## A History of Analysis |

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### Contents

Precursors of Differentiation and Integration | 41 |

Newtons Method and Leibnizs Calculus | 73 |

Algebraic Analysis in the 18th Century | 105 |

The Origins of Analytic Mechanics in the 18th Century | 137 |

The Foundation of Analysis in the 19th Century | 155 |

Complex Function Theory 17801900 | 213 |

Theory of Measure and Integration from Riemann to Lebesgue | 261 |

Foundations of Analysis | 291 |

A Historical Overview to circa 1900 | 325 |

A Historical Survey | 355 |

The Origins of Functional Analysis | 385 |

409 | |

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### Common terms and phrases

18th century algebraic analytic analytic functions applied arbitrary Archimedes arithmetic axiom axiomatic Berlin Bernoulli Bolzano calculus of variations Cantor Cauchy Cauchy's Chapter circle cissoid coefficients concept condition considered construction continuous function convergence corresponding curve d'Alembert Darboux Dedekind defined definition dense sets derivative Descartes Dirichlet discontinuous divergent series domain Elements equal Euler example existence expression finite fluxions formula Fourier Fourier series Frege functional analysis fundamental Gauss geometrical given Greek Hankel Hilbert Huygens idea infinitely small infinitesimal integral interval Johann Johann Bernoulli Lagrange Lagrange's later Lebesgue lectures Leibniz limit linear magnitudes mathematical mathematicians memoir method modern natural numbers Newton notion obtained Oeuvres paper Paris partial differential equations power series principle problem proof proved published quadrature quantities ratio rational numbers real numbers Riemann rigorous Schooten sequence set theory solution surface tangent theorem uniform convergence variable Weierstrass zero

### Popular passages

Page 9 - Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Page 7 - A ratio is a sort of relation in respect of size between two magnitudes of the same kind. 4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

Page 3 - Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be hft some magnitude which will be less than the lesser magnitude set out.

Page 11 - If all points of a straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, there exists one and only one point which produces this division of all the points into two classes, this division of the straight line into two parts.

Page 8 - A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line; 9 And when the lines containing the angle are straight, the angle is called rectilineal.