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Conceptual Frameworks on the Relationship Between Physics–Mathematics in the Newton Principia Geneva Edition (1822)

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Abstract

The aim of this paper is twofold: (1) to show the principal aspects of the way in which Newton conceived his mathematical concepts and methods and applied them to rational mechanics in his Principia; (2) to explain how the editors of the Geneva Edition interpreted, clarified, and made accessible to a broader public Newton’s perfect but often elliptic proofs. Following this line of inquiry, we will explain the successes of Newton’s mechanics, but also the problematic aspects of his perfect geometrical methods, more elegant, but less malleable than analytical procedures, of which Newton himself was one of the inventors. Furthermore, we will also consider the way in which Newtonianism was spread before in England and afterwards on continental Europe. In this respect the Geneva Edition plays a fundamental role because of its complete apparatus of notes, and because it appeared only thirteen years after the publication of the third edition of the Principia (1726). Finally, we will also confront some problems connected to the metaphysics of calculus. Therefore, the case of Newton is one of those in which, starting from mathematics applied to physics, it is possible to connect an impressive series of fundamental arguments such as the role of mathematics in science; the comparison between Newton’s geometrical methods and analytical methods; the way in which Newtonianism was spread as well as the philosophical implications of Newton’s mathematical concepts.

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Fig. 1

Source: BnF Public Domain | https://gallica.bnf.fr

Fig. 2

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Fig. 4

Source: Google Books | Public Domain. a: Proposition VI. Theorem V. In a space void of resistance, if a body revolves in any orbit about an immoveable centre and in the least time describes any arch just then nascent; and the versed sine of that arch is supposed to be drawn, bisecting the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc, will be as the versed sine directly [will be as the versed fine directly] and the square of the time inversely. (Newton, [1739–1742]1822, Prop. VI, Th. V, I, p. 79; Author's Italics; translation from [1726]1729, p. 68).) (Newton, [1739–1742]1822, Prop. VI, Th. V, I, p. 81 [pp. 79–82])

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Notes

  1. Thomas Le Seur and François Jacquier were not Jesuits, but belonged to the Gallicana Minimorum Familia. They were Minim Friars (Cfr. Jacquier, 1755; Seur & Jacquier, 1768, Seur et al., 1743).

  2. He belonged to no congregation (Calandrini, 1722; Calandrini & Serres, 1727).

  3. Sometime spelled as ‘‘l’Hospital’’, and sometimes—because of French spelling—the silent ‘‘s’’ is removed and replaced with a circumflex. Even if the former spelling is still used in English where there is no circumflex accent, due historical methodological reasons, we prefer let his original name.

  4. Since 2014, for Oxford University Press, we have been engaged to provide a critical translation (into English) of this whole apparatus as expounded in the GE. No translation exists of the notes–commentaries of the GE. Besides the translation of the notes, the variants of the notes existing in the 1739–1742 and 1760 editions will be highlighted. Therefore, 5 volumes are expected within 2022. This will clarify the nature, purposes and structure of all the notes/commentaries (apparatus) added to Newton’s text by the commentators, so providing a useful guide to understand the relevance of Newton's mathematical methods within his physics.

  5. For sake of brevity, we have omitted detailing the methodological interchange between Huygens and Newton (Cfr. Huygens, 1673, 1690, 1897; Nauenberg, 1994a, b, 1996).

  6. With regard to Newton’s biography we only mention the fundamental: Westfall ([1983]1995). For sake of brevity and just to mention the following accredited secondary literatures on Newton, the Principia and Newtonianism: Westfall (1958), Koyré (1965), Westfall (1958, 1971), Cohen (1990), Newton (1972, 1999), Cohen and Smith (2002), Chandrasekhar (1969, 1995), Bertoloni Meli (1993, 2006), Brackenridge (1995), De Gandt (1995), Force and Hutton (2004), Force and Popkin (1999), Guerlac (1981), Guicciardini (1989, 1998, 1999, 2002, 2004a, b, 2015), King-Hele and Hall (1988), Pulte and Mandelbrot (2011), Purrington and Durham (1990), Hall (1978), Rupert Hall (1999), Châtelet (1759), Clarke ([1730]1972), Desaguilers (1717), Gregory (1702), Keill (1701), McLaurin ([1748]1971), Pemberton (1728), Rouse Ball ([1893]1972), Whiston (1707), Wright (1833), Hiscock (1937), Wightman (1957), Eagles (1977), Agassi (1978), Agostino (1988), Ahnert (2004), Allen (1998), Arthur (1995), Axtell (1965), Baillon (2004), Barber (1979), Barker (2006), Beaver (1987), Berggren and Goldsein (1987), Biagioli (1998, 1999), Boss (1972), Bourdieu ([1975]1999), Bricker and Hughes (1990), Briggs (1983), Brockliss (1992), Calinger (1968, 1969), Casini (1988), Champion (1999), Clark (1992, 1997), Clark et al. (1997), Craig (1963), Crasta (1989), Coudert (1999), Cunningham and Williams (1993), Dear (1987, 1995, 1998), Elliott (2000), Fellmann (1988), Ferrone (1982), Force (1985; 20041983), Friesen (2006), Gascoigne (1988), Gaukroger (1986), Goldish (1998, 1999), Golinski (1998), Guerlac (1981), Guerrini (1985), Hampson (1981), Hankins (1990), Harman (1988), Harrison (1995), Haycock (2004), Heidarzadeh (2006), Heimann and McGuire (1971), Henry (1992), Hessen (1931), Hutton (2004a, b), Iliffe (2004), Iltias (1977), Jackson (1994), Jacob (1976, 1977, 1978), Leshem (2003), Lord (2000), Lüthy (2000), Lynn (1997), Malet (1990), Mandelbrote (2004a, b), Marcialis (1989), Markley (1999), Mazzotti (2004), McMullin (1978), Montgomery (2000), Munby (1952), Osler (2004), Pagden (1988), Pater (1994), Phemister (1993), Phillipson (1981), Porter (1981), Porter and Teich (1992), Rattansi (1981), Rousseau and Porter (1980), Ruderman (1997), Schama (1981), Smolinski (1999), Snobelen (1997, 1998, 2004), Stewart (1992, 2004), Taylor (1981), Teich (1981), Thijssen (1992), Wall Van der (2004), Whaley (1981), Wigelsworth (2003), Yolton (1990, 1994), Young (2004), Zinsser (2001, 2003), Aiton (1964, 1965, 19721962), Harper (2001), Di Salle (2006); and our works listed below.

  7. ''L'un part de ce qu'il entend nettement pour trouver la cause de ce qu'il voir. L'autre part de ce qu'il voir pour en trouver la cause, soit claire, soit obscure'' (Fontenelle Le Bouvier, 1728b, p. 24; the French version is also enclosed in the end of Fontenelle Le Bouvier, 1728a, p. 11). See the quotation in another French edition (Fontenelle Le Bouvier, 1728c, p. 17).

  8. On the centre of gravity also related to the science of weights see: Pisano (2007, 2011, 2013b), Pisano and Capecchi (2016) and Pisano and Bussotti (2012).

  9. In the note 212 (Figs. 4, 5) because the time and the distance PQ tend to 0 Newton’s Proposition VI—and its related corollaries—introduces a mathematical expression of central force (centripetal) by means of a differential form (Ivi, pp. 81–82). We note that this form is presented as a relationship between the traced segments (Fig. 4). In the successive Scholium (Newton, [1739–1742]1822, notes 213–216, pp. 81–82) the commentators explain this relationship between this centripetal physical force and its applied mathematical interpretation; that is a differential form. Thus, they provide the right form of the centripetal force in order to make explicit its correlated (Ibidem, note 214). For additional details on this analysis see Bussotti and Pisano (2014a; see also 2014b).

  10. The works cited are: Daniel Bernoulli, Sur le Flux et Reflux de la Mer (1740), McLaurin: De causa physica fluxus et refluxus maris (1740) and Euler, Inquisitio physica in causam fluxus ac refluxus maris (1740). (Newton, [1739–1742]1822, I, pp. 101–341; Cfr. Euler, [1911, 1913, 1917]1941; Euler, 1752–1753, 1754–1755, 1756–1757, 1760–1761, 1765, 1770, 1773, 1732–1738).

  11. “[…] punctis P et Q coeuntibus […]” (Newton, [1739–1742]1822, I, p. 106; Newton, Id., 1729, p. 76).

  12. With respect to Newton's infinitesimal quantities, we also dealt with the interpretation of his lineola such as the quantity whose length is given and that Newton employs in his demonstration. In Motte's words it represents a very small line (Newton, [1726]1729, I, p. 165). The Latin expression is “lineola(m)” (Newton, [1726] [1739–1742]1822, I, p. 237).

  13. Leibniz (1684a, b), Bussotti and Pisano (2017), D’Alembert, Lazare Carnot, Lagrange, Mach, and Poincaré, critically perceived these ambiguous arguments, also mathematically described by Newton, as uncertainties (Pisano, 2013d, 2017, 2020; Pisano et al., 2020; Pisano & Capecchi, 2013).

  14. See also Lazare Carnot's the théorie de la compensation des erreurs so-called à la Berkeley: “[…] deux erreurs [two errors]” are made, which nevertheless are algebraically annulled in the end, and “[…] by a compensation of errors: which compensation, however, is a necessary and certain consequence of the operations of the calculus.” (Carnot, 1813, pp 12–15, pp 30–34, p. 189). For an historical account on the subject, see Gillispie and Pisano, 2014, Chaps 6 and 9; Pisano et al., 2020, 2021; see also Carnot, 1786, 1803a, b, 1813).

  15. We are conscious that the identification of fluxions with derivatives is not completely correct because in Newton’s time the concept of function was not yet completely clear, and derivatives are strictly connected to functions. However, for sake of brevity, and in modern terms, we expound the reasoning using the concept of function and derivative. In any case, the core of Berkeley’s methodological criticism is not modified.

  16. For the structure of the notes inserted in the GE, we refer to: Bussotti and Pisano (2014a, b), Guicciardini (2015) and Pisano and Bussotti (2016a, b, 2020).

  17. It is necessary to point out that ours is only an outline of a complicated history we are writing, only to pose Newton’s work in an historical perspective. As a matter of fact, the history of ellipsoid attraction is multi-faceted and many authors contributed to it. For a complete history up to the 1830s, we refer to Todhunter (Todhunter, 1873). We do not consider the problem of celestial bodies’ shapes and of the figure of equilibrium of a fluid mass. We refer to three fundamental publications on this subject: once again to Todhunter (Ibidem) which is a detailed text on the history of this topic from Newton to Laplace, but which, in fact, also includes Ivory’s (1809), Gauss’ and Poisson’s results, to Chandrasekhar (1969), where the reader can find all the mathematical details of these problems and which is also enriched by an historical introduction, and to Greenberg (1995). An interesting paper where many historical and conceptual observations are present is Poincaré (1892).

  18. Lagrange (1774, 1769, 1770, [1773]1755, 1777, 1867–1897), Pisano (2013a) and Pisano and Capecchi (2013).

  19. This memoire by Laplace was read in 1784 at the Académie Royale des Sciences. It appears in the issue 1782 (but published in 1785) of the Mémoires. To avoid confusion, we will refer to it as Laplace 1784 (Laplace, [1782, 1784]1785). A good paper on the contributions given by Legendre and Laplace to the theory of the ellipsoid attraction is Pecot (1993). See also Laplace (1799, [1782, 1784]1785).

  20. We mean it was not yet completely achieved from a conceptual point of view. It is well known that the modern notion of function was defined by Johann Peter Gustav Lejeune Dirichlet (1805–1859) in 1837, but a concept can be used and treated correctly before a formal definition. In this case, the editors of the GE, but more in general the mathematicians around the 1740s were close to the concept of function, but had not yet fully conceived it.

  21. There are several manners to show that \(y^{2} = \frac{{t^{2} }}{{s^{2} }} \times (2sp + 2sx - p^{2} - 2px - x^{2} )\) is the equation of an ellipse. We chose the following one. To simplify, be the abscissa x rather than p + x. Thence, the equation will be \(y^{2} = \frac{{t^{2} }}{{s^{2} }} \times (2sx - x^{2} )\). Now pose \(\frac{{t}^{2}}{{s}^{2}}=\frac{p}{d};2s=p\), so that the equation gets the form \({y}^{2}=2px-\frac{p}{d}{x}^{2}\) which is the equation of an ellipse referred to a diameter and to the tangent at the extremity of such a diameter.

  22. Newton's mathematical method applied to physics (Mechanics) raised as an integrated mathematics-physics, a sort of a new unique discipline which worked with geometry, mathematics and physics. After Newton’s mechanics, particularly in nineteenth century, for some analytical theories only, this integration was emphasised. The physics and mathematics worked as a unique discipline, not mathematical applications to physics and vice versa, but physics mathematics as a new approach to physical and mathematical integrated studies (and above all so different from successive and modern mathematical physics (Cfr. Pisano's works)). It was a structured discipline with its own hypotheses, methods of proofs, and with an internal coherent logic. New methodological approaches were conceived to solve physical problems (in their background) where an object can be both physical and mathematical quantity (1st novelty) and measurement is not a priority or a prerogative (2nd novelty). However, it is a coherent and valid physical science. One can think, for example, to reversibility in thermodynamics as interpreted by infinitesimal analyses: each point is a quasistatic point that is a quasistatic thermodynamic process that happens infinitely slowly. In effect, it is far from a measurable physics because it is not a real process, but such processes can be approximated by performing them very slowly. In other words, it is typical of intellectual dynamics of infinitesimal analysis applied to an idealistic physics; but the power of methods, application and results are formidable and undisputable.

  23. Any historiographic thesis had to be substantiated by evidences. In our previous papers (see References section below) we have offered many instances of historical and foundational analysis of the GE.

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    Raffaele Pisano

  2. DIUM, Udine University, Udine, Italy

    Paolo Bussotti

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Pisano, R., Bussotti, P. Conceptual Frameworks on the Relationship Between Physics–Mathematics in the Newton Principia Geneva Edition (1822). Found Sci 27, 1127–1182 (2022). https://doi.org/10.1007/s10699-021-09820-2

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