# A Simple Non-Euclidean Geometry and Its Physical Basis: An by Basil Gordon (auth.), Basil Gordon (eds.) By Basil Gordon (auth.), Basil Gordon (eds.)

There are many technical and well known bills, either in Russian and in different languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, some of that are indexed within the Bibliography. This geometry, also referred to as hyperbolic geometry, is a part of the necessary material of many arithmetic departments in universities and lecturers' colleges-a reflec­ tion of the view that familiarity with the weather of hyperbolic geometry is an invaluable a part of the historical past of destiny highschool academics. a lot consciousness is paid to hyperbolic geometry through university arithmetic golf equipment. a few mathematicians and educators fascinated about reform of the highschool curriculum think that the necessary a part of the curriculum should still contain parts of hyperbolic geometry, and that the not obligatory a part of the curriculum should still contain an issue relating to hyperbolic geometry. I The wide curiosity in hyperbolic geometry is no surprise. This curiosity has little to do with mathematical and clinical purposes of hyperbolic geometry, because the functions (for example, within the thought of automorphic capabilities) are particularly really expert, and usually are encountered by means of only a few of the numerous scholars who carefully learn (and then current to examiners) the definition of parallels in hyperbolic geometry and the certain positive factors of configurations of traces within the hyperbolic aircraft. The important reason behind the curiosity in hyperbolic geometry is the real truth of "non-uniqueness" of geometry; of the life of many geometric systems.

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Example text

The vector a is assumed constant, it follows, just as before, that VI =vi +a, V2=V;+a, and VI- V2=vi -V;. lIICf. Galileo's "Dialogues" , pp. l7l-172. 23 I. What is mechanics? ;;J_ _ _oooO(a(t)) A ...... 21 So far we have concerned ourselves with two-dimensional (more accurately, plane-parallel) motions affecting points A(x,y) of some plane xOy. However, nothing prevents us from restricting ourselves to even simpler, rectilinear motions, where we need only consider motions of points A =A(x) of some fixed line o.

Since the endpoints A', B', C',... belong to the image I' of I, the endpoints Ai,B;,C;, ... belong to the image I; of II' Hence I; is a line parallel to 1'. This proves that the shear (la) maps the line II onto the line I;. (b) The proof of (b) is implicit in the proof of (a). 2 2It is easy to show that if I and l' form (Euclidean) angles a and a', respectively, with the x-axis, then tana'=tana+v. 36 I. Distance and Angle; Triangles and Quadrilaterals Figure 27 (c) The equality C' D' / A' B' = CD / AB follows directly from Figure 27; its proof is left to the reader.

Consequently these quantities are not comparable; knowing that two intersecting lines form an angle of 30 0 , and two parallel lines are 15 cm apart (cf. Fig. 29), we cannot say that one of these two deflections is larger than the other. We also note that the distance (4) between lines is defined only if the angle (3) between them is zero, and that two lines coincide if and only if they form an angle 8 equal to zero and the distance d between them is zero. " '-1 (5) it equals the signed length of the projection PP I of the segmentAAI on the x-axis (Fig. 