: Preface of the editor of the German edition


10-26-2019, 05:05 AM
Geometry (from the Greek word for measuring the Earth, the modern scientific discipline of which is now called geodesy), branch of science which
deals with regular patterns, shapes and solids, was one of the first human
attempts, after counting, to concern themselves with the emerging science
mathematics. This is evident from the spirals on megalithic graves, incisions
in stone and patterns on clay fragments.
In this book, you will learn how geometry has developed over the millennia
from these earliest origins in distant times and much more. Geometry is
an indispensible aid for building and surveying, and became an axiomatic
science of plane and spatial shapes in Ancient Greece. It served as a basis
for astronomical observations and calculations, for Islamic decorative art, and
the building of medieval Christian cathedrals. Furthermore we will look at the
discovery of perspective and its application in Renaissance art, at the disputes
regarding the Euclidean parallel postulate, the discovery of non-Euclidean
geometries in the 19th century, and, finally, the theory of infinite-dimensional
spaces and contemporary computer graphics.
This book is edited by the project group History of Mathematics at the
University of Hildesheim as part of the series Vom Z ahlstein zum Computer
(From Pebbles to Computers). Other titles in this series published by Springer
Publishing Heidelberg are: 4000 Jahre Algebra (4000 Years of Algebra) [Alten et al. 2003], and 6000 Jahre Mathematik (6000 Years of Mathematics)
[Wußing, in two volumes 2008/09]. To the series From Pebbles to Computers
two video films have been produced (University of Hildesheim): Mathematik
in der Geschichte Altertum (Mathematics in History Antiquity) [Wesem uller-Kock/Gottwald 1998] and Mathematik in der Geschichte Mittelalter (Mathematics in History Middle Ages) [Wesem uller-Kock/Gottwald
2004]. Following multiple reprints and the second edition in 2004 we now
present the third edition of 5000 Jahre Geometrie including new research
results on circular ditches in the Stone Age and the Nebra Sky Disk, as well
as many illustrations in colour.
In this book, we will reflect on the development of geometry as part of our
cultural history over the course of five millennia. Both authors have succeeded
in portraying the origins and growth of this branch of mathematics, which is
often thought of as dry and jejune, in a tremendously lively manner. They uncover the origins and impulses for the development of geometric notions and
methods, and present how they are related to historical events and personal
fates. Moreover, they describe the applications of geometrical knowledge and
methods in other areas and the interdependencies that resulted from them.
Finally, they emphasize their importance for other disciplines.
At the heart of this book series is portraying the history of mathematics as
an integral part of the history of mankind, particularly as a fundamental part
of our cultural heritage. Both authors have done justice to this task in an
impeccable manner. They have depicted the genesis of geometry and its inVVI
terlacing with cultural developments in other areas, such as literature, music,
architecture, visual arts and religion, by a standard far higher than usual in
mathematical-historical presentations. They also describe the implications of
geometrical findings and methods for other areas. As such, the authors also
deal far more extensively than usual with the development of geometry in
other cultures, mainly in the ancient oriental cultures, in Islamic countries,
as well as in India, China, Japan and the old American cultures. Tables at
the beginning of each chapter give an overview of important political and
cultural events of each cultural area and era dealt with. Tables at the end
summarise the main geometrical contents of each chapter in note form.
Moreover, the authors compare views of ancient and medieval mathematicians with modern mathematical findings and link those to contemporary
mathematics and related sciences, for example, references to computer sciences regarding the description of Euclids algorithmic accomplishment.
Furthermore, they highlight the specifications of geometrical examinations of
different eras and cultural areas and the changes in content, methods and
approaches geometry has faced as a proto-physics within three-dimensional
or even infinite-dimensional spaces. They discuss the relationship of geometry with other branches of mathematics, for instance with algebra, analysis,
and stochastics. Refreshing asides with biographical highlights and references
to unexpected relations, as well as text excerpts in the appendix, bring this
book to life.
Chapters 1 through 4, with the exception of sub-chapter 2.3 (Euclid), were
written by Dr. Christoph J. Scriba, professor emeritus for the history of the
natural sciences in the former Institute for History of Natural Sciences, Mathematics and Engineering at the University of Hamburg. Euclids accomplishments and the development of geometry in modern times from Chapters 5
through 8 were described by Dr. Peter Schreiber, professor for geometry and
the foundations of mathematics at the University of Greifswald.
We are also grateful to the authors for numerous illustrations and the texts
for the appendixes. The figures that have been added to support geometrical
theorems that are not referenced were drawn by the authors themselves.
They also thought of the summarising problems for every sub-chapter at
the end of each chapter (cf. Introduction). They often differ from ordinary
tasks in regard to type and size and also vary in level of difficulty. Thus,
solving them requires very different background knowledge, as well as the
use of secondary literature at times. Hence, to solve some of the problems
of Chapters 1 through 4, you will mainly need knowledge gained in junior
high school, while other problems will require highschool knowledge, whereas
some problems to Chapters 5 through 8 demand insight into notions and
methods taught at university. This is due to the nature of the subject, since
mathematics has grown more and more complex and difficult over the course
of the centuries and understanding modern mathematics usually assumes
knowledge of the mathematics of past eras. Therefore, you will occasionally
find hints to solutions within the text and also the literature. However, the
Preface of the editor of the German editionVII
solutions themselves have not been included in the appendix to avoid the
following: first, we do not want you to look up the solutions too quickly;
second, the solutions most often are not the result of calculations, but require
the description of approaches for solving the problem at hand or retracing
more or less extensive considerations.
All this has been done intentionally in order to attract as large a readership
as possible. Cursory readers or those that are in a hurry should not simply
skip the problems, since they include many interesting historical remarks and
additions to the text, which is why reading the problems carefully will benefit
everyone. The extensive bibliography and index of names invite the reader
to study further.
I thank both authors sincerely for the multifaceted and intensive work in
particular their dedication to setting new accents with this book integrating
geometry in cultural history and composing many interesting problems.
I further express my gratitude to my colleagues Dauben, Flachsmeyer, Folkerts, Grattan-Guinness, Kahle, L uneburg, Nadenık und Wußing for their
scholarly advice and critical reviewing and thank H. Mainzer for advice on
historical details and Lars-Detlef Hedde (University of Greifswald), Thomas
Speck and Sylvia Voß (University of Hildesheim) for converting the manuscripts, illustrations and figures into printable electronic formats.
Moreover, I wish to thank media educator Anne Gottwald, who helped us
clear the licensing for printing the illustrations, and each publisher for authorising the printing rights.
I also remain grateful to the director of the Centre for Distance Learning and
Extension Studies (ZFW), Prof. Dr. Erwin Wagner, the present and former
directors of the Institute for Mathematics and Applied Computer Science,
Prof. Dr. F orster and Prof. Dr. Kreutzkamp, the deans Prof. Dr. Schwarzer
and Prof. Dr. Ambrosi and the administration of the University of Hildesheim.
Last but not least, I wish to thank the members of the project group History
of Mathematics of ZFW: the historian of mathematics Dr. Alireza Djafari
Naini and the media expert and sociologist Heiko Wesem uller-Kock, for the
great and intensive teamwork while planning and preparing this book. I express my gratitude to Springer Publishing Heidelberg for taking my requests
into account and the excellent design of this book.
I hope that this volume will inspire many readers to study the history of
mathematics more intensively, and to learn about the background of the origins and incredibly exciting development of geometrical notions and methods.
Hopefully, this will result in the reader viewing geometry not just as a mathematical discipline or as an indispensible aid for architects, robot engineers
and scientists, but also as a valuable part of our culture that we encounter
everywhere and that makes the world in which we live so much richer.
On behalf of the project group
Hildesheim, August 2009 Heinz-Wilhelm Alten