Earlier in this book, we saw that there was a close connection between movement and curves. A moving object could be made to leave an inked trail on the paper; this curve then gave us a record of how the object had moved. By passing the curve past a narrow slit, we could again see the movement of the point rising and falling. Alternatively, we could make a cam to the shape of the curve. (See pages 13 and 14.)

The curve is a complete record of the motion. Anything that can be said about the motion can be deduced by examining the curve.

Until now, we have spoken mainly in terms of motion. We have thought of s′ as measuring the velocity of a moving object. But the velocity of the object at any moment must somehow or other be shown by the shape of the corresponding curve. So it ought to be possible to interpret s′ as describing some geometrical property of the curve.

We have already touched on this question twice (pages 13 and 22). We came to the conclusion that the velocity of the object was related to the steepness of the curve. So s′ should measure the steepness of a curve. The general idea here is clear enough. If s′ is large, say s′ = 100, we should be dealing with a very steep curve; if s′ is small, say s′ = ¼, the curve should be not very steep. If s′ = 0, the curve should be flat.