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Über dieses Buch

The present volume provides a fascinating overview of geometrical ideas and perceptions from the earliest cultures to the mathematical and artistic concepts of the 20th century. It is the English translation of the 3rd edition of the well-received German book “5000 Jahre Geometrie,” in which geometry is presented as a chain of developments in cultural history and their interaction with architecture, the visual arts, philosophy, science and engineering.

Geometry originated in the ancient cultures along the Indus and Nile Rivers and in Mesopotamia, experiencing its first “Golden Age” in Ancient Greece. Inspired by the Greek mathematics, a new germ of geometry blossomed in the Islamic civilizations. Through the Oriental influence on Spain, this knowledge later spread to Western Europe. Here, as part of the medieval Quadrivium, the understanding of geometry was deepened, leading to a revival during the Renaissance. Together with parallel achievements in India, China, Japan and the ancient American cultures, the European approaches formed the ideas and branches of geometry we know in the modern age: coordinate methods, analytical geometry, descriptive and projective geometry in the 17th an 18th centuries, axiom systems, geometry as a theory with multiple structures and geometry in computer sciences in the 19th and 20th centuries.

Each chapter of the book starts with a table of key historical and cultural dates and ends with a summary of essential contents of geometr

y in the respective era. Compelling examples invite the reader to further explore the problems of geometry in ancient and modern times.

The book will appeal to mathematicians interested in Geometry and to all readers with an interest in cultural history.

From letters to the authors for the German language edition

I hope it gets a translation, as there is no comparable work.

Prof. J. Grattan-Guinness (Middlesex University London)

"Five Thousand Years of Geometry" - I think it is the most handsome book I have ever seen from Springer and the inclusion of so many color plates really improves its appearance dramatically!

Prof. J.W. Dauben (City University of New York)

An excellent book in every respect. The authors have successfully combined the history of geometry with the general development of culture and history. …

The graphic design is also excellent.

Prof. Z. Nádenik (Czech Technical University in Prague)



1. The beginnings of geometrical representations and calculations

Long before writing was developed, mankind may have realised and systematically used geometrical structures. Nature offers the eye multiple curved lines, and a blade of grass or a tree trunk can symbolise the thought of a straight line as well as the idea of a circle (as a cross-section). When weaving or braiding we generate simple two-dimensional patterns, which then are purposely modified or also replicated as decoration on clay pots.
Christoph J. Scriba, Peter Schreiber

2. Geometry in the Greek-Hellenistic era and late Antiquity

Generally speaking, the Greeks are accepted to be the founders of the natural sciences, in other words, of rational explanations of natural phenomena based on principles and systems. At the same time, it was they who systemised and accounted for rules and instructions passed on (partially by the Oriental cultures) for counting, measuring and solving equations by means of a selfdeveloped logic. These were summarised into a system of theories, which made them the founders of scientific mathematics.
Christoph J. Scriba, Peter Schreiber

3. Oriental and old American geometry

Before we deal with the European Middle Ages in the following chapter, we will present an overview in this chapter of the development of geometry in the oriental countries up to the 15th century, in Japan until the end of its seclusion in 1868, and in the old American cultures.
Christoph J. Scriba, Peter Schreiber

4. Geometry in the European Middle Ages

The time from the collapse of the Roman Empire as the result of the migration of nations (“Völkerwanderung”) up until the Renaissance shall be summarised here as the era of the European Middle Ages. From the aspect of development within the mathematical, natural-scientific realm we end here with the beginning of the 15th century and not only with the discovery of America. As a result, the two “newcomers” of mathematics, Nikolaus von Kues (Nicholas of Cusa) and Regiomontanus will open the next period.
Christoph J. Scriba, Peter Schreiber

5. New impulses for geometry during the Renaissance

The following time period of around 230 years between approx. 1400 and approx. 1630 is usually referred to as the Renaissance (i.e., rebirth, namely of Antiquity) in the history of science, although this is not entirely accurate and cannot be reconciled with the art-historical periods.
Christoph J. Scriba, Peter Schreiber

6. The development of geometry in the 17 th and 18 th centuries

From about 1630 until approximately 1800 (when mathematics yet again took a profound turn due to reasons we will discuss later on), those scholars who dealt with the development of mathematics in a manner that – from today’s perspective – was significant for its historical development, were small in number and easy to identify. Generally speaking, they were in contact with one another.
Christoph J. Scriba, Peter Schreiber

7. New paths of geometry in the 19 th century

At the turn from the 18th to the 19th century, both the character of mathematics and its external conditions changed fundamentally. The industrialisation beginning in 1770 was the general background for technological development. However, political turmoil in Europe caused by the French Revolution and the following Napoleonic Wars, which conveyed civil ideas to almost every corner of Europe, also contributed greatly to the changes.
Christoph J. Scriba, Peter Schreiber

8. Geometry in the 20 th century

If it was difficult for the 19th century to fit the wealth of geometrical tendencies into a limited number of sub-chapters, it will be hardly possible to do so analogously for the 20th century and would not serve the purpose of this book.
Christoph J. Scriba, Peter Schreiber


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