A History of Greek Mathematics, Volume II · From Aristarchus to Diophantus by Heath, Sir Thomas -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Contents XII. Aristarchus of Samos XIII. Archimedes

Traditions

(α) Astronomy (β) Mechanics

Summary of Main Achievements Character of Treatises List of Works Still Extant Traces of Lost Works The Text of Archimedes Contents of the Method On the Sphere and Cylinder, I, II

Cubic Equation Arising Out of 11.4

(i) Archimedes’s Own Solution (ii) Dionysodorus’s solution (iii) Diocles’s Solution of Original Problem

Measurement of a Circle On Conoids and Spheroids On Spirals On Plane Equilibriums, I, II The Sand-Reckoner (Psammites or Arenarius) The Quadmture of the Parabola On Floating Bodies, I, II

The Problem of the Crown

Other Works

(α) The Cattle-Problem (β) On Semi-Regular Polyhedra (γ) The Liber Assumptorum (δ) Formula for Area of Triangle

Eratosthenes

Measurement of the Earth

XIV. Conic Sections. Apollonius of Perga

A. History of Conics Up to Apollonius

Discovery of the Conic Sections by Menaechmus

Menaechmus’s Probable Procedure

Works by Aristaeus and Euclid

‘Solid Loci’ and ‘Solid Problems’

Aristaeus’s Solid Loci Focus-Directrix Property Known to Euclid

Proof from Pappus

Propositions Included in Euclid’s Conics Conic sections in Archimedes

B. Apollonius of Perga

The Text of the Conics Apollonius’s Own Account of the Conics

Extent of Claim to Originality Great Generality of Treatment

Analysis of the Conics Book I

Conics Obtained in the Most General way from Oblique Cone New Names, ‘Parabola’ ‘Ellipse’ ‘Hyperbola’ Fundamental Properties Equivalent to Cartesian Equations Transition to New Diameter and Tangent at its Extremity First Appearance of Principal Axes

Book II Book III Book IV Book V

Normals as Maxima and Minima Number of Normals from a Point. Propositions Leading Immediately to Determination of Evoluta of Conic Construction of Normals

Book VI Book VII Other Works by Apollenius

(α) On the Cutting off of a Ratio , Two Books (β) On the Cutting-off of an Area , Two Books (γ) On Detemninate Section, Two Books (δ) On Contacts or Tangencies, Two Books (ε) Plane Loci, Two Books (ζ) (Vergings or Inclinations), Two Books (η) Comparison of Dodecahedron with Icosahedron (θ) General Treatise (ι) On the Cochlias (κ) On Unordered Irrationals (λ) On the Burning-mirror (μ)

Astronomy

XV. The Successors of the Great Geometers

Nicomedes Diodes Perseus Isoperimetric Figures. Zenodorus Hypsicles Dionysodorus Posidonius Geminus

Attempt to Prove the Parallel-Postulate On Meteotvlogica of Posidonius Introduction to the Phaenomena Attributed to Geminus

XVI. Some Handbooks

Cleomedes, De motu Circulari Nicomachus Theon of Smyrna, Expositio Rerum Mathematicarum ad Legendum Platonem Utilium

XVII. Trigonometry: Hipparchus, Menelaus, Ptolemy

Theodosius Works by Theodosius Contents of the Sphaerica

No Actual Trigonometry in Theodosius

The Beginnings of Trigonometry Hipparchus

The Work of Hipparchus

First Systematic use of Trigonometry Table of Chords Menelaus

The Sphaerica of Menelaus

(α) ‘Menelaus’s Theorem’ for the Sphere (β) Deductions from Menelaus’s Theorem (γ) Anharmonic Property of Four Great Circles Through One Point (δ) Propositions Analogous to Eucl. VI. 3

Claudius Ptolemy

The (Arab. Almagest)

Commentaries Translations and Editions Summary of Contents

Trigonometry in Ptolemy

(α) Lemma for Finding Sin 18° and Sin 36° (β) Equivalent of (γ) ‘Ptolemy’s Theorem’, Giving the Equivalent of (δ) Equivalent of (ε) Equivalent of (ζ) Method of Interpolation Based on Formula (η) Table of Chords (θ) Further use of Proportional Increase (ι) Plane Trigonometry in Effect Used

Spherical Trigonometry : Formulae in Solution of Spherical Triangles The Analemma The Planisphaerium The Optics A mechanical work, Attempt to Prove the Parallel-Postulate

XVIII. Mensuration: Heron of Alexandria, Pages

Controversies as to Heron’s Date Character of Works List of Treatises Geometry

(α) Commentary on Euclid’s Elements (β) The Definitions

Mensuration

The Metrica, Geometrica, Stereometrica, Geodaesia, Mensurae Contents of the Metrica

Book I. Measurement of Areas

(α) Area of Scalene Triangle

Proof of formula

(β) Method of Approximating to the Square Root of a Non-Square Number (γ) Quadrilaterals (δ) Regular Polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 Sides (ε) The Circle (ζ) Segment of a Circle (η) Ellipse, Parabolic Segment, Surface of Cylinder, Right Cone, Sphere and Segment of Sphere

Book II. Measurement of Volumes

(α) Cone, Cylinder, Parallelepipedi Prism), Pyramid and Frustum (β) Wedge-Shaped Solid (γ) Frustum of Cone, Sphere, and Segment Thereof (δ) Anchor-Ring or Tore (ε) The Two Special Solids of Archimedes’s ‘Method’ (ζ) The Five Regular Solids

Book III. Divisions of Figures

Approximation to the Cube Root of a Non-Cube Number Quadratic Equations Solved in Heron

Indeterminate Problems in the Geometrica

The Dioptra The Mechanics

Aristotle’s Wheel The Parallelogram of Velocities Motion on an Inclined Plane On the Centre of Gravity The Five Mechanical Powers Mechanics in Daily Life : Queries and Answers Problems on the Centre of Gravity, &c.

The Catoptrica

Heron’s Proof of Equality of Angles of Incidence and Reflection

XIX. Pappus of Alexandria

Date of Pappus Works (commentaries) Other than the Collection The Synagoge or Collection

(α) Character of the Work ; Wide Range (β) List of Authors Mentioned (γ) Translations and Editions (δ) Summary of Contents Book III. Section (1). On the Problem of the Two Mean Proportionals

Section (2). The Theory of Means Section (3). The ‘Paradoxes’ of Erycinus Section (4). The Inscribing of the Five Regular Solids in a Sphere

Book IV. Section (1). Extension of Theorem of Pythagoras

Section (2). On Circles Inscribed in the (‘Shoemaker’s Knife’) Sections (3), (4). Methods of Squaring the Circle and Trisecting any Angle

(α) The Archimedean Spiral (β) The Conchoid of Nicomedes (γ) The Quadratrix (δ) Digression : A Spiral on a Sphere Trisection (or Division in Any Ratio) of Any Angle

Section (5). Solution of the of Archimedes, On Spirals, Prop. 8, by Means of Conics

Book V. Preface on the Sagacity of Bees

Section (1). Isoperimetry After Zenodorus Section (2). Comparison of Volumes of Solids Having Their Surfaces Equal. Case of Sphere Section (3). Digression on Semi-Regular Solids of Archimedes Section (4). Propositions on the Lines of Archimedes, On the Sphere and Cylinder Section (5). Of Regular Solids with Surfaces Equal, that is Greater which has More Faces

Book VI

Problem Arising Out of Euclid’s Optics

Book VII. On the ‘Treasury of Analysis’

Definition of Analysis and Synthesis List of Works in the ‘Treasury of Analysis’ Description of the treatises Anticipation of Guldin’s Theorem Lemmas to the Different Treatises

(α) Lemmas to the Sectio Rationis and Sectiospatii of Apollonius (β) Lemmas to the Determinate Section of Apollonius (γ) Lemmas on the of Apollonius (δ) Lemmas on the on Contacts of Apollonius (ε) Lemmas to the Plane Loci of Apollonius (ζ) Lemmas to the Porisms of Euclid (η) Lemmas to the Conics of Apollonius (θ) Lemmas to the Surface Loci of Euclid (ι) An Unallocated Lemma

Book VIII. Historical Preface

The Object of the Book On the Centre of Gravity The Inclined Plane Construction of a Conic Through Five Points Given two Conjugate Diameters of an Ellipse, to Find the Axes Problem of Seven Hexagons in a Circle Construction of Toothed Wheels and Indented Screws

XX. Algebra: Diophantus of Alexandria

Beginnings Learnt from Egypt ‘Hau’ -Calculations Arithmetical Epigrams in the Greek Anthology Indeterminate Equations of First Degree Indeterminate Equations of Second Degree Before Diophantus Indeterminate Equations in Heronian Collections Numerical Solution of Quadratic Equations Works of Diophantus The Arithmetica

The Seven Lost Books and Their Place Relation of ‘Porisms’ to Arithmetica Commentators from Hypatia Downwards Translations and Editions Notation and Definitions

Sign for Unknown ( = x) and its Origin Signs for Powers of Unknown &c The Sign for Minus and its Meaning

The Methods of Diophantus

I. Diophantus’s Treatment of Equations

(A) Determinate Equations

(1) Pure Determinate Equations (2) Mixed Quadratic Equations (3) Simultaneous Equations Involving Quadratics (4) Cubic Equation

(B) Indeterminate Equations

(a) Indeterminate Equations of the Second Degree

(1) Single Equation (2) Double Equation

1. Double Equations of First Degree 2. Double Equations of Second Degree

(b) Indeterminate Equations of Degree Higher than Second

(1) Single Equations (2) Double Equations

II. Method of Limits III. Method of Approximation to Limits

Porisms and Propositions in the Theory of Numbers

(α) Theorems on the Composition of Numbers as the Sum of Two Squares (β) On Numbers Which are the Sum of Three Squares (γ) Composition of Numbers as the Sum of Four Squares

Conspectus of Arithmetica, with Typical Solutions The Treatise on Polygonal Numbers

XXI. Commentators and Byzantines

Serenus

(α) On the Section of a Cylinder (β) On the Section of a Cone

Tlieon of Alexandria

Commentary on the Syntaxis Edition of Euclid’s Elements Edition of the Optics of Euclid

Hypatia Porphyry. Iamblichus Proclus

Commentary on Euclid, Book I

(α) Sources of the Commentary (β) Character of the Commentary

Hypotyposis of Astronomical Hypotheses Commentary on the Republic

Marinus of Neapolis Domninus of Larissa Simplicius

Extracts from Eudemus

Eutocius Anthemius of Tralles

On Burning-Miirors

The Papyrus of Akhmim Geodaesia of’ Heron the Younger’ Michael Psellus Georgius Pachymeres Maximus Planudes

Extraction of the Square Root Two Problems

Manuel Moschopoulos Nicolas Rhabdas

Rule for Approximating to Square Root of a Non-Square Number

Ioannes Pediasimus Barlaam Isaac Argyrus

Appendix. On Archimedes’s Proof of the Subtangent-Property of a Spiral Index of Greek Word English Index

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