Time in History: Views of Time From Prehistory to the Present Day

by G. J. Whitrow

  2. calculate the remainder d, when 19a + 24 is divided by 30;

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  (3) calculate the remainder e, when 2b + 4(c - 1) + 6d is divided by 7;

  (4) if the sum d + e does not exceed 9, Easter occurs on the day in March given by d + e + 22, but if d + e exceeds 9, Easter occurs on the day in April given by d + e - 9.

  For example, for the year 1988, we have N = 88, a = 12, b = 0, c = 4, d = 12, e = 0, and hence Easter falls on 3 April.

  Unfortunately, Gauss's neat solution fails to give correct results for some years after 4200, and so in 1817 the problem was further inves- tigated by the French astronomer Jean-Baptiste Delambre ( 1749- 1822). Sixty years later a thorough re-examination of the problem was made by Samuel Butcher, Bishop of Meath.1 In 1876 a New York correspondent had sent, without proof, to the weekly scientific periodical Nature a rule for the determination of Easter which, unlike Gauss's rule, is subject to no exceptions.2 Butcher showed that this rule followed from Delambre's analytical solution. For any given year n the rule is as follows:

  DivideByQuotientRemainder

  n 19 a

  n 100 b c

  b 4 d e

  b + 8 25 f

  b - f + 1 3 g

  19a + b - d - g + 15 30 h

  c 4 i k

  32 + 2e + 2i - h - k 7 l

  a + 11h + 22l 451 m

  h + l - 7m + 114 31 p q

  The number of the month in which Easter falls is given by p and the day of the month by q + 1. For example, for the year 1988 this calculation yields p = 4, q = 2, and hence we conclude that Easter Day that year is 3 April. The earliest day on which Easter can occur is 22 March and the latest 25 April. Uspensky and Heaslet provide an elementary mathematical discussion of calendrical problems, including the calculation of Easter.3

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  References

  Preface

  1 G. J. Whitrow, Tlte Natural Philosophy of Time ( London and Edinburgh,; Nelson, 1961; Oxford: Clarendon Press; 2nd edn., 1980).

  2 S. Toulmin and J. Goodfield, The Discovery of Time ( London: Hutchinson, 1965; Harmondsworth: Penguin Books, 1967).

  3 R. Wendorff, Zeit und Kultur: Geschichte des Zeitbewusstseins in Europa ( Wiesbaden: Westdeutscher Verlag, 1980).

  4 H. Trivers, The Rhythm of Being: A Study of Temporality ( New York: Philosophical Library, 1985), part III: 'Time and History'.

  Chapter 1. Awareness of Time

  1 R. Wallis, Le Traps, quatrieme dimension de lesprit ( Paris: Flammarion, 1966), 51 ff.

  2 J. Piaget, The Child's Conception of Time, trans. A. J. Pomerans ( London: Routledge & Kegan Paul, 1969).

  3 A. E. Wessmann and B. S. Gorman, The Personal Experience of Time ( New York: Plenum Press, 1977), 8.

  4 E. Michaud, Essai sur lorganisation de la connaissance entre 10 et 14 ans ( Paris: Vrin, 1949).

  5 P. M. Bell, "'Sense of time'", New Scientist ( 15 May 1975), 406.

  6 C. Ralling, "'A vanishing race'", Listener ( 16 July 1959), 87.

  7 W. Koehler, The Mentality of Apes, trans. from 2nd rev. edn. Ella Winter ( Harmondsworth: Penguin Books, 1957), 234.

  8 S. Walker, Animal Thought ( London: Routledge & Kegan Paul, 1983), 190.

  9 B. L. Whorf, Language, Thought and Reality, ed. J. B. Carroll ( Cambridge, Mass.: MIT Press, 1956), 57-64.

  10Ibid. 58.

  11 S. C. McCluskey, "'The astronomy of the Hopi Indians'", Journal for the History of Astronomy, 8 ( 1977), 174-95.

  12 E. E. Evans-Pritchard, Witchcraft, Oracles and Magic among the Azande ( Oxford: Clarendon Press, 1937), 347.

  13 E. E. Evans-Pritchard, The Nuer: A Description of the Modes of Livelihood and Political Istitutions ofa Nilotic People ( Oxford: Clarendon Press, 1940), 103.

  14Ibid. 105.

  15Ibid. 108.

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  Chapter, 2 Describing Time

  1 E. H. Lenneberg, Biological Foundations of Language ( New York: Wiley, 1968), 106.

  2 C. M. Bowra, "'Some aspects of speech'", in In General and Particular ( London: Weidenfeld & Nicolson, 1966), 14.

  3 R. E. Passingham, "'Broca's area and the origin of human vocal skill'", Phil. Trans. Roy. Soc. ( London), B292 ( 1981), 167-75.

  4 G. Steiner, After Babel. Aspects of Language and Translation ( Oxford University Press, 1975), 157.

  5 S. Fleischman, The Future in Thought and Language ( Cambridge University Press, 1982), 50.

  6 Steiner, op. cit. (above, n. 4), 139.

  7 Whitrow, Natural Philosophy of Time ( 2nd edn.; preface, n. 1), 174 ff.

  8 M. P. Nilsson, Primitive Time-reckoning (Lund: C. W. K. Gleerup, 1920), 9-10.

  Chapter 3 Time at the Dawn of History

  1 P. Radin, Primitive Man as Philosopher, 2nd edn. ( New York: Dover, 1957).

  2Ibid. 244.

  3 A. Marshack, "'Some implications of the Palaeolithic symbolic evidence for the origins of language'", Current Anthropology, 17 ( 1976), 274.

  4 R. S. Solecki, "'Shanidar IV, a Neanderthal flower burial in northern Iraq'", Science, 190 ( 1975), 880.

  5 D. C. Heggie, Megalithic Science: Ancient Mathematics and Astronomy in North-west Europe ( London: Thames & Hudson, 1981).

  6 S. G. F. Brandon, Time and Mankind ( London: Hutchinson, 1951), 33.

  7 H. Frankfortet al., Before Philosophy ( Harmondsworth: Penguin Books, 1949), 35.

  8 O. Neugebauer, The Exact Sciences in Antiquity ( Providence, RI: Brown University Press, 1957), 81.

  9 H. E. Winlock, "'The origin of the ancient Egyptian calendar'", Proc. Amer. Phil. Soc., 83 ( 1940), 447.

  10 J. H. Breasted, "'The beginnings of time-measurement and the origins of our calendar'", in Time and its Mysteries, Series I ( New York University Press, 1936), 80.

  11 T. G. H. James, An Introduction to Ancient Egypt ( London: British Museum Publications, 1979), 125.

  12 Neugebauer, loc. cit. (above, n. 8).

  13 S. N. Kramer, The Sumerians ( Chicago and London: University of Chicago Press, 1963), 328.

  14 J. G. Gunnell, Political Philosophy and Time ( Middleton, Conn.: Wesleyan University Press, 1968), 40.

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  15 E. Voegelin, The Ecumenic Age (vol. 4 of Order and History) ( Baton Rouge: Louisiana State University Press, 1980), 84.

  16 G. Contenau, Everyday Life in Babylon and Anyria, trans. K. R. and A. R. MaxweH-Hyslop ( London: Edward Arnold, 1954), 213.

  17 N. K. Sanders, The Epic of Gilgamesh ( Harmondsworth: Penguin Books, 1960), 104.

  18 D. Pingree, "'Astrology'", in P. P. Wiener (ed.), Dictionary of the History of Ideas ( New York: Scribner, 1973), i. 118.

  19 A. Sachs, "'Babylonian horoscopes'", Journal of Cutteifortn Studies, 6 ( 1952), 49.

  20 Seneca, Nat. Quaest. 111. 29. 1 ( London: Heinemann, 1971), 286.

  21 O. Neugebauer, "'The history of ancient astronomy: problems and methods'", Publications of the Astronomical Society of the Pacific, 58 ( 1946), no. 340, 33.

  22 O. Neugebauer, A History of Ancient Mathematical Astronomy ( Berlin: Springer Vertag, 1975). i. 4.

  23Ibid. ii. 593.

  24 R. C. Zaelmer, Daum and Twilight of Zoroastrianism ( London: Weidenfeld & Nicolson, 1961), 55.

  25 R. C. Zaehner, Zunpan: A Zoroastrian Dilemma ( Oxford: Clarendon Press, 1955), 410.

  26 S. G. F. Brandon, Creation Legends of the Ancient Near East ( London: Hodder & Stoughton, 1963), 206.

  27 W. Hartner, "'The Young-Avestan and Babylonian calendars and the ante- cedents of precession'", Journal for the History of Astronomy, 10 ( 1979), 1-22.

  28 S. H. Taqizadah, Old Iranian Calendars ( London: Royal Asiatic Society, 1938).

  29 E. Yarshater, "'Time-reckoning'", in Cambridge History of Iran ( Cambridge University Press, 1982), ii. 790.

  Chapter 4. Time in Classical Antiquity

  1 Gunnell op. cit. (ch. 3, n. 14), 15.

  2 F. M. Cornford, From Religion to Philosophy ( London: Edward Arnold, 1912), 181.

  3 W. K. C. Guthrie, "'The religion and mythology
of the Greeks'", in The Cambridge Ancient History, rev. edn. ( Cambridge University Press, 1961), ii, ch. 40, 39-40.

  4 H. Lloyd Jones, The Justice of Zeus ( Berkeley: University of California Press, 1971), 5-6 and 166-7 n. 23.

  5 W. Jaeger, The Theology of the Early Greek Philosophers, trans. E. S. Robinson ( London: Oxford University Press, 1967), 35.

  6 Whitrow, Natural Philosophy of Time (ch. 2, n. 7), 190-200.

  7 Nemesius, Bishop of Emesa, in E. Bevan, Later Greek Religion ( London: Dent, 1927), 30-1.

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  8 L. Edelstein, The Idea of Progress in Classical Antiquity ( Baltimore: Johns Hopkins University Press, 1967), xxi.

  9 R. Drews, The Greek Accounts of Eastern History ( Cambridge, Mass.: Harvard University Press, 1973), 35-6.

  10 M. I. Finley, "'Thucydides the moralist'", in Aspects of Antiquity ( Harmondsworth: Penguin Books, 1977), 53.

  11 A. Momighano, "'The place of Herodotus in the history of historiography'", in Studies in Historiography ( London: Weidenfeld & Nicolson, 1966), 130.

  12 J. de Romilly, Time in Greek Tragedy ( Ithaca: Cornell University Press, 1968), 5-6.

  13 E. R. Dodds, "'Progress in classical antiquity'", in P. P. Wiener (ed.), Dictionary of the History of Ideas ( New York: Scribner, 1973), iii. 633.

  14 A . Momigliano, "'Time in ancient historiography'", in History and Theory, 1966, Suppl. 6 ( 'History and the concept of time'), 10.

  15 W. K. C. Guthrie, In the Beginning: Some Greek Views on the Origin of Life and the Early State of Man ( London: Methuen, 1957), 65.

  16 Momigliano, 'Time in ancient historiography' (above, n. 14), 13.

  17 P. Duhem, Le Système du monde ( Paris: Hermann, 1954), ii (new edn.), 299.

  18 Alexander of Aphrodisias, On Destiny. Addressed to the Emperors, trans. A. Fitzgerald ( London: Scholaris Press, 1931), 25.

  19 Kramer, op. cit. (ch. 3, n. 13), 262.

  20 W. K. C. Guthrie, A History of Greek Philosophy ( Cambridge University Press, 1969), iii. 82.

  21 Guthrie, In the Beginning (above, n. 15), 79.

  22 Guthrie, A History of Greek Philosophy (above, n. 20), iii. 292.

  23 J. V. Noble and D. J. de Solla Price, "'The water-clock in the Tower of Winds'", Amer. J. Archaeol., 72 ( 1968), 345-55.

  24 T. C. Vriezen, The Religion of Ancient Israel, trans. H. Hoskins ( London: Lutterworth Press, 1969), 243.

  25 O. Cullmann, Christ and Time, trans. F. V. Filson ( London: SCM Press, 1951), 51.

  26 Gunnell, op. cit. (ch. 3, n. 14), 75.

  27 G. W. Trompf, The Idea of Historical Recurrence in Western Thought ( Berkeley: University of California Press, 1979), 134.

  28 W. O. E. Oesterley, The Evolution of the Messianic Idea ( London: Isaac Pitman & Sons, 1908), 206.

  29 Gunnell, op. cit. (ch. 3, n. 14), 63-4.

  30 H. Frankfort, Kingship and the Gods: A Study of Near Eastern Religion and the Integration of Society and Nature ( Chicago: University of Chicago Press, 1978; Phoenix edn.), 343-4.

  31 E. Voegehn, Israel and Revelation (vol. 1 of Order and History) ( Baton Rouge: Louisiana State University Press, 1956).

  32 G. Van Seters, In Searrh of History. Historiography in the Ancient World andthe Origins of Biblical History

  -197-

  the Origins of Biblical History ( New Haven and London: Yale University Press, 1983), 241.

  33 H. Webster, Rest Days. A Study in Early Law and Morality ( New York: Macmillan, 1916), 252.

  34Ibid. 254.

  35 Vriezen, op. cit. (above, n. 24), 234.

  36 M. Testuz, Les Idtes religieuses du Livre des Jubilées ( Geneva: Droz; Paris: Minard, 1960), 136.

  37 L. Casson, Travel in the Ancient World ( London: Allen & Unwin, 1974), 155.

  38 R. Syme, The Roman Revolution ( London: Oxford University Press, 1960), 315-16.

  39 J. T. Shotwell, The History of History ( New York: Columbia University Press, 1939), 301.

  40 E. R. Curtius, European Literature and the Latin Middle Ages, trans. W. R. Trask ( New York: Pantheon Books, 1953), 252.

  41 Lucretius, The Nature of the Universe, trans. R. E. Latham ( Harmondsworth: Penguin Books, 1951), 40-1.

  42 P. Brown, The World of Latte Antiquity. From Marcus Aurelius to Muhammad ( London: Thames & Hudson, 1971), 62.

  43Ibid.

  44 H.-C. Puech, "'Gnosis and time'", in Man and Time. Papers from the Eranos Yearbooks ( London: Roudledge & Kegan Paul, 1958), 61.

  45 F. Cumont, The Mysteries of Mithra, trans. T. J. McCormack ( Chicago: Open Court, 1903), 1.

  46Ibid. 39.

  47 M. J. Vennaseren, "'A magical time god'", in J. R. Hinnells (ed.), Mithraic Studies: Proceedings of the First International Congress of Mithraic Studies, 1971 ( Manchester University Press, 1975), 451.

  48 E. A. Budge Wallis, Orisis and the Egyptian Resurrection ( London: Philip Lee Warner, 1911), i. 60.

  49 Vermaseren, op. cit. (above, n. 47), 456.

  50 S. Sambursky and S. Pines, The Concept of Time in Late Neoplatonism ( Jerusalem: Israel Academy of Sciences and Humanities, 1971), 11.

  51 J. F. Callahan, Four Views of Time in Ancient Philosophy ( Cambridge, Mass.: Harvard University Press, 1948), 124.

  52 C. N. Cochrane, Christianity and Classical Culture. A Study of Thought and Action from Augustus to Augustine ( London: Oxford University Press, 1974), 186.

  53 E. Frank, Philosophical Understanding and Religious Truth ( New York: Oxford University Press, 1945), 68.

  54 J. Baillie, The Belief in Progress ( Cambridge University Press, 1951), 76.

  55 O. Pedersen, "The ecclesiastical calendar and the life of the Church'", in G. V.,Coyne, M. A. Hoskin, and O. Pedersen (eds.), Gregorian Reform of

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  the Calendar ( Vatican City: Pontifica Academica Scientiarum, 1983), 22.

  56 Frank, op. cit. (above, n. 53), 70.

  57 R. L. Poole, "'The beginning of the year in the middle ages'", in Studies in Chronology and History ( Oxford: Clarendon Press, 1934), 1-27.

  58 E. J. Bickerman, Chronology of the Ancient World ( London: Thames & Hudson, 1968), 77.

  59 F. K. Ginzel, Handbuch der Chronologie, vol. iii ( Leipzig: Hinrichs, 1914), 115.

  60 F. H. Colson, The Week: An Essay on the Origin and Development of the Seven-day Cycle ( Cambridge University Press, 1926).

  61 Bickerman, op. cit. (above, n. 58), 61.

  62 H. I. Marrou, A History of Education in Antiquity, trans. G. Lamb ( London: Sheed & Ward, 1956), 148.

  63 Cochrane, op. cit. (above, n. 52), 330-1.

  64 G. Teres, "'Time computations and Dionysius Exiguus'", Journal for the History of Astronomy, 15 ( 1984), 177-88.

  Chapter5 Time in the Middle Ages

  1 R. W. Southern, Medieval Humanism and Other Studies ( Oxford: Blackwell, 1970), 3.

  2 M. L.W. Laistner, "'The library of the Venerable Bede'", in A. Hamilton Thompson (ed.), Bede, His Life, Times, and Writings: Essays in Commemoration of the Twelfth Centenary of his Death ( Oxford: Clarendon Press, 1935), 238.

  3 A. Bryant, A History of Britain and the British People, vol. 1: Set in a Silver Sea ( London: Collins, 1984), 29.

  4 Bede, The Ecclesiastical History of the English Nation ( London: Dent, 1935; Everyman edn.), 152.

  5 R. L. Poole, "'Imperial influences on the forms of Papal documents'", in Studies in Chronology and History ( Oxford: Clarendon Press, 1934), 178.

  6 J. A. Burrow, The Ages of Man: A Study in Medieval Writing and Thought ( Oxford: Clarendon Press, 1986), 29-30.

  7 Southern, op. cit. (above, n. 1), 158.

  8Ibid. 162.

  9 C. H. Haskins, Studies in the History of Medieval Science, 2nd edn. ( Cambridge, Mass.: Harvard University Press, 1927), 117; see also Southern, op. cit. (above, n. 1), 166-7; and Bodleian MS. Auct. F.1.9, fo. 90.