THE GEOMETRY OF MOTION 



201 



easy to show that this turning of two Hnes CA and CB 

 into new positions CA' and CB' may also be attained by 

 turning the body about a certain line of direction CO 

 through a certain angle.^ Thus the manner in which we 

 conceive change of aspect to be described and measured 



1 This may be proved by the aid of elementary geometry in the following 

 manner : — 



Let the triangle CBA be displaced into the position CB'A'. Join the 

 points A, A' and B, B', and let the mid-points of AA' and BB' be M and N 

 respectively. Through C and M draw a plane perpendicular to AA' and 

 through C and N a plane perpendicular to BB'. These two planes meet in a 

 line passing through C, since C is common to them both. Let O be any 

 point in this line, and join it to M and N, them OM and ON are respectively 

 perpendicular to AA' and BB'. In the triangles AOM, A'OM, AM and 

 A'M are equal, OM is common, and the angles at M are right, hence it 

 follows by Euclid i. 4 that the third sides OA and OA' are equal. For 



Fig. 4. 



precisely similar reasons it follows that OB and OB' are equal. Hence the 

 three distances of O from the angles of the triangle ABC are equal to its 

 distances from the three angles of the triangle A'B'C respectively. Thus the 

 two tetrahedrons with summits at O and having bases ABC and A'B'C 

 respectively are equal in every respect, for all their edges are equal each to 

 each. One of them may thus be looked upon as the other in a changed 

 position. They have, however, the same edge OC. Hence one tetra- 

 hedron may be moved into the position of the other by rotating it through a 

 certain angle about the edge OC. That is to say, the triangle CBA may be 

 turned into the position CB'A' by rotating it through a certain angle — the 

 angle between the planes BOC and B'OC — about the line OC. 



