THE GEOMETRY OF MOTION 215 



the help of Fig. 9 it tells us the exact time the train takes 

 to reach, not only every station, but every position what- 

 ever between either terminus ; or, on the other hand, it 

 tells us the exact position for every time up to 38 minutes 

 after leaving Aldgate. How far has the train got in 26 

 minutes, for example ? To answer this we must scale off 

 along the horizontal line, or time-axis, 26 eighths of an 

 inch ; we must then draw a vertical line, striking our curve 

 in the point M ; a horizontal through M strikes the verti- 

 cal line of stations, or distance-axis, at the point N between 

 Praed Street and Bayswater, and a scale divided into -fths 

 of an inch applied to P^^N tells us how many furlongs the 

 train is beyond Praed Street. An inverse process will show 

 us the time to any chosen position on the distance-axis. 

 Our geometrical time-table, or time-chart, as we shall call 

 it, thus gives us a good deal more information than 

 Bradshaw. It is further clear that such a time-chart can 

 be drawn in conception for every point-motion, and that, 

 taken in conjunction with a map of the path, it fully 

 describes the most complex point- motion. Hence the 

 fundamental problem in such motions is to ascertain the 

 map and the time-chart.^ 



^ 10. — Steepness and Slope 



If we examine the time-chart we see that there is a 

 considerable difference in its steepness at different points, 

 and other motions would give us curves with still greater 

 variations in this respect. We observe that if we lessen 

 the time between two stations, say P^^ and P^^, we must 

 shift the line Qj/^ towards Q^q^jq, and the result is that 

 the curve becomes steeper between Q^^, and O^^. On the 

 other hand, if we lessen the space traversed in a given 

 time the curve becomes less steep and ultimately quite 

 horizontal if the train stops at a station. Thus tJie 

 steepness of the time-chart curve corresponds in some manner 



1 The time-chart has been generally attributed to Galilei ; I do not know 

 on what authority. A speed-chart occurs in his Discorsi, but I do not think 

 there is anything that could be called a time-chart. 



