Projects
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Book Projects
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A History of Egyptian Mathematics - An Introduction (in preparation for Princeton University Press) |
| Chapter: "Egyptian Mathematics" in the Source Book of Non Western Mathematics (in preparation for Princeton University Press) |
| Chapter: "Traditions and myths in the historiography of Egyptian mathematics" in the Oxford Handbook of the History of Mathematics (in preparation for Oxford University Press) |
Research Projects
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| Demotic Mathematics - Traditions, Development, Transmission |
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Excerpt from the Introduction
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The study of Egyptian mathematics has found the interest of modern scholars
since almost 200 years. Some try to find the foundations of the impressive
building efforts---still evident today in remains of pyramids, temples, and
tombs all over Egypt---which were executed under the auspices of Egyptian
pharaohs. Others are drawn to the subject by a fascination with mathematics
and its earliest foundations: Egypt and Mesopotamia were the first cultures
to develop sophisticated mathematical systems. The way these were organized,
their distinct characteristic features which in some respect differ greatly
from our modern mathematical habits, in others, however, are surprisingly
similar to what we do today, appeals to mathematicians as well as students
interested in the history of their subject. But it is also relevant---and
due to the extant sources maybe most interesting of all the above mentioned
groups yet---to those who study practical aspects of daily life in ancient
Egypt. Then, as today, mathematics was needed in daily life. Consequently
the mathematical system developed in pharaonic Egypt was practically oriented,
designed to satisfy the needs of bureaucracy. Thus it is "more than mathematics"
that we can find in Egyptian mathematical sources. We can learn about the
practical backgrounds which required and shaped the evolving mathematical
knowledge by reading mathematical texts---they inform us about exchange of
bread and beer, the distribution of rations, work-rates of different professions
and other aspects of daily life. Thereby the information found in mathematical
texts complements the evidence from administrative documents and archaeological
finds. |
The first publications about mathematics in pharaonic Egypt appeared at
the beginning of the 19th century, after the now famous Rhind mathematical
papyrus had made its way into the collections of the British Museum, and
thereby made itself available for study. Until today it constitutes our
most important source and its initial publication in 1877 was followed by
two further editions in 1923 and 1927, as well as numerous articles and some
monographs about Egyptian mathematics---most of which are based on the contents
of the Rhind mathematical papyrus. The amount of available literature is
indeed rather astonishing, if it is taken into account that the only sources
on which all studies of Egyptian mathematics are founded are four papyri,
a leatherroll a wooden board and two ostraca---all of them mostly from the
Middle Kingdom (2055--1650 BC) and five more papyri (as well as some ostraca)
from the Ptolemaic and Roman Periods (332 BC--395 AD), more than a thousand
years later than the earlier texts. Furthermore, the Ptolemaic and Roman
sources (the number of which is likely to increase from unpublished texts
in museums as well as new finds) were rarely taken into consideration. Studies
of Egyptian mathematics mostly concentrated on the earlier material---of
which half of the papyri actually consits of a number of fragments only.
In spite of all these limitations, the sources still allow us some insights,
and are sufficient material for a number of studies as is proven by publications
of the last 125 years. Moreover, we seem to be far from an exhaustive exploitation
of these sources. Due to developments in the field of history of mathematics
of the last 30 years, it has only recently become obvious that many "statements"
about Egyptian mathematics which were made a long time ago, and which have
since then been accepted as "truths", are in fact in serious need of reassessment. |
At the beginning of the 20th century---a period in which research on Egyptian
mathematics reached its first b(l)oom, questions of {\it how} the ancient
mathematical knowledge related to later developments (e.g. Greek geometry)
and our modern mathematical system determined research on ancient sources.
The unfamiliar way in which mathematical operations were expressed were translated
into their "modern equivalents"---unfortunately without ascertaining that
the same mathematical concepts are used in both mathematical cultures. It
is exactly the insight that there is not only "one mathematics" developed
at different times differently but resting on the same foundation on which
then more and more is built. Rather, mathematics is very much dependant on
the culture in which it is created that has lead to the new trend in history
of mathematics characterized by 'close readings' of sources. |
We are now at the beginning, or possibly the end of the beginning, of a
new stage in our knowledge---and this book aims to give an account of what
we know by now, as well as the direction of studies that are currently undertaken.
Along with this, I hope to reduce the amount of speculations about Egyptian
science. The scarce sources have not been (and almost certainly never will
be) able to answer every question asked today. Unfortunately, this has encouraged
rather speculative theories founded on practically no evidence. In fact only
the lack of evidence enabled speculations of this kind. Classical examples
are the way the pyramids were aligned, and built, as well as the way Egyptian
mathematical knowledge was discovered. The honest answer to these questions
(at least at the present moment) is simply "we don't know". However, for
some of these questions at least some possibilities can be discussed, others
may almost certainly never be answered. With the same certainty there will
always be efforts and speculative theories to do so. While it may sometimes
be true that "a wrong idea is better than none", it needs to be ascertained
to keep the actual available evidence in the picture. By presenting the available
mathematical sources this book aims to encourage readers in critically judging
speculations about Egyptian science. |
The improvement in history of mathematics which has influenced recent studies
of Egyptian mathematics, is paralleled by a better understanding of Egyptian
languages and culture as well. The mathematical texts have not yet benefitted
from this development. The Rhind mathematical papyrus was last edited by
a group of mathematicians (!) in 1927/28; its {\it editio princeps} remains
the second edition prepared by the Egyptologist (and former mathematician)
Thomas Eric Peet in 1923. The Moscow mathematical papyrus was only edited
once in 1930. Since then several parts of both texts have been used and reworked;
however, an edition of these two major sources in the philological quality
of Egyptological studies has not yet been attempted. The third largest source,
the Lahun mathematical fragments, have just been reedited within the new
edition of the Lahun papyri. It is to be hoped that other sources will eventually
follow. |
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Abstract
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Until fairly recently, studies of Egyptian mathematics have followed rather
traditional paths. In most cases, scholars have not dealt with the primary
source material in a substantial way. Instead, their work has drawn on the
published editions of preserved primary sources (prepared more than sixty
years ago). |
There are only two major sources of Egyptian mathematical texts, the Rhind
and Moscow mathematical papyri. It is rather surprising therefore, that only
the Rhind mathematical papyrus is available in an English translation. The
most easily available book is the reprint of the edition by Chace, Bull,
Manning of 1927/29, printed in the 1970s. This, however, contains only a
so-called "free" translation, which is rather an interpretation of the source
text than a translation. As for the Moscow papyrus, the only edition was done
in 1930 by Wasili Struve. He translated it into German. The source book
by Marshall Clagett unfortunately does not fill this gap, but gives an English
rendering of the text in the style of a "free" translation. |
The proposed source book is a first step to improve this situation. It will
include major examples from the two most famous texts. In addition, texts
that are less known in the history of mathematics will be incorporated. These
are on the one hand administrative and literary texts contemporary to the
Hieratic mathematical texts. These further sources allow us to see "mathematics
in practice" and the reputation of mathematics in Egyptian society, while
the mathematical texts present only the pedagogical side. On the other hand,
the Demotic mathematical texts, which show Egyptian mathematics almost 1000
years later, shall be included in the examples of this source book. Apart
from their publication in the 1970s, these texts have for the most part
gone unnoticed in the history of mathematics. |
The translations provided in this book will fulfill the standards of modern
Egyptology. Using the examples of the source book, students will be able
to gain access to the individual sources. Taking administrative and literary
texts into account, as indicated above, will enable a deeper understanding
of the individual mathematical problems in particular and the role of mathematics
in Egyptian society in general. |
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Abstract
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Research into Egyptian mathematics began in the second half of the 19th century (the
Mathematical Papyrus Rhind was edited for the 1st time in 1877). Most sources were
published by 1930, and their editions represent outstanding pioneering work. However, since
then fundamental changes in the historiography of mathematics as well as significant
development in our knowledge of Egyptian philology and culture require a critical
reassessment of these early works. A close look at more recent secondary literature indicates
that this reassessment has often not been done. Instead, a number of “opinions” that are
outdated by modern standards, or have since been proven to be wrong, have remained in the
literature unchallenged, and, over time, have acquired the status of “truths”. This chapter will
look at instances of such “mythical truths” (e.g. the so-called Horus-eye fractions,
cumbersome Egyptian fraction reckoning, stagnation in the development of Egyptian
mathematics), explain why they are not valid, and indicate possible reasons for their
persistence in modern literature.
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| Egyptian mathematics has traditionally been seen as having
reached the height of its development in the early Middle Kingdom, remaining
nearly "unchanged" from that time forth. This assessment, however, is based
on a very small number of Hieratic texts that span a period of at most 200
years. Yet to challenge this established view, a serious reassessment of the
source material is needed. Drawing on my previous work on the Hieratic mathematical
problem texts, I plan to extend the scope of my studies of Egyptian mathematics
by taking up the Demotic mathematical problem texts. |
The Demotic mathematical texts, not all of which have been published, date
from more than 1000 years later than their Hieratic predecessors. Like the
Hieratic texts, however, they are written in an algorithmic style, i.e. as
a series of problem statements followed by step-by-step instructions leading
to their solution. A careful study of the original sources, most of which
are held in the British Museum, should reveal the most important algorithms
employed in the Demotic problem texts. These results can then be compared
with the algorithms used in the Hieratic problem texts, as established in
my doctoral thesis. |
In a 1972 study of Demotic mathematical papyri, Richard Parker claimed to
discern a Mesopotamian influence in some of the problems, an assertion based
on the occurrence of identical problems in Demotic and Babylonian mathematical
texts [1]. This alone, however, does not necessarily indicate evidence of
a transmission of mathematical knowledge, as has been emphasized by James
Ritter [2]. To draw conclusions of this kind one must analyse the problem-solving
techniques employed by the mathematicians working in these respective cultures.
Examining their procedures for solving problems in terms of an algorithmic
analysis enables us to make a detailed comparison of the respective mathematical
methods they used. It is not to be expected that such an analysis will reveal
that Mesopotamian and Egyptian mathematicians employed identical algorithms;
after all they utilized different arithmetical concepts (for instance a sexagesimal
number system in Mesopotamia versus the Egyptian decimal system without
place value). Nevertheless, a comparison of the individual algorithms should
reveal similarities or differences in the underlying strategies used to
solve similar problems. This study should therefore help to determine whether
or not a significant transmission of mathematical knowledge took place between
these two cultures. |
I intend to write a book about Demotic mathematics focusing on the questions
indicated above. After a close textual study of the Demotic sources, the
majority of which are in the British Museum, I will establish the algorithms
found in these texts. I then plan to compare the Demotic techniques with
the Hieratic and Mesopotamian procedures. During the final stage of this
project my focus will be to prepare a monograph presenting the results of
this study. In addition to its intrinsic importance as the first detailed
study of the Demotic mathematical texts since 1972, this project will complement
my earlier work on the Hieratic material to form part of my wider study of
Egyptian mathematics. |
References
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| [1] Richard A. Parker, Demotic Mathematical Papyri, Providence,
RI: Brown University Press and London: Lund Humphries 1972, p. 6. |
[2] Jim Ritter, "Measure for Measure: Mathematics in Egypt and Mesopotamia".
In: Michel Serres (ed.), A History of Scientific Thought. Elements of a History
of Science: 44-72. Oxford and Cambridge, Mass.: Blackwell Publishers Inc,
pp. 44-72. |
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| © 2001-2009 Annette Imhausen |
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