In 1638, Galileo published his analysis of several situations. This included the famous "interrupted pendulum". This can be described (in modernized language) as conservatively converting potential energy to kinetic energy and back again. However, Galileo did not explain the process in modern terms and had not understood the modern concept either. The German Gottfried Wilhelm Leibniz during 1676-1689 attempted a mathematical formulation of the kind of energy which is connected with "motion" (kinetic energy). Leibniz noticed that in many mechanical systems (of several masses, "mi" each with velocity "vi" ),