THE GEOMETRY OF MOTION 211 



which, put into words, reads : Step from O' the directed 

 step O'O, and then take the directed step OPg, and the 

 spot finally reached will be the same as if the directed 

 step O'Pg had been taken from O'. The reader must be 

 careful not to confuse this geometrical addition with 

 ordinary arithmetical addition. For example, if OO' 

 were eight furlongs, O'P^ ten furlongs, and OPg twelve 

 furlongs, then we appear at first sight to have : — 



8+12 = 10, 



and this is deemed absurd. But it is only absurd to the 

 arithmetician. For the geometrician 8, 12, and 10 may 

 be the lengths of directed steps, and he knows that, if he 

 follows a directed step of 8 furlongs by one of 1 2, he 

 may really have got only ten furlongs from his original 

 position. How, then, is the arithmetician limited ? 

 Why, obviously we must suppose him incapable of 

 stepping out in all directions in space, we must tie him 

 down to motion along one and the same straight line. 

 In this case a step of 8 followed by one of 1 2 will 

 always make a step of 20, as arithmetic teaches us it 

 should do. Briefly, the freedom of the geometrician con- 

 sists in his power of turning corners. 



Let us now go back a little and note that the 

 geometrical addition of steps, O'O + OP^ — O'P^, may 

 be represented in a slightly different manner. Let 

 us draw the line O'A parallel to OPg and P^A parallel to 

 00', then we are said to complete the parallelogram on 

 O'O and OP,, the line O'Pg joining two opposite angles is 

 termed a diagonal, and we have the following rule : 

 Complete the parallelogram on two steps, and its diagonal 

 will measure a single step equivalent to the sum of the 

 other two. This rule is termed addition by the parallelo- 

 gram law, and we see that the steps by which we measure 

 relative position, or displacements, obey this law. In 

 itself it is the same thing as geometrical addition. Its 

 importance lies in the fact that all the conceptions of the 

 geometry of motion, displacements, velocities, spins, and 

 accelerations may be represented as steps and can be 



