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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 Euclid’s “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. Euclid’s 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, N´aden´ık 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

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 Euclid’s “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. Euclid’s 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, N´aden´ık 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