# The Elements of Non-Euclidean Geometry by J. Coolidge

By J. Coolidge

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Extra info for The Elements of Non-Euclidean Geometry

Example text

Bound. dihedral angle. We may, by following the analogy of the plane, define null, and re-entrant dihedral angles. The definition of the dihedral angles of a tetrahedron will also be immediately straight, evident. A plane perpendicular to the edge of a dihedral angle will cut the faces in two half-lines perpendicular to the edge. The interior (exterior) angle of these two shall be called a plane angle of the interior (exterior) dihedral angle. Theorem 43. Two plane angles of a dihedral angle are con- gruent.

E. one of the mutually vertical external angles) is l) 1 C' + 4LC'i> 1 l)s ), and 1 Dt congruent to H D AAB D (S : > . 2 is the difference between and as and as a limiting position, the 2 2 approach 2 and //2 at every point in angles determined by J5 2 2, 2 H ^ABD. , ^S^D^ B D D , %-B^H^ D space decrease together towards a null angle as a limit. JSZ) will become, and remain infinitesimal. and ^AGI) is Lastly, the difference between which will, by our previous reasoning, become infinitesimal finitesimal.

Instead of the ratio of their measures, distances, such a ratio simply ~ Let us then take the isosceles . XY and write birectangular quadrilateral A'ABB', the right angles having and B. Let us imagine that and A' are is on a fixed line at a very small distance from A. Let G be the middle point of (AB), and let the A A their vertices at fixed points, while B AB at G meet (A'B') at C", which, by the middle point of (AB). Now, by III. '), or (CG') extended beyond Csv'/ ,0 < -AC -L \77ir -rt But . n i A -= =-=r AB AC C", for jA.