5000 Years of Geometry: Mathematics in History and Culture
Format: PDF / Kindle (mobi) / ePub
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)
circumference.) The Babylonians also dealt with circle segments cut oﬀ from the circle through a chord c (cf. Illus. 1.2.5), whose height (the line segment, which stands perpendicularly on the middle of the chord between chord and circumference), also called sagitta s, was calculated with diameter d and the chord c according to the formula s= 1 (d − 2 d2 − s2 ). (1.2.11) The chord or segment base c was calculated according to c= d2 − (d − 2s)2 . (1.2.12) (see Problem 1.2.6). We are dealing
administration, citizens took part in public life to a greater extent. These towns became the centres of classic Greek culture and science. The peripheral areas of the Mediterranean Sea and the Black Sea were also Hellenised due to the founding of colonies. The Ionian era is particularly known for the ﬁrst great natural philosophers of all time: Thales, Anaximander and Anaximenes. It is here that we ﬁnd the origins of European thinking and, also, of the deductive method in mathematics, which
function when investigating inﬁnitesimal problems (e.g., squaring the parabola). The only text of his that was entirely dedicated to geometry dealing with semi-regular polyhedra – was lost. His treatise ‘On spirals’ [Archimedes c] serves as an example of how Archimedes used geometrical considerations in his research. Archimedes dedicated this particular work to the mathematician Dositheus. Hereby, his introduction was written as a letter to Dositheus and referred to other works and proofs that
the formula that is to be proven. The following method of proof is determined by the fact that the four quantities are represented geometrically and expressed appropriately. Extend the triangle side CDB beyond B by line segment AF until H. As a result (why?): CH · OD = Φ and, thus, Φ2 = CH 2 · OD 2 . Now construct the perpendicular OL on OC (which intersects BC in K), the perpendicular BL on BC and ﬁnally link C with L. Why do points C, O, B, L lie on one circle now and, consequently, COB + CLB =
3.4.6). A special accomplishment of Arabic mathematics is the calculation of constant π in al-K¯ash¯ı’s ‘Treatise on the circumference’, which he concluded in 1424. Al-K¯ ash¯ı had set himself the ambitious aim to calculate π so exactly that the error concerning a circumference, the diameter of which is 600 000 Earth diameters, would not exceed a hair’s breadth. He contemplated using a regular quadrilateral, the side of which fulﬁls the inequation a < 6084 for a circle with a radius of 60. His