The Shape of the Great Pyramid

By Roger Herz-Fischler

Who has not seen a picture of the Great Pyramid of Egypt, massive in size but deceptively simple in shape, and not wondered how that shape was determined?

Starting in the late eighteenth century, eleven main theories were proposed to explain the shape of the Great Pyramid. Even though some of these theories are well known, there has never been a detailed examination of their origins and dissemination. Twenty years of research using original and difficult-to-obtain source material has allowed Roger Herz-Fischler to piece together the intriguing story of these theories. Archaeological evidence and ancient Egyptian mathematical texts are discussed in order to place the theories in their proper historical context. The theories themselves are examined, not as abstract mathematical discourses, but as writings by individual authors, both well known and obscure, who were influenced by the intellectual and social climate of their time.

Among results discussed are the close links of some of the pyramid theories with other theories, such as the theory of evolution, as well as the relationship between the pyramid theories and the struggle against the introduction of the metric system. Of special note is the chapter examining how some theories spread whereas others were rejected.

This book has been written to be accessible to a wide audience, yet four appendixes, detailed endnotes and an exhaustive bibliography provide specialists with the references expected in a scholarly work.

• Language: English
• Publisher: Wilfrid Laurier University Press
• Release date: Oct 21, 2009
• ISBN: 9781554587032

Roger Herz-Fischler

Roger Herz-Fischler teaches mathematics at Carleton University.

Book preview

The Shape of

the Great Pyramid

The three large pyramids at Giza. From L. Borchardt, Egypt: Architecture, Landscape, Life of the People (New York: Westermann, 1929[?]).

The Shape of the Great Pyramid

Roger Herz-Fischler

O you Great Ennead which is in Ōn, make the King’s endure, make this pyramid of this King and this construction of his endure for ever, just as the name of Atum who presides over the Great Ennead endures. As the name of Shu, Lord of the Upper Mnst in Ōn, endures, so may the King’s name endure, and so may this pyramid of his and this construction of his endure likewise for ever.

—Utterance 601 [Fifth Dynasty], The Ancient Egyptian Pyramid Texts, [Faulkner, 1969, 247]

This book has been published with the help of a grant from the Humanities and Social Sciences Federation of Canada, using funds provided by the Social Sciences and Humanities Research Council of Canada. We acknowledge the financial support of the Government of Canada through the Book Publishing Industry Development Program for our publishing activities.

Canadian Cataloguing in Publication Data

Herz-Fischler, Roger, 1940-

The shape of the Great Pyramid

Includes bibliographical references and index.

ISBN 0-88920-324-5

1. Great Pyramid (Egypt) – Miscellanea. 2. Weights and measures – Egypt – Miscellanea. I. Title.

© 2000 Wilfrid Laurier University Press

Waterloo, Ontario N2L 3C5

Cover design by Leslie Macredie, using a photograph of the Great

Pyramid from L. Borchardt, Egypt: Architecture, Landscape,

Life of the People (New York: Westermann, 1929[?]). Back

cover visual: the Star Cheops.

Printed in Canada

The Shape of the Great Pyramid has been produced from a manuscript supplied in camera-ready form by the author.

All rights reserved. No part of this work covered by the copyrights hereon may be reproduced or used in any form or by any means—graphic, electronic or mechanical—without the prior written permission of the publisher. Any request for photocopying, recording, taping or reproducing in information storage and retrieval systems of any part of this book shall be directed in writing to the Canadian Reprography Collective, 214 King Street West, Suite 312, Toronto, Ontario M5H 3S6.

Pour

Eliane, Mychèle, Seline et Rachel

Für

Issak, Teddy und Morris Fischleiber

Julie Sommer

Joseph und Artur Höllander

die uns zu früh verliessen

I o pas mie sons fie, ni roso sons espino, ni nouse sons crubil—Provençal proverb

TABLE OF CONTENTS

Acknowledgements

Introduction

PART I.         THE CONTEXT

Chapter 1. Historical and Architectural Context

Chapter 2. External Dimensions and Construction

Surveyed Dimensions

Angle of Inclination of the Faces

Egyptian Units of Measurement

Building and Measuring Techniques

Chapter 3. Historiography

Early Writings on the Dimensions

Modern Historiographers

PART II.         ONE PYRAMID, MANY THEORIES

Diagrams

Chapter 4. A Summary of the Theories

Terminology, Notation, Observed Dimensions

Definitions of the Symbols – Observed Values

A Comparison of the Theories

Chapter 5. Seked Theory

The Mathematical Description of the Theory

Seked Problems in the Rhind Papyrus

Archaeological Evidence

Early Interpretations of the Rhind Papyrus

Petrie

Borchardt

Philosophical and Practical Considerations

Chapter 6. Arris = Side

The Mathematical Description of the Theory

Herodotus (vth century)

Greaves (1641)

Paucton (1781)

Jomard (1809)

Agnew (1838)

Fergusson (1849)

Beckett (1876)

Howard, Wells (1912)

Chapter 7. Side: Apothem = 5:4

The Mathematical Description of the Theory

Plutarch’s Isis and Osiris

Jomard (1809)

Perring (1842)

Ramée (1860)

Chapter 8. Side: Height = 8:5

The Mathematical Description of the Theory

Jomard (1809)

Agnew (1838)

Perring (1840?)

Röber (1855)

Ramée (1860)

Viollet-le-Duc (1863)

Garbett, (1866)

A. X., (1866)

Brunés (1967)

Chapter 9. Pi-theory

The Mathematical Description of the Theory

Egyptian Circle Calculations

Agnew (1838)

Vyse (1840)

Chantrell (1847)

Taylor (1859)

Herschel (1860)

Smyth (1864)

Petrie (1874)

Beckett (1876)

Proctor (1877)

Twentieth-Century Authors

Chapter 10. Heptagon Theory

The Mathematical Description of the Theory

Fergusson (1849)

Texier (1934)

Chapter 11. Kepler Triangle Theory

The Mathematical Description of the Theory

Kepler Triangle and Equal Area Theories

Kepler Triangle, Golden Number, Equal Area

Röber (1855)

Drach, Garbett (1866)

Jarolimek (1890)

Neikes (1907)

Chapter 12. Side: Height = Golden Number

The Mathematical Description of the Theory

Röber (1855)

Zeising (1855)

Misinterpretations of Röber

Choisy (1899)

Chapter 13. Equal Area Theory

The Mathematical Description of the Theory

The Passage from Herodotus

Agnew (1838)

Taylor (1859)

Herschel (1860)

Thurnell (1866)

Garbett (1866)

Smyth (1874)

Hankel (1874)

Beckett and Friend (1876)

Proctor (1880)

Ballard (1882)

Petrie (1883)

Twentieth-Century Authors

Chapter 14. Slope of the Arris = 9/10

The Mathematical Description of the Theory

William Petrie (1867)

James and O’Farrell (1867)

Smyth (1874)

Beckett (1876), Bonwick (1877), Ballard (1882)

Flinders Petrie (1883)

Texier (1939)

Lauer (1944)

Chapter 15. Height: Arris = 2:3

The Mathematical Description of the Theory

Unknown (before 1883)

Chapter 16. Additional Theories

PART III.         CONCLUSIONS

Chapter 17. Philosophical Considerations

Chapter 18. Sociology of the Theories – A Case Study: The Pi-theory

The Social and Intellectual Background in Victorian Britain

Relationship of the Pi-theory to Other Topics

A Profile of the Authors

Chapter 19. Conclusions

The Sociology of the Theories

What Was the Design Principle?

APPENDICES

Appendix 1. An Annotated Bibliography

Appendix 2. Tombal Superstructures: References and Dimensions

Appendix 3. Egyptian Measures

Appendix 4. Egyptian Mathematics

Appendix 5. Greek and Greek-Egyptian Measures

NOTES

BIBLIOGRAPHY/INDEX

Symbols

The symbols for the quantities directly related to a pyramid are shown on the diagrams at the beginning of Chapter 4.

Cover

The aerial photograph of the Great Pyramid on the cover is taken from a wonderful 1929 work by the Egyptologist Ludwig Borchardt entitled Egypt: Architecture, Landscape, Life of the People. This book of text and photographs shows not only the monuments of Egypt, but also the people, their dwellings and their habitat.

Acknowledgements

Il fait bon ne rien savoir: l’on apprend toujours.

—French Proverb [Dournon, 1993, 296]

One often finds authors thanking their spouses either for their support or direct help. While I have to thank my wife Eliane in both these categories there is much more to her influence upon me. The reader should keep in mind that my studies were in engineering and very abstract mathematics. While I had taken a few general courses in English literature, the universe of the humanities was completely foreign to me. After our marriage in 1964, I began to look over her shoulder as she pursued her studies in French literature. Her ability to work from many sources and to analyze texts amazed me and I have always considered her to be the intellectual in the family. My true apprenticeship in the humanities started out in a way consistent with my abilities; I typed her papers, sought out books and references for her and worked on mastering the language of Molière, Racine and Thomas Corneille. The next step came in the summer of 1970 when she worked on her doctoral thesis¹ at the Bibliothèque Nationale in Paris and under her guidance I became an expert on the bibliography of Thomas Corneille. I believe that it was because of all this that when I started doing my research on the pyramid theories and golden numberism in 1975-76, I did not feel that I was in a strange world.

I would like to take this opportunity to publicly thank the unsung heroes of the academic world: the librarians and especially the interlibrary loan librarians. My particular thanks go to two particularly wonderful interlibrary loans librarians, Terry Sulymko and Callista Kelly.

Many other people have been of assistance to me and I have mentioned some of them in the notes. Two of the referees of the manuscript made some very pertinent criticisms and their comments have resulted in a major revision of the original manuscript. I wish to thank these referees for the time and effort that they devoted to my work.

I am especially appreciative of the support of Sandra Woolfrey, former director of Wilfrid Laurier University Press, and in particular her willingness to publish my books, which are not only different from those usually published by the Press, but which do not easily fit into precise categories.

Caroline Gowdy-Williams, Eliane Herz-Fischler and Andrew Williams read the text before it was sent to Wilfrid Laurier University Press. David Millman provided important editorial assistance and Jeff Coughlin drew the diagrams.

Many improvements were suggested—and errors eliminated—by the fine staff at Wilfrid Laurier University Press, in particular Sandra Woolfrey, Carroll Klein, Heather Blain-Yanke and Steve Izma. The cover layout is by Leslie Macredie.

The research for this work was done with the aid of a research grant from the Interdisciplinary Committee of the Social Sciences and Humanities Research Council of Canada. Publication was made possible by a grant from the Canadian Federation of the Humanities Aid to Scholarly Publishing Programme. Without the support of Wilfrid Laurier University for its fine press, publication would not have been possible. It is a source of pleasure for me that there still exist public organizations and universities that support the humanities and especially interdisciplinary works such as mine which sit on the border between the humanities and the sciences.

Introduction

As regards the unit of Egyptian measure, the notion that it could possibly have been based upon geometry is now probably universally abandoned. Such a supposition … was suited to the extraordinary excitement of the European mind in the 18th century.—Bunsen, [1854, 29]

Perhaps no other structure built by humans has attracted as much attention as the Great Pyramid of Egypt. Its size, with a base of 230 m and a height of 147 m, is not the sole cause of awe. The setting, on the edge of the desert and overlooking the Nile valley, only adds to its impressiveness, while the complex system of passages, chambers and blockage points, and the yet to be found tomb of the Pharoah Khufwey (Cheops), have added an aura of mystery. The present work is devoted to what at first glance would appear to be a rather innocent question, "What was the geometrical¹ basis, if any, that was used to determine the shape of the Great Pyramid?" However, as the reader can ascertain from its size, there is much more to this book than just giving a mathematical description of a well-known monument from antiquity. In order to better describe its contents, it is necessary to explain its origins.

In 1972 I was asked to teach a mathematics course for first-year students of architecture. Since I was essentially free² to choose the topics for the course, I decided to introduce some material dealing with the use of mathematical proportions in architecture. Among the material that I came across was a statement in Ghyka’s 1927 book, L’Esthétique des proportions dans la nature et dans les arts, concerning a putative text by the ancient Greek historian Herodotus. This ancient text, it was claimed, explained the shape of the Great Pyramid. Indeed, it seemed from the numbers that were presented by Ghyka, that theory and observation were in concordance with one another. Not having any reason to doubt what I had read, I presented the theory, over a period of three years, to my classes. It was only later, when I began to write a mathematics textbook for students of architecture,³ that I tried to locate the quotation by Herodotus. This proved to be impossible, for the putative statement by Herodotus simply did not exist; the only description in the Histories of Herodotus which dealt with the dimensions of the Great Pyramid bore little prima facie resemblance to what Ghyka had written.

My curiosity was piqued and thus began a long, tortuous and complicated investigation into the theories that had been proposed concerning the shape of the Great Pyramid. I would come across a new theory and then try to trace it back to its origins, sometimes via comments of others, but most often by working backwards through bibliographic references. Thus, what started out as a factual, historical study became a more involved and multi-faceted project. I became interested, not only in the theories and their history as such, but also in what I refer to as the sociological aspect of these theories; namely how these theories originated, how they were propagated and why some theories survived, whereas others passed into oblivion. This aspect eventually led me back to the Victorian era and to relationships—hardly anticipated at the beginning of my research—between the pyramid theories, and, among other topics, the theory of evolution and the struggle against the introduction of the metric system. Another question also presented itself for, as will be seen, several of the theories gave results which, from a practical viewpoint, were indistinguishable from the observed values. I was thus led to consider philosophical questions related to the acceptance of theories.

The present work is the result of my research and reflection. My basic approach in this book is the same as that in my A Mathematical History of Division in Extreme and Mean Ratio, my articles in art and architectural history and my forthcoming The Golden Number, i.e., keep reading and backtracking through the literature, be skeptical of secondary sources, go off on interesting side tracks,⁴ and avoid all preconceived theoretical approaches to the subject matter. Above all I believe in letting the material that one finds shape the book rather than writing a book that shapes the material.⁵

The book is divided in three parts which correspond in general terms to the historical and physical background to the theories, the theories themselves, and an overview.

In Part 1, Chapter 1 provides the historical and contextual background for the book. I have summarized, while at the same time giving references for those readers who wish to read more detailed discussions, the early history of Egypt and the development of the pyramid. Appendix 1 provides a further, annotated, bibliography of various topics related to the pyramids. Appendix 2 provides a table, together with references, of the dimensions of early pyramids and other tombal superstructures. To my knowledge the set of references to writings on the dimensions and angles of the pyramids is the most complete one available. Chapter 2 begins with the surveyed dimensions of the Great Pyramid and the estimated original angle of inclination of the triangular sides. This is followed by brief discussions of how the Egyptians measured, what their units of measurements were, and what is known of their building techniques. Appendix 3 provides more detailed information on Egyptian units of measure. Chapter 3 is historiographical in nature, and considers previous studies of the theories of the shape of the Great Pyramid.

The second part of the book begins with diagrams which illustrate the different ways in which the shape of a pyramid can be defined and gives the terminology employed in the rest of the book. Part 2 begins in Chapter 4 with a comparative table of the theories and the angles of inclinations of the faces which correspond to these theories. I also point out parallels between certain of the theories. Then follows, in Chapters 5 through 15, the historical and sociological developments of the eleven theories that are known to me. The presentation is in chronological order, with respect to the first known appearance of the theory. The one exception is the seked theory of Chapter 5, for which the theoretical basis is an ancient Egyptian text. I thus presented this theory first, even though a formal connection with the Great Pyramid was not stated until 1922.

Each chapter begins with a brief mathematical description, in simple trigonometric or geometric language, of the theory in question. The first note of each section contains a complete list of the angles, lengths etc. associated with this theory. The formulae for computing these quantities are given in the notes to Chapter 4. The rest of each chapter is then a mixture of historical and sociological material, including a description of the mathematical approach of different authors.

Several of the Chapters in Part 2 contain special material, which I felt was necessary for a proper understanding of the background of the theory. Chapter 5 includes archaeological evidence related to the seked theory, as well as a discussion of the pyramid problems in the Rhind Papyrus. Similarly Chapter 9 discusses what the Egyptians knew about circle calculations. Other aspects of Egyptian mathematics are summarized in Appendix 4. The text of Herodotus cited above in connection with the book by Ghyka, and which constitutes the historical basis for two of the theories, is discussed in Chapter 6, with Appendix 5 providing a technical background for Greek and Greek-Egyptian systems of measures. Chapter 7 contains a discussion of another ancient text which has formed the theoretical basis for the discussions of various authors, namely Plutarch’s Isis and Osiris in which the 3-4-5 triangle is related to these Egyptian gods. Chapter 16 presents some additional material which, while never appearing as formal theories of the shape of the Great Pyramid, is of interest in the context of this book.

Part 3 begins with a discussion of philosophical matters related to the theories. One notes immediately that there are only very small differences between the angles resulting from the theories and the observed value of the angle of inclination of the faces. Since the correct theory cannot be determined on the basis of numerical accuracy—or to look at the matter in another way, cannot be rejected on the basis of a discrepancy between theory and observation—philosophical questions arise as to when we can, or should, accept or reject a theory. Chapter 17 proposes some criteria related to the acceptance of theories.

Chapter 18 is devoted to a case study of the sociology of the pi-theory. As we shall see, the pi-theory is a true theory of Victorian Britain and so we have a very special opportunity to observe the conditions which give rise to a theory and cause it to be widely disseminated. The first section of Chapter 18 discusses the social and intellectual background in Victorian Britain which gave rise to the pi-theory and led to its widespread dissemination. The next section deals with the four topics of great interest in that period with which the pi-theory was associated: the squaring of the circle, units of measure, the Bible, and the theory of evolution. The last section deals with the authors themselves. By means of specialized biographical sources, I have made an analysis of the background, occupation and interests of the nine principal Victorian authors who wrote on the pi-theory. I hope that the reader will find the maze of interconnected external influences and people as fascinating as I did.

Chapter 19 contains my conclusions. The first section deals with my observations as to how theories propagate and in particular why certain theories flourished whereas others essentially disappeared. The second section of Chapter 19 returns to the question, What was the geometrical basis that was used to determine the shape of the Great Pyramid?

The bibliography contains some 315 items. Since many of the primary and secondary sources are very difficult to locate or obtain, I have indicated with each bibliographic entry, except for very common twentieth-century material, the library that was kind enough to lend me the material. For certain bibliographical entries, I have added comments or references to other works so as to aid future researchers. Since this is to a large extent a book about books and articles, I felt that it would be more useful to the reader to have an index to an author’s individual books rather than just having an index with only the names of the authors. Thus the bibliography also serves as the index, with the location of the discussion of a book or article being given at the end of the bibliographic entry. The detailed table of contents provides another entry to the authors and topics discussed.

* * *

Cost considerations prevented the inclusion of various photographs, diagrams etc. discussed in this book. These will be posted on a web site attached to the Wilfrid Laurier University Press web site. When this book went to press the Press’s site was being completely revised and it was thus not possible to give a precise address. It is suggested that reader start at the site:

https://www.wlupress.wlu.ca/

Part One

The Context

Chapter 1

Historical and Architectural Context

Much lesse can I assent to the opinion … [that the pyramids] were built by the Patriarch Joseph, Receptacles, and Granaries of the seven plentifull yeares. For, besides that this figure is most improper for such a purpose, a Pyramid being the least capacious of any Regular Mathematicall body, the straightnesse, and fewnesse of the roomes within (the rest of the building being one solid, and intire fabrick of stone) doe utterly overthrow this conjecture.—Greaves, Pyramidographia or a Description of the Pyramids in Ægypt, [Greaves, 1646, 2]

Ancient Egyptian history, from the beginning of the Early Dynastic Period under Mēnēs (c. -3100¹) until the conquest by Alexander the Great in -332, is divided into thirty-one dynasties. The three large pyramids of Giza, and in particular the Great Pyramid, all date from the Fourth Dynasty which began c. -2620.² For the purpose of chronological comparison I note that the Babylonian king Hammurabi lived C.-1800 and that various indications suggest that the Exodus from Egypt described in the Bible took place in the XIIIth century.³

The Giza pyramids, and in particular the Great Pyramid, do not represent, whether considered from the viewpoint of form, purpose or building techniques, an isolated phenomenon. Rather, when viewed in context, they represent one phase of a continuing development of Egyptian architecture and, more precisely, of funerary architecture. At least as early as the First Dynasty, the kings of Egypt started constructing necropolises. They are located in a north-south strip, approximately ninety kilometres in length, and go from Abu Rawash (approximately 10 kilometres north of Giza) in the north, to Meidum in the south. These necropolises contain both royal and non-royal tombs of various sizes. Where the kings decided to build their tombs was apparently influenced by such factors as the general suitability of the site and the availability of stone for the construction, as well as the geographical relationship to royal palaces and other cemeteries. That the site at Giza was one of the royal necropolises is shown by the various inscriptions, sarcophagi, human remains and other objects that have been found.⁴ There does not appear to be any contemporary evidence that the structures served any other purpose, e.g., as astronomical observatories or as recorders of historical events. Thus, from this viewpoint, Giza is just one necropolis among several; what distinguishes the site at Giza is the size of the larger of the pyramids located there.

The development of the tombal superstructures in the first four dynasties took place in several stages. The following is a brief outline of the historical process; for more detailed references see Appendices 1 and 2.

Early Structures

These structures either had some sort of wooden roof, or a simple cover of stones held together by plaster. The later introduction of brickand Abydos; that of ‘Aha measures approximately 42 m × 16m and another grave in Nakada measures approximately 53 m × 26 m.

Mastabas

A further stage was the development of the mastaba, which is an oblong shaped structure made from either brick or stone with sloping sides and a flat top. The sides of the mastabas rise very sharply, with the angle of inclination of the sides being in the 74° to 81° range. Some of these mastabas were of the stepped type, i.e., they consisted of layers of several simple mastaba type structures.

Stepped Pyramids

, was started, but never completed, by Sekhemkhet, another king of the Third Dynasty. In turn, this latter pyramid is close in style to the layer pyramid at Zawyet-el-‘Aryan, which is thus apparently also from the Third Dynasty.

The Pyramid at Meidum

The first true pyramid, of which only ruins remain, is the pyramid at Meidum (about 50 km south of Memphis). Indirect evidence associates it with Śnofru (Sneferu), founder of the Fourth Dynasty and the father of Khufwey, the builder of the Great Pyramid. This pyramid may however date from the end of the Third Dynasty. Although its final shape is that of a true pyramid, it was built up from a stepped pyramid. The slope of the faces is approximately the same as that of the Great Pyramid (51°51′).

The Bent Pyramid at Dashûr

The so-called Bent Pyramid (or Double Pyramid etc.) obtained its name from the change in the angle of inclination part of the way up. The lower part rises more steeply than the Great Pyramid (54°24′ vs. 51°51′), but then the upper half has a much smaller angle of inclination (43°21′). It is known from inscriptions to have been built by Śnofru.

The North (Red) Pyramid at Dashûr

This is the first true pyramid planned from the beginning as a pyramid. It too is known to have been built by Śnofru. Thus the name Śnofru is associated with the first three true pyramids. The angle of inclination of this pyramid is 43° 36′, which is very close to that of the upper part of the Bent Pyramid and much smaller than that of the Great Pyramid.

The Great Pyramid

From excavations in the surrounding graveyards and the appearance of the name of Khufwey (Greek name: Cheops⁶) on a stone inside the pyramid,⁷ the ancient association of the Great Pyramid with this second king of the Fourth Dynasty has been confirmed. Above, I gave the date -2620 as the approximate beginning of the Fourth Dynasty. Thus since Śnofru, the first king, appears to have reigned approximately thirty years, I shall take the round number -2600 as the approximate starting date for the construction of the Great Pyramid.⁸

Other Pyramids at Giza

In addition to the Great Pyramid there are two other large pyramids on the Giza plateau. The second pyramid is that of Kha‘frē (Ra‘kha‘ef⁹;" Greek name: Chephren) the fourth king of the Fourth Dynasty. It is almost as large as the Great Pyramid (sides: 215 m vs. 230 m; height 143 m vs. 147 m), but the angle of inclination is greater (53° 10′ vs. 51°51′). The pyramid of Kha‘frē sits on higher ground and thus it appears to be larger than the Great Pyramid. The other large pyramid on the Giza plateau is that of Menkaurē (Greek name: Mycerinus), but with sides of 108 m, it is much smaller than those of Khufwey and Kha‘frē

The Giza plateau also has seven smaller pyramids associated with queens of the pharaohs.

Other Fourth Dynasty Pyramids

In addition to the two or three pyramids of Śnofru and the pyramids on the Giza plateau, there are two other pyramids associated with the Fourth Dynasty. The first is at Abu Rawash and it is known to have been built by Djedefre (Ra’djedef) a son of Khufwey who ruled briefly before Kha‘frē. Very little remains and it may never have been completed. There is also a very large unfinished pyramid at Zawyet-el-‘Aryan (south of Giza) which has been associated with a son of Djedefre named Baka (Nebkare, Bicheris).

. . . . .

In summary, the true pyramid age started with Meidum at the end of the Third Dynasty or the beginning of the Fourth. The final shape of this pyramid was very close to that of the Great Pyramid. Following this we have the two pyramids, definitely attributed to Śnofru, whose faces, at least in the upper half in the case of the Bent Pyramid, slope up at a much smaller angle than those of the Great Pyramid. Then comes the Great Pyramid, followed by those of Kha‘frē and Menkaurē the former rising slightly more sharply than the Great Pyramid, while the second rises at approximately the same angle. Aside from the smaller pyramids at Giza and the two mentioned in the last paragraph, there are no other known true pyramids that can be assigned to the Fourth Dynasty. While true pyramids continued to be built in later dynasties, none attained the size or perfection of the Giza group.

Chapter 2

External Dimensions and Construction

Whereas Chapter 1 was devoted to a general introduction to the pyramids, Chapter 2 is devoted to details concerning the Great Pyramid and its construction. The first two sections deal with the modern values for the measurements and angle of inclination of the Great Pyramid and with the difficulties involved in obtaining these values. These are followed by sections dealing with our knowledge of ancient Egyptian units of measurement and techniques of construction. The reason that I have included these topics is that without a discussion of the practical aspects of Egyptian architecture the theories would remain in a purely abstract setting.

Surveyed Dimensions

When discussing the various theories concerning the shape of the Great Pyramid one of the things that will interest us is how closely the values predicted by these theories agree with the observed values. Thus the first question that we must consider is what indeed are, or rather what were, the dimensions of the Great Pyramid. Given the simple shape one would think that this would be an easy enough question to answer, but the answer turns out to be not so straightforward.

In his 1922 book the Egyptologist Borchardt¹ spoke of the difficulties and inexactitudes associated with past surveys of the Great Pyramid. At his request a new survey, headed by Cole, was carried out in 1925. The results of the survey are described in strictly surveying terms by Cole,² and with drawings, photographs and archaeological details by Borchardt.³ The survey was not an easy task because of the preliminary excavations required, the broken or missing stones and also because of the way the Great Pyramid was built.⁴ This is how Cole⁵ describes the construction of the pyramid and the aim of the survey:

The construction of the Pyramid on the outside was as follows: The desert was cleared down to solid rock and on this rock was built a pavement which was accurately levelled. The actual base of the Pyramid was laid out on this pavement leaving about 40 centimetres width of pavement all round the bottom edge of the casing blocks. This width is, however, not exactly the same on all four sides, it being 38 centimetres on the western side, 42 centimetres on the northern side, and 48 centimetres on the eastern side, at the places where it could be measured. At the four corners of the Pyramid the rock was cut away, giving a greater depth for the foundation of these points. These excavation are rectangular in form and are called the corner sockets.

The purpose of this survey is to determine as exactly as possible the exact size, shape and orientation of the original base of the Pyramid on the pavement.

The details mentioned by Cole are important because surveys done prior to that of Petrie in 1880-82 were not accurate. This was due to insufficient excavation and also because of a lack of understanding of the physical relationship between the underlying rock, the pavement made of dressed stone and the casing stones of the pyramid. In particular a misinterpretation of the role of the corner sockets meant that the early calculations of the length of the sides of the pyramid were usually based on the position of the extreme outer edges of the corner sockets.⁶

The values obtained by Cole⁷ for the lengths of the four sides were: north 230.253 m, south 230.454 m, east 230.391 m and west 230.357 m.⁸ The error estimates range from 12 mm on the north side to 6 cm on the west. I shall thus define 230.4 m, which is the mean of these values rounded to the first decimal place, to be the observed length of the side of the Great Pyramid.

Cole does not give a value for the height, presumably because most of the summit and the outer casing stones of the Great Pyramid are missing.⁹ I note however that if we use the observed length of the side 230.4 m in connection with the observed angle of inclination of 51.844°, to be discussed in the next section, then we can conclude that the original height was approximately 146.6 m.¹⁰ Any discussion of the height of the Great Pyramid brings up the matter of whether or not the Egyptians used height as one of the dimensions while planning the pyramid and whether they were capable of accurately determining the height of a completed pyramid. These and related questions will be discussed in Chapter 5. The orthogonality of the corners and the orientation of the sides will be discussed below.

Angle of Inclination of the Faces

In order to be able to compare the various theories concerning the shape of the Great Pyramid with a fixed numerical value, I have decided to establish an observed value for the angle of inclination of the triangular faces of the pyramid. This angle will be denoted by the Greek letter α.

Petrie¹ had measured the angle of inclination of the faces of the Great Pyramid using portions of casing blocks which were either still in place or on the ground. Among the values that he gives is 51°52′ ± 02″, which he obtained as the average of five casing stones found amomg the rubble on the north side. Petrie writes: On the whole we probably cannot do better than to take 51° 52′ ± 02″ as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side. ² In his table Petrie also gives