Motion and genetic definitions in the sixteenth-century Euclidean tradition (Q2122126)

This is an exploration of genetic definitions and, by extension, of motion in the framework of 16th-century commentaries on the geometrical books of Euclid's \textit{Elements}. The emphasis is on the ontological and epistemological status of genetic definitions and of motions. A \textit{genetic definition} or a \textit{definition by genesis} refers to a definition that characterises a geometrical object, be it the line, the plane, the circle or the sphere, by its mode of generation instead of a definition by properties. In this sense, a genetic definition of the line would define the line as the result of the flow or the motion of a point (instead of ``breadthless length''). It starts with a wonderful, deeply informative chapter on the history of motion in geometry and the controversies that surrounded it from antiquity to the early modern era, in which the difference between ῥύσις and κίνησις (\textit{fluxus} and \textit{motus}), flow and motion, plays a central role. It then moves on to an in-depth study of the treatment of genetic definitions and motion in six commentaries on Euclid's \textit{Elements} published between 1536 and 1574 in France, England, and Italy, written by Oronce Fine (1494--1555) (Paris, 1536), Jacques Peletier (1517--1582) (Lyon, 1557), François de Foix, Count of Candale (1512--1594) (Paris, 1566), Henry Billingsley (d. 1606) (London, 1570), Federico Commandino (1509--1575) (Pesaro, 1572), and Christopher Clavius (1538--1612) (Rome, 1574). Some additional sources by these authors are also considered, as is John Dee's (1527--1608/1609) preface for Billingsley's English translation and commentary of Euclid. These authors were chosen for ``their conspicuous and significant use of genetic definitions when commenting on Euclid's definitions'' (p.\ 14). The positions of each of the above-named commentators are first analyzed in a chapter devoted to each, and then in a chapter offering a comparative view. The last chapter is devoted to later developments from the 17th century, in particular to the work of Descartes, not only in his \textit{Géométrie}, but also in the unpublished treatise \textit{Le monde}.