## Continuous GeometryIn his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice system |

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### Contents

Foundations and Elementary Properties | 1 |

Independence | 8 |

Perspectivity and Projectivity Fundamental Properties | 16 |

Perspectivity by Decomposition | 24 |

Distributivity Equivalence of Perspectivity and Projectivity | 32 |

Properties of the Equivalence Classes | 42 |

Dimensionality | 54 |

PART II | 61 |

Relations Between the Lattice and its Auxiliary Ring | 160 |

Further Properties of the Auxiliary Ring of the Lattice | 168 |

Special Considerations Statement of the Induction to be Proved | 177 |

Treatment of Case I | 191 |

Preliminary Lemmas for the Treatment of Case II | 197 |

Completion of Treatment of Case II The Fundamental Theorem | 199 |

Perspectivities and Projectivities | 209 |

Inner Automorphisms | 217 |

Theory of Ideals and Coordinates in Projective Geometry | 63 |

Theory of Regular Rings | 69 |

APPENDIX 1 | 82 |

APPENDIX 2 | 84 |

APPENDIX 3 | 90 |

Order of a Lattice and of a Regular Ring | 93 |

Isomorphism Theorems | 103 |

Projective Isomorphisms in a Complemented Modular Lattice | 117 |

Definition of LNumbers Multiplication | 130 |

Addition of LNumbers | 136 |

The Distributive Laws Subtraction and Proof that the LNumbers Form a Ring | 151 |

Properties of Continuous Rings | 222 |

RankRings and Characterization of Continuous Rings | 231 |

PART III | 239 |

Center of a Continuous Geometry | 240 |

APPENDIX 1 | 245 |

APPENDIX 2 | 259 |

Transitivity of Perspectivity and Properties of Equivalence Classes | 264 |

Minimal Elements | 277 |

LIST OF CHANGES from the 193537 Edition and comments on the text | 283 |

297 | |