## Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th-19th Century France and GermanyThis volume is, as may be readily apparent, the fruit of many years’ labor in archives and libraries, unearthing rare books, researching Nachlässe, and above all, systematic comparative analysis of fecund sources. The work not only demanded much time in preparation, but was also interrupted by other duties, such as time spent as a guest professor at universities abroad, which of course provided welcome opportunities to present and discuss the work, and in particular, the organizing of the 1994 International Graßmann Conference and the subsequent editing of its proceedings. If it is not possible to be precise about the amount of time spent on this work, it is possible to be precise about the date of its inception. In 1984, during research in the archive of the École polytechnique, my attention was drawn to the way in which the massive rupture that took place in 1811—precipitating the change back to the synthetic method and replacing the limit method by the method of the quantités infiniment petites—significantly altered the teaching of analysis at this first modern institution of higher education, an institution originally founded as a citadel of the analytic method. |

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### Contents

8 | |

14 | |

Cardanos refutation of the rule of signs | 44 |

relative position of opposite segments | 77 |

the four quadrants | 82 |

the hyperbola in the quadrants | 83 |

Early Modern Times | 157 |

lesion of continuity | 180 |

Reproduction of Carnots figure in his 1785 memoir | 341 |

Reproduction of Carnots operating with the limitsymbol | 345 |

Content structure of Lacroixs lecture course on analysis 1804 | 388 |

Example of de Pronys tableaux in his Mécanique Philosophique | 392 |

Cauchys Compromise Concept | 427 |

Believing Myself Infallible | 441 |

Contents of Cauchys lecture course on analysis 181617 | 459 |

Reproduction of Dirksens operating with double limits | 473 |

jump discontinuities and continuity | 181 |

continuous and discontinuous lines | 243 |

area of curves | 246 |

intermediate values on lines | 249 |

Culmination of Algebraization and Retour du Refoulé | 257 |

the graphs of the circle parabola and hyperbola 2745 | 277 |

the four quadrants | 290 |

The Renaissance of the Synthetic | 295 |

From the Perspective of Mathematical | 309 |

Limit to the Infinitely Small | 334 |

Development of Pure Mathematics in PrussiaGermany | 481 |

Diesterweg interpreting directions in figures | 518 |

number lines as coordinate axes in 1829 | 527 |

Conflicts Between Confinement to Geometry | 567 |

Summary and Outlook | 601 |

Appendix 619 | 618 |

References | 631 |

671 | |

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Conflicts Between Generalization, Rigor, and Intuition: Number Concepts ... Gert Schubring No preview available - 2010 |

### Common terms and phrases

algebra already Ampère analysis analytic application approach arithmetic Arnauld assumed basic concept Berlin Bézout Bolzano Carnot Cauchy Cauchy’s Chapter concept of limit concept of negative concept of number concept of quantity Condillac context contrast curves d’Alembert definition Descartes differential and integral differential calculus Dirksen discussed École polytechnique edition Encyclopédie epistemological equations Euler explicitly expression Fontenelle formulated foundations France function geometry Hausen hence indivisibles infiniment petits infinitely large infinitely small quantities infinitesimal calculus integral calculus introduced Johann Bernoulli Karsten Kästner L’Huilier Lacroix Lagrange latter’s law of continuity Lazare Carnot Leibniz limit method MacLaurin magnitudes Malebranche mathematicians mathematics means minus multiplication Nachlass negative numbers negative quantities negative solutions Newton operations opposite quantities Paris positive and negative positive numbers presented Prestet principle problem published quantities ibid ratio reflection Reyneau rigor rule of signs subtraction textbook theorem theory values variable zero