## The Elements of Non-Euclidean Plane Geometry and Trigonometry |

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### Other editions - View all

The Elements of Non-Euclidean Plane Geometry and Trigonometry Horatio Scott Carslaw No preview available - 2022 |

The Elements of Non-Euclidean Plane Geometry and Trigonometry Horatio Scott Carslaw No preview available - 2013 |

### Common terms and phrases

a₁ acute angle angle of parallelism angular points assumption Axiom axis b₁ bisects c₁ common perpendicular Concentric Limiting-Curves construction cosh b cosh defect denoted draw the perpendicular ds˛ dy˛ element of area Elliptic Geometry equation Equidistant-Curve equivalent Euclid Euclidean Geometry finite number follows formulae fundamental circle Gauss given line greater Hilbert Hilbert's Axiom Hyperbolic Geometry Hyperbolic Plane hypothenuse Hypothesis ideal point II(p infinite intersect Let it cut Let the Limiting-Curve lowest order m₁ measure of area middle points nominal line nominal points Non-Euclidean Geometry not-intersecting lines obtain Obtuse Angle ordinary point Parallel Postulate Pasch's Axiom pencil Plane Geometry point at infinity polygons produced proof prove quadrilateral with three right angles right-angled triangle Saccheri's Quadrilateral sech segment side sinh Spherical Trigonometry tanh tanha theorems three right angles triangle ABC triangle is equal

### Popular passages

Page 3 - In any triangle if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.

Page 131 - The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180° and less than 540°. (gr). If A'B'C' is the polar triangle of ABC...

Page 1 - Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Page 85 - From this it will follow that if there are two triangles which have a side of the one equal to a side of the other, and the...

Page 2 - That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles...

Page 19 - If I commenced by saying that I am unable to praise this work (by John), you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years.

Page 2 - A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.

Page 2 - ... the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.

Page 174 - What then, are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar coordinates false. One geometry cannot be more true than another; it can only be more convenient.

Page 27 - Titel : Appendix scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem ; adjecta ad casum falsitatis, quadratura circuli geometrica. Auctore JOHANNE BOLYAI de eadem *), Geometrarum in exercitu Caesareo Regio Austriaco Castrensium Capitaneo.