## The Historical Development of the CalculusThe 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 con 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 ap 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 treat 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 ap propriately titled in its original edition-The Concepts of the Calculus than in its reprinting. |

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### Indice

1 | |

Area and the Method of Exhaustion | 16 |

Archimedes | 29 |

The Volume and Surface Area of a Sphere | 42 |

Solids of Revolution | 62 |

References | 75 |

The Arab Connection | 81 |

The Analytic Geometry of Descartes and Fermat | 95 |

The Reversion of Series | 204 |

Applications of Integration by Substitution | 210 |

Arclength Computations | 217 |

The Calculus and the Principia Mathematica | 224 |

References | 230 |

The Characteristic Triangle | 239 |

Transmutation and the Arithmetical Quadrature | 245 |

The Invention of the Analytical Calculus | 252 |

Arithmetical Quadratures | 109 |

Early Tangent Constructions | 122 |

The Relationship Between Quadratures and Tangents | 138 |

Arithmetic and Geometric Progressions | 151 |

References | 164 |

The Calculus According to Newton | 189 |

The Chain Rule and Integration by Substitution | 196 |

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algebraic analysis analytical applied Archimedes arithmetic base binomial series calculus Cauchy Chapter circle circumscribed coefficients computations concept cone continuous function convergence corresponding curve cylinder defined definition denote derivative Descartes difference differential element ellipse equal equation Euclid's Elements Eudoxus Euler example ExERCISE Fermat Figure finite fluxions follows formula function f geometric magnitudes geometric series given gives Greek mathematics height Hence indivisibles infinite series infinitely small infinitesimal inscribed integral interpolation interval Lebesgue Lebesgue integrable Leibniz length limit line segment logarithms method method of exhaustion motion Napier Newton notation obtain ordinate parabola polynomial positive integer problems proof pyramids quadrature quantities radius ratio rational numbers real numbers rectangle regular polygon result Riemann Riemann integrable rigorous root sequence solution sphere square subintervals substitution surface T. L. Heath Table tangent line techniques termwise tion variable vector velocity volume Wallis x-axis