Conceptual frameworks from Newton’s <i>Principia</i> and Newton’s <i>Principia Geneva Edition</i> : case studies and analysis of the problem of isochronism, <i>Propositions XLVIII–LIII</i>

Generally speaking, isochronous timing events (and the related consequent problem of isochronism) describes a sequential set of events that may be defined isochronous if the events (and/or the interplay between two or more events) follow recurrently within certain given time-periodical parameters. In other words, it is a property of a physical system where a sequence is at equal time intervals; it also applies to variations in other measurable quantities within the same physical system. Our scientific-historical research is dedicated to the propositions Newton devoted to the problem of isochronism property (at that time) in his Principia and to the notes added by the editors of the Geneva edition (hereafter GE) to such propositions. Isochronism was a problem that had interested scientists at least since Galileo's studies in the late 16th and early 17th century. Huygens gave the definitive solution, discovering the cycloid to be the isochronous curve. However, Newton improved the theory of isochronism in a fundamental way: he inserted such a theory within the picture of his rational mechanics, considered the normal cycloid as a limit case of hypocycloids and epicycloids and specified the nature of force for the tautochrone to be a cycloid. On this last topic, the editors of the GE added an interesting note to Newton's text, thus reaching results that were only implicit in his masterpiece; we analyse one such note-it concerns advanced research in physics and mathematics. Nonetheless, the editors, as usual, added a numerous series of notes to Newton's theory of isochronism, most of which are dedicated to the explanation of Newton's text, although some of them regard, as the one mentioned above, advanced research and others, the history of physics. We offer a broad picture of these notes. They will be useful for the reader to understand the tone of the editors' intervention. Taking into account our numerous publications on Newton and Newton's GE topics, for the historical context of GE and other physical-mathematical analyses on the subject, we refer the reader to consult our list of publications at the end of this article. This article is part of the theme issue 'Newton, Principia, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.