Ancient Mathematics: History of Mathematics in Ancient Greece and Hellenism

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