The historical development of the calculus
C. H. Edwards (Author)
The calculus has served for three centuries as the principal quantitative language of Western science. In the course of its genesis and evolution some of the most fundamental problems of mathematics were first conƯ fronted and, through the persistent labors of successive generations, finally resolved. Therefore, the historical development of the calculus holds a special interest for anyone who appreciates the value of a historical perspective in teaching, learning, and enjoying mathematics and its apƯ plications. My goal in writing this book was to present an account of this development that is accessible, not solely to students of the history of mathematics, but to the wider mathematical community for which my exposition is more specifically intended, including those who study, teach, and use calculus. The scope of this account can be delineated partly by comparison with previous works in the same general area. M.E. Baron's The Origins of the Infinitesimal Calculus (1969) provides an informative and reliable treatƯ ment of the precalculus period up to, but not including (in any detail), the time of Newton and Leibniz, just when the interest and pace of the story begin to quicken and intensify. C.B. Boyer's well-known book (1949, 1959 reprint) met well the goals its author set for it, but it was more apƯ propriately titled in its original edition-The Concepts of the CalculusƯ than in its reprinting
1 online resource (xiii, 368 pages) : illustrations
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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
Bibliographic Level Mode of Issuance: Monograph
"With 150 illustrations."
English
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