Selections illustrating the history of greek mathematics. From Thales to Euclid 1

Title page Preface 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 Educatiun (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 numhers (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 theoem 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. PHOCLUS's SUMMARY V. THALES VI. PYTHAGOHEAN GEOMETRY (a) General (b) Sum of the angles of a triangle (c) "Pythagoras's theorem" (d) The application of areas186 (e) The irrational (f) The £ive 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) Diodes: the Cissoid (iv) Menaechmus: solution by conics (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 hy means of a verging (c) Direct solutions hy means of conics 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 incommensurables (iv) Method of exhaustion (v) Regular solids (c) The Data (d) The Porisms (e) The Conics (f) The Surface Loci (g) The Optics