THE GEOMETRY OF MOTION 217 



is then termed the mean slope of the portion AC of the 

 curve, because, however the steepness may vary from A 

 to C, the final result CB in AB could have been attained 

 by the uniform average slope of AC. 



But this idea of mean slope does not settle the actual 

 steepness of the curve, say, at the point A. Now let the 

 reader imagine that the curve AC is a bent piece of wire, 

 and the chord AC a straight piece of wire ; further, he 

 must suppose small rings placed about both wires at A 

 and C. In conception we will suppose the wires to be 

 indefinitely thin, so that they approach as closely as we 

 please to the geometrical ideals of curve and line. Then 

 the ring A being held firmly at A on the curved wire, let 

 the ring C be moved along the curved wire towards A, 

 As it moves, the straight wire slips first into the position 

 AC', and ultimately, when the ring C reaches A, takes up 

 the position AT. In this position the straight line is 

 termed the tangent to the curved line at the point A. 

 As the slope of AC or AC' measures the mean steepness 

 of the curve from A to C, or from A to C', so does the 

 slope of the chord in its limiting position of touching 

 line, or tangent, measure the mean steepness of an in- 

 definitely small part of the curve about A. The slope of 

 the tangent is then said to measure the steepness of the 

 curve at A. It is clear that in this notion of measuring 

 the mean for a vanishingly small length of curve we are 

 dealing with a conception which is invaluable as a method 

 of description. It represents, however, a limit which, no 

 more than a curve or line, can be attained in perceptual 

 experience. 



^11 . — Speed as a ^lope. Velocity 



Having now reached a conception by aid of which we 

 can measure the steepness of a curve at any point — 

 namely, by the slope of the tangent at that point — we 

 may return to the curve of our time-chart and ask what 

 we are to understand by its slope. Turning to Fig. 10, 

 we observe that the mean slope of the portion 0^0^ of 



