# 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

which gives

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

so

Therefore,

so

and

which is what we wanted to show.