210 



THE GRAMMAR OF SCIENCE 



any instant of the motion is given by the step OP. In 

 order that the reader may have a clejirer conception of 

 what we are considering, we will suppose the motion to 

 take place in one plane, and conceptualise certain every- 

 day perceptions. We will suppose O to be a point taken 

 as the conceptual limit of Charing Cross, P to be the point 

 which marks the conceptual motion of translation of a 

 train on the Metropolitan Railway, and the curve in Fig, 9 

 to be a conceptual map of the same railway to the scale 

 of about one furlong to the ^jyth of an inch. The points 

 Pj, Pq, P3, . . . P^g mark the successive stations between 

 Aldgate and South Kensington. Any step like OP^ will 



O * f,- 

 O'SI PAULS ^ P| 



OOHARIfJG CROSS 



^ Pi P 

 ,?-'^'*~^®SOUTH KENSINGTON 



accurately determine a certain position of the train 

 relative to Charing Cross. The reader must notice an 

 important result about these steps. Suppose we had 

 been determining the position of P^, relative to O^' — say 

 St. Paul's — instead of O. We see at once that there are 

 two ways of describing the position of P^ relative to O'. 

 We might either say, step the directed step O'Pg, or, 

 again, step first from O' to O, and then step from O to 

 Pg. These two latter steps lead to exactly the same final 

 position as the former single step. Now science is not 

 only an economy of thought, but, what is almost the 

 same thing, an economy of language. Hence we require 

 a shorthand mode of expressing this equivalence in final 

 result of two stepping operations. This is done as 

 follows : — 



0'0 + OPg = 0'Pg, 



