How was the law os conservation of energy discovered?

A very simple experiment of dropping a body of mass m from a height h above the ground level.
At the top the body has only Potential energy and no kinetic energy. Hence mgh is the total.
When the body falls freely and at the time of touching the ground level the potential will be taken as zero but the kinetic will be come 1/2 m v². V² can be proved to be 2gh and hence KE = mgh. So the total is mgh. If we have any doubt about the total energy on the way we can check that too and can be derived to have the same as mgh. So the energy seems conserved all the time.