Abstract

We trace the history of atmospheric refraction from the ancient Greeks up to the time of Kepler. The concept that the atmosphere could refract light entered Western science in the second century B.C. Ptolemy, 300 years later, produced the first clearly defined atmospheric model, containing air of uniform density up to a sharp upper transition to the ether, at which the refraction occurred. Alhazen and Witelo transmitted his knowledge to medieval Europe. The first accurate measurements were made by Tycho Brahe in the 16th century. Finally, Kepler, who was aware of unusually strong refractions, used the Ptolemaic model to explain the first documented and recognized mirage (the Novaya Zemlya effect).

© 2005 Optical Society of America

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  1. A. Lejeune, Recherches sur la Catoptrique Grecque (Académie Royale de Belgique, Brussels, 1957), p. 3.
  2. A. M. Smith, Ptolemy’s Theory of Visual Perception: an English Translation of the Optics (American Philosophical Society, Philadelphia, 1996), p. 19. This second century work was organized according to the three divisions.
  3. M. R. Cohen, I. E. Drabkin, A Source Book in Greek Science (Harvard University, Cambridge, Mass., 1958), p. 200.
  4. Ref. 3, p. 93.
  5. Aristotle, Works: Translated into English, 12 vols., W. D. Ross, ed. (Clarendon, Oxford, 1908–1952), vol. III (Meteorologica), pp. 373b12–373b13.
  6. P. Ver Eecke, Euclide: L’Optique et la Catoptrique (Albert Blanchard, Paris, 1959). It is considered doubtful if the Catoptrica were actually written by Euclid; the lowest layer in this work stems from Euclid’s time, but subsequent editing added several more recent layers.
  7. H. E. Burton, “The optics of Euclid,” J. Opt. Soc. Am. 35, 357–372 (1945).
    [CrossRef]
  8. The one short statement on refraction in Euclid’s Catoptrica is considered to be a later addition; see Ref. 6, p. 99.
  9. T. L. Heath, A History of Greek Mathematics, 2 vols. (Clarendon, Oxford, 1921).
  10. Archimedes appears to be the originator of the coin in the cup experiment: a coin is placed in the bottom of a cup, out of sight to an observer who is looking into the cup at an angle. When the cup is filled with water, refraction makes the coin visible. See Ref. 9, Vol. 1, p. 444. This is the statement referred to in Note 8.
  11. This assumption also appears in Euclid’s Catoptrica,probably within the lowest layer, which predates Archimedes.
  12. Ref. 1, p. 179.
  13. Pliny, Natural History, 10 vols., translated by H. Rackham, Loeb Classical Library (Heinemann, London, 1938), vol. I, p. 207.
  14. O. Neugebauer, A History of Ancient Mathematical Astronomy, 2 Vols. (Springer-Verlag, Berlin, 1975), pp. 274–298.
    [CrossRef]
  15. G. J. Toomer, “Hipparchus,” Dictionary of Scientific Biography, C. Coulston Gillispie, ed. (Scribner’s, New York, 1990), Suppl. I, pp. 207–224.
  16. Cleomedes, On the Circular Motion of the Heavenly Bodies, in Ref. 3, p. 284. Cleomedes took this part of his work from Posidonius.
  17. There is some controversy about when Cleomedes lived: The consensus is first century A.D. although others say second or even fourth century. The later dates appear unlikely, because he never refers to the well known second century work of Ptolemy.
  18. Ref. 3, p. 284.
  19. Ref. 9, vol. II, p. 236.
  20. H. E. Ross, “Cleomedes (ca. 1st century A.D.) on the celestial illusion, atmospheric enlargement, and size–distance invariance,” Perception 29, 863–871 (2000).
    [CrossRef]
  21. Hipparchus probably, and Posidonius certainly, tried to explain the Moon illusion.
  22. Ref. 20, p. 864.
  23. “Posidonius,” Dictionary of Scientific Biography, Ref. 15. The writings of Posidonius have not survived; we know of his work primarily through Cleomedes himself.
  24. Ref. 9, vol. II, pp. 235–238. That Posidonius studied the Moon illusion is independently confirmed by Strabo (see Ref. 20, p. 864).
  25. D. C. Lindberg, Theories of Vision from al-Kindi to Kepler (University of Chicago, Chicago, Ill., 1976).
  26. Ptolemy’s Almagest, translated and annotated by G. J. Toomer (Springer-Verlag, New York, 1984).
  27. R. Wolf, Geschichte der Astronomie (R. Oldenbourg, München, 1877), p. 88.
  28. O. Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (American Institute of Physics, Molville, New York, 1993), p. 25.
  29. Ref. 2, p. 45. Ptolemy’s refraction measurements appear in many places, e.g., in Ref. 42, cited below.
  30. Interestingly, if we expand Snell’s Law in a Taylor series and truncate it after the quadratic term, we get a remarkably good agreement with the quadratic function implicit in Ptolemy’s data. This occurs, however, only if the expansion is done about the midpoint of the data; Wilk expands about the origin and gets a poor fit. See S. R. Wilk, “Claudiuś Ptolemy’s law of refraction,” Op. Photon. News 1510, 14–17 (2004).
  31. Ref. 2, p. 242.
  32. H. E. Ross, G. M. Ross, “Did Ptolemy understand the moon illusion?,” Perception 5, 377–385 (1976).
    [CrossRef] [PubMed]
  33. Ref. 32, p. 379.
  34. Ref. 2, p. 49.
  35. R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses.” Isis 81, 464–491 (1990).
    [CrossRef]
  36. Ibn Sahl applied his insight to the analysis of lenses; in particular, he solved the problem of the anaclastic lens: a paraboloid lens that will bring parallel incident rays to a perfect focus.
  37. His full name was Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham. While he is known to most as Alhazen, scholars now believe that this should be written as Alhacen.
  38. Ref. 35, p. 491.
  39. Ref. 25, p. 209.
  40. F. Risner, ed., Opticae thesaurus. Alhazeni Arabis libri septem, nuncprimum editi. Eiusdem liber De Crepusculis et nubium ascensionibus. Item Vitellonis Thuringopoloni libri X, reprint of the 1572 edition; introduction by D. C. Lindberg (Johnson Reprint Corp., New York, 1972).
  41. Ref. 2, p. 56.
  42. E. Grant, ed., A Source Book in Medieval Science (Harvard University, Cambridge, Mass., 1974), p. 420.
  43. Ref. 40, Alhazen, Book VII, Chap. 2.
  44. With respect to lenses, Alhazen’s contribution was small. Risner’s edition (Ref. 40) contains no discussion of lenses (in spite of an erroneously inserted diagram showing a burning sphere). Alhazen wrote a separate work entitled On the Burning Sphere, which considered only spherical lenses. See E. Wiedemann, “Über die Brechung des Lichtes in Kugeln nach Ibn al Haitam und Kamâl al Dîn al Fârisî,” Sitzungsber. Phys.-med. Sozietät Erlangen 42, 15–58 (1910). He certainly did not develop a rigorous geometric theory of lenses, as Rashed (Ref. 35) has claimed. Grant (Ref. 42) states that the burning sphere was as close as any medieval scholar came to a study of lenses.
  45. Ref. 40, Alhazen, Book VII, Chaps. 15 and 16.
  46. Ref. 2, Ptolemy Book V, Sects. 23–30.
  47. The vertical case follows from Ptolemy, but he never stated it explicitly for separation between stars. Alhazen, on the other hand, never concerned himself with the shift in apparent elevation of stars.
  48. Ref. 40, E. Alhazen, Book VII, Chap. 55.
  49. A. I. Sabra, “Psychology vs mathematics: Ptolemy and Alhazen on the moon illusion,” in Mathematics and its Applications to Science and Natural Philosophy in the Middle AgesE. Grant, J. E. Murdoch, eds. (Cambridge University, Cambridge, 1987).
  50. J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922), Part 1.
  51. A. I. Sabra, “The authorship of Liber de crepusculis, an eleventh-century work on atmospheric refraction,” Isis 58, 77–85 (1967). He calls it a work of atmospheric refraction, which it is not.
    [CrossRef]
  52. Ref. 40, Liber de crepusculis, p. 288. This edition also gives the height of the atmosphere as 52000 paces (p. 287). The Roman pace was 5 ft. (1.7 m), but its length varied over the centuries.
  53. A. M. Smith, “Ptolemy, Alhacen, and Ibn Mu’adh and the problem of atmospheric refraction,” Centaurus 45, 100–115 (2003).
    [CrossRef]
  54. Ref. 40, Introduction by D. Lindberg, pp. vii–xiii.
  55. J. Burchardt, Witelo: filosofo della natura del XIII sec.: una biografia (Polskiei Akademii Nauk, Wroclaw, 1984).
  56. Ref. 40, Introduction by D. Lindberg, p. xiii.
  57. Ref. 40, Introduction by D. Lindberg, p. xx.
  58. Ref. 42, pp. 424–426.
  59. Ref. 40, Witelo, Book X, Chap. 8.
  60. Ref. 40, Introduction by D. Lindberg, p. xxi.
  61. D. C. Lindberg, John Pecham and the Science of Optics: Perspectiva Communis (University of Wisconsin, Madison, 1970). Pecham’s book presented Alhazen’s conclusions in a reduced and more readable form, while preserving their central ideas.
  62. Ref. 40, Introduction by D. Lindberg, p. xxiii.
  63. C. Bruhns, Die astronomische Strahlenbrechung in ihrer historischen Entwickelung (Voigt & Günther, Leipzig, 1861), pp. 13 and 14.
  64. The Air Almanac 1987 (United States Naval Observatory, Washington, D.C., 1986).
  65. Ref. 25, pp. 185–186.
  66. M. Caspar, Kepler, translated by C. D. Hellman (Abelard-Schuman, New York, 1959).
  67. W. H. Donahue (transl.), Optics: Paralipomena to Witelo & Optical Part of Astronomy, by Johannes Kepler (Green Lion Press, Santa Fe, N.M., 2000).
  68. J. Kepler, Ad Vitellionem Paralipomena, quibus Astronomiae Pars Optica Traditur (Apud Claudium Marnium & Haeredes Ioannis Aubrii, Frankfurt, 1604).
  69. Ref. 67, p. 149.
  70. Mästlin saw the eclipse on 7 July 1590 (Old Style).
  71. In a later work, Kepler stated that the solar parallax could not be more than 1’.
  72. Ref. 67, p. 144.
  73. Ref. 67, p. 141.
  74. G. de Veer, Waerachtige Beschryvinge van drie seylagiën ter werelt noyt soo vreemt ghehoort (Claes Claesz, Amsterdam, 1598). A Latin version also appeared in 1598.
  75. G. de Veer (English translation), The true and perfect description of three voyages, so strange and woonderfull that the like has neuer been heard of before (T. Pauier, London, 1609).
  76. To quote Kepler, “The story is familiar to everybody, of the journey of the Netherlanders …”; Ref. 67, p. 151.
  77. S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).
  78. S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
    [CrossRef]
  79. W. H. Lehn, “The Novaya Zemlya effect: an arctic mirage,” J. Opt. Soc. Am. 69, 776–781 (1979).
    [CrossRef]
  80. The effect is actually refraction confined to a very narrow zone; on a larger scale, it looks just like reflection.
  81. G. H. Liljequist, “Refractive phenomena in the polar atmosphere,” Scientific Results, Norwegian-British-Swedish Antarctic Expedition 1949–1952, Vol. 2, Part 2 (Oslo University, Oslo, 1964). His observation took place on 1 July 1951, when the Sun was 4.3° below the horizon.
  82. Ref. 67, p. 96
  83. Ref. 14, Book VII. His star tables are given to a resolution of one sixth of a degree.
  84. S. van der Werf, “Het astrolabium,” Cornelis Douwes 160, 20–21 (2004).
  85. K. Ferguson, The Nobleman and His Housedog: Tycho Brahe and Johannes Kepler: the Strange Partnership that Revolutionised Science (Review, London, 2002), p. 126. On one instrument at least, degrees were divided into six parts; then Tycho used a slanted pattern of 10 dots to subdivide each of these into minutes.

2004

Interestingly, if we expand Snell’s Law in a Taylor series and truncate it after the quadratic term, we get a remarkably good agreement with the quadratic function implicit in Ptolemy’s data. This occurs, however, only if the expansion is done about the midpoint of the data; Wilk expands about the origin and gets a poor fit. See S. R. Wilk, “Claudiuś Ptolemy’s law of refraction,” Op. Photon. News 1510, 14–17 (2004).

S. van der Werf, “Het astrolabium,” Cornelis Douwes 160, 20–21 (2004).

2003

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

A. M. Smith, “Ptolemy, Alhacen, and Ibn Mu’adh and the problem of atmospheric refraction,” Centaurus 45, 100–115 (2003).
[CrossRef]

2000

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

H. E. Ross, “Cleomedes (ca. 1st century A.D.) on the celestial illusion, atmospheric enlargement, and size–distance invariance,” Perception 29, 863–871 (2000).
[CrossRef]

1990

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses.” Isis 81, 464–491 (1990).
[CrossRef]

1979

1976

H. E. Ross, G. M. Ross, “Did Ptolemy understand the moon illusion?,” Perception 5, 377–385 (1976).
[CrossRef] [PubMed]

1967

A. I. Sabra, “The authorship of Liber de crepusculis, an eleventh-century work on atmospheric refraction,” Isis 58, 77–85 (1967). He calls it a work of atmospheric refraction, which it is not.
[CrossRef]

1945

1910

With respect to lenses, Alhazen’s contribution was small. Risner’s edition (Ref. 40) contains no discussion of lenses (in spite of an erroneously inserted diagram showing a burning sphere). Alhazen wrote a separate work entitled On the Burning Sphere, which considered only spherical lenses. See E. Wiedemann, “Über die Brechung des Lichtes in Kugeln nach Ibn al Haitam und Kamâl al Dîn al Fârisî,” Sitzungsber. Phys.-med. Sozietät Erlangen 42, 15–58 (1910). He certainly did not develop a rigorous geometric theory of lenses, as Rashed (Ref. 35) has claimed. Grant (Ref. 42) states that the burning sphere was as close as any medieval scholar came to a study of lenses.

Alhazen,

Ref. 40, Alhazen, Book VII, Chaps. 15 and 16.

Ref. 40, Alhazen, Book VII, Chap. 2.

Alhazen, E.

Ref. 40, E. Alhazen, Book VII, Chap. 55.

Aristotle,

Aristotle, Works: Translated into English, 12 vols., W. D. Ross, ed. (Clarendon, Oxford, 1908–1952), vol. III (Meteorologica), pp. 373b12–373b13.

Bruhns, C.

C. Bruhns, Die astronomische Strahlenbrechung in ihrer historischen Entwickelung (Voigt & Günther, Leipzig, 1861), pp. 13 and 14.

Burchardt, J.

J. Burchardt, Witelo: filosofo della natura del XIII sec.: una biografia (Polskiei Akademii Nauk, Wroclaw, 1984).

Burton, H. E.

Caspar, M.

M. Caspar, Kepler, translated by C. D. Hellman (Abelard-Schuman, New York, 1959).

Cleomedes,

Cleomedes, On the Circular Motion of the Heavenly Bodies, in Ref. 3, p. 284. Cleomedes took this part of his work from Posidonius.

Cohen, M. R.

M. R. Cohen, I. E. Drabkin, A Source Book in Greek Science (Harvard University, Cambridge, Mass., 1958), p. 200.

Davidson, W. P. S.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

de Veer, G.

G. de Veer, Waerachtige Beschryvinge van drie seylagiën ter werelt noyt soo vreemt ghehoort (Claes Claesz, Amsterdam, 1598). A Latin version also appeared in 1598.

G. de Veer (English translation), The true and perfect description of three voyages, so strange and woonderfull that the like has neuer been heard of before (T. Pauier, London, 1609).

Drabkin, I. E.

M. R. Cohen, I. E. Drabkin, A Source Book in Greek Science (Harvard University, Cambridge, Mass., 1958), p. 200.

Exner, F.

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922), Part 1.

Ferguson, K.

K. Ferguson, The Nobleman and His Housedog: Tycho Brahe and Johannes Kepler: the Strange Partnership that Revolutionised Science (Review, London, 2002), p. 126. On one instrument at least, degrees were divided into six parts; then Tycho used a slanted pattern of 10 dots to subdivide each of these into minutes.

Gingerich, O.

O. Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (American Institute of Physics, Molville, New York, 1993), p. 25.

Heath, T. L.

T. L. Heath, A History of Greek Mathematics, 2 vols. (Clarendon, Oxford, 1921).

Kepler,

To quote Kepler, “The story is familiar to everybody, of the journey of the Netherlanders …”; Ref. 67, p. 151.

Kepler, J.

J. Kepler, Ad Vitellionem Paralipomena, quibus Astronomiae Pars Optica Traditur (Apud Claudium Marnium & Haeredes Ioannis Aubrii, Frankfurt, 1604).

Kepler, Johannes

W. H. Donahue (transl.), Optics: Paralipomena to Witelo & Optical Part of Astronomy, by Johannes Kepler (Green Lion Press, Santa Fe, N.M., 2000).

Können, G. P.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

Lehn, W. H.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

W. H. Lehn, “The Novaya Zemlya effect: an arctic mirage,” J. Opt. Soc. Am. 69, 776–781 (1979).
[CrossRef]

Lejeune, A.

A. Lejeune, Recherches sur la Catoptrique Grecque (Académie Royale de Belgique, Brussels, 1957), p. 3.

Liljequist, G. H.

G. H. Liljequist, “Refractive phenomena in the polar atmosphere,” Scientific Results, Norwegian-British-Swedish Antarctic Expedition 1949–1952, Vol. 2, Part 2 (Oslo University, Oslo, 1964). His observation took place on 1 July 1951, when the Sun was 4.3° below the horizon.

Lindberg, D.

Ref. 40, Introduction by D. Lindberg, p. xiii.

Ref. 40, Introduction by D. Lindberg, pp. vii–xiii.

Ref. 40, Introduction by D. Lindberg, p. xxi.

Ref. 40, Introduction by D. Lindberg, p. xx.

Ref. 40, Introduction by D. Lindberg, p. xxiii.

Lindberg, D. C.

D. C. Lindberg, John Pecham and the Science of Optics: Perspectiva Communis (University of Wisconsin, Madison, 1970). Pecham’s book presented Alhazen’s conclusions in a reduced and more readable form, while preserving their central ideas.

D. C. Lindberg, Theories of Vision from al-Kindi to Kepler (University of Chicago, Chicago, Ill., 1976).

F. Risner, ed., Opticae thesaurus. Alhazeni Arabis libri septem, nuncprimum editi. Eiusdem liber De Crepusculis et nubium ascensionibus. Item Vitellonis Thuringopoloni libri X, reprint of the 1572 edition; introduction by D. C. Lindberg (Johnson Reprint Corp., New York, 1972).

Neugebauer, O.

O. Neugebauer, A History of Ancient Mathematical Astronomy, 2 Vols. (Springer-Verlag, Berlin, 1975), pp. 274–298.
[CrossRef]

Pernter, J. M.

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922), Part 1.

Pliny,

Pliny, Natural History, 10 vols., translated by H. Rackham, Loeb Classical Library (Heinemann, London, 1938), vol. I, p. 207.

Ptolemy,

Ref. 2, Ptolemy Book V, Sects. 23–30.

Rashed, R.

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses.” Isis 81, 464–491 (1990).
[CrossRef]

Ross, G. M.

H. E. Ross, G. M. Ross, “Did Ptolemy understand the moon illusion?,” Perception 5, 377–385 (1976).
[CrossRef] [PubMed]

Ross, H. E.

H. E. Ross, “Cleomedes (ca. 1st century A.D.) on the celestial illusion, atmospheric enlargement, and size–distance invariance,” Perception 29, 863–871 (2000).
[CrossRef]

H. E. Ross, G. M. Ross, “Did Ptolemy understand the moon illusion?,” Perception 5, 377–385 (1976).
[CrossRef] [PubMed]

Sabra, A. I.

A. I. Sabra, “The authorship of Liber de crepusculis, an eleventh-century work on atmospheric refraction,” Isis 58, 77–85 (1967). He calls it a work of atmospheric refraction, which it is not.
[CrossRef]

A. I. Sabra, “Psychology vs mathematics: Ptolemy and Alhazen on the moon illusion,” in Mathematics and its Applications to Science and Natural Philosophy in the Middle AgesE. Grant, J. E. Murdoch, eds. (Cambridge University, Cambridge, 1987).

Smith, A. M.

A. M. Smith, “Ptolemy, Alhacen, and Ibn Mu’adh and the problem of atmospheric refraction,” Centaurus 45, 100–115 (2003).
[CrossRef]

A. M. Smith, Ptolemy’s Theory of Visual Perception: an English Translation of the Optics (American Philosophical Society, Philadelphia, 1996), p. 19. This second century work was organized according to the three divisions.

Steenhuisen, F.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

Toomer, G. J.

G. J. Toomer, “Hipparchus,” Dictionary of Scientific Biography, C. Coulston Gillispie, ed. (Scribner’s, New York, 1990), Suppl. I, pp. 207–224.

van der Werf, S.

S. van der Werf, “Het astrolabium,” Cornelis Douwes 160, 20–21 (2004).

van der Werf, S. Y.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

Ver Eecke, P.

P. Ver Eecke, Euclide: L’Optique et la Catoptrique (Albert Blanchard, Paris, 1959). It is considered doubtful if the Catoptrica were actually written by Euclid; the lowest layer in this work stems from Euclid’s time, but subsequent editing added several more recent layers.

Wiedemann, E.

With respect to lenses, Alhazen’s contribution was small. Risner’s edition (Ref. 40) contains no discussion of lenses (in spite of an erroneously inserted diagram showing a burning sphere). Alhazen wrote a separate work entitled On the Burning Sphere, which considered only spherical lenses. See E. Wiedemann, “Über die Brechung des Lichtes in Kugeln nach Ibn al Haitam und Kamâl al Dîn al Fârisî,” Sitzungsber. Phys.-med. Sozietät Erlangen 42, 15–58 (1910). He certainly did not develop a rigorous geometric theory of lenses, as Rashed (Ref. 35) has claimed. Grant (Ref. 42) states that the burning sphere was as close as any medieval scholar came to a study of lenses.

Wilk, S. R.

Interestingly, if we expand Snell’s Law in a Taylor series and truncate it after the quadratic term, we get a remarkably good agreement with the quadratic function implicit in Ptolemy’s data. This occurs, however, only if the expansion is done about the midpoint of the data; Wilk expands about the origin and gets a poor fit. See S. R. Wilk, “Claudiuś Ptolemy’s law of refraction,” Op. Photon. News 1510, 14–17 (2004).

Witelo,

Ref. 40, Witelo, Book X, Chap. 8.

Wolf, R.

R. Wolf, Geschichte der Astronomie (R. Oldenbourg, München, 1877), p. 88.

App. Opt.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” App. Opt. 42, 379–389 (2003).
[CrossRef]

Centaurus

A. M. Smith, “Ptolemy, Alhacen, and Ibn Mu’adh and the problem of atmospheric refraction,” Centaurus 45, 100–115 (2003).
[CrossRef]

Cornelis Douwes

S. van der Werf, “Het astrolabium,” Cornelis Douwes 160, 20–21 (2004).

Isis

A. I. Sabra, “The authorship of Liber de crepusculis, an eleventh-century work on atmospheric refraction,” Isis 58, 77–85 (1967). He calls it a work of atmospheric refraction, which it is not.
[CrossRef]

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses.” Isis 81, 464–491 (1990).
[CrossRef]

J. Opt. Soc. Am.

Ned. Tijdschr. Natuurkd.

S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, “Waerachtighe Beschryvinghe van het Nova-Zembla-Effect,” Ned. Tijdschr. Natuurkd. 66, 120–126 (2000).

Op. Photon. News

Interestingly, if we expand Snell’s Law in a Taylor series and truncate it after the quadratic term, we get a remarkably good agreement with the quadratic function implicit in Ptolemy’s data. This occurs, however, only if the expansion is done about the midpoint of the data; Wilk expands about the origin and gets a poor fit. See S. R. Wilk, “Claudiuś Ptolemy’s law of refraction,” Op. Photon. News 1510, 14–17 (2004).

Perception

H. E. Ross, G. M. Ross, “Did Ptolemy understand the moon illusion?,” Perception 5, 377–385 (1976).
[CrossRef] [PubMed]

H. E. Ross, “Cleomedes (ca. 1st century A.D.) on the celestial illusion, atmospheric enlargement, and size–distance invariance,” Perception 29, 863–871 (2000).
[CrossRef]

Sitzungsber. Phys.-med. Sozietät Erlangen

With respect to lenses, Alhazen’s contribution was small. Risner’s edition (Ref. 40) contains no discussion of lenses (in spite of an erroneously inserted diagram showing a burning sphere). Alhazen wrote a separate work entitled On the Burning Sphere, which considered only spherical lenses. See E. Wiedemann, “Über die Brechung des Lichtes in Kugeln nach Ibn al Haitam und Kamâl al Dîn al Fârisî,” Sitzungsber. Phys.-med. Sozietät Erlangen 42, 15–58 (1910). He certainly did not develop a rigorous geometric theory of lenses, as Rashed (Ref. 35) has claimed. Grant (Ref. 42) states that the burning sphere was as close as any medieval scholar came to a study of lenses.

Other

Ref. 40, Alhazen, Book VII, Chaps. 15 and 16.

Ref. 2, Ptolemy Book V, Sects. 23–30.

The vertical case follows from Ptolemy, but he never stated it explicitly for separation between stars. Alhazen, on the other hand, never concerned himself with the shift in apparent elevation of stars.

Ref. 40, E. Alhazen, Book VII, Chap. 55.

A. I. Sabra, “Psychology vs mathematics: Ptolemy and Alhazen on the moon illusion,” in Mathematics and its Applications to Science and Natural Philosophy in the Middle AgesE. Grant, J. E. Murdoch, eds. (Cambridge University, Cambridge, 1987).

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922), Part 1.

Ref. 32, p. 379.

Ref. 2, p. 49.

Ref. 2, p. 242.

Ibn Sahl applied his insight to the analysis of lenses; in particular, he solved the problem of the anaclastic lens: a paraboloid lens that will bring parallel incident rays to a perfect focus.

His full name was Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham. While he is known to most as Alhazen, scholars now believe that this should be written as Alhacen.

Ref. 35, p. 491.

Ref. 25, p. 209.

F. Risner, ed., Opticae thesaurus. Alhazeni Arabis libri septem, nuncprimum editi. Eiusdem liber De Crepusculis et nubium ascensionibus. Item Vitellonis Thuringopoloni libri X, reprint of the 1572 edition; introduction by D. C. Lindberg (Johnson Reprint Corp., New York, 1972).

Ref. 2, p. 56.

E. Grant, ed., A Source Book in Medieval Science (Harvard University, Cambridge, Mass., 1974), p. 420.

Ref. 40, Alhazen, Book VII, Chap. 2.

Hipparchus probably, and Posidonius certainly, tried to explain the Moon illusion.

Ref. 20, p. 864.

“Posidonius,” Dictionary of Scientific Biography, Ref. 15. The writings of Posidonius have not survived; we know of his work primarily through Cleomedes himself.

Ref. 9, vol. II, pp. 235–238. That Posidonius studied the Moon illusion is independently confirmed by Strabo (see Ref. 20, p. 864).

D. C. Lindberg, Theories of Vision from al-Kindi to Kepler (University of Chicago, Chicago, Ill., 1976).

Ptolemy’s Almagest, translated and annotated by G. J. Toomer (Springer-Verlag, New York, 1984).

R. Wolf, Geschichte der Astronomie (R. Oldenbourg, München, 1877), p. 88.

O. Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (American Institute of Physics, Molville, New York, 1993), p. 25.

Ref. 2, p. 45. Ptolemy’s refraction measurements appear in many places, e.g., in Ref. 42, cited below.

The one short statement on refraction in Euclid’s Catoptrica is considered to be a later addition; see Ref. 6, p. 99.

T. L. Heath, A History of Greek Mathematics, 2 vols. (Clarendon, Oxford, 1921).

Archimedes appears to be the originator of the coin in the cup experiment: a coin is placed in the bottom of a cup, out of sight to an observer who is looking into the cup at an angle. When the cup is filled with water, refraction makes the coin visible. See Ref. 9, Vol. 1, p. 444. This is the statement referred to in Note 8.

This assumption also appears in Euclid’s Catoptrica,probably within the lowest layer, which predates Archimedes.

Ref. 1, p. 179.

Pliny, Natural History, 10 vols., translated by H. Rackham, Loeb Classical Library (Heinemann, London, 1938), vol. I, p. 207.

O. Neugebauer, A History of Ancient Mathematical Astronomy, 2 Vols. (Springer-Verlag, Berlin, 1975), pp. 274–298.
[CrossRef]

G. J. Toomer, “Hipparchus,” Dictionary of Scientific Biography, C. Coulston Gillispie, ed. (Scribner’s, New York, 1990), Suppl. I, pp. 207–224.

Cleomedes, On the Circular Motion of the Heavenly Bodies, in Ref. 3, p. 284. Cleomedes took this part of his work from Posidonius.

There is some controversy about when Cleomedes lived: The consensus is first century A.D. although others say second or even fourth century. The later dates appear unlikely, because he never refers to the well known second century work of Ptolemy.

Ref. 3, p. 284.

Ref. 9, vol. II, p. 236.

Ref. 40, Liber de crepusculis, p. 288. This edition also gives the height of the atmosphere as 52000 paces (p. 287). The Roman pace was 5 ft. (1.7 m), but its length varied over the centuries.

K. Ferguson, The Nobleman and His Housedog: Tycho Brahe and Johannes Kepler: the Strange Partnership that Revolutionised Science (Review, London, 2002), p. 126. On one instrument at least, degrees were divided into six parts; then Tycho used a slanted pattern of 10 dots to subdivide each of these into minutes.

A. Lejeune, Recherches sur la Catoptrique Grecque (Académie Royale de Belgique, Brussels, 1957), p. 3.

A. M. Smith, Ptolemy’s Theory of Visual Perception: an English Translation of the Optics (American Philosophical Society, Philadelphia, 1996), p. 19. This second century work was organized according to the three divisions.

M. R. Cohen, I. E. Drabkin, A Source Book in Greek Science (Harvard University, Cambridge, Mass., 1958), p. 200.

Ref. 3, p. 93.

Aristotle, Works: Translated into English, 12 vols., W. D. Ross, ed. (Clarendon, Oxford, 1908–1952), vol. III (Meteorologica), pp. 373b12–373b13.

P. Ver Eecke, Euclide: L’Optique et la Catoptrique (Albert Blanchard, Paris, 1959). It is considered doubtful if the Catoptrica were actually written by Euclid; the lowest layer in this work stems from Euclid’s time, but subsequent editing added several more recent layers.

The effect is actually refraction confined to a very narrow zone; on a larger scale, it looks just like reflection.

G. H. Liljequist, “Refractive phenomena in the polar atmosphere,” Scientific Results, Norwegian-British-Swedish Antarctic Expedition 1949–1952, Vol. 2, Part 2 (Oslo University, Oslo, 1964). His observation took place on 1 July 1951, when the Sun was 4.3° below the horizon.

Ref. 67, p. 96

Ref. 14, Book VII. His star tables are given to a resolution of one sixth of a degree.

Ref. 40, Introduction by D. Lindberg, pp. vii–xiii.

J. Burchardt, Witelo: filosofo della natura del XIII sec.: una biografia (Polskiei Akademii Nauk, Wroclaw, 1984).

Ref. 40, Introduction by D. Lindberg, p. xiii.

Ref. 40, Introduction by D. Lindberg, p. xx.

Ref. 42, pp. 424–426.

Ref. 40, Witelo, Book X, Chap. 8.

Ref. 40, Introduction by D. Lindberg, p. xxi.

D. C. Lindberg, John Pecham and the Science of Optics: Perspectiva Communis (University of Wisconsin, Madison, 1970). Pecham’s book presented Alhazen’s conclusions in a reduced and more readable form, while preserving their central ideas.

Ref. 40, Introduction by D. Lindberg, p. xxiii.

C. Bruhns, Die astronomische Strahlenbrechung in ihrer historischen Entwickelung (Voigt & Günther, Leipzig, 1861), pp. 13 and 14.

The Air Almanac 1987 (United States Naval Observatory, Washington, D.C., 1986).

Ref. 25, pp. 185–186.

M. Caspar, Kepler, translated by C. D. Hellman (Abelard-Schuman, New York, 1959).

W. H. Donahue (transl.), Optics: Paralipomena to Witelo & Optical Part of Astronomy, by Johannes Kepler (Green Lion Press, Santa Fe, N.M., 2000).

J. Kepler, Ad Vitellionem Paralipomena, quibus Astronomiae Pars Optica Traditur (Apud Claudium Marnium & Haeredes Ioannis Aubrii, Frankfurt, 1604).

Ref. 67, p. 149.

Mästlin saw the eclipse on 7 July 1590 (Old Style).

In a later work, Kepler stated that the solar parallax could not be more than 1’.

Ref. 67, p. 144.

Ref. 67, p. 141.

G. de Veer, Waerachtige Beschryvinge van drie seylagiën ter werelt noyt soo vreemt ghehoort (Claes Claesz, Amsterdam, 1598). A Latin version also appeared in 1598.

G. de Veer (English translation), The true and perfect description of three voyages, so strange and woonderfull that the like has neuer been heard of before (T. Pauier, London, 1609).

To quote Kepler, “The story is familiar to everybody, of the journey of the Netherlanders …”; Ref. 67, p. 151.

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Figures (13)

Fig. 1
Fig. 1

Refraction.

Fig. 2
Fig. 2

Cleomedes’ model for the paradoxical eclipse.

Fig. 3
Fig. 3

Magnification.

Fig. 4
Fig. 4

Ptolemy.

Fig. 5
Fig. 5

Ptolemy’s model for refraction in air.

Fig. 6
Fig. 6

Alhazen.

Fig. 7
Fig. 7

Twilight model of Ibn Mu’adh.

Fig. 8
Fig. 8

Witelo’s model for the Moon illusion.

Fig. 9
Fig. 9

Tycho Brahe.

Fig. 10
Fig. 10

Tycho’s measured refraction in air.

Fig. 11
Fig. 11

Kepler.

Fig. 12
Fig. 12

Kepler’s model for multiple reflections.

Fig. 13
Fig. 13

Novaya zemlya effect.

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