3rd edition The updated new edition of the classic and comprehensive guide to the history of mathematics For more than forty years, A History of Mathematics has been the reference of choice for those looking to learn about the fascinating history of humankind s relationship with numbers, shapes, and patterns. Foreword. Preface to the First Edition. Preface to the Second Edition. 1: Traces. Concepts and Relationships. Early Number Bases. Number Language and Counting. Spatial Relationships. 2: Ancient Egypt. The Era and the Sources. Numbers and Fractions. Arithmetic Operations. Heap Problems. Geometric Problems. Slope Problems. Arithmetic Pragmatism. 3: Mesopotamia. The Era and the Sources. Cuneiform Writing. Numbers and Fractions; Sexagesimals. Positional Numeration. Sexagesimal Fractions. Approximations. Tables. Equations. Measurements: Pythagorean Triads. Polygonal Areas. Geometry as Applied Arithmetic. 4: Hellenic Traditions. The Era and the Sources. Thales and Pythagoras. Numeration. Arithmetic and Logistic. Fifth Century Athens. Three Classical Problems. Incommensurability. Paradoxes of Zeno. Deductive Reasoning. Democritus of Abdera. Mathematics and the Liberal Arts. The Academy. Aristotle. 5: Euclid of Alexandria. Alexandria. Lost Works. Extant Works. The Elements. 6: Archimedes of Syracuse. The Siege of Syracuse. On the Equilibriums of Planes. On Floating Bodies. The Sand Reckoner. Measurement of the Circle. On Spirals. Quadrature of the Parabola. On Conoids and Spheroids. On the Sphere and Cylinder. Book of Lemmas. Semiregular Solids and Trigonometry. The Method. 7: Apollonius of Perge. Works and Tradition. Lost Works. Cycles and Epicycles. The Conics. 8: Cross Currents. Changing Trends. Eratosthenes. Angles and Chords. Ptolemy s Almagest. Heron of Alexandria. Decline of Greek Mathematics. Nichomachus of Gerasa. Diophantus of Alexandria. Pappus of Alexandria. The End of Alexandrian Dominance. Proclus of Alexandria. Boethius. Athenian Fragments. Byzantine Mathematicians. 9: Ancient and Medieval China. The Oldest Known Texts. The Nine Chapters. Rod Numerals. The Abacus and Decimal Fractions. Values of Pi. Thirteenth Century Mathematics. 10: Ancient and Medieval India. Early Mathematics in India. The Sulbasutras. The Siddhantas. Aryabhata. Numerals. Trigonometry. Multiplication. Long Division. Brahmagupta. Indeterminate Equations. Bhaskara. Madhava and the Keralese School. 11: The Islamic Hegemony. Arabic Conquests. The House of Wisdom. al Khwarizmi. `Abd Al Hamid ibn Turk. Thabit ibn Qurra. Numerals. Trigonometry. Abu l Wefa and Al Karkhi. Al Biruni and Alhazen. Omar Khayyam. The Parallel Postulate. Nasir al Din al Tusi. Al Kashi. 12: The Latin West. Introduction. Compendia of the Dark Ages. Gerbert. The Century of Translation. Abacists and Algorists. Fibonacci. Jordanus Nemorarius. Campanus of Novara. Learning in the Thirteenth Century. Archimedes Revived. Medieval Kinematics. Thomas Bradwardine. Nicole Oresme. The Latitude of Forms. Infinite Series. Levi ben Gerson. Nicholas of Cusa. Decline of Medieval Learning. 13: The European Renaissance. Overview. Regiomontanus. Nicolas Chuquet s Triparty. Luca Pacioli s Summa. German Algebras and Arithmetics. Cardan s Ars Magna. Rafael Bombelli. Robert Recorde. Trigonometry. Geometry. Renaissance Trends. François Viète. 14: Early Modern Problem Solvers. Accessibility of Computation. Decimal Fractions. Notation. Logarithms. Mathematical Instruments. 15: The Infinite and Italian Curves. Infinitesimal Methods: Stevin. Johannes Kepler. Galileo s Two New Sciences. Bonaventura Cavalieri. Evangelista Torricelli. 16: Analysis, Synthesis, and Numbers. Mersenne s Communicants. Descartes. Fermat s Loci. Gregory of St. Vincent. Theory of Numbers. Gilles Persone de Roberval. Girard Desargues and Projective Geometry. Blaise Pascal. Philippe de Lahire. Georg Mohr. Pietro Mengoli. Frans van Schooten. Jan de Witt. Johann Hudde. René François de Sluse. Christiaan Huygens. 17: Newton and British Techniques. John Wallis. James Gregory. Nicolaus Mercator and William Brouncker. Barrow s Method of Tangents. Newton. Abraham De Moivre. 18: Leibniz and Continental Methods. Leibniz: Early Career and Travels. The Bernoulli Family. Tschirnhaus Transformations. Solid Analytic Geometry. Michel Rolle and Pierre Varignon. The Clairauts. Mathematics in Italy. The Parallel Postulate. Divergent Series. 19: Euler. Life of Euler. Notation. Foundation of Analysis. Logarithms and the Euler Identities. Differential Equations. Probability. Theory of Numbers. Textbooks. Analytic Geometry. The Parallel Postulate: Lambert. 20: Pre to Post Revolutionary France. Men and Institutions. The Committee on Weights and Measures. D?Alembert. B zout. Condorcet. Lagrange. Monge. Carnot. Laplace. Legendre. Aspects of Abstraction. Paris in the 1820s. Fourier. Cauchy. Diffusion. 21: Gauss. Nineteenth Century Overview. Gauss: Early Work. Number Theory. Reception of the Disquisitiones Arithmeticae. Astronomy. Gauss s Middle Years. Differential Geometry. Gauss s Later Work. Gauss s Influence. 22: Geometry. The School of Monge. Projective Geometry: Poncelet and Chasles. Synthetic Metric Geometry: Steiner. Synthetic Nonmetric Geometry: von Staudt. Analytic Geometry. Non Euclidean Geometry. Riemannian Geometry. Spaces of Higher Dimensions. Felix Klein. Post Riemannian Algebraic Geometry. 23: Algebra. Introduction. British Algebra and the Operational Calculus of Functions. Boole and the Algebra of Logic. Augustus De Morgan. William Rowan Hamilton. Grassmann and Ausdehnungslehre. Cayley and Sylvester. Linear Associative Algebras. Algebraic Geometry. Algebraic and Arithmetic Integers. Axioms of Arithmetic. 24: Analysis. Berlin and Göttingen at Mid Century. Riemann in Göttingen. Mathematical Physics in Germany. Mathematical Physics in English Speaking Countries. Weierstrass and Students. The Arithmetization of Analysis. Dedekind. Cantor and Kronecker. Analysis in France. 25: Poincaé and Hilbert. Turn of the Century Overview. Poincar . Hilbert. 26: Twentieth Century Legacies: Pre 1930. General Overview. Integration and Measure. Functional Analysis and General Topology. Algebra. Differential Geometry and Tensor Analysis. Probability. Bounds and Approximations. 27: Twentieth Century Legacies: Post 1930. The 1930s and World War II. Homological Algebra and Category Theory. Bourbaki. Algebraic Geometry. Logic and Computing. 28: Recent Trends. Overview. The Four Color Conjecture. Classification of Finite Simple Groups. Fermat s Last Theorem. Poincaré s Query. Future Outlook. References. General Bibliography. Index.