in the opening lines to book ii of war and peace, tolstoy offered the following insight on the continuity of motion:

This is almost exactly what Aristoxenus did. As he conceived the problem of melodic motion or, as he put it, the motion of the singing voice, it was only by taking infinitesimally small units for measurement and attaining to the art of integrating them – that is, finding the sum of these infinitesimals – that he was he able to arrive at the laws of melodic consecution. The common ratio commanding the infinitely small basic elements of his progression was one-twelfth. In working toward this goal, Aristoxenus contrived to break down the mathematically logical opposition between magnitude (megethos) and multitude (plēthos), in effect, between geometry and arithmetic. The very nature of his method invited him to believe that he could achieve his goal without altering the nature of scientific reasoning. Directly inspired by his knowledge of Pythagorean theory, he discovered a completely new way of dealing with musical intervals: he substituted arithmetic for geometry. It was necessary for him to do this if he wished to extend mathematically rigorous methods to problems where no quantity was involved.