By Edoardo Sernesi

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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" [15], 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.