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.
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, 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.
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.
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.
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. 32, p. 379.
Ref. 2, p. 49.
Ref. 2, p. 242.
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.
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.