A Newtonian tale details on notes and proofs in Geneva edition of Newton's Principia

Abstract

Based on our research regarding the relationship between physics and mathematics in HPS, and recently on Geneva Edition of Newton's Philosophiae Naturalis Principia Mathematica (1739–42) by Thomas Le Seur (1703–70) and François Jacquier (1711–88), in this paper we present some aspects of such Edition: a combination of editorial features and scientific aims. The proof of Proposition XLIII is presented and commented as a case study.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Among commentaries to Newton's text, foremost is Madame du Châtelet's because of its notable importance and strict connection to Newton's Principia (that is, Keill 1701; Gregory 1702; Whiston 1707; Desaguilers 1717; Clarke [1730] 1972; McLaurin [1748] 1971; Pemberton 1728; Châtelet 1759; Wright 1833; Rouse Ball [1893] 1972; Chandrasekhar 1995). In the context among the first commentators of the Principia, the role of David Gregory was important (Hiscock 1937; Wightman 1957; Eagles 1997; Bussotti and Pisano 2014b).

² The first Geneva edition (1739–42) was divided into three tomes and the tomus tertius was divided into two parts. However, because of editorial reasons, our publication project with the Oxford University Press will follow the edition of Glasgow (1822), which is divided into four volumes. Because of this we have used the expression ‘four volume’ in the running text.

³ With regard to Le Seur's and Jacquier's works, in particular as far as the damage of St Peter's Dome in 1743 is concerned, an important paper is Schlimme 2006. The author mentions 14 works written on this subject in 1743 (the list presented by Schlimme on the works on the damages of the dome of St Peter between 1680 and 1767 specifies 21 works). See Montègre 2006, 2011. As to specific indications on the commentators, see Bussotti and Pisano 2014a, 37; Bussotti and Pisano 2014b, 421–423; Guicciardini 2014.

⁴ Clairault criticized the treatise by Calandrini (Guicciardini 2014, 22–23).

⁵ On the problem of the asterisk indicating the notes, see Bussotti and Pisano 2014b. Particularly, a note with an asterisk was not by Calandrini (Guicciardini 2014, ft 77).

⁶ In what follows we will indicate this edition simply as Newton 1822.

⁷ Although no ambiguity is possible on the fact that the commentators were Minims, the Geneva edition is often called the ‘Jesuit Edition’. It is likely this misunderstanding derives from what mistakenly the typographers of the Glasgow edition wrote. For, we read: ‘Our intention was to publish the edition of Le Seur and Jacquier, belonging to the Jesuit Society, in an integral form, with their commentaries, correcting the mistakes that here and there could have occurred’ (Newton 1822, I, page VIII, our translation; Bussotti and Pisano 2014a, 37, 2014b, section 3.1).

⁸ For the most important physical and mathematical (that is, nature of infinitesimals) questions see: Bussotti and Pisano 2014a, 2014b. Significant works include: Arthur 1995; De Gandt 1995; Guicciardini 2014.

⁹ The Tentamen (Leibniz [1689] [1880] 1971) is one of the first essays in which differential equations are explicitly used in physics (Bussotti and Pisano 2017; Bussotti 2015).

¹⁰ We do not enter into the different meanings of the term analytic. We use here this term simply to indicate that the results by Newton were transcribed using explicitly the language of mathematical analysis, that is differential equations.

¹¹ We are referring to physicists—only mentioning the most famous, such as the Jakob e Johann Bernoulli, Pierre Varignon (1654–1722) and Guillaume François Antoine, Marquis de l’Hôpital (1661–1704), Willem Jacob 's Gravesande (1688–1742), Abraham de Moivre (1667–1754), John Keill (1671–1721), Stirling (1692–1770) and Colin McLaurin (1698–1759). Let us think that starting from the end of 1720s to the beginning of the 1730s in the eighteenth century, the works by Euler (1707–83) began to ‘invade’ the scientific community, changing the face of physics. Clairaut (1713–65) was also active in the 1730s.

¹² Newton did not use only geometrical-infinitesimal methods in the Principia. He resorted to the development in infinite series. However, while methods such as the infinite series became standard methods, for example, in astronomy, the geometrical-infinitesimal methods were typical of Newton. Because of this, they are so important and fascinating from a mathematical-historical point of view.

¹³ As to apsides-line's movement in Kepler, see the chapter XLII of the Astronomia Nova, van Dyck and Caspar 1937–2009, III, 275–282 and Epitome Astronomiae Copernicanae, Book IV, Part III, van Dyck and Caspar 1937–2009, VII, 340–342. See also Stephenson, 1987, 1994, 112–114. Recently on Kepler's Corpus and force conceptualization see: Pisano and Bussotti 2016a.

¹⁴ Newton, 1729, 1, p. 177. Latin text: ‘Efficiendum est ut corpus in trajectoria quacunque circa centrum virium revolvente perinde moveri possit, atque corpus aliud in eadem trajectoria quiescente’, Newton, 1822, I, 258.

¹⁵ This corollary (Newton 1729, 1, 70 and Newton [1726] [1739–42] 1822, 1, 80) explains how—given a curvilinear figure as trajectory and the centre of the force—it is possible to find the centripetal force under whose action a body moves along the given trajectory.

¹⁶ The concept of an infinitesimal given time is typical in Newton. This time is neither an actual nor a potential infinitesimal in the classical sense of this expression (Bussotti and Pisano 2014b; Pisano and Bussotti 2016b).

¹⁷ In the lemmas II, III, and IV of the Principia (Newton 1729, 1, 42–45; Newton 1822, 1, 45–48), Newton introduces the elements of integral calculus—to use a modern expression—useful in the course of his work. In particular (Newton 1729, 1, 44–45; Newton [1726] [1739–42] 1822, I, 48) the corollary to Lemma IV establishes that: ‘If two quantities of any kind are any how divided into an equal number of parts: and those parts, when their number is augmented and their magnitude diminishes in infinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order: the whole quantities will be one to the other in that same given ratio.’ (Newton 1729, 44).

¹⁸ It is well known that Proposition II inverts Proposition I and shows that if a body moves in a curvilinear trajectory and, with the radius-vector directed toward an immobile or moving of rectilinear uniform motion point, describes area proportional to the times, then it is pushed by a centripetal force tending to the same point.

¹⁹ ‘Vide Varignonium Legem vis centripetae in trajectoria Vpn determinantem, in Comm, Paris 1705’ (Newton 1822, 259; our translation).