234 THE GRAMMAR OF SCIENCE 



and the motion of O' relative to O known, we require to 

 find the motion of P relative to O'. Let P^, P (Fig. i8) 

 be two successive positions of P relative to O, and O' , 

 O'^ the corresponding positions of O'. Then O'^P^ is the 

 first and O'^P^ is the second step, measuring the position 

 of P relative to O'. From O'^ draw O'^P'^ parallel and 

 equal to O'^P^, then O'^P^ and O'^P, give the relative 

 motion of P with regard to O^, and the relative displace- 

 ment in the given interval is P^P'a- Now draw O' O 

 parallel and equal to O'p, then O'^O, and 0\p, or 

 O'jOg, give the relative positions of O with regard to O'. 



Fig. i8. 



But by the equality of opposite sides of parallelograms 

 OO2 equals 0',0'^, equals P.P'^. Hence P^P'. is equal to 

 the displacement of O relative to O'. " But in the 

 geometry of steps (p. 210) : — 



P P' =P P -1-P P' 



12 -^ 1^ 2 ' 2^ 2' 



or in words : the displacement of P relative to O' is 

 equal to the displacement of P relative to O added 

 geo7netrically to the displacement of O relative to O'. 

 Now this result is true, however large or small these 

 displacements may be, and these displacements divided 

 by the number of units in the interval of time which 

 is the same for all of them, represent the mean velocities 

 in this interval. Hence we conclude that : the mean 



