212 THE GRAMMAR OF SCIENCE 



shown to obey the parallelogram law : that is to say, we 

 add together velocities, spins, or accelerations geometrically 

 and not arithmetically. Although the space at our 

 disposal may not admit of our demonstrating this result 

 for all the conceptions of kinematics,^ the reader will do 

 well to bear it in mind, as it is an important principle 

 to which we shall have occasion again to refer. 



^ 9. — TJie Time-Chart 



Hitherto we have been considering how the position 

 of the point P relative to O might be determined at each 

 instant of time. We want, however, to know how the 

 position changes, and how this change is to be described 

 and measured. In order to do this we must consider how 

 the displacement OPg, for example, changes to the 

 displacement OP^. In our geometrical shorthand : 

 OP„ = OP(, + PgP^, and the step PgP^ measures the change 

 of position. We want, then, to ascertain a fitting measure 

 of the manner in which this change varies with the time. 

 To enable the reader better to conceive our purpose we 

 will try to turn into geometry a column of Bradshaw, or, 

 more definitely, a portion of a time-table of the Metro- 

 politan Railway, corresponding to the stations marked in 

 Fig. 9. Down the left-hand side of Fig. 10 are placed 

 the names of the stations represented in Fig. 9 by the 

 points P^, P^, Pg, P^, . . . Pjg. These are placed, as in 

 Bradshaw, against a vertical line, but we will somewhat 

 improve on his arrangement. He puts the stations at 

 equal distances below each other, and gives no hint as to 

 the distance between each pair of them. Now we will 

 place them at such distances along the vertical from each 

 other that every ^^^th of an inch represents a furlong, or 

 ^ths of an inch represents a mile, so that an inch-scale 

 applied to the vertical ought theoretically to determine 

 the parliamentary fare between any two stations. In the 

 next place, we will place off (or plot off, as it is termed) 



' For proofs see Clifford's Elements of Dynamic, " Velocities," p. 59, 

 "Spins," pp. 123-4. 



