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\mni
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\centerline{ \bfsixteen   The Shape of the Great Pyramid}%        
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\centerline{ \bffourteen Roger Herz-Fischler \(2000\)}
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\centerline{ \bffourteen Introduction}
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\nin
Perhaps no other structure built by humans has at\-tracted 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 block\-age points, and the yet to be found
tomb of the Pharoah Khu\-fwey (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, \lq What
was the {\iteleven  geometrical\/}$^{1}$ basis, if any, that was used to
determine the shape of the Great Pyramid?\hglue 1.0pt \rq\ 
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.
\mni%
%%
In 1972 I was asked to teach a mathematics course for first-year
 students of architecture. Since I was
 essentially free$^{2}$  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, {\iteleven  L'Esth\ae tique
 des proportions dans la nature et dans les arts}, concerning a
 putative text by the ancient Greek historian He\-ro\-dot\-us. 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 \lq theory\rq\  and \lq observation\rq\  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
Hero\-dot\-us. This proved to be impossible, for the putative
statement by He\-ro\-dot\-us simply did not exist;  the only
description in the {\iteleven  Histories\/} of He\-ro\-dot\-us which dealt
with the dimensions of the Great Pyramid bore little  prima
facie resemblance to what Ghyka had written.
%
\mni
%
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 {\iteleven 
history\/} as such, but also in
what I refer to as the {\iteleven  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 {\iteleven  philosophical\/} questions
related to the acceptance of theories.
%
\mni
%
The present work is the result of my research and reflection. My
basic approach in this book is the same as that in my {\iteleven  A
Mathematical History of Division in Extreme and Mean Ratio}, my
articles in art and architectural history and my forthcoming {\iteleven 
 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
\lq approaches\rq\  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.$^{5}$
\mni
%
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.
%
\mni
%
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 tom\-ball 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.
%
\mni
%
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 {\iteleven  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.
%
\mni
%
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.
%
\mni%
%
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 {\iteleven  seked\/} theory, as well as a discussion of the pyramid
problems in the {\iteleven  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 He\-ro\-dot\-us cited above in connection with the book
by Ghyka, and which constitutes the  \lq historical\rq\  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 Plu\-tarch's {\iteleven  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.
%
\mni
%
Part 3 begins with a discussion of {\iteleven  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.
%
\mni
%
Chapter 18 is devoted to a case study of the {\iteleven  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 \lq squaring
of the circle\rq\ , 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.
%
\mni
%
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, \lq What
was the geometrical basis that was used to determine the shape of
the Great Pyramid?\hglue 1.0pt \rq\gl. 
%
\mni
%
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.
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