Accumulating Insights: The Problem of Motion Along Broken Chords Driving Conceptual Development

The chapter discusses how Galileo’s attempt to approximate motion along an arc by naturally accelerated motion along paths composed of a series of inclined planes, in or around 1602—and thus, as he believed, in due course pendulum motion—bestowed him with a number of new results. These results were to have repercussions far beyond the narrower context of the investigation into the relation between swinging and rolling from which they had originally ensued. Most of the insights achieved, propositions formulated and methods developed in that period, indeed became core assets of the new science of motion and found their way into the Discorsi more than 35 years later. It is, in particular, demonstrated how, as a by-product of his attempts to construct a proof for the so-called law of the broken chord, Galileo developed a new technique for rendering the kinematics of motion combining the representation of spacial and temporal aspects of motion in just one single diagram. In this period Galileo, in particular, scrutinized the especially challenging problem of finding the geometrical configuration under which the time of motion along a path allowed to vary in a geometrically defined manner became minimal. The problem remained unsolvable, yet his attempts at a solution resulted in the formulation of a set of new propositions complete with proof that were later included, many of them almost verbatim, in the Discorsi.