The physics–mathematics interplay in Newton <i>Principia</i> Geneva Edition (1822): a new case study on the three-body problem, <i>Proposition LXVI–Theorem XXVI</i>

The interplay between physics and mathematics with history deals with the role played by the relationship between physical and mathematical objects within a scientific theory. Taking into account the rigour of proof in mathematics and of the measurability/reproducibility of experiments in physics, a question arises: how can we read the interplay between physics and mathematics as a historical category of inquiry in order to analyse the development of a physical-mathematical theory, e.g., with respect to a physical theory, or with respect to a purely mathematical one (inspired by physical phenomena)? The role played by the interplay between physical and mathematical objects in a scientific theory is also specified by definitions, suppositions, theorems, diagrams, calculus, etc. Therefore, taking into account our interest in inquiring physics-mathematics relationships with the history of science, in this paper, we first propose a methodological account in order to show how the relationships between physics and mathematics with history work, and second, we use it in order to analyse Newton's three-body problem within his physical and mathematical backgrounds. But, how does the Geneva Edition description of this differ from how this was described by Newton? In detail, we analyse the case study of Proposition LXVI as presented and discussed in Newton's Principia Geneva Edition (Newton ([1726] [1739-1742] 1822, Book I), which, with its 22 corollaries, is the longest proposition of Newton's masterpiece. It deals with the three-body problem (a physical-mathematical interplay, also a relationship). Therefore, our historical-scientific analysis concerns the explanation of the Principia Geneva Edition's notes added by the three editors to Newton's treatment of the three-body problem as novel, as a separate step after Newton's treatment (and before Poincaré's statement). This article is part of the theme issue 'Newton, Principia, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.