Original from Digitized by UNIVERSITY OF MICHIGAN AKTES SCIENTIA VERITAS t^r\r^rt\o Original from Digitized by VjOUglC UNIVERSITY OF MICHIGAN t^r\r^rt\o Original from Digitized by VjOUglC UNIVERSITY OF MICHIGAN t^r\r^rt\o Original from Digitized by VjOUglC UNIVERSITY OF MICHIGAN t^r\r^rt\o Original from Digitized by VjOUglC UNIVERSITY OF MICHIGAN t^r\r^rt\o Original from Digitized by VjOUglC UNIVERSITY OF MICHIGAN THE RHIND MATHEMATICAL PAPYRUS BRITISH MUSEUM IOO57 AND IOO58 PHOTOGRAPHIC FACSIMILE, HIEROGLYPHIC TRANSCRIPTION, TRANSLITERATION, LITERAL TRANSLATION, FREE TRANSLATION, MATHEMATICAL COMMENTARY, AND BIBLIOGRAPHY IN TWO VOLUMES MATHEMATICAL ASSOCIATION OF AMERICA OBERLIN, OHIO, U. S. A. 1927 Original from Digitized by UNIVERSITY OF MICHIGAN THE RHIND MATHEMATICAL PAPYRUS BRITISH MUSEUM IOO57 AND IOO58 VOLUME I FREE TRANSLATION AND COMMENTARY BY ARNOLD BUFFUM CHACE Chancellor, Brown Unhersity WITH THE ASSISTANCE OF HENRY PARKER MANNING Associate Professor of Mathematics, Brown University, Retired BIBLIOGRAPHY OF EGYPTIAN MATHEMATICS BY RAYMOND CLARE ARCHIBALD Professor of Mathematics, Brown University MATHEMATICAL ASSOCIATION OF AMERICA OBERLIN, OHIO, U. S. A. 1927 Original from Digitized by UNIVERSITY OF MICHIGAN 7 / When the manuscript of this treatise approached comple- tion, Doctor Chace requested the Mathematical Associa- tion of America to undertake its publication, generously providing the means necessary to assure its appearance in a form commensurate with its importance. Upon learning of the desire of the Trustees to contribute to the success of the undertaking in every way possible, Doctor Chace made the further offer to donate to the Association a certain number of copies, adding the suggestion that these be then sold at a price considerably below the cost of publication. This was done with the desire not merely to assist finan- cially an organization of which he was a charter member and to which he has always contributed with great liberal- ity, but to make it possible for the individual and institu- tional members to purchase the work much more reasonably than would have been possible had it been simply a com- mercial product. The proceeds from its sale are to constitute an endowment of the Association known as the Arnold BuFFUM Chace Fund. The Association takes pleasure in making these facts known, and at the same time in expressing its high appre- ciation of this notable work of a well-trained scholar in mathematics, of the interest which he has always shown in the cultivation of the science in this country, and of his generosity in connection with this publication. Raymond Clare Archibald David Eugene Smith Publication Committee on behalf of The Mathematical Association of America Original from UNIVERSITY OF MICHIGAN fflcmorp of tnp Biife E-LizA Greene Chace WITHOUT WHOSE ENTHUSIASTIC ENCOURAGEMENT AND ASSISTANCE I SHOULD NEVER HAVE COMPLETED THIS WORK Original from UNIVERSITY OF MICHIGAN — 1 _ Original from Digitized by VjUU^lC UNIVERSITY OF MICHIGAN : PREFACE DURING the past fifteen years the preparation of the work now presented to the public has occupied my leisure hours, in an other- wise active business life. In 1910 my wife and I made a trip to Egypt and became much interested in its monuments and literature. The British Museum copies of the Book of the Dead and of the Rhind Mathematical Papyrus were acquired two years later and I soon com- menced an intensive study of the latter. This was inspired by my in- terest in the history of mathematics and in the ancient Egyptian civilization. The pleasure derived from such study has been great and especially have I enjoyed the intimate knowledge which I thereby gained regard- ing the reasoning power of those ancient people. I venture to suggest that if one were to ask for that single attribute of the human intellect which would most clearly indicate the degree of civilization of a race, the answer would be, the power of close reasoning, and that this power could be best determined in a general way by the mathematical skill which members of the race displayed. Judged by this standard the Egyptians of the nineteenth century before Christ had a high degree of civilization. The great advance in our present knowledge of the Egyptian language would of itself have justified the publication in 1923 of Professor Peet's scholarly work on the Rhind papyrus, marking such a notable advance over the remarkable translation and commentary by Eisenlohr forty- six years earlier. It seemed to me, however, that there was room for yet another work on this subject intended for mathematicians and the general public, rather than for Egyptologists who will find philological matters fully discussed by Professor Peet. In my volumes the new con- tributions may be summarized as follows 1. A mathematical commentary, in the preparation of which I have been greatly assisted by Doctor Manning who, while associated with me in this work during the past seven years, has displayed scholarly insight and exactness; 2. The very elaborate and notable critical Bibliography of Egyptian Mathematics contributed by Professor Archibald and including refer- Original from UNIVERSITY OF MICHIGAN ence to the literature of over fifty documents dating from 3500 B.C. to about 1000 A.D.; 3. The photographic reproduction of the Rhind mathematical papyrus; 4. The presentation of the original hieratic text, the hieroglyphic transcription, the transliteration, and the literal translation in such a manner that even the reader untutored in Egyptian may pick out in the original those signs corresponding to the phrases in the free trans- lation. Particularly in this part of the work, during the past two years, have I regarded myself as fortunate in securing the cooperation of Doctor Bull. Without such aid I could scarcely have hoped to escape severer strictures than will now be meted out to me by Egyptologists. My grateful thanks are due to the Trustees of the British Museum for their courtesy in supplying me with photographs of the Rhind Papyrus, and in granting me permission to reproduce them. In connec- tion with this arrangement, the kindness displayed and information furnished by Mr. S. R. K. Glanville, of the staff in the Department of Egyptian and Assyrian Antiquities of the British Museum, were highly appreciated. The New York Historical Society has been most generous in placing its fragments of the Rhind papyrus at our disposal for study, arrange- ment, and photographing, and in permitting the publication of the results. To Professor D. E. Smith, of Columbia University, who, with Professor Archibald, was appointed by the Mathematical Association of America to assist in seeing my work through the press, I am greatly indebted for encouragement and valuable suggestions. I wish also to express thanks for assistance to Professor T. Eric Peet, of the University of Liverpool, to Dr. G. A. Reisner, of the Boston Museum of Fine Arts, and to Doctor Caroline Ransom Williams of Toledo, Ohio. I hope that the distribution of my work by the Mathematical Asso- ciation of America may lead to a deeper interest in the contributions of the golden period of Egyptian science. A. B. C. Original from UNIVERSITY OF MICHIGAN CONTENTS PREFACE INTRODUCTION Egyptian Arithmetic 3-30 Numbers and operations; Special processes emphasized in the Rhind papyrus; Table of the division of 2 by odd numbers; Division of the numbers 1-9 by 10; Multiplication by certain fractional expressions; Divisions by a fractional number; Separation in given proportions; Other arithmetical problems; Arithmetical and geometrical progressions. Egyptian Measures 31-34 Measures of capacity; Measures of area and length. i Egyptian Geometry 35-38 Determination of volumes; Determination of areas; Pyramids, seked. The Methods and Aims of the Egyptian Mathematician . 3^43 His methods; His mistakes; His theoretical interest. Note on the Egyptian Calendar and Egyptian Chronology 43-46 Note About Words for Grain and Other Kinds of Food . 46-47 FREE TRANSLATION AND COMMENTARY Title, Date and Scribe 49 Chapter I, Egyptian Arithmetic 50-85 I. Table of the division of 2 by odd numbers; II. Problems 1-6: Table of the division of the numbers 1-9 by 10; III. 7-20: Multiplication by certain fractional e.xpressions ; IV. 21-23: Problems in completion; V. 24-29: 'Aha' or quantity problems; VI. 30-34: Division by a frac- tional expression ; \"1I. 35-38: Division of a A«ia/; VIII. 39-40: Division of loaves. Arithmetical progression. Chapter II, Egyptian Geometry 86-99 I. Problems 41-46: Volumes; II. 47: Division of too hekat; III. 48- 55: Areas; IV. 56-60: Pyramids, seked. Chapter III, Miscellaneous Problems 100-119 Problems 61-84; Numbers 85-87. BIBLIOGRAPHY Prefatory Note, Survey of Literature, Chronolgical List of Documents Discussed, Index of Authors, Subject Index 121-206 GENERAL INDEX TO VOLUME I . 207-210 Original from Digitized by UNIVERSITY OF MICHIGAN — 1 _ Original from Digitized by VjUU^lC UNIVERSITY OF MICHIGAN \ I INTRODUCTION^ The Rhind Mathematical Papyrus Wcis found at Thebes in the ruins of a small building near the Ramesseum. It was purchased in 1858 by A. Henry Rhind and after his death came into the possession of the British Museum. It is a copy that was made by a scribe named A'h- mosd cmd begins with a rather ambitious statement of what the author proposes to do. The date at which this copy was made is indicated in the old Egyptian method by the year of a certain pharaoh named 'A-user-Re',^ who has been identified as one of the Hyksos dynasty, living approximately 1650 B.C. The scribe further says that it is in likeness to older writings, of the time of king Ne-ma'et-Re' (Amen- em-h^t III) who reigned from 1849 to 1801 B.C.' The papyrus is written in hieratic * and originally it was a single roll nearly 18 feet long and about 13 inches high, but it came to the British Museum broken apart, and with a number of fragments missing, the most important of which have been found in the possession of the New York Historical Society.