The So-called Seqed and Scientific cradle of the angle θ in ancient Egypt

Volume 3, Issue 2, April 2018     |     PP. 86-105      |     PDF (4728 K)    |     Pub. Date: April 24, 2018
DOI:    2405 Downloads     19214 Views  

Author(s)

Ayman Waziry, Associate Professor, Department of Egyptology, Faculty of Archaeology, Fayoum University, Main building of Faculty of Archeology, Postcode: 63514, Al-Fayoum City, Fayoum Governorate , Egypt

Abstract
Science is a matter of asking for information by specifying the difference between fact and opinion. Fact in a scientific context is a generally accepted reality, but still open to scientific inquiry, as opposed to an absolute truth, which is not a part of science. The hypotheses and theories are generally based on objective inferences, unlike opinions, which are generally based on subjective influences. Therefore and hence we can say ‘the facts versus the opinions’. In determining the factual length of an ancient Egyptian cubit or the true value of a Seqed, as discussed in the Rhind and Moscow Mathematical Papyruses, one will notice confusing differences. Varying opinions have been advanced, e.g., by Carter & Gardiner, Iversen, Hayes, Budge, Noblecourt, Lorenzen, Müller, Gardiner, Stricker, Gay, Legon, Gay & Shute, Naguib, Roik, Shaffer, and the Encyclopaedia Britannica. Possibly, though, also ancient Egyptian sources might have offered different values. The question now is how to measure a Seqed (sQd) in ancient Egypt accurately? And is the value of the Seqed or the angle θ correct applying the rules of modern trigonometry? The question arises as to whether the problems of the Seqed are actually mirrored in the ancient Egyptian architecture methods and construction techniques?

Keywords
Mathematics, Dimensions, Digits, Seqed, Tangent Angle, Cotangent

Cite this paper
Ayman Waziry, The So-called Seqed and Scientific cradle of the angle θ in ancient Egypt , SCIREA Journal of Mathematics. Volume 3, Issue 2, April 2018 | PP. 86-105.

References

[ 1 ] Waziry, A., (2016). Evidences on Harbingers of Mensuration Methodology in Ancient Egyptian Mathematics and Geometry, International Journal of Scientific & Engineering Research, Vol. 7, Issue 12,pp.1624-1638. 19
[ 2 ] Eisenlohr, A., (1877). Ein Mathematisches Handbuch der alten Aegypter (Papyrus Rhind des British Museum), 2 Vols. Hinrichs, Leipzig. (Digitized 30 Jan 2008 by the University of Michigan).
[ 3 ] Herz-Fischler, R., (2000). The Shape of the Great Pyramid, Wilfrid Laurier University Press, Canada.
[ 4 ] Petrie, W.M.F., (1877). Inductive Metrology, Saunders, London.
[ 5 ] Carter, H.& Gardiner ,A.H., (1917).The tomb of Ramesses IV and the Turin plan of a royal tomb, JEA, Vol. 4 , pp. 130-158.
[ 6 ] Weeks, K.R., (1979).The Berkeley Map of the Theban Necropolis: Report of the Second Season, 1979, University of California, Berkeley.
[ 7 ] Iversen, E., (1955). Canon and Proportions in Egyptian Art, Sidgwick & Jackson Publishers, London.
[ 8 ] Hayes, W., (1957). Varia from the Time of Hatshepsut, MDAIK, Vol. 15, pp.78-90.
[ 9 ] Budge, E.A.W., (1960). An Egyptian Hieroglyphic Dictionary, with an Index of English Words, King List and Geographical List with Indexes, List of Hieroglyphic Characters, Coptic and Semitic Alphabets, etc. Vol. I. A-Kha, Frederick Ungar Publishing Co., New York.
[ 10 ] Desroches-Noblecourt, Ch., (1965). Life and Death of a Pharaoh Tutankhamen, Doubleday & Company press, New York.
[ 11 ] Lorenzen, E., (1970). Along the line where columns are set. Book II continuing Technological Studies in Ancient Metrology, Translated by John R.B. Gosney, Nyt Nordisk Forlag, Arnold Busck, Copenhagen.
[ 12 ] Müller, H.W., (1973). Der Kanon in der ägyptischen Kunst, Der "Vermessene" Mensch. Anthropometrie in Kunst und Wissenschaft, München, pp. 9-31.
[ 13 ] Gardiner, A. H., (1973).Egyptian Grammar, 3rd. ed. Cambridge University Press, London.
[ 14 ] Stricker, B. H., (1976). De Maat der dingen, B.V. Noord-Hollandsche Uitgeversmaatschappij, Amsterdam /Mededelingen der Koninklijke Nederlandse Maatschappij van Wetenschappen, Afd. Letterkunde/ Amsterdam, Vol. 39.No.7,pp.1-36.
[ 15 ] Gillings, R., (1982). Mathematics in the Time of the Pharaohs, Dover Publications, USA.
[ 16 ] Robins, G., (1985). Standing Figures in the Late Grid System of the 26th Dynasty, SAK, Vol. 12, pp. 101-116.
[ 17 ] Gay, R., & Shute, C.D., (1985). Mathematical Bases of Ancient Egyptian Architecture and Graphic Art, Historia Mathematica, Vol. 12, pp.107-122.
[ 18 ] Legon, J.A.R., (1988). The Design of the Pyramid of Khufu, DE, Vol. 12, pp. 41-48.
[ 19 ] Gay, R. & Shute, C.D., (1990). Irrational Numbers and Pyramids, DE, Vol. 18, pp.43-53.
[ 20 ] Naguib, V.,(1991).The Rod (nbj) and Its Use in Egyptian Architecture, GM, Vol. 121, pp. 101-110.
[ 21 ] Kanawati, N., (1986). The Rock Tombs of El Hawawish, Vol. V: The Cemetery of Akhmim, Macquarie Ancient History Association Publisher, Sydney.
[ 22 ] Bard, K.A., & Shubert, S.B., (Eds.) (1999). Akhmim, Encyclopedia of the Archaeology of Ancient Egypt, Routlege/Taylor & Francis Group, London & New York, pp.124-127.
[ 23 ] Thompson, E.M., (2001). A Study of the Architecture of the Cemetery of El-Hawawish at Akhmim in Upper Egypt in the Old Kingdom, Thesis submitted for the Degree of Master of Arts within Macquarie University, Division of Humanities, School of Ancient History,Sydney.
[ 24 ] Arnold, D., (1991). Building in Egypt: Pharaonic Stone Masonry, Oxford University Press, Oxford.
[ 25 ] Lauer, J.P., (1930). Étude sur quelques monuments de la IIIe dynastie (Pyramide à degrésde Saqqarah), ASAE, Vol. 30, pp.137-146.
[ 26 ] Lauer, J.P., (1931).Étude sur quelques monuments de la IIIe dynastie, ASAE, Vol. 31,pp.49-64. 20
[ 27 ] Roik, E., (1993). Das Längenmasssystem im Alten Ägypten, Christian-RosenkreuzVerlag, Hamburg.
[ 28 ] Legon, J.A.R., (1996).The Cubit and the Egyptian Canon of Art, DE, Vol. 35, pp.61-76.
[ 29 ] Shaffer, B.S., (1996). Treasures of the Ancients, C.F.I. Press, U.S.A.
[ 30 ] Merriam-Webster, (2016). Merriam-Webster’s International Encyclopædia Britannica since 1828-2016. Enc. Britannica Inc. http://corporate.britannica.com (at July 2016).
[ 31 ] Petrie, W.M.F., (1883).The Pyramids and Temples of Gizeh, Field & Tuer, The Leadenhall Press, London.
[ 32 ] Edwards, I.E.S., (1986). The Pyramids of Egypt: Revised Edition, Penguin Books,London.
[ 33 ] Petrie, W.M.F., (1892). Medium, David Nutt publisher, London.
[ 34 ] Lightbody, D., (2008). Egyptian Tomb Architecture: The Archaeological Facts of Pharaonic Circular Symbolism. British Archaeological Reports International Series,S1852, pp. 22–27.
[ 35 ] Petrie, W.M.F., (1940). Wisdom of the Egyptians, British School of Archaeology in Egypt and B. Quaritch Ltd., London.
[ 36 ] Erman, E., & Grapow, H., (1971). Wörterbuch der ägyptischen Sprache, 7 Vols. Akademie Verlag Publishing, Berlin.
[ 37 ] Hannig, R., (2005). Groβes Hand Wörterbuch, Ägyptische Deutsch, Die Sprache der Pharaonen Marburger Edition, Marburg – Germany.
[ 38 ] Borger, R., (2005). Ägyptische Wörterbuch II; Mittleres Reich und Zweite Zwischenzeit, der mich das Systematische Arbeitenlehrte, Marburg University Edition, Marburg –Germany.
[ 39 ] Faulkner, R. O., (1991) A concise dictionary of Middle Egyptian, Griffith Institute Ashmolean Museum Press, Oxford.
[ 40 ] Dickson, P., (2006), Dictionary of Middle Egyptian in Gardiner Classification Order, Creative Commons Press, USA.
[ 41 ] Peet, T.E., (1923). The Rhind Mathematical Papyrus British Museum 10057 and 10058: introduction, transcription, translation and commentary, Univ. Pr. of Liverpool; Hodder & Stoughton, London.
[ 42 ] Gunn, B. & Peet, T.E., (1929). Four geometrical problems from the Moscow mathematical papyrus, JEA, Vol. 15, pp.167-185.
[ 43 ] Robins, G., (1987). The Rhind Mathematical Papyrus: an ancient Egyptian text, London.
[ 44 ] Imhausen, A., (2003). Ägyptische Algorithmen: eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden.
[ 45 ] Borchardt, L., (1913). Das Grabdenkmal des Königs S'aḥu-Re, Vol. II, Ausgrabungen derDeutschen Orient-Gesellschaft,Verlag Hinrichs, Leipzig.
[ 46 ] Sethe, K., (1908-1910). Die Altägyptischen Pyramidentexte, 2 Vols. J. C. Hinrichs'sche Buchhandlung, Leipzig.
[ 47 ] De Buck, A., (1935-1961). The Egyptian Coffin Texts, 7 Vols. University of Chicago Press, Chicago.
[ 48 ] Lesko, L., & Lesko, B., (Eds.). (2004). A dictionary of Late Egyptian, Vol.2, 2nd ed.,Providence, RI : B.C. Scribe Publications, Providence – USA.
[ 49 ] Sethe, K., (1903). Urkunden des Alten Reichs, Vol.1, Leipzig.
[ 50 ] Steindorff, G., (1955-1961). Urkunden des äegyptischen Altertums, IV, Sethe, K.,Urkunden der 18 Dynastie, 1-16, Berlin, 1914; Helck, W., Urkunden der 18 Dynastie, 17- 22, Berlin .
[ 51 ] Kitchen, K. A., (1970). Ramesside inscriptions: historical and biographical, 6 Vols. B.H. Blackwell, Oxford.
[ 52 ] Helck, W., (2000). Die Prophezeiung des Nfr.tj, KÄT, Vol. 2, Wiesbaden.
[ 53 ] Helck, W., (1977). Die Lehre für König Merikare, KÄT, Vol.7, Wiesbaden. 21
[ 54 ] Goyon, G., (1936). Les travaux de Chou et les tribulations de Geb d'après le naos 2248d'Ismaïlia, Kêmi, Vol. 6, pp. 1- 42.
[ 55 ] Quibell, J.E., (1911-1912). Excavations at Saqqara (1905-1914), 6 Vols. Impr. del'Institut français d'archéologie orientale, Le Caire.
[ 56 ] Griffith, F. LI., (1889). The inscriptions of Siût and Dêr Rîfeh, Trübner Publisher, London.
[ 57 ] Davies, N. de G., (1900-1901). The Mastaba of Ptahhetep and Akhethetep at Saqqara, 2 Vols. The Archeological Survey of Egypt, Egyptian Exploration Fund, London.
[ 58 ] Wreszinski, W., (1913). Der Papyrus Ebers, J.C. Hinrichs'sche Buchhandlung, Leipzig.
[ 59 ] Davies, N. de G., (1922-1923). The Tomb of Puyemrê at Thebes, 2 Vols. Publications of the Metropolitan Museum of Art, New York.
[ 60 ] Baillet, J., (1906) Les noms de l'esclave en égyptien, Rec. Trav., Vol. 28, pp.113-131.
[ 61 ] Möller,G.,(1909-1936). Hieratische Paläographie, 4 Vols. J.C. Hinrichs'sche Buchhandlung, Leipzig.
[ 62 ] Vycichl, W., (1973). Zwei ägyptische Nomina der hamito-semitischen Nominalklasse katul. Berichtigung der bisherigen Lesung von Hm.t "Frau" und qd.t"Kite, Gewicht von 9,1 Gramm", ZÄS, Vol. 99, 2, pp. 135-139.
[ 63 ] Helck, W., & Otto, E., (1999). Maße und Gewichte, In: Kleines Lexikon der Ägyptologie, Harrassowitz, Wiersbaden, cols.178-180.
[ 64 ] Vleeming, S.P., (1980). Maße und Gewichte in den demotischen Texten, In: Helck, W., Westendorf, W., (Eds.), Lexikon der Ägyptologie, Vol. 3, Harrassowitz, Wiesbaden, cols. 1209-1214.
[ 65 ] Kamel, A., (2000). Maße und Gewichte, In: Wissenschaft im Alten Ägypten, Kemet, Vol.4, Berlin, pp.38-40.
[ 66 ] Wilson, P., (1997). A Ptolemaic Lexikon. A Lexicographical Study of the Texts in the Temple of Edfu, Uitgeverij Peeters en Departement Oosterse Studies, Leuven.
[ 67 ] Devauchelle, D., (1986). 1D: deben or kite? Enchoria, Vol.14, pp.157-158.
[ 68 ] Eran, A., (1985). The Old-Egyptian Weight-Unit "Deben" - Its Reality, its Dispersion and its Late Echo, Acta Metrologiae Historicae, Vol. 14, pp.94-116.
[ 69 ] Raymond, W., (1924). La "kite" d'or de Byblos, Revue égyptologique, Vol. 2, pp. 21-37.
[ 70 ] Spiegelberg, W., (1923). [kd.t] als Bezeichnung der Kite, ZÄS, Vol. 58, p.154.
[ 71 ] Kammerzell, F., (1994). Zur Etymologie des ägyptischen Zahlworts "4", Lingua Aegyptia, Vol.4, pp.165-189.
[ 72 ] Crum, W. E., (1962). A Coptic Dictionary, Clarendon Press, Oxford.
[ 73 ] Černý, J., (1976). Coptic Etymological Dictionary, Cambridge University Press, Cambridge.
[ 74 ] Pierce, R.H., (1965). Notes on Obols and Agios in Demotic Papyri, JEA, Vol.51, pp. 155- 159.
[ 75 ] Plumley, J.M., (1948). An introductory Coptic Grammar; Sahidic Dialect, Home and Van Thal, London.
[ 76 ] Westendorf, W., (1977). Koptisches Handwörterbuch. Bearbeitet auf Grund des Koptischen Handwörterbuchs Wilhelm Spiegelbergs, Carl Winter Universitätsverlag, Heidelberg.
[ 77 ] Till, W.C., (1955). Koptische Grammatik; Saïdischer Dialekt mit Bibliographie, Lesestücken und Wörterverzeichnissen, Otto Harrassowitz, Leipzig.
[ 78 ] Grieshammer, R., (1970). Das Jenseitsgericht in den Sargtexten, Otto Harrassowitz, Ägyptologische Abhandlungen, Herausgegeben von Wolfgang Helck und Eberhard Otto, Vol. 20, Wiesbaden, pp. 1-188.
[ 79 ] Seeber, Ch., (1976). Untersuchungen zur Darstellung des Totengerichts im Alten Ägypten, Deutscher Kunstverlag, MÄS/ Münchner Ägyptologische Studien herausgegeben von Hans Wolfgang Müller, Vol.35, München/Berlin, pp.1- 271. 22
[ 80 ] Rossi, C., (2003). Architecture and Mathematics in Ancient Egypt, Cambridge University Press, London
[ 81 ] Verner, M., (2002). The Pyramids; The Mystery, Culture, and Science of Egypt's Great Monuments Grove Press, New York.
[ 82 ] Lehner, M., (1999). The Complete pyramids, London, Thames and Hudson Press.
[ 83 ] Lehner, M., (1985). A contextual approach to the Giza pyramids, Archiv fur Orientforsch, Vol. 31, pp.136-158.
[ 84 ] Jeffreys, D., (1998). The Topography of Heliopolis and Memphis: some cognitive aspects, Beitrage zur Kulturgeschichte Ägyptens, Rainer Stadelmann gewidmet, Mainz, pp.63-71.
[ 85 ] Baines, J., & Malek, J., (1984). The Cultural Atlas of the World: Ancient Egypt, Oxford.
[ 86 ] Goyon, J.L., (1977). The secrets of the pyramids’ builders, Pygmalion Press, Paris.
[ 87 ] Barta, M., (2005). Location of Old Kingdom pyramids in Egypt, Cambridge Arch. Journal, Vol. 15, pp.177-191.
[ 88 ] Stadelmann, R., (1991). Die ägyptischen Pyramiden, Vom Ziegelbau zum Weltwunder: P. von Zabern Press, Mainz.
[ 89 ] Spence, K., (2000). Ancient Egyptian Chronology and the Astronomical Orientation of Pyramids, Nature, Vol. 408, 16 November, pp. 320-324.
[ 90 ] Magli, G., (2010). Topography, astronomy and dynastic history in the alignments of the pyramid fields of the Old Kingdom, Mediterranean Archaeology and Archaeometry, Vol. 10, No.2, pp. 59-74.
[ 91 ] Wakefield, S.P., (2016). A Study of Ancient Unified Numerical and Metrological Systems, Mountains of the Moon, The Stone-Cored Pyramids of Aereia, Last update 13th July 2016, Chapter 3, Figs. 3-6.
[ 92 ] Bryn, O.J., (2010). Retracing Khufu's Great Pyramid; The "Diamond Matrix" and the number 7, Nordic Journal of Architectural Research, Vol. 22, No.1/2, pp.135-144, Figs.1-3.
[ 93 ] Nickel, J. D., (2009). The Proto-Trigonometry of the Pyramids, In: Biblical Christian World View, pp.1-2 www.biblicalchristianworldview.net (at 22/7/2016).
[ 94 ] Maor, E., (1998).Trigonometric Delights, Princeton University Press, Princeton.
[ 95 ] Faulkner, R. O., (1933). The Papyrus Bremner-Rhind, British Museum, No. 10188, Bibliotheca Aegyptiaca, Vol.3, pp. 1-10.
[ 96 ] Faulkner, R. O., (1937). The Bremner-Rhind Papyrus III, JEA, Vol. 23, pp.166-185.
[ 97 ] Spalinger, A., (1990). The Rhind Mathematical Papyrus as a Historical Document, SAK, Vol. 17, pp.295-337.
[ 98 ] Schott, S., (1952). Astronomie und Mathematik, Wissenschaftliche Literatur, In: Brunner, H., Kees, H., Morenz, S., & Spiegel, J., (Eds.), Handbuch der Orientalistik, 2.Vols. E. J.Brill, Leiden, pp.170-176.
[ 99 ] Struve, V., (1930). Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau / hrsg. und kommentiert von W. W. Struve unter Benutzung einer hieroglyphischen Transkription von B. A. Turajeff, Berlin, J. Springer, XII, pp.1-197, figs.1-10 /Quellen und Studien zur Geschichte der Mathematik; Abteilung A, Quellen 1, pp. 120-123, Nos. 14,18.
[ 100 ] Toomer, G. J., (1971). Mathematics and Astronomy, The Legacy of Egypt, Second Ed.,Edited by J.R. Harris, At the Clarendon Press, Oxford, pp. 27-54.
[ 101 ] Möller, G., (1913). Die Beiden Toten Papyrus Rhind des Museums zu Edinburg, Leipzig.
[ 102 ] https://www.math.washington.edu/~greenber/slope.gif (at 15/7/2016).
[ 103 ] https://www.math.washington.edu/~greenber/slope.html (at 15/7/2016).
[ 104 ] h'p://www.mathorigins.com/image%20grid.html (at 15/7/2016).
[ 105 ] De Young, G., (2009). Diagrams in ancient Egyptian geometry Survey and assessment, Historia Mathematica, Vol.36, pp.321–373, fig.2.23
[ 106 ] Clagett, M., (1999). Ancient Egyptian Science: A Source Book. Vol. 3: Ancient Egyptian Mathematics, Memoirs of the American Philosophical Society 232, American Philosophical Society, Philadelphia.
[ 107 ] Magli, G., (2013). Architecture, Astronomy and Sacred Landscape in Ancient Egypt, Cambridge University Press, New York.
[ 108 ] Kaiser, W., (1956). Zu den Sonnenheiligtümern der 5 Dynastie, MDAIK, Vol.14, pp.104-116.
[ 109 ] Magdolen, D., (2000). On the Orientation of Old-Kingdom Royal Tombs. In: Bárta, M., & Krejcí, J., (Eds.), Abusir and Saqqara in the Year 2000, Prague: Academy of Sciences of the Czech, Oriental Institute, pp. 491-498.
[ 110 ] Stadelmann, R., (2000). Userkaf in Saqqara und Abusir, In: Saqqara in the Year 2000, Archiv Orientální Supplementa, Vol. 9, Prague, pp. 529-542.
[ 111 ] Goedicke, H., (2001). Abusir - Saqqara – Giza, In: Bárta, M., & Krejcí, J., (Eds.), Abusir and Saqqara in the Year 2000, Prague: Academy of Sciences of the Czech, Oriental Institute, pp.397-412.
[ 112 ] Nuzzolo, M., (2007). The Sun Temples of the Vth Dynasty: A Reassessment, SAK, Vol.36, pp. 217-247.
[ 113 ] Hohenkerk, C.Y., Yallop, B.D., Et al., (2005). Celestial Reference Systems, In: Seidelmann, P.K., (Ed.) Explanatory Supplement to the Astronomical Sausalito, University Science Books Press, USA, pp. 99-100.
[ 114 ] Mueller, I., (1969). Spherical and Practical Astronomy as applied to Geodesy, New York, pp.58-59.
[ 115 ] Neugebauer, O., (1950). The Alleged Babylonian Discovery of the Precession of the Equinoxes, Journal of the American Oriental Society, Vol. 70, No.1 , pp.1-2.
[ 116 ] Neugebauer, O., (1980). On the Orientation of the Pyramids, Centaurs, An International Journal of History of Science and its Cultural Aspects ,Vol.24, Issue.1, USA, Published Online at 26 July,2007,pp.1-3, Fig.2.
[ 117 ] Isler, M., (1989). An Ancient Method of Finding and Extending Direction, JARCE, Vol. 26, pp.191-200.
[ 118 ] Couprie, D. L., (2011). Heaven and Earth in Ancient Greek Cosmology from Thales to Heraclides Ponticus, New York, Figs. 2.13; 2.15.
[ 119 ] Trimble, V., (1964). Astronomical investigation concerning the so-called Air-shafts of Cheops's Pyramid, Mitteilungen des Instituts für Orientforschung, Vol.10, No. 2/3, pp.183-187.
[ 120 ] Bauval, R., (1990). The seeding of the star-gods: a fertility rite inside Cheop's Pyramid? ,Discussion in Egyptology, Vol.16, pp. 21-25.
[ 121 ] Bauval, R., (1993). Cheop's Pyramid: A new dating using the latest astronomical Data, Discussions in Egyptology, Vol. 26, pp. 5-6.
[ 122 ] Bruck, M.T., (1995). Can the Great Pyramid be astronomically dated? , Journal of the British Astronomical Association, Vol.105, No. 4, pp. 161-164.
[ 123 ] Imhausen, A., (2016). Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, Oxford.
[ 124 ] Laskar, J., (1986). Erratum - Secular Terms of Classical Planetary Theories Using the Results of General Theory, Astronomy and Astrophysics, Vol.157, pp.59-70, and Vol. 164, No.2, p.437.
[ 125 ] Edvardsson, S., Karlsson, K. G., and Engholm, M., (2002). Accurate spin axes and solar system dynamics: Climatic variations for the earth and mars, Astronomy and Astrophysics, Vol. 384, No.2, pp.689-701.