## Non-Euclidean Geometry"Non-Euclidean Geometry is a history of the alternate geometries that have emerged since the rejection of Euclid s parallel postulate. Italian mathematician ROBERTO BONOLA (1874 1911) begins by surveying efforts by Greek, Arab, and Renaissance mathematicians to close the gap in Euclid s axiom. Then, starting with the 17th century, as mathematicians began to question whether it was actually possible to prove Euclid s postulate, he examines non-Euclidean predecessors Saccheri, Lambert, Legendre, W. Bolyai, Wachter, and Thibaut, and non-Euclidean founders Gauss, Schweikart, Taurinus, Lobachevski, and J. Bolyai. He concludes with a look at later developments in non-Euclidean geometry. Including five appendices and an index of authors, Bonola s Non-Euclidean Geometry is a useful reference guide for students of mathematical history." |

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### Contents

1 | |

II | 9 |

III | 12 |

IV | 22 |

V | 44 |

VI | 51 |

VII | 55 |

VIII | 60 |

XXVII | 176 |

XXVIII | 177 |

XXIX | 181 |

XXX | 184 |

XXXI | 192 |

XXXII | 195 |

XXXIII | 200 |

XXXIV | 206 |

### Other editions - View all

Non-Euclidean Geometry: A Critical and Historical Study of Its Development Roberto Bonola Limited preview - 1955 |

### Common terms and phrases

absolute Acute Angle analytical angle of parallelism Appendix argument axioms Beltrami Bolyai circle of inversion Clifford's Surface common perpendicular constant curvature construction corresponding cosh deduced definition demonstration distance equal equation equidistant Euclid's Postulate Euclidean Geometry Euclidean hypothesis Euclidean plane Fifth Postulate figures Foncenex formula fundamental circle Gauss geodesic given Horocycle hyperbolic Hyperbolic Geometry Ideal Geometry Ideal Length Ideal Line Ideal Points Ideal Segment imaginary improper points infinite intersect Johann Lambert Leipzig line at infinity Lobatschewsky Math memoir metrical middle point Non-Euclidean Geometry obtain Obtuse Angle ordinary orthogonal Parallel Postulate pencil plane geometry point at infinity Postulate of Archimedes preceding Proclus projective geometry proof Prop properties proposition prove quadrilateral radius regard result Riemann's right angles right-angled triangle Saccheri sheaf sides space sphere straight line surfaces of constant Taurinus theorem theory of parallels transformations translation trigono

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Page 1 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.