If I ask the train personnel, ‘Conductor, does the Cambridge railway station pass by this train?’ I’m being a bit strange, but not wrong, because velocities are relative. However, saying, ‘Conductor, does the Cambridge railway station stop at this train?’ makes no sense in our Universe.

Because position is relative, we must use the change of position in order to describe motion. This change we call velocity. But velocity is relative as well; therefore we must use the change of velocity in order to describe motion. This change we call acceleration. This is, in fact, an observable, as every cyclist knows who has had to brake for a traffic light.

The word is a bit specialized, because in physics we use the word ‘acceleration’ for every kind of change in velocity: an increase or a decrease of speed as well as a change in direction are all called by this name. Velocity has a direction and a magnitude, so that a change of direction counts as an acceleration too, even if the speed (number of metres travelled per second along a path) remains the same.

That is precisely the case with Galileo's circular motion. The speed remains constant, but the direction changes steadily. The question then is: how does that feel? In a brilliant sequence of arguments, Huygens equated the acceleration of uniform circular motion to the steady acceleration of a falling object. Thus, he could relate the pull which is felt on the string of a slingshot directly to the acceleration of gravity.

From Galileo's work, Huygens knew that the speed of a falling object increases linearly with time. If the amount of acceleration is arbitrarily set to 1, and an object starts at speed zero, then in 2 units of time it reaches speed 2. The mean speed in that interval is then (0+2)/2=1. The mean speed in the next interval is 1+2=3, so that the mean velocity follows the sequence of odd numbers: 1, 3, 5, 7… at successive instants of time. The distances travelled at these times are then the sums of the numbers: 1, 1+3, 1+3+5,… which add up to 1, 4, 9, 16… These are all square numbers: 1×1, 2×2, 3×3, 4×4…, from which it follows that the distance travelled by a falling object increases quadratically with time.