The Historical Development of the Calculus

1 Area Number and Limit Concepts in Antiquity. - Babylonian and Egyptian Geometry. - Early Greek Geometry. - Incommensurable Magnitudes and Geometric Algebra. - Eudoxus and Geometric Proportions. - Area and the Method of Exhaustion. - Volumes of Cones and Pyramids. - Volumes of Spheres. - References. - 2 Archimedes. - The Measurement of a Circle. - The Quadrature of the Parabola. - The Area of an Ellipse. - The Volume and Surface Area of a Sphere. - The Method of Compression. - The Archimedean Spiral. - Solids of Revolution. - The Method of Discovery. - Archimedes and Calculus?. - References. - 3 Twilight Darkness and Dawn. - The Decline of Greek Mathematics. - Mathematics in the Dark Ages. - The Arab Connection. - Medieval Speculations on Motion and Variability. - Medieval Infinite Series Summations. - The Analytic Art of Viète. - The Analytic Geometry of Descartes and Fermat. - References. - 4 Early Indivisibles and Infinitesimal Techniques. - Johann Kepler (15711630). - Cavalieri's Indivisibles. - Arithmetical Quadratures. - The Integration of Fractional Powers. - The First Rectification of a Curve. - Summary. - References. - 5 Early Tangent Constructions. - Fermat's Pseudo-equality Methods. - Descartes' Circle Method. - The Rules of Hudde and Sluse. - Infinitesimal Tangent Methods. - Composition of Instantaneous Motions. - The Relationship Between Quadratures and Tangents. - References. - 6 Napier's Wonderful Logarithms. - John Napier (15501617). - The Original Motivation. - Napier's Curious Definition. - Arithmetic and Geometric Progressions. - The Introduction of Common Logarithms. - Logarithms and Hyperbolic Areas. - Newton's Logarithmic Computations. - Mercator's Series for the Logarithm. - References. - 7 The Arithmetic of the Infinite. - Wallis' Interpolation Scheme and Infinite Product. -Quadrature of the Cissoid. - The Discovery of the Binomial Series. - References. - 8 The Calculus According to Newton. - The Discovery of the Calculus. - Isaac Newton (16421727). - The Introduction of Fluxions. - The Fundamental Theorem of Calculus. - The Chain Rule and Integration by Substitution. - Applications of Infinite Series. - Newton's Method. - The Reversion of Series. - Discovery of the Sine and Cosine Series. - Methods of Series and Fluxions. - Applications of Integration by Substitution. - Newton's Integral Tables. - Arclength Computations. - The Newton-Leibniz Correspondence. - The Calculus and the Principia Mathematica. - Newton's Final Work on the Calculus. - References. - 9 The Calculus According to Leibniz. - Gottfried Wilhelm Leibniz (16461716). - The BeginningSums and Differences. - The Characteristic Triangle. - Transmutation and the Arithmetical Quadrature of the Circle. - The Invention of the Analytical Calculus. - The First Publication of the Calculus. - Higher-Order Differentials. - The Meaning of Leibniz' Infinitesimals. - Leibniz and Newton. - References. - 10 The Age of Euler. - Leonhard Euler (17071783). - The Concept of a Function. - Euler's Exponential and Logarithmic Functions. - Euler's Trigonometric Functions and Expansions. - Differentials of Elementary Functions à la Euler. - Interpolation and Numerical Integration. - Taylor's Series. - Fundamental Concepts in the Eighteenth Century. - References. - 11 The Calculus According to Cauchy Riemann and Weierstrass. - Functions and Continuity at the Turn of the Century. - Fourier and Discontinuity. - Bolzano Cauchy and Continuity. - Cauchy's Differential Calculus. - The Cauchy Integral. - The Riemann Integral and Its Reformulations. - The Arithmetization of Analysis. - References. - 12 Postscript: TheTwentieth Century. - The Lebesgue Integral and the Fundamental Theorem of Calculus. - Non-standard AnalysisThe Vindication of Euler?. - References. Language: English