Classics of Mathematics

Introduction
____From A Mathematician's Apology - G. H. HARDY
____From "Mathematics as an Element in the History of Thought" - ALFRED NORTH WHITEHEAD
I. Protomathematics in the Late Age of Stone and in Ancient Mesopotamia and Egypt
____Introduction
________1. From The Exact Sciences in Antiquity - O. NEUGEBAUER
________2. From the A'h-mose or Rhind Papyrus - JAMES R. NEWMAN
________3. Problem No. 14 of the Moscow Papyrus - BATTISCOMBE GUNN AND T. ERIC PEET
________4. The Ancient Hebrews and Protomathematics Based on the Old Testament of the Bible
II. The Rise of Theoretical Mathematics in Ancient Greece
____Introduction
____Proclus
________Biography of Proclus
________5. ____From "The Catalogue of Geometers" - PROCLUS
____Pythagoras and His Followers
________Biography of Pythagoras of Samos
________6. From On Marvels 6 - APOLLONIUS PARADOXAGRAPHUS
________7. From Book VII of the Elements: Definitions - EUCLID
________8. From Prior Analytics i.23 (Irrationality of the Square Root of 2) - ARISTOTLE
________9. From On Nicomachus's Introduction to Arithmetic (Arithmetic, Geometric, and Harmonic Means) - IAMBLICHUS
________10. From Commentary on Ptolemy's Harmonics - PORPHYRY
________11. From Metaphysics A5 - ARISTOTLE
________12. From Commentary on Euclid i (Sum of the Angles of a Triangle) - PROCLUS
________13. From Convivial Questions viii. 2. (Pythagoras's Theorem) - PLUTARCH
________14. From On Slips in Greetings 5 - LUCIAN
________15. From Elements X. Scholium (The Irrational or Incommensurable) - EUCLID
________16. From Elements X. Definitions (Commensurable and Incommensurable) - EUCLID
____Hippocrates of Chios
________Biography of Hippocrates of Chios
________17. From Commentary on Aristotle's Physics A 2 - PHILOPONUS
________18. From Commentary on Aristotle's Physics A 2 (Quadrature of Lunules, the Crescent-shaped Figures Between Two Intersecting Arcs of Circles) - SIMPLICIUS
____Plato
________Biography of Plato
________19. From the Republic VI.510 (Approach to Mathematics) - PLATO
________20. From the Republic VII.522-28 (The Quadrivium) - PLATO
________21. From the Timaeus 53-56 - PLATO
____Eudoxus
________Biography of Eudoxus
________22. From Book V of the Elements. Definitions and Propositions 1, 7, 9, and 10 (Theory of Proportions) - EUDOXUS-EUCLID
________23. From Book XII.2 of the Elements (Method of Approximation, the So-called Method of Exhaustion) - EUCLID
____Aristotle
________Biography of Aristotle
________24. From Posterior Analytics i.10 (First Principles or Theory of Statements) - ARISTOTLE
________25. From the Metaphysics (1066-67) (The Infinite, the Essence of the Archimedean Postulate) - ARISTOTLE
________26. From the Metaphysics (1068b-1069a) and Physics (230a-240a) (On the Continuous and Zeno's Paradoxes) - ARISTOTLE
III. Mathematics in the West During Hellenistic and Roman Times
____Introduction
____Euclid
________Biography of Euclid
________27. From Book I of the Elements: Definitions, Postulates, Axioms, and Propositions 1-13 - EUCLID
________28. From Book I of the Elements: Propositions 27-32 (Theory of Parallels) - EUCLID
________29. From Book I of the Elements: Proposition 47 (Pythagorean Theorem) - EUCLID
________30. From Book VII of the Elements'. Propositions 1 and 2 (Euclidean Algorithm) - EUCLID
________31. From Book IX of the Elements: Propositions 14 (Fundamental Theorem in the Theory of Numbers), 20 (Infinitude of Primes), and 25-30
________32. From Book X of the Elements: Propositions 1-3 - EUCLID
____Archimedes
________Biography of Archimedes
________33. From Sphere and Cylinder I: Greeting and Assumptions - ARCHIMEDES
________34. From Sphere and Cylinder I: Propositions 33 and 34 (Surface and Volume of a Sphere) - ARCHIMEDES
________35. Measurement of a Circle: Propositions 1-3 (Approximation of PI Using in Essence Upper and Lower Limits) - ARCHIMEDES
________36. From Quadrature of the Parabola: Introduction and Propositions 17-24 - ARCHIMEDES
________37. From The Equilibrium of Planes I: Propositions 6 and 7 (Principle of the Lever) - ARCHIMEDES
________38. From The Sand-Reckoner: Introduction and Section on Large Numbers - ARCHIMEDES
________39. From The Cattle Problem (Indeterminate Analysis) - ARCHIMEDES
____Eratosthenes
________Biography of Eratosthenes
________40. From Introduction to Arithmetic /: Chapters XII and XIII (Sieve for Finding Primes) - NICOMACHUS OF GERASA
________41. From On the Circular Motion of the Heavenly Bodies i. 10.52 (Estimate of the Circumference of the Earth) - CLEOMEDES
____Apollonius
________Biography of Apollonius
________42. From Conics: Introduction to Book One - APOLLONIUS
________43. From Conics: Propositions 7 and 11 (A Novel Method of Construction of Sections) - APOLLONIUS
____Claudius Ptolemy
________Biography of Claudius Ptolemy
________44. From the Syntaxis or Almagest i (Trigonometry: Table of Sines) - CLAUDIUS PTOLEMY
____Diophantus
________Biography of Diophantus
________45. From the Dedication to Arithmetica (Algebraic Notation) - DIOPHANTUS
________46. From Arithmetica (Origins of Number Theory) - DIOPHANTUS
IV. Arabic Primacy with Chinese, Indian and Maya Contributions
____Introduction
____(Muhammad ibn Musa) al-Khwarizmi
________Biography of (Muhammad ibn Musa) al-Khwarizmi
________47. From The Book of Algebra and Almucabola (Quadratic Equations in Algebra: Verbal Form) - AL-KHWaRIZMi
____'Umar al-Khayyami
________Biography of 'Umar al-Khayyami
________48. From the Algebra - 'UMAR AL-KHAYYAMI
____Marquis Zhang Cang
________Biography of Marquis Zhang Cang
________49. From a Description of the Nine Chapters on the Mathematical Art (Jiuzhang suanshu)
____Bhaskara II
________Biography of Bhaskara II
________50. From Lilavati (Arithmetic and Geometry) - BHASKARA II
________51. From Bijagnita (Algebra) - BHASKARA II
____Maya Civilization and Mathematics
________Commentary
________52. From The Ancient Maya - SYLVANUS GRISWOLD MORLEY
________52a. The Mathematical Notation of the Ancient Maya - MICHAEL P. CLOSS
V. The Medieval-Renaissance-Reformation Periods in Europe
____Introduction by Joseph E. Brown, Rensselaer Polytechnic Institute
____Leonardo of Pisa
________Biography of Leonardo of Pisa
________53. From Liber abbaci (The Rabbit Problem) - LEONARDO OF PISA
____Nicole Oresme
________Biography of Nicole Oresme
________54. From De configurationibus (The Latitude of Forms) - NICOLE ORESME
________55. From Questiones super geometriam Euclidis (The Latitude of Forms) - NICOLE ORESME
____Girolamo Cardano
________Biography of Girolamo Cardano
________56. From the Ars Magna - GIROLAMO CARDANO
____Francois Viete
________Biography of Francois Viete
________57. From In artem analyticem isagoge (The New Algebra) - FRANCOIS VIETE
____Simon Stevin
________Biography of Simon Stevin
________58. From De Thiende (Decimal Fractions) - SIMON STEVIN
____John Napier
________Biography of John Napier
________59. From Mirifici logarithmorum canonis constructio (Logarithms) - JOHN NAPIER
VI. The Scientific Revolution at Its Zenith (1620-1720)
____Introduction
____Algebra, Analytic Geometry, and Arithmetic
________Biography of Rene Descartes
________60. From the Regulae (Rule IV: "In search for the truth of things a method is indispensable") - RENE DESCARTES
________61. From Discours de la Methode (Four Fundamental Rules of Logic) - RENE DESCARTES
________62. From La geometrie (1637) (Theory of Equations) - RENE DESCARTES
________63. From La geometrie (1637) (The Principle of Nonhomogeneity) - RENE DESCARTES
________Biography of Pierre de Fermat
________64. From a Letter to Bernard Frenicle de Bessy (October 10, 1640) (Are Numbers of the Form 2^(n+1) Prime When n = 2^t?) - PIERRE DE FERMAT
________65. From Two Letters of February 1657 [Challenge to Mathematicians: Find an Infinity of Integer Solutions for the "Pell" Equation (that is, x^2 - Ay^2 = 1), where A may be any Nonsquare Integer] - PIERRE DE FERMAT
________Biography of Blaise Pascal
________66. From Traite du triangle arithmetique... (The So-called Pascal Triangle) - BLAISE PASCAL
____Origins of Infinitesimal Calculus
________Biography of Johannes Kepler
________67. From Nova stereometria doliorum vinariorum (1615) (Integration Methods) - JOHANNES KEPLER
________Biography of Galileo Galilei
________68. From Two New Sciences (1638) (Paradoxes of Infinity: The Relationship Between Points and Lines, the Order of an Infinity, Infinitesimals, the Concept of a Continuum) - GALILEO GALILEI
________69. From "On the Transformation and Simplification of the Equations of Loci" (ca. 1640) (Integration) - PIERRE DE FERMAT
________70. From "On a Method for the Evaluation of Maxima and Minima" (... His algorithm was subsequently developed into the method of the "characteristic triangle," dx, dy, and ds.) - PIERRE DE FERMAT
________71. From "On the Sines of a Quadrant of a Circle" (1659) (... Pascal's paper partially rejects indivisibles and presages the indefinite integral.) - BLAISE PASCAL
____The Discovery of Differential and Integral Calculus
________Biography of Gottfried Wilhelm Leibniz
________72. From "A New Method for Maxima and Minima as Well as Tangents, Which Is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This (1684) (Differential Calculus) - GOTTFRIED WILHELM LEIBNIZ
________73. From "Supplementum geometriae dimensoriae ... , "in Acta Eruditorum (1693) (The Fundamental Theorem of Calculus) - GOTTFRIED WILHELM LEIBNIZ
________Biography of Isaac Newton
________74. From Specimens of a Universal [System of] Mathematics (written ca. 1684) - ISAAC NEWTON
________75. From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) - ISAAC NEWTON
________76. From a Letter to Henry Oldenburg on a General Method for Finding Quadratures (October 24, 1676) - ISAAC NEWTON
________77. From Principia Mathematica (1687) (Prime and Ultimate Ratios: The Theory of Limits) - ISAAC NEWTON
________78. From the Introduction to the Tractatus de quadratura curvarum (1704) - ISAAC NEWTON
____The Bernoullis
________Biography of Jakob Bernoulli
________79. From Ars Conjectandi (1713) (The Law of Large Numbers) - JAKOB BERNOULLI
________Biography of Johann Bernoulli
________80. From "The Curvature of a Ray in Nonuniform Media" (1697) (The Brachistochrone) - JOHANN BERNOULLI
VII. The Age of Enlightenment and the French Revolution (1720-1800)
____Introduction
____Elaboration and Criticism of Infinitesimal Analysis
________Biography of Brook Taylor
________81. From Methodus Incrementorum Directa et Inversa (1715) (The Taylor Series) - BROOK TAYLOR
________Biography of George Berkeley
________82. From The Analyst (1734) (Criticism of the Foundations of Calculus) - GEORGE BERKELEY
________Biography of Colin Maclaurin
________83. From Treatise of Fluxions (1742) (On Series and Extremes) - COLIN MACLAURIN
________Biography of Jean Le-Rond d'Alembert
________84. From "Differential," Encyclopedie, Vol. 4 (1754) (On Limits) - JEAN D'ALEMBERT
________Biography of Leonhard Euler
________85. From Introductio in analysin infinitorum I (1748) (Functions, Logarithms, and Trigonometry) - LEONHARD EULER
________Biography of Joseph-Louis Lagrange
________86. From "Attempt at a New Method for Determining the Maxima and Minima of Indefinite Integral Formulas" (1760-61) (The Calculus of Variations) - JOSEPH-LOUIS LAGRANGE
________87. From Mathematical Thought from Ancient to Modem Times (1990) (Taylor Series with Remainder) - MORRIS KLINE
____Topology, Number Theory, and Probability
________88. From the Problem of the Seven Bridges of Konigsberg (1736) (The Origins of Topology) - LEONHARD EULER
________89. From "Theorems on Residues Obtained by the Division of Powers" (1758/ 59) (Number Theory: Power Residues) - LEONHARD EULER
________90. From "Demonstrations of Certain Arithmetical Theorems" (1738) (A proof of Fermat's great theorem-x^n + y^n = z^n has no positive integral solutions for n > 2 for the case n = 4.) - LEONHARD EULER
________Biography of Pierre-Simon Laplace
________91. From Essais philosophique sur les probabilites (1814) (The Theory of Probability) - PIERRE-SIMON LAPLACE
VIII. The Nineteenth Century
____Introduction by Helena M. Pycior, University of Wisconsin-Milwaukee
____Algebra
________Biography of Carl Friedrich Gauss
________92. From "New Proof of the Theorem That Every Integral Rational Algebraic Function of One Variable Can Be Decomposed into Real Factors of the First or Second Degree" (1799) - CARL FRIEDRICH GAUSS
________Biography of Niels (Henrik) Abel
________93. From a Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) - NIELS ABEL
________Biography of Evariste Galois
________94. The Testamentary Letter Sent to Auguste Chevalier (May 29, 1832) (Group Theory and Abelian Integrals) - EVARISTE GALOIS
________Biography of William Rowan Hamilton
________95. From Elements of Quaternions (1866) (On Quaternions a Generalization of Complex Numbers) - WILLIAM ROWAN HAMILTON
________Biography of George Boole
________96. From An Investigation of the Laws of Thought (1854) (The Joining of Algebra and Logic) - GEORGE BOOLE
____Non-Eudidean Geometries
________Biography of Nikolai Ivanovich Lobachevsky
________97. From The Theory of Parallels ( 1840) (Hyperbolic Geometry) - NIKOLAI IVANOVICH LOBACHEVSKY
________Biography of (Georg Friedrich) Bernhard Riemann
________98. From "On the Hypotheses Which Lie at the Foundations of Geometry" (1854) (Elliptic Geometry and a Distinction Between Boundlessness and Infinitude of Straight Lines) - BERNHARD RIEMANN
____The Development and Arithmetization of Mathematical Analysis
________Biography of (Jean-Baptiste-) Joseph Fourier
________99. From Joseph Fourier 1768-1830 - I. GRATTAN-GUINNESS
________100. From Theorie analytique de la chaleur (1822) - JOSEPH FOURIER
________101. From "On the Continuity of Functions Defined by Power Series" (1826) (The Binomial Series; Convergence of Power Series) - NIELS ABEL
________Biography of Augustin-Louis Cauchy
________102. From Cours d'analyse de l'Ecole Royale Polytechnique (1821) - AUGUSTIN-LOUIS CAUCHY
________103. From Resume des lecons ... sur le calcul infinitesimal (1823) (On the Derivative as a Limit) - AUGUSTIN-LOUIS CAUCHY
________104. From Resume des lecons ... sur le calcul infinitesimal (1823) (First Rigorous Proof About Derivatives) - AUGUSTIN-LOUIS CAUCHY
________Biography of Karl (Theodor Wilhelm) Weierstrass
________105. From Lectures on the Differential Calculus (1861) - KARL WEIERSTRASS
________106. From a Letter to Hermann Amandus Schwarz (May 5, 1875) - KARL WEIERSTRASS
________107. From Encounters with Mathematics (1977) (Riemann on Physics and Partial Differential Equations) - LARS GARDING
____Number Theory, Set Theory, and Symbolic Logic
________108. From Carl Friedrich Gauss: A Biography (1970) (A Discussion of the Disquisitiones arithmeticae, Including Congruences and the Fundamental Theorem of Arithmetic) - TORD HALL
________Biography of Ernst Eduard Kummer
________109. From "On the Theory of Complex Numbers" (1847) (Theory of Ideal Prime Factors) - ERNST EDUARD KUMMER
________Biography of (Julius Wilhelm) Richard Dedekind
________110. From Stetigkeit und irrationale Zahlen (1872) (Continuity, Irrational Numbers, and Dedekind Cuts) - RICHARD DEDEKIND
________111. From Was sind und was sollen die Zahlen? (1888) (Simply Infinite Systems) - RICHARD DEDEKIND
________Biography of Georg (Ferdinand) Cantor
________112. From Grundlagen einer Allgemeinen Mannigfaltigkeitslehre (1883) (Fundamental Series) - GEORG CANTOR
________113. From a Letter to Richard Dedekind (1899) (Transfinite Cardinal Numbers and Set Theory) - GEORG CANTOR
________Biography of (Friedrich Ludwig) Gottlob Frege
________114. From Begrijfsschrift (1879) (Symbolic Logic) - GOTTLOB FREGE
________115. From Die Grundlagen der Arithmetik (1884) (Definition of Number in Logical Terms) - GOTTLOB FREGE
________Biography of Giuseppe Peano
________116. From Arithmetices principia (1889) (Set of Axioms for Integers) - GIUSEPPE PEANO
________Biography of Bertrand (Arthur William) Russell
________117. From Introduction to Mathematical Philosophy (1919) (The Definition of Number) - BERTRAND RUSSELL
IX.____The Early Twentieth Century to 1932
____Introduction by Helena M. Pycior, University of Wisconsin-Milwaukee
____Creativity and the Paris Problems
________Biography of (Jules-) Henri Poincare
________118. From Science and Method (1908) - HENRI POINCARE
________Biography of David Hilbert
________119. From "Mathematical Problems: Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900" (Paris Problems and the Formalist Program) - DAVID HILBERT
________Biography of Ernst (Friedrich Ferdinand) Zermelo
________120. From a Letter to David Hilbert (September 24, 1904) - ERNST ZERMELO
____Foundational Crisis and Undecidability
________121. From Correspondence Between Bertrand Russell and Gottlob Frege (June 16 and 22, 1902) (Russell Paradox and the Logicist School) - BERTRAND RUSSELL AND GOTTLOB FREGE
________122. From a Letter to Jean van Heijenoort Concerning the Publication of the Russell-Frege Correspondence (November 23, 1962) - BERTRAND RUSSELL
________123. From Principia Mathematica (1910) (Russell's Solution to the Burali-Forti Paradox and to His Own) - ALFRED NORTH WHITEHEAD AND BERTRAND RUSSELL
________Biography of Luitzen E(gbertus) J(an) Brouwer
________124. From "Intuitionism and Formalism" (1912) - L. E. J. BROUWER
________125. From "Intuitionistic Reflections on Formalism" (1927) - L. E. J. BROUWER
________Biography of Kurt Godel
________126. From "Einige metamathematische Resultate uber Entscheidungsdefinitheit und Widerspruchsfreiheit" (Some Metamathematical Results on Completeness and Consistency, 1930) - KURT GODEL
________127. From "Uber formal unentscheidbare Satze der Principia mathematica und verwandter Systeme I (1931) (Incompleteness Theorem: Undecidability) - KURT GODEL
________128. From "Uber Vollstandigkeit und Widerspruchsfreiheit" ("On Completeness and Consistency," 1931) - KURT GODEL
Selected Topics: The Development of General Abstract Theories
____129. From the Preface to "Analysis Situs" (1895) (Algebraic Topology) - HENRI POINCARE
____130. Poincare and Topology - P. S. ALEKSANDROV
____131. From An Introduction to the Foundations and Fundamental Concepts of Mathematics (1966) - HOWARD EVES AND CARROLL V. NEWSOM
____Biography of Henri (Leon) Lebesgue
____132. From "The Development of the Integral Concept" (1926) - HENRI LEBESGUE
____Biography of (Amalie) Emmy Noether
____133. Proof of a Fundamental Theorem in the Theory of Algebras (1932) - R. BRAUER (KONIGSBERG), H. HASSE (MARBURG), AND E. NOETHER (GOTTINGEN)
____Biography of George David Birkhoff
____134. From "Proof of the Ergodic Theorem" (1931) - GEORGE DAVID BIRKHOFF
INDEX