Author | Edwards, C. H. author |
---|---|
Title | The Historical Development of the Calculus [electronic resource] / by C. H. Edwards |
Imprint | New York, NY : Springer New York, 1979 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6230-5 |
Descript | XII, 368 p. online resource |
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 Viรจte -- The Analytic Geometry of Descartes and Fermat -- References -- 4 Early Indivisibles and Infinitesimal Techniques -- Johann Kepler (1571โ{128}{147}1630) -- Cavalieriโ{128}{153}s Indivisibles -- Arithmetical Quadratures -- The Integration of Fractional Powers -- The First Rectification of a Curve -- Summary -- References -- 5 Early Tangent Constructions -- Fermatโ{128}{153}s Pseudo-equality Methods -- Descartesโ{128}{153} Circle Method -- The Rules of Hudde and Sluse -- Infinitesimal Tangent Methods -- Composition of Instantaneous Motions -- The Relationship Between Quadratures and Tangents -- References -- 6 Napierโ{128}{153}s Wonderful Logarithms -- John Napier (1550โ{128}{147}1617) -- The Original Motivation -- Napierโ{128}{153}s Curious Definition -- Arithmetic and Geometric Progressions -- The Introduction of Common Logarithms -- Logarithms and Hyperbolic Areas -- Newtonโ{128}{153}s Logarithmic Computations -- Mercatorโ{128}{153}s Series for the Logarithm -- References -- 7 The Arithmetic of the Infinite -- Wallisโ{128}{153} 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 (1642โ{128}{147}1727) -- The Introduction of Fluxions -- The Fundamental Theorem of Calculus -- The Chain Rule and Integration by Substitution -- Applications of Infinite Series -- Newtonโ{128}{153}s Method -- The Reversion of Series -- Discovery of the Sine and Cosine Series -- Methods of Series and Fluxions -- Applications of Integration by Substitution -- Newtonโ{128}{153}s Integral Tables -- Arclength Computations -- The Newton-Leibniz Correspondence -- The Calculus and the Principia Mathematica -- Newtonโ{128}{153}s Final Work on the Calculus -- References -- 9 The Calculus According to Leibniz -- Gottfried Wilhelm Leibniz (1646โ{128}{147}1716) -- The Beginningโ{128}{148}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 Leibnizโ{128}{153} Infinitesimals -- Leibniz and Newton -- References -- 10 The Age of Euler -- Leonhard Euler (1707โ{128}{147}1783) -- The Concept of a Function -- Eulerโ{128}{153}s Exponential and Logarithmic Functions -- Eulerโ{128}{153}s Trigonometric Functions and Expansions -- Differentials of Elementary Functions ร la Euler -- Interpolation and Numerical Integration -- Taylorโ{128}{153}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 -- Cauchyโ{128}{153}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 Analysisโ{128}{148}The Vindication of Euler? -- References