Whatever Happened to Swinging and Rolling: Faint Echoes and a Late Insight

Confronted with overwhelming problems, in or shortly after 1602, Galileo abandoned his ardent investigation into the relation of swinging and rolling without, however, entirely giving up on his approach. The fundamental idea of his early research, namely, to approximate motion along an arc by motion along polygonal paths made up of a series of inclined planes, still resonates in the Discorsi, albeit in but one argument. By means of this argument, Galileo intended to demonstrate that the brachistochrone, i.e., the curve between two points along which motion is completed in least time, was an upright arc. It is argued that this was a late attempt by Galileo to make his earlier considerations regarding swinging and rolling bear fruit and that he had reason to surmise that his argument was flawed. The chapter, moreover, discusses how, stimulated by his perusal of Giovanni Battista Baliani’s De motu naturali gravium solidorum published in 1638, Galileo briefly resumed his work on swinging and rolling. Based on the understanding he himself had gained in his own earlier investigation into the same problem, Galileo drafted a critique of Baliani’s approach as part of which, for the first time, he devised a proof of the law of the pendulum. Galileo had thus accomplished in part what he had sought to achieve some 35 years earlier, and the new proof was indeed earmarked for inclusion in a second edition of the Discorsi.