I. Introductory——

(a) Mathematics and its divisions—

(i) Origin of the name

(ii) The Pythagorean quadrivium

(iii) Plato's scheme

(iv) Logistic

(v) Later classification

(b) Mathematics in Greek Education

(c) Practical calculation—

(i) Enumeration by fingers

(ii) The abacus

II. Arithmetical Notation And The Chief Arithmetical Operations—

(a) English notes and examples

(b) Division

(c) Extraction of the square root

(d) Extraction of the cube root

III. Pythagorean Arithmetic—

(a) First principles

(b) Classification of numbers

(c) Perfect numbers

(d) Figured numbers—

(i) General

(ii) Triangular numbers

(iii) Oblong and square numbers

(iv) Polygonal numbers

(v) Gnomons of polygonal numbers

(e) Some properties of numbers—

(i) The " sieve " of Eratosthenes

(ii) Divisibility of squares

(iii) A theorem about cube numbers

(iv) A property of the pythmen

(f) Irrationality of the square root of 2

(g) The theory of proportion and means—

(i) Arithmetic, geometric and harmonic means

(ii) Seven other means

(iii) Pappus's equations between means

(iv) Plato on means between two squares or two cubes

(v) A theorem of Archytas

(h) Algebraic equations—

(i) Side- and diameter-numbers

(ii) The " bloom " of Thymaridas

IV. Proclus's Summary

V. Thales

VI. Pythagorean Geometry—

(a) General

(b) Sum of the angles of a triangle

(c) " Pythagoras's theorem "

(d) The application of areas

(e) The irrational

(f) The five regular solids

VII. Democritus

VIII. Hippocrates Of Chios—

(a) General

(b) Quadrature of lunes

(c) Two mean proportionals

IX. Special Problems—

1. Duplication of the Cube —

(a) General

(b) Solutions given by Eutocius—

(i) " Plato "

(ii) Heron

(iii) Diocles

(iv) Menaechmus : solution by conies

(v) Archytas : solution in three dimensions

(vi) Eratosthenes

(vii) Nicomedes : the Conchoid

2. Squaring of the Circle—

(a) General

(b) Approximation by polygons—

(i) Antiphon

(ii) Bryson

(iii) Archimedes

(c) Solutions by higher curves—

(i) General

(ii) The Quadratrix

3. Trisection of an Angle—

(a) Types of geometrical problems

(b) Solution by means of a verging

(c) Direct solutions by means of conies

X. Zeno Of Elea

XI. Theaetetus—

(a) General

(b) The five regular solids

(c) The irrational

XII. Plato—

(a) General

(b) Philosophy of mathematics

(c) The diorismos in the Meno

(d) The nuptial number

(e) Generation of numbers

XIII. Eudoxus of Cnidos—

(a) Theory of proportion

(b) Volume of pyramid and cone

(c) Theory of concentric spheres

XIV. Aristotle—

(a) First principles

(b) The infinite

(c) Proof differing from Euclid's

(d) Mechanics—

(i) Principle of the lever

(ii) Parallelogram of velocities

XV. Euclid—

(a) General

(b) The Elements—

(i) Foundations

(ii) Theory of proportion

(iii) Theory of incommemurables

(iv) Method of exhaustion

(v) Regular solids

(c) The Data

(d) The Porisms

(e) The Conies

(f) The Surface Loci

(g) The Optics

(h) A Pre-Eudoxan Theory of Proportion

(i) The Golden Section