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(24201-3) $9.95 GREAT EXPERIMENTS IN PHYSICS, Morris H. Shamos. (25346-5) $10.95 ALL ABOUT LIGHTNING, Martin A. Uman. (25237-X) $6.95 THE COMPLEAT STRATEGYST: BEING A PRIMER ON THE THEORY OF GAMES OF STRATEGY, J. D. Williams. (25101-2) $7.95 FAMous PROBLEMS OF GEOMETRY AND How To SOLVE THEM, Benjamin Bold. (24297-8) $4.95 SIDELIGHTS ON RELATIVm; Albert Einstein. (24511-X) $3.95 (continued on back flap) Mathematics in the Time of the Pharaohs The scribe Ka-Irw-Khufu of the Fourth Dynasty, excavated at Giza. The statue is now owned by the Cairo Museum. Copyright Roger \Vood, London. Mathematics inthe Time of the Pharaohs RICHARD J. GILLINGS Dover Publications, Inc. NEW YORK Copyright a 1972 by The Massachusetts Institute of Technology. Appendix 15 copyright o 1981 by Academic Press, Inc. All rights reserved under Pan American and International Copy- right Conventions. Published in Canada by General Publishing Company, Ltd., 30 1.esmill Road, Don Mills. Toronto, Ontario. This Dover edition, first published in 1982, is an unabridged, slightly corrected republication of the work first published in 1972. It is reprinted by special arrangement with The MIT Press, 28 Carleton Street, Cambridge, MA 02142, publisher of the original edition. Appendix 15 originally appeared in Historia Mathematica 8 (1981), pp. 456-459, and is presented here by permission of Academic Press, Inc. Manufactured in the United States of America Dover Publications, Inc. 31 East 2nd Street Mineola, N.Y. 11501 Library of Congress Cataloging In Publication Data Gillings. Richard J. Mathematics in the time of the pharaohs. Reprint. Originally published: Cambridge, Mass.: MIT Press, 1972. Bibliography: p. Includes index. 1. Mathematics, Egyptian. I. Title. [QA27.E3G52 1982] 510' .932 81-19445 ISBN 0-486-24315-X (pbk.) AACR2 Dedicated to 0. Neugebauer and A. Sachs, in friendship and gratitude I wish I knew as much as I thought I knew 10 years ago 0. Neugebaucr, 1953 PREFACE xi 1 Introduction I 2 Hieroglyphic and Hieratic Writing and Numbers 4 3 The Four Arithmetic Operations 11 ADDITION AND SUBTRACTION 11 MULTIPLICATION 16 DIVISION19 FRACTIONS 20 4 The Two-Thirds Table for Fractions24 PROBLEMS 61 AND 61B OF THE RHIND MATHEMATICAL PAPYRUS 29 TWO-THIRDS OF AN EVEN FRACTION 32 AN EXTENSION OF RMP 61B AS THE SCRIBE MAY HAVE DONE IT 33 EXAMPLES FROM THE RHIN'D MATHEMATICAL PAPYRUS OF THE TWO-THIRDS TABLE 34 5 The G Rule in Egyptian Arithmetic39 FURTHER EXTENSIONS OF THE G. RULE 42 6 The Recto of the Rhind Mathematical Papyrus45 THE DIVISION OF 2 BY THE ODD NUMBERS 3 TO 10145 CONCERNING PRIMES 52 FURTHER COMPARISON OF THE SCRIBE'S AND THE COMPUTER'S DECOMPOSITIONS 53 7 The Recto Continued 71 EVEN NUMBERS IN THE RECTO: 2 = 13 71 MULTIPLES OF DIVISORS IN THE RECTO 74 TWO DIVIDED BY THIRTY-FIVE: THE SCRIBE DISCLOSES HIS METHOD 77 Problems in Completion and the Red Auxiliaries81 USE OF THE RED AUXILIARIES OR REFERENCE NUMBERS 81 AN INTERESTING OSTRAC'.ON A6 viiiContents 9 The Egyptian Mathematical Leather Roll89 THE FIRST GROUP 95 THE SECOND GROUP 95 THE THIRD GROUP 96 THE FOURTH GROUP 98 THE NUMBER SEVEN 99 LINE 10 OF THE FOURTH GROUP 102 THE FIFTH GROUP 102 I0 Unit-Fraction Tables104 UNIT.FRACFION TABLES OF THF RHIND MATHEMATICAL PAPYRUS I06 PROBLEMS 7 TO 20 OF THE RHIND MATHEMATICAL PAPYRUS109 U Problems of Equitable Distribution and Accurate Measurement120 DIVISION OF THE NUMBERS I TO 9 BY 10120 CUTTING UP OF LOAVES 123 SALARY DISTRIBUTION FOR THE PERSONNEL OF THE TEMPLE OF I -IAH IN 124 12 Pesu Problems 128 EXCHANGE OF LOAVES OF DIFFER _NT P SUS132 11 Areas and Vnlumes 137 THE AREA OF A RFCTANGI.E137 THE AREA OF A TRIANGLE 138 THE AREA OF A CIRCLE139 THE VOLUME OF A CVI_INDRlCAI_ GRANARY I4fi THE DETAILS OF KAHUN IV, 3 151 14 Equations of the First and Second Degree154 THE FIRST GROUP 154 SIMILAR PROBLEMS FROM OTHER PAPYRI156 THE SECOND AND THIRD GROUPS 158 EQUATIONS OF THE SECOND DEGREE 161 Contentsix KAHUN LV, 4162 SUGGESTED RESTORATION OF MISSING LINES OF KAHUN LV, 4, AND MODERNIZATION OF OTHERS I63 15 Geometric and Arithmetic Progressions166 GEOMETRIC PROGRESSIONS- PROBLEM 79 OF THE RHIND MATHEMATICAL PAPYRUS 166 ARITHMETIC PROGRESSIONS: PROBLEM 40 OF THE RHIND MATHEMATICAL PAPYRUS 170 PRORTFM 64 OF THE Rf-ITND \fATHFMfATIrAl_ PAPYRUS173 KAHUN IV, 3176 16 "Think of a Number" Problems 181 PROBLEM 28 OF THE RHIND MATHEMATICAL PAPYRUS 181 PROBLEM 29 OF THE RHIND MATHEMATICAL PAPYRUS 183 17 Pyramids and Truncated Pyramids185 THE SEKED OF A PYRAMID 185 THE VOLUME OF A TRUNCATED PYRAMID 187 is The Area of a Semicylinder and the Area of a Hemisphere194 19 Fractions of a Hekat202 20 Egyptian Weights and Measures207 21 Squares and Square Roots214 22 The Reisner Papyri: The Superficial Cubit and Scales of Notation218 APPENDIX 1 The Nature of Proof 232 APPENDIX 2 The Egyptian Calendar235 xContents APPENDIX 3 Great Pyramid Mysticism237 APPENDIX 4 Regarding Morris Kline's Views in Mathematics, A Cultural Approach240 APPENDIX 5 The Pythagorean Theorem in Ancient Egypt242 APPENDIX 6 The Contents of the Rhind Mathematical Papyrus243 APPFNDiX 7 The Contents of the Moscow Mathematical Papyrus246 APPENDIX 8 A Papyritic Memo Pad248 APPENDIX 9 Horus-Eye Fractions in Terms of Hinu: Problems 80, 81 of the Rhind Mathematical Papyrus249 APPENDIX 10 The Egyptian Equivalent of the Least Common Denominator251 APPENDIX I I A Table of Two-Term Equalities for Egyptian Unit Fractions254 APPENDIX 12 Tables of Hieratic Integers and Fractions, Showing Variations255 APPENDIX 1 Chronology260 APPENDIX 14 A Map of Egypt266 APPENDIX 15 The Egyptian Mathematical Leather Roll 267 BIBLIOGRAPHY 269 INDEX 279 PREFACE In 1811, Thomas Cooper, professor of chemistry at Dickinson College, Carlisle, Pennsylvania, and a Britisher by birth, said at his opening lecture, "The history of an art or science is a proper introduction to the study of it; as giving a clear and concise view of the manner in which improvements have been effected; as furnishing due caution against future errors by exhibiting the mistakes of superior minds of olden times, and as rendering merited honor to those who have benefitted mankind by their discoveries." I have often quoted these words in my own lectures on ancient Egyptian mathematics, but wherever and whenever I have differed from recognized authorities in this field of study, like Eisenlohr, Gillain, Griffith, Sethe, Peet, Struve, Vogel, Chace, Neugebauer, Van der Waerden, and Bruins, I have done so only after due caution and consideration, yet nevertheless with firm and proper conviction. For it was only after carefully examining their translations, their opinions, their solutions and comments, their criticisms, and especially their statements wherein I judged they may have erred, that I have been enabled, I hope, to achieve a truer overall picture of the mathematics of the ancient Egyptians. In other words, I have been privileged to stand on the shoulders of savants and scholars, and the experience has been profoundly rewarding. R.J. Gillings University of New South Wales, and Turramurra, Australia, 1971 Mathematics in the Time of the Pharaohs 1 INTRODUCTION One of the oddest of all the phenomena which come to the attention of students of the history of mathematics is that logarithms were invented by Napier more than a decade before Descartes first conceived the idea of using indices in algebra. This oddity becomes even more striking when we observe that mathematical textbooks today introduce the subject of logarithms by a preliminary study of the index laws of algebra, which is pedagogically perhaps the very best way to do it. Chrono- logically, therefore, the expected order of invention seems to have been reversed; things are the wrong way round! A second oddity of the history of mathematics was brought to light when the Babylonian clay tablet Plimpton 322 (museum number, Columbia University, New York) was translated by Neugebauer and Sachs in 1945.