A History of Mathematics.

XV, 717 p.: Ill. Lediglich der Einband ist leicht berieben, sonst ein gutes und sauberes Exemplar / Only the cover is slightly rubbed, otherwise a good and clean copy. - Contents Chapter I. Primitive Origins 1 The concept of number. 2 Early number bases. 3 Number language and the origin of counting. 4 Origin of geometry. Chapter II. Egypt 1 Early records. 2 Hieroglyphic notation. 3 Ahmes papyrus. 4 Unit fractions. 5 Arithmetic operations. 6 Algebraic problems. 7 Geometrical problems. 8 A trigonometric ratio. 9 Moscow papyrus. 10 Mathematical weaknesses. Chapter III. Mesopotamia 1 Cuneiform records. 2 Positional numeration. 3 Sexagesimal fractions. 4 Fundamental operations. 5 Algebraic problems. 6 Quadratic equations. 7 Cubic equations. 8 Pythagorean triads. 9 Polygonal areas. 10 Geometry as applied arithmetic. 11 Mathematical weaknesses. Chapter IV. Ionia and the Pythagoreans 1 Greek origins. 2 Thales of Miletus. 3 Pythagoras of Samos. 4 The Pythagorean pentagram. 5 Number mysticism. 6 Arithmetic and cosmology. 7 Figurate numbers. 8 Proportions. 9 Attic numeration. 10 Ionian numeration. 11 Arithmetic and logistic. Chapter V. The Heroic Age 1 Centers of activity. 2 Anaxagoras of Clazomenae. 3 Three famous problems. 4 Quadrature of lunes. 5 Continued proportions. 6 Hippias of Ellis. 7 Philolaus and Archytas of Tarentum. 8 Duplication of the cube. 9 Incommensurability. 10 The golden section. 11 Paradoxes of Zeno. 12 Deductive reasoning. 13 Geometrical algebra. 14 Democritus of Abdera. Chapter VI. The Age of Plato and Aristotle 1 The seven liberal arts. 2 Socrates. 3 Platonic solids. 4 Theodorus of Cyrene. 5 Platonic arithmetic and geometry. 6 Origin of analysis. 7 Eudoxus of Cnidus. 8 Method of exhaustion. 9 Mathematical astronomy. 10 Menaech- mus. 11 Duplication of the cube. 12 Dinostratus and the squaring of the circle. 13 Autolycus of Pitane. 14 Aristotle. 15 End of the Hellenic period. Chapter VII. Euclid of Alexandria 1 Author of the Elements. 2 Other works. 3 Purpose of the Elements. 4 Definitions and postulates. 5 Scope of Book I. 6 Geometrical algebra. 7 Books III and IV. 8 Theory of proportion. 9 Theory of numbers. 10 Prime and perfect numbers. 11 Incommensurability. 12 Solid geometry. 13 Apocrypha. 14 Influence of the Elements. Chapter VIII. Archimedes of Syracuse I The siege of Syracuse. 2 Law of the lever. 3 The hydrostatic principle. 4 The Sand-Reckoner. 5 Measurement of the circle. 6 Angle trisection. 7 Area of a parabolic segment. 8 Volume of a paraboloidal segment. 9 Segment of a sphere. 10 On the Sphere and Cylinder. 11 Book of Lemmas. 12 Semiregular solids and trigonometry. 13 The Method. 14 Volume ofa sphere. 15 Recovery of the Method. Chapter IX. Apollonius of Perga 1 Lost works. 2 Restorations of lost works. 3 The problem of Apollonius. 4 Cycles and epicycles. 5 The Conics. 6 Names of the conic sections. 7 The double-napped cone. 8 Fundamental properties. 9 Conjugate diameters. 10 Tangents and harmonic division. 11 The three-and-four-line locus. 12 Intersecting conics. 13 Maxima and minima, tangents and normals. 14 Similar conics. 15 Foci of conics. 16 Use of coordinates. Chapter X. Greek Trigonometry and Mensuration 1 Early trigonometry. 2 Aristarchus of Samos. 3 Eratosthenes of Cyrene. 4 Hipparchus of Nicaea. 5 Menelaus of Alexandria. 6 Ptolemy’s Almagest. 7 The 360 degree circle. 8 Construction of tables. 9 Ptolemaic astronomy. 10 Other works by Ptolemy. 11 Optics and astrology. 12 Heron of Alexandria. 13 Principle of least distance. 14 Decline of Greek mathematics. Chapter XI. Revival and Decline of Greek Mathematics 1 Applied mathematics. 2 Diophantus of Alexandria. 3 Nicomachus of Gerasa. 4 The Arithmetica of Diophantus. 5 Diophantine problems. 6 The place of Diophantus in algebra. 7 Pappus of A lexandria. 8 The Collection. 9 Theorems of Pappus. 10 The Pappus problem. 11 The Treasury of Analysis. 12 The Pappus-Guldin theorems. 13 Proclus of Alexandria. 14 Boethius. 15 End of the Alexandrian period. 16 The Greek Anthology. 17 Byzantine mathematicians of the sixth century. Chapter XII. China and India 1 The oldest documents. 2 The Nine Chapters. 3 Magic squares. 4 Rod numerals. 5 The abacus and decimal fractions. 6 Values of pi. 7 Algebra and Horner’s method. 8 Thirteenth-century mathematicians. 9 The arithmetic triangle. 10 Early mathematics in India. 11 The Sulvasutras. 12 The Siddhantas. 13 Aryabhata. 14 Hindu numerals. 15 The symbol for zero. 16 Hindu trigonometry. 17 Hindu multiplication. 18 Long division. 19 Brahmagupta. 20 Brahmagupta’s formula. 21 Indeterminate equations. 22 Bhaskara. 23 The Lilavati. 24 Ramanujan. Chapter XIII. The Arabic Hegemony 1 Arabic conquests. 2 The House of Wisdom. 3 Al-jabr. 4 Quadratic equations. 5 The father of algebra. 6 Geometric foundation. 7 Algebraic problems. 8 A problem from Heron. 9 Abd al-Hamid ibn-Turk. 10 Thabit ibn-Qurra. 11 Arabic numerals. 12 Arabic trigonometry. 13 Abu’l-Wefa and al- Karkhi. 14 Al-Biruni and Alhazen. 15 Omar Khayyam. 16 The parallel postulate. 17 Nasir Eddin. 18 Al-Kashi. CONTENTS Chapter XIV. Europe in the Middle Ages 1 From Asia to Europe. 2 Byzantine mathematics. 3 The Dark Ages. 4 Alcuin and Gerbert. 5 The century of translation. 6 The spread of Hindu-Arabic numerals. 7 The Liber abaci. 8 The Fibonacci sequence. 9 A solution of a cubic equation. 10 Theory of numbers and geometry. 11 Jordanus Nemor- arius. 12 Campanus of Novara. 13 Learning in the thirteenth century. 14 Medieval kinematics. 15 Thomas Bradwardine. 16 Nicole Oresme. 17 The latitude of forms. 18 Infinite series. 19 Decline of medieval learning. Chapter XV. The Renaissance 1 Humanism. 2 Nicholas of Cusa. 3 Regiomontanus. 4 Application of algebra to geometry. 5 A transitional figure. 6 Nicolas Chuquet’s Triparty. 7 Luca Pacioli’s Summa. 8 Leonardo da Vinci. 9 Germanic algebras. 10 Cardan’s Ars magna. 11 Solution of the cubic equation. 12 Ferrari’s solution of the quartic equation. 13 Irreducible cubics and complex numbers. 14 Robert Recorde. 15 Nicholas Copernicus. 16 Georg Joachim Rheticus. 17 Pierre de la Ramée. 18 Bombelli’s Algebra. 19 Johannes Werner. 20 Theory of perspective. 21 Cartography. Chapter XVI. Prelude to Modern Mathematics 1 François Viète. 2 Concept of a parameter. 3 The analytic art. 4 Relations between roots and coefficients. 5 Thomas Harriot and William Oughtred. 6 Horner’s method again. 7 Trigonometry and prosthaphaeresis. 8 Trigonometric solution of equations. 9 John Napier. 10 Invention of logarithms. 11 Henry Briggs. 12 Jobst Biirgi. 13 Applied mathematics and decimal fractions. 14 Algebraic notations. 15 Galileo Galilei. 16 Values of pi. 17 Reconstruction of Apollonius’ On Tangencies. 18 Infinitesimal analysis. 19 Johannes Kepler. 20 Galileo's Two New Sciences. 21 Galileo and the infinite. 22 Bonaventura Cavalieri. 23 The spiral and the parabola. Chapter XVII. The Time of Fermat and Descartes 1 Leading mathematicians of the time. 2 The Discours de la méthode. 3 Invention of analytic geometry. 4 Arithmetization of geometry. 5 Geometrical algebra. 6 Classification of curves. 7 Rectification of curves. 8 Identification of conics. 9 Normals and tangents. 10 Descartes’ geometrical concepts. 11 Fermat's loci. 12 Higher-dimensional analytic geometry. 13 Fermat’s differentiations. 14 Fermat’s integrations. 15 Gregory of St. Vincent. 16 Theory of numbers. 17 Theorems of Fermat. 18 Gilles Persone de Roberval. 19 Evangelista Torricelli. 20 New curves. 21 Girard Desargues. 22 Projective geometry. 23 Blaise Pascal. 24 Probability. 25 The cycloid. Chapter XVIII. A Transitional Period 1 Philippe de Lahire. 2 Georg Mohr. 3 Pietro Mengoli. 4 Frans van Schooten. 5 Jan de Witt. 6 Johann Hudde. 7 René François de Sluse. 8 The pendulum clock. 9 Involutes and evolutes. 10 John Wallis. 11 On Conic Sections. 12 Arithmetica infinitorum. 13 Christopher Wren. 14 Wallis’ formulas. 15 James Gregory. 16 Gregory’s series. 17 Nicolaus Mercator and William Brouncker. 18 Barrow’s method of tangents. Chapter XIX. Newton and Leibniz 1 Newton’s early work. 2 The binomial theorem. 3 Infinite series. 4 The Method of Fluxions. 5 The Principia. 6 Leibniz and the harmonic triangle. 7 The differential triangle and infinite series. 8 The differential calculus. 9 Determinants, notations, and imaginary numbers. lO The algebra of logic. 11 The inverse square law. 12 Theorems on conics. 13 Optics and curves. 14 Polar and other coordinates. 15 Newton’s method and Newton’s parallelogram. 16 The Arithmetica universalis. 17 Later years. Chapter XX. The Bernoulli Era 1 The Bernoulli family. 2 The logarithmic spiral. 3 Probability and infinite series. 4 L’Hospital’s rule. 5 Exponential calculus. 6 Logarithms of negative numbers. 7 Petersburg paradox. 8 Abraham de Moivre. 9 De Moivre’s theorem. 10 Roger Cotes. 11 James Stirling. 12 Colin Maclaurin. 13 Taylor’s series. 14 The Analyst controversy. 15 Cramer’s rule. 16 Tschirnhaus transformations. 17 Solid analytic geometry. 18 Michel Rolle and Pierre Varignon. 19 Mathematics in Italy. 20 The parallel postulate. 21 Divergent series. Chapter XXI. The Age of Euler 1 Life of Euler. 2 Logarithms of negative numbers. 3 Foundation of analysis. 4 Infinite series. 5 Convergent and divergent series. 6 Life of d’Alembert. 7 The Euler identities. 8 D’Alembert and limits. 9 Differential equations. 10 The Clairauts. 11 The Riccatis. 12 Probability. 13 Theory of numbers. 14 Textbooks. 15 Synthetic geometry. 16 Solid analytic geometry. 17 Lambert and the parallel postulate. 18 Bezout and elimination. Chapter XXII. Mathematicians of the French Revolution 1 The age of revolutions. 2 Leading mathematicians. 3 Publications before 1789. 4 Lagrange and determinants. 5 Committee on Weights and Measures. 6 Condorcet on education. 7 Monge as administrator and teacher. 8 Descriptive geometry and analytic geometry. 9 Textbooks. 10 Lacroix on analytic geometry. 11 The Organizer of Victory. 12 Metaphysics of the calculus and geometry. 13 Géométrie de position. 14 Transversals. 15 Legendre’s Geometry. 16 Elliptic integrals. 17 Theory of numbers. 18 Theory of functions. 19 Calculus of variations. 20 Lagrange multipliers. 21 Laplace and probability. 22 Celestial mechanics and operators. 23 Political changes. Chapter XXIII. The Time of Gauss and Cauchy 1 Early discoveries by Gauss. 2 Graphical representation of complex numbers. 3 The fundamental theorem of algebra. 4 The algebra of congruences. 5 Reciprocity and frequency of primes. 6 Constructible regular polygons. 7 Astronomy and least squares. 8 Elliptic functions. 9 Abel’s life and work. 10 Theory of determinants. 11 Jacobians. 12 Mathematical journals 13 Complex variables. 14 Foundations of the calculus. 15 Bernhard Bolzano. 16 Tests for convergence. 17 Geometry. 18 Applied mathematics. Chapter XXIV. The Heroic Age in Geometry 1 Theorems of Brianchon and Feuerbach. 2 Inversive geometry. 3 Poncelet’s projective geometry. 4 Pliicker’s abridged notation. 5 Homogeneous coordinates. 6 Line coordinates and duality. 7 Revival of British mathematics. 8 Cayley’s «-dimensional geometry. 9 Geometry in Germany. 10 Lobachevsky and Ostrogradsky. 11 Non-Euclidean geometry. 12 The Bolyais. 13 Riemannian geometry. 14 Spaces of higher dimension. 15 Klein’s Erlanger Programm. 16 Klein’s hyperbolic model. Chapter XXV. The Arithmetization of Analysis 1 Fourier series. 2 Analytic number theory. 3 Transcendental numbers. 4 Uneasiness in analysis. 5 The Bolzano-Weierstrass theorem. 6 Definition of real number. 7 Weierstrassian analysis. 8 The Dedekind “cut”. 9 The limit concept. 10 Gudermann’s influence. 11 Cantor’s early life. 12 The “power” of infinite sets. 13 Properties of infinite sets. 14 Transfinite arithmetic. 15 Kronecker’s criticism of Cantor’s work. Chapter XXVI. The Rise of Abstract Algebra 1 The Golden Age in mathematics. 2 Mathematics at Cambridge. 3 Peacock, the “Euclid of algebra.” 4 Hamilton’s quaternions. 5 Grassmann and Gibbs. 6 Cayley’s matrices. 7 Sylvester’s algebra. 8 Invariants of quadratic forms. 9 Boole’s analysis of logic. 10 Boolean algebra. 11 De Morgan and the Peirces. 12 The tragic life of Galois. 13 Galois theory. 14 Field theory. 15 Frege’s definition of cardinal number. 16 Peano’s axioms. Chapter XXVII. Aspects of the Twentieth Century 1 The nature of mathematics. 2 Poincare’s theory of functions. 3 Applied mathematics and topology. 4 Hilbert’s problems. 5 Godel’s theorem. 6 Transcendental numbers. 7 Foundations of geometry. 8 Abstract spaces. 9 The foundations of mathematics. 10 intuitionism, formalism, and logicism. 11 Measure and integration. 12 Point set topology. 13 Increasing abstraction in algebra. 14 Probability. 15 High-speed computers. 16 Mathematical structure. 17 Bourbaki and the “New Mathematics.” General Bibliography Appendix: Chronological Table Index. ISBN 9780691023915