Calculating Napier's logarithm
We’ll show that
The point moves at a constant speed of , so we have Since moves at a speed that is proportional to the distance left to travel we have From this we see that
for some constant
We can work out the value of using our initial conditions. At the start, the point still needs to travel the whole length of the line segment , which is . Therefore . The point hasn’t gone anywhere yet, so
Plugging this into the expression above gives
which is what we wanted to show.