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 (1571-1630) -- 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 (1550-1617) -- 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 (1642-1727) -- 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 (1646-1716) -- The Beginning-Sums 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 (1707-1783) -- 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: The Twentieth Century -- The Lebesgue Integral and the Fundamental Theorem of Calculus -- Non-standard Analysis-The Vindication of Euler? -- References.