## Noneuclidean Tesselations and Their Groups. (Pure & Applied by Wilhelm Magnus

By Wilhelm Magnus

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54) A,' dA1 (a, b, c, d real), A3' = C2Al which has the determinant (ad - bc)2. Wc may assume (ad - bC)2 = 1. 53) has an interior which is defined as the set of points from which no (real) tangent to the conic exists. 54). Now we are able to introduce the Definition of the hyperbolic space Kz. 55). The selfmappings with ad - bc = +1 are called proper. 56) hi' = Al, A,' = -A,, h3' = A3 as a coset representative. 57) x = Az/Aa, X ' 4-y2 = A,/& + define a mapping of the upper half of the complex z = x i y plane onto the interior of the conic A1A3 - 82' = 0, which is one-one and has the following properties.

17 There exists exactly one element in r* which maps a ray r onto a given ray r' so that a given sidr of r is mapped onto a given side of r'. If r coincides with r', then the element,of r*which exchanges the sides of r leaves fixed all points of r and of the line 1 of which it is a part. We call this element the rejlectim in 1. An element 7 of the subgroup r of r* is uniquely determined by a fixed ray and its image under the action of 7. r is called the group of proper (noneuclidean or hyperbolic) motions.

Then there exists a t least one circle C* orthogonal to C1, CZ,and Co. 5 Notes on Elliptic and Spherical Geometry 37 Proof: If two of the circles touch, we can map them onto parallel lines, and the lemma becomes obvious. Assume then that the three circles are disjoint in pairs. We map Ca onto the real axis. The circles orthogonal to the maps C1' and Cz' of C1 and CZall go through two common points PI and P z . The centers of these orthogonal circles lie therefore on the (euclidean) line 1 orthogonal to the line PlPz and bisecting the interval PIP2.