2i8 THE GRAMMAR OF SCIENCE 



the curve corresponding to the transit from King's Cross 

 to Gower Street is Q^vi in O^ni, or since Qjn is equal to 

 P^ P,, and Q^m to t^L, it is P^P, in t^L. But P^P^ is, in a 

 certain scale, the number of miles between the two 

 stations, and t^t^ is, in another scale, the number of 

 minutes between the two stations. Thus the slope, which 

 with one interpretation is a certain rise in a certain 

 horizontal length, is with another interpretation a certain 

 number of miles in a certain number of minutes. Now a 

 certain number of miles in a certain number of minutes 

 is exactly what we understand by the mean or average 

 speed of the train between King's Cross and Gower 

 Street ; the train has increased its distance from Aldgate 

 by so many miles in so many minutes. The manner 

 in which change of distance is taking place during any 

 finite time is thus determined by the slope of the corre- 

 sponding chord of the time-chart. The average rate of 

 change of distance, or the mean speed for any given interval, 

 is thus recorded by the slopes of these chords. 



It is clear, however, that by varying the length of the 

 chord QfO-, — by bringing Q^ nearer to Q^, for example — 

 we shall obtain different mean speeds for different lengths 

 of the journey after passing King's Cross. The shorter 

 we take the time the steeper becomes in this case the 

 chord, the greater the mean speed. The conception of a 

 limit to this mean speed is then formed ; namely, the 

 mean speed for a vanishingly small time after leaving 

 King's Cross, and this mean speed is defined as the actual 

 speed of passing King's Cross. We see at once that the 

 actual speed will be measured by the slope of the tangent 

 to the time-chart at Qg, for this tangent is, according to 

 our definition, the limit to the chord. Thus the actual 

 speed at each instant of the motion is determined by the 

 steepness at the corresponding point of the time-chart, and 

 it is measured in miles per minute by the slope of the 

 tangent at that point. We thus find that our time-chart 

 is not only like Bradshazv, a time-table, but is also a 

 diagram of the varying speed of the train throughout its 

 journey. 



