The volume contains a comprehensive and problem-oriented presentation of ancient Greek mathematics from Thales to Proklos Diadochos. Exemplarily, a cross-section of Greek mathematics is offered, whereby also such works of scientists are appreciated in detail, of which no German translation is available. Numerous illustrations and the inclusion of the cultural, political and literary environment provide a great spectrum of the history of mathematical science and a real treasure trove for those seeking biographical and contemporary background knowledge or suggestions for lessons or lectures. The presentation is up-to-date and realizes tendencies of recent historiography.
In the new edition, the central chapters on Plato, Aristotle and Alexandria have been updated. The explanations of Greek calculus, mathematical geography and mathematics of the early Middle Ages have been expanded and show new points of view. A completely new addition is a unique illustrated account of Roman mathematics. Also newly included are several color illustrations that successfully illustrate the book's subject matter. With more than 280 images, this volume represents a richly illustrated history book on ancient mathematics.
Author(s): Dietmar Herrmann
Publisher: Springer
Year: 2023
Language: English
Pages: 461
Preface
Contents
1 Introduction
1.1 To the Content of the Book
1.2 On the State of Mathematical Historical Research
References
2 How Greek Science Began
2.1 The Origins of Mathematics
2.2 The Greek Numeral Symbols
2.3 The Greek School
Further Reading
3 Thales of Miletus
3.1 Mathematical Work
3.2 More Reports on Thales
Further References
4 Pythagoras and the Pythagoreans
4.1 Pythagoras of Samos
4.2 The Pythagoreans
4.3 Mathematical Discoveries of the Pythagoreans
4.4 Figured Numbers
4.5 Pythagoras’ Theorem
4.6 Pythagorean Number Triplets
4.7 Heronian Triangles and Applications
4.8 Pythagoras and Music
Further References
5 Hippocrates of Chios
5.1 Quadratures According to Alexander of Aphrodisias
5.2 Quadratures According to Eudemos
References
6 Athens and the Academy
6.1 Athens
6.2 The Academy
6.3 The Mathematicians of the Academy
6.4 Eudoxus of Knidus
6.5 Theodorus of Cyrene
6.6 Theaitetos of Athens
References
7 Plato
7.1 Plato’s Most Beautiful Triangles
7.2 From the Book Menon
7.3 Platonic Solids
7.4 Plato’s Lambda
7.5 The Role of Mathematics in Plato
Further References
8 Aristotle and the Lyceum
8.1 Life and Work of Aristotle
8.2 Mathematics with Aristotle
References
9 Alexandria
9.1 The Library
Further References
10 Euclid of Alexandria
10.1 From Book I of the Elements
10.2 From Book II of the Elements
10.3 The Circle Theorems in Book III
10.4 Perfect and Friendly Numbers
10.5 The Euclidean Algorithm
10.6 Euclid’s Theorem on Primes
10.7 The Parallel Axiom
10.8 Equivalent Postulates to the Parallel Axiom
10.9 Book of Area Divisions
Further References
11 The Classical Problems of Greek Mathematics
11.1 Incommensurability
11.2 The Constructability According to Euclid
11.3 Angle Trisection
11.3.1 Constructions for angle trisection
11.4 Squaring the Circle
11.5 Doubling the Cube
11.6 Constructability of the Pentagon
11.7 Constructibility of the Heptagon
11.8 Quadratures of Lunes
11.9 The Division in extreme and mean ratio (EMR)
Further References
12 Archimedes of Syracuse
12.1 On the Centers of Gravity
12.2 Problem of the Broken Chord
12.3 The Regular Heptagon
12.4 The Book of Measuring the Circle
12.5 From the Book of Spirals
12.6 The Book of Lemmata
12.7 The Quadrature of the Parabola
12.8 The Palimpsest
12.9 The Stomachion
12.10 The Method, Theorem 2
12.11 Archimedes’ Tomb Figure
12.12 More Works by Archimedes
References
13 Eratosthenes of Cyrene
13.1 Eratosthenes as a Mathematician
13.2 Eratosthenes as a Geographer
References
14 Conic Sections
14.1 The Parabola
14.2 The Ellipse
14.3 Hyperbola
References
15 Apollonius of Perga
15.1 From Book III of the Conica
15.2 Apollonius’ Theorem
15.3 The Touching Problem of Apollonius
15.4 The Theorem of Apollonius
References
16 Beginnings of Trigonometry
16.1 Aristarchus of Samos
16.2 Hipparchus of Nicaea
16.3 Menelaus’ Theorem
16.4 Applications in Geography
Further References
17 Heron of Alexandria
17.1 From the Definitions
17.2 From Metrica and Geometrica
17.3 From the Stereometrica
17.4 Heron’s Formula
17.5 Cube Doubling by Heron
17.6 Area of the Regular Pentagon
17.7 Root Calculation Among the Greeks
17.8 Other Works by Heron
References
18 Claudius Ptolemy
18.1 Trigonometry in the Almagest
18.2 Ptolemy’s Theorem
18.3 The Addition Theorem
18.4 Construction of the Pentagon According to Ptolemy
18.5 Construction of the 15-gon
18.6 The Geographical Work
References
19 Nicomachus of Gerasa
19.1 From the Arithmetica
19.2 Proportions and Averages
19.3 The Nikomachos Theorem
19.4 From the Commentary of Lamblichos
References
20 Theon of Smyrna
20.1 The Side or Diagonal Numbers
20.2 Geometric Interpretation
20.3 Number Theory
References
21 Diophantus of Alexandria
21.1 From Diophantus’ Book I and II
21.2 From Diophantus’ Book III to V
21.3 From Diophantus’ Book VI
21.4 From Diophantus’ Books in Arabic
21.5 To the Mathematics of Diophantus’
21.6 Linear Diophantine Equation
21.7 Outlook
21.7.1 The Problem of Congruent Numbers
21.7.2 Representability by the Sum of Two Cubes
21.7.3 Representability by the Sum of Three Cubes
21.7.4 Representability by the Sum of Two Squares
Further References
22 Pappus of Alexandria
22.1 From Book VII of the Collectio
22.2 Pappos’ Rule
22.3 Pappos’ Touching Problem
22.4 Pappos’ Theorem
22.5 The Complete Quadrilateral
22.6 Harmonic Division
22.7 The Four-Line Problem
22.8 More Problems of Pappos
References
23 Theon of Alexandria
23.1 Life and Work
23.2 Hypatia of Alexandria
References
24 Proclus Diadochus
24.1 The Mathematician's Catalogue of Eudemus
24.2 More Important Quotes
References
25 Roman Mathematics
25.1 Calculating with Roman Numerals
25.2 Mathematical Examples from Literature
25.3 The Roman school
25.4 The Role of Roman Surveyors
25.5 From the Corpus Agrimensorum
References
26 The Legacy of Hellenistic Mathematics
26.1 Mathematics in Byzantium
26.1.1 From the Greek Anthology (Anthologia Graeca)
26.2 Outlook on the Latin Early Middle Ages
26.2.1 Boethius
26.2.2 Isidore of Seville
26.3 Greek legacy in Islam
References
Literatur
