THE GEOMETRY OF MOTION 219 



There are one or two points about speed which the 

 reader will find it useful to bear in mind. In the first 

 place, speed is a numerical quantity, it is equal to a slope, 

 the unit of which is one vertical unit in or/^r one horizontal 

 unit ; thus the speed unit is one space unit in or per one 

 time unit — for example, one mile per minute. Secondly, 

 unless the time-chart has a straight line for its curve, the 

 speed must continually change its magnitude from one 

 point to another of the path. If the curve of the time- 

 chart be a straight line the speed is said to be tmiform, 

 otherwise it is called variable. Lastly, looking back at 

 the map of the path (Fig. 9, p. 210), we see that the 

 bearing of the motion as well as the speed varies from 

 point to point of the path. Remembering our definition 

 of tangent we see that the direction of the motion at P is 

 along the tangent at P, and further it has a sense — for 

 example, the motion is from Pg to P^ and not from P^ to 

 P,. Now we see that the change in the motion is of two 

 kinds : change in magnitude, or change in speed, and 

 change in bearing. In order to trace this change still 

 more clearly we form a new conception, namely, that of 

 speed with a certain bearing, and this combination of 

 speed and bearing we term velocity. To fully describe 

 the velocity, say at the position Pg, we must therefore 

 combine speed and bearing ; the speed is the slope of 

 the tangent at Qg (Fig. 10, p. 213), and, when the units 

 of time and space have been chosen, it is solely a number; 

 the bearing is the direction of the tangent to the path at 

 Pg (Fig. 9) together with the sense, namely, from Pg to 

 P^. Like displacement, velocity can accordingly be re- 

 presented by a step, the magnitude of the step measures 

 the speed, the direction of the step shows the direction 

 of the motion, and the arrow-head gives the sense of the 

 motion. 



§ 12. — The Velocity Diagram or Hodograph. Acceleration 



Now, as it is awkward to have to turn to two different 

 figures — the map of the path and the time-chart — in order 



