nova, 161.Google Scholar
  32. 36.
    OC13, 244–247.Google Scholar
  33. 37.
    OC13, 246–253.Google Scholar
  34. 38.
    Hug29, 151–167.Google Scholar
  35. 39.
    OC13, 198–199.Google Scholar
  36. 40.
    OC13, 252n1. See below, section 3.1.2.Google Scholar
  37. 42.
    OC13, 258–261.Google Scholar
  38. 43.
    OC1, 280; 301–303; 321–322. Huygens did not pin much faith in Van Schooten’s proposal.Google Scholar
  39. 45.
    Van Helden, Invention, 35–36; Galileo, Sidereus nuncius, 3–4 (Van Helden’s introduction).Google Scholar
  40. 46.
    De Waard, Uitvinding, 105–225; Van Helden, Invention, 20–25.Google Scholar
  41. 47.
    OC13, 436–437.Google Scholar
  42. 48.
    Van Helden, Invention, 21, 36.Google Scholar
  43. 49.
    Van Helden, Invention, 26; Galileo, Sidereus nuncius, 6, 9 (Van Helden’s Introduction).Google Scholar
  44. 50.
    Van Helden, “Galileo and the telescope”, 153–157.Google Scholar
  45. 51.
    Galileo, Sidereus nuncius, 94 (Van Helden’s Conclusion).Google Scholar
  46. 52.
    Galileo, Sidereus nuncius, 37–39.Google Scholar
  47. 53.
    Kepler, Conversation, [19–21].Google Scholar
  48. 54.
    Kepler, Dioptrice, dedication (KGW4, 331). “... circaque eam alij de palma primae inventionis certarent, alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est causarum lege geometrica demonstrandarum, quibus tam exoptati, tam jucunda varietate multiplices effectus inniterentur.”Google Scholar
  49. 55.
    Straker, “Kepler’s theory of pinhole images”, 276–278.Google Scholar
  50. 56.
    Cited and translated in: Straker, “Kepler’s theory of pinhole images”, 278.Google Scholar
  51. 57.
    Straker, “Kepler’s theory of pinhole images”, 275–276; 280–282.Google Scholar
  52. 59.
    Dupré points out Risner’s programmatic discussion of the science of optics in the preface to the edition which constitute an important, yet still little studied, agenda for seventeenth-century optics. Dupré, Galileo, the Telescope, 54.Google Scholar
  53. 60.
    See Lindberg, “Laying the foundations”, 14–29.Google Scholar
  54. 61.
    Alhacen, Optics I, 68 (book 1, section 17) and 77 (book 1, section 46).Google Scholar
  55. 63.
    Dupré, Galileo, the telescope, 31.Google Scholar
  56. 64.
    Kepler, Paralipomena, 4 (KGW2, 16). “⋯ hae tenebrae sint Astronomorum oculi, hi defectus doctrinae sint abundantia, hi naevi mentes mortalium preciosissimis pictu ris illustrent.” Translation Donahue, Optics, 16.Google Scholar
  57. 65.
    Kepler, Paralipomena, 201 (KGW2, 181). “Itaque non oportet nos ad restotas respicere, sed ad rerum singular puncta,⋯” Translation Donahue, Optics, 217.Google Scholar
  58. 66.
    Rosen, “The invention of eyeglasses”, 13–46.Google Scholar
  59. 67.
    Lindberg, “Optics in 16th century Italy” 136–141. Maurolyco had preceded Kepler in his analysis of the pinhole image: Lindberg, “Optics in 16th century Italy”, 132–135; Lindberg, “Laying the foundations”.Google Scholar
  60. 68.
    Kepler, Paralipomena, 200–202 (KGW2, 181–183).Google Scholar
  61. 69.
    Malet, “Kepler and the telescope” offers a detailed discussion of Dioptrice, without however presenting it as a part of the ‘optical part of astronomy’.Google Scholar
  62. 70.
    Kepler, Dioptrice, dedication (KGW4, 331).Google Scholar
  63. 71.
    Kepler, Dioptrice, 11 (KGW4, 363).Google Scholar
  64. 72.
    Kepler, Dioptrice, 12–15 (KGW4, 363–367).Google Scholar
  65. 73.
    Kepler, Dioptrice, 45–49 (KGW4, 388–393).Google Scholar
  66. 74.
    Kepler, Dioptrice, 16–18 (KGW4, 367–369).Google Scholar
  67. 75.
    Kepler, Dioptrice, dedication (KGW4, 335).Google Scholar
  68. 76.
    Kepler, Dioptrice, 21–24 (KGW4, 371–372).Google Scholar
  69. 77.
    Kepler, Dioptrice, 35–42 (KGW4, 381–387).Google Scholar
  70. 78.
    Kepler, Dioptrice, 42–43 (KGW4, 387–388).Google Scholar
  71. 79.
    A possible source of inspiration may have come from the analogous configuration of the eye and a convex spectacle glass, as the eye acts as a convex lens does. See also Malet, “Kepler and the telescope”, 119–120.Google Scholar
  72. 80.
    OC1, 6 (Stampioen’s list of recommended readings spans pages 5–10) and OC6, 215.Google Scholar
  73. 81.
    Della Porta’s account of refraction by spheres and lenses in De refractione is discussed in Lindberg, “Optics in 16th century Italy”, 143–146.Google Scholar
  74. 82.
    Della Porta, De Telescopio, 113–114.Google Scholar
  75. 83.
    Della Porta, De telescopio, 141–142.Google Scholar
  76. 84.
    Compare Lindberg, “Optics in 16th century Italy”, 146–147.Google Scholar
  77. 85.
    Ronchi, “Refractione au Telescopio”, 56 and 34. “They know nothing of perspective.” and “... and it pleases me that the idea of the telescope in a tube has been mine;...”Google Scholar
  78. 86.
    Pedersen, “Sagredo’s optical researches”, 144–148.Google Scholar
  79. 87.
    KGW4, “Nachbericht”, 476.Google Scholar
  80. 88.
    Dupré, Galileo, the Telescope, chapters 4 to 6 in particular.Google Scholar
  81. 90.
    Rashed, “Pioneer”, 478–486.Google Scholar
  82. 91.
    For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250.Google Scholar
  83. 92.
    Gaukroger, Descartes, 138–146. Dupré, Galileo, the Telescope, 53–54.Google Scholar
  84. 94.
    Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.”Google Scholar
  85. 95.
    Descartes, AT6, 155–160.Google Scholar
  86. 96.
    Descartes, AT6, 165. “Des figures que doivent avoir les corps transparens pour detourner les rayons par refraction en toutes les façons qui servent a la veuë”Google Scholar
  87. 97.
    Descartes, AT6, 82–83. Ribe, “Cartesian optics” offers an enlightening account of the artisanal roots of La Dioptrique.Google Scholar
  88. 98.
    Descartes, AT6, 82.Google Scholar
  89. 99.
    OC10, 402–403. “Mr. des Cartes n’a connu quel seroit ľeffet de ses Lunettes hyperboliques, et en a presumè incomparablement plus qu’il ne devoit. n’entendant pas assez cette Theorie de la dioptrique, ce qui paroit par sa demonstration très mal bastie des Telescopes.”Google Scholar
  90. 100.
    Stampioen, Wis-konstigh ende reden-maetigh bewys, 58. “... mijn Knecht Ondersoeck sal hem eens een beter Verre-kijcker sonder cirkeltjes daer toe weten te drayen:... Maer niettemin ť geen dese Mathematicien al over 6 Iaren belooft heeft te doen, blijft nog on-vol-daen.”Google Scholar
  91. 101.
    Stroud, Minute, 20; Prins, “Hobbes on light and vision”, 129–132. On Hobbes’ derivation of the sine law, see section 5.2.1.Google Scholar
  92. 102.
    Lohne, ”Geschichte des Brechungsgesetzes”, 166.Google Scholar
  93. 104.
    Barrow, Lectiones, [82–83].Google Scholar
  94. 105.
    Compare Shapiro, “The Optical Lectures”, 130 & 133–134.Google Scholar
  95. 106.
    Compare Shapiro, “The Optical Lectures”, 150–151; and Malet, “Isaac Barrow”, 286.Google Scholar
  96. 107.
    Shapiro, “The Optical Lectures”, 149–150.Google Scholar
  97. 108.
    Barrow, Lectiones, [168].Google Scholar
  98. 109.
    Newton, Optical Papers 1, 427.Google Scholar
  99. 111.
    Halley, “Instance”, 960.Google Scholar
  100. 112.
    See: Albury, “Halley, Huygens, and Newton”, 455–457.Google Scholar
  101. 113.
    Galileo, Sidereus nuncius, 112–113 and 92–93 (Van Helden’s conclusion). See Dupré, Galileo and the telescope, 175–178.Google Scholar
  102. 114.
    Schuster, “Descartes opticien” and Van Berkel, “Descartes’ debt”.Google Scholar
  103. 115.
    Beeckman, Journal, II, 209–211; 294–296. For lens grinding see down, page 57.Google Scholar
  104. 116.
    For the second idea see Beeckman, Journal, II, 367–368. For a later consideration see for example: III, 296.Google Scholar
  105. 117.
    Beeckman, Journal, II, 296; 357.Google Scholar
  106. 118.
    Beeckman, Journal, III, 109–110.Google Scholar
  107. 119.
    Van Helden, Measure, 118–119.Google Scholar
  108. 120.
    Compare Dear, Discipline and Experience, 210–216.Google Scholar
  109. 121.
    Van Helden, “Astronomical telescope”, 26–32. See also below section 3.1.1.Google Scholar
  110. 122.
    Rigaud, Correspondence, 46: “This is that admirable secret, which, as all other things, appeared when it pleased the All Disposer, at whose direction a spider’s line drawn in an opened case could first give me by its perfect apparition, when I was with two convexes trying experiments about the sun, the unexpected knowledge.”Google Scholar
  111. 123.
    McKeon, “Les débuts I”, 258–266.Google Scholar
  112. 124.
    Old Corr 3, 293: “... prendre les diametres du soleil, de la lune et des planetes par une methode que nous avons, Monsieur Picard et moy, que ie croy la meilleure de toutes celles que ľon a pratiquer Jusques a present,...”Google Scholar
  113. 125.
    McKeon, “Les débuts I”, 266–269.Google Scholar
  114. 126.
    McKeon, “Les débuts I”, 286. In Micrographia (1665) Hooke had suggested that a scale may be inserted into the focal plane of telescopes. Hooke, Micrographia, 237.Google Scholar
  115. 127.
    OC21, 348–351.Google Scholar
  116. 128.
    Van Helden, Measure, 120–121.Google Scholar
  117. 129.
    OC21, 352–353.Google Scholar
  118. 130.
    McKeon, “Les débuts I”, 286; Van Helden, Measure, 118.Google Scholar
  119. 131.
    McKeon, “Renouvellement”, 122.Google Scholar
  120. 132.
    McKeon, “Renouvellement”, 126.Google Scholar
  121. 133.
    Flamsteed, Gresham lectures, 34–39 (Forbes’s introduction).Google Scholar
  122. 134.
    Old Corr 9, 326–327.Google Scholar
  123. 135.
    Old Corr 10, 520.Google Scholar
  124. 136.
    Old Corr 4, 448.Google Scholar
  125. 137.
    Van Helden, “Huygens and the astronomers”, 156-157; Van Helden, Measure, 127–129.Google Scholar
  126. 138.
    Flamsteed, Gresham lectures, 154.Google Scholar
  127. 139.
    Flamsteed, Gresham lectures, 119 & 132. Flamsteed later deleted the part between brackets.Google Scholar
  128. 140.
    Flamsteed, Gresham lectures, 120–127.Google Scholar
  129. 141.
    Flamsteed, Gresham lectures, 136.Google Scholar
  130. 142.
    Flamsteed, Gresham lectures, 140–143.Google Scholar
  131. 143.
    Flamsteed, Gresham lectures, 40; 146n2 (Forbes’ introduction).Google Scholar
  132. 144.
    Flamsteed, Gresham lectures, 8–9; 40 (Forbes’ introduction).Google Scholar
  133. 145.
    Flamsteed, Gresham lectures, 39 (Forbes’ introduction).Google Scholar
  134. 146.
    Flamsteed, Gresham lectures, 149.Google Scholar
  135. 147.
    Flamsteed, Gresham lectures, 4–5 (Forbes’ introduction).Google Scholar
  136. 148.
    Molyneux, Dioptrica nova, (Admonition to the reader).Google Scholar
  137. 149.
    Molyneux mentioned Kepler, Cavalieri, Hérigone, Dechales, Fabri, Gregory and Barrow.Google Scholar
  138. 150.
    Molyneux, Dioptrica nova, (Admonition to the reader).Google Scholar
  139. 151.
    Molyneux, Dioptrica nova, 19–23.Google Scholar
  140. 152.
    Molyneux, Dioptrica nova, 20.Google Scholar
  141. 153.
    Molyneux, Dioptrica nova, 22.Google Scholar
  142. 154.
    Molyneux, Dioptrica nova, 9.Google Scholar
  143. 155.
    Molyneux, Dioptrica nova, 24. From the preceding it will be clear, that following Molyneux’s line of thought this distance should be zero, for both points are by definition the same.Google Scholar
  144. 156.
    Molyneux, Dioptrica nova, 36–38.Google Scholar
  145. 157.
    Molyneux, Dioptrica nova, 38.Google Scholar
  146. 158.
    Picolet, “Correspondence”, 38–39.Google Scholar
  147. 159.
    «... peuuent aussi estre sujets a certaines refractions qu’il faut bien connoistre.” Quoted in McKeon, “Renouvellement”, 126–128. It is found in: A. Ac. Sc., Registres, t. 3, fol 156 ro — 164 vo spéc. 157 vo.Google Scholar
  148. 160.
    Blay, “Travaux de Picard”, 329–332. Blay cites several references.Google Scholar
  149. 161.
    Blay, “Travaux de Picard” 343. “Ce que nous venons ďexpliquer touchant la construction des lunettes ďapproche, n’est que par rapport α ľusage que ľon en fait dans les instruments qui servent α ľobserver,...”Google Scholar
  150. 162.
    Divers Ouvrages de Mathematique et de Physique, par Messieurs de ľAcademie Royale des Sciences (1693), 375–412.Google Scholar
  151. 163.
    OC13, “Avertissement”, 7.Google Scholar
  152. 164.
    Blay, “Travaux de Picard”, 340.Google Scholar

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