# The Infinite in the Finite

## Contents

1. Symphonies of stone
2. The pyramid builders
3. The Theban Mysteries
4. Babylon
Babylonian mathematics
5. The Middle Kingdom
Chinese mathematics
6. The Achaeans
7. A world made of numbers
Pythagoras
The shapes of numbers
The regular polyhedra
Euler’s number
Polyhedra in the world
8. The thoughts of Zeus
The symmetries of polygons
The symmetry groups of the regular polyhedra
9. The philosopher’s criticism
Geometry
The Peloponnesian War
Socrates
Plato
Aristotle
Aristotle’s logic
The Stoics construct the truth
10. The Elements of Euclid
Euclid’s dream
Similar triangles
The angles of triangles
The area of a triangle
Pythagoras’ theorem
Triangles in circles
11. An island interlude
12. Proportion
The geometrical solution of aha problems
The theory of proportion
The construction of regular polygons
The uses of proportion
A problem of maxima
13. The Divine Archimedes
Archimedes
The measurement of a circle
The method of exhaustion
The surface area of a sphere
The volume of a sphere
The volume of a cone
Archimedes’ principle
The Rancher’s Dilemma
14. Apollonius the great geometer
Apollonius
Apollonius’ Conics
The three conic sections
Tangents to conic sections
The property of the parabola used by Archimedes
The centres of conics
The foci of a conic
Reflection properties of conic sections
The focal construction of conics
15. The science of numbers
Pythagorean numerology
Prime numbers
Irrational numbers
Pythagorean triples
Patterns of primes
16. The School of Alexandria
Alexandria
Heron
Diophantus
Pappus
The last of the Greeks
The Eudemian summary
17. The dark subcontinent of India
The Aryans
Sanskrit and the Hindu numerals
Hindu astronomy
The mathematics of Brahmagupta and Mahavira
A pearl of number theory
18. The contribution of Islam
The conquests of the Arabs
Trigonometry
The uses of trigonometry
The geometry of the sphere
The gnomon curve
Algebra
The summation of powers of integers
Spain under Islam
• Bibliography
• Index