Abstract
This paper examines an annotation in Newton’s hand found by H. W. Turnbull in David Gregory’s papers in the Library of the Royal Society (London). It will be shown that Gregory asked Newton to explain to him how the trajectories of a body accelerated by an inverse-cube force are determined in a corollary in the Principia: an important topic for gravitation theory, since tidal forces are inverse cube. This annotation opens a window on the more hidden mathematical methods which Newton deployed in his magnum opus. The received view according to which the Principia are written in a geometric style with no help from calculus techniques must be revised.
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Notes
“Newton solved what was called afterwards for a short time ‘the direct Kepler problem’ (‘le problème direct’): given a curve (e.g. an ellipse) and the centre of attraction (e.g. the focus), what is the law of this attraction if Kepler’s second law holds? The ‘problème inverse’ (today: the ‘problème direct’) was attacked systematically only later, first by Jacob Hermann, then solved completely by Johann Bernoulli in 1710 and following Bernoulli by Pierre Varignon” (Speiser 1996, 103). However, as we shall see, in writing his Principia Newton successfully applied his method of fluxions to the inverse problem for inverse-cube forces. Whether he could do the same for inverse-square forces is an open question. However, Newton’s procedure expounded in an annotation to Gregory dated 1694, and which I analyse in this paper, when applied to inverse-square forces leads to a quadrature that Newton could perform. The quadrature occurs in the second catalogue of curves (ordo octavus) of De methodis (Newton 1967–1981, 3, 252), and a similar quadrature is employed in the solution of Corollary 2, Proposition 91, Book 1: for details, see Guicciardini (2009, 282–90).
The original is MS 210 (Royal Society Library, London). There are also three other copies: in Christ Church (Oxford), in the University Library (Edinburgh), and in the Gregory Collection of the University of Aberdeen.
The original Latin text, of which we translate only a few relevant passages, reads: “Secundus Tractatus Methodum suam Quadraturarum continebit quae rem istam mire augebit et promovebit [...] Huic subjungit tabulas pro diversis formis et gradibus figurarum usque ad ordinem decimum [...] Item alias tabulas ad usque classem undecimum ubi spatia non quadrabilia cum coni sectionibus comparantur. Innituntur quaedam abstrusiora in Philosophia sua hactenus edita ut Corol: 3 prop. XLI et Corol: 2 prop. XCI.” Newton (1967–1981), 7, p. 197. It is very interesting that Gregory refers to tables in which “spaces which cannot be squared are compared with conic sections,” since, as we shall see below, these are the quadrature techniques that allow the most difficult of the quadratures implied in Corollary 3.
This annotation was found by W. H. Turnbull who published it in the third volume of Newton’s correspondence (1668–1694, 348–54). It was later included by D. T. Whiteside in Newton (1967–1981, 6, 437–8). Bruce Brackenridge provides a thorough analysis of Newton’s annotation for Gregory in Brackenridge (2003). Herman Erlichson discusses Corollary 3 in Erlichson (1994). Whiteside’s commentary in Newton (1967–1981, 6, 352–6) is important. I am deeply indebted to the above works. Another similar, but just sketched, annotation concerning Corollary 3 is at the top of folio Add. 3960.13: 223r (Cambridge University Library) and is edited in Newton (1967–1981), 6, 435–7. This annotation might have been written just before the one found in the Library of the Royal Society that I discuss in this paper.
Indeed, in the July 1694 memorandum Gregory wrote: “Most of what in early May of 1694 he [Newton] had corrected or altered in his own copy has been corrected or altered at the respective places in my own copy or in my notes.” English translation by Turnbull in Newton (1668–1694), 386. In the pasted sheet on p. 28 of the Notæ , Gregory not only copied Newton’s annotation, but added a few interesting remarks on a method of quadrature he claimed to have found independently from Newton and that was printed in the second volume of Wallis’s Opera in 1693. See Guicciardini (1999), 183; Wallis (1693), 378.
In the third edition (1726) of the Principia, the two curves to the right meet at point a and one of the two curves has an asymptote. This is not relevant for the present paper. I prefer to consider the text that was in front of Gregory’s eyes and that was discussed with Newton in 1694.
Newton imagines that a second body falls from a rest position A so that during the vertical fall AV it acquires the initial given speed with which the first body is projected at V. Allowing ourselves the use of modern symbolism, Newton states that \(v_r^2=2 ABFD =2\int _{ CD =r}^{ CA }F\text {d}\rho =2\int _{r}^{ CV =r_0}F\text {d}\rho +2\int _{r_0}^{ CA }F\text {d}\rho =2\int _{r}^{r_0}F\text {d}\rho +v_0^2\).
In the modified diagram printed in the third edition (1726) of the Principia, the trajectory associated with the auxiliary hyperbola is more clearly drawn as spiralling towards C.
The simplest approach is via the so-called Binet formula and by employing the exponential form of the trigonometric and hyperbolic functions. See Lange and Pierrus (2010), 237–43. The approach I choose here is interesting because of its similarities with Newton’s.
A positive root is chosen. The negative root corresponds to a time reversal \(t\rightarrow -t\): if r(t) is a solution also \(r(-t)\) is a solution (depending on initial conditions).
We note that in Cases 1 and 2, \(\theta =\theta _0\) corresponds to \(r=r_0=\sqrt{(h^2-2\alpha )/2E}\) and \(r=r_0=\sqrt{(2\alpha -h^2)/2|E|}\), respectively. In Case 3, \(\theta =\theta _0\) corresponds to the direction of an asymptote. In Cases 4 and 5, \(\theta =0\) corresponds to \(r=r_0\). Asymptotes for Cases 1 and 5 are easily calculated. For example, in Case 5, an asymptote occurs for \(\theta =-h/(r_0\sqrt{2E})\).
A solution using the sine function is also possible, but this does not generate a new family of trajectories because the sine function can be converted to a cosine by a shift of the polar coordinate. This, of course, is not true for the hyperbolic functions occurring in Cases 2 and 3 below.
In Cases 3, 4, and 5, the particle spirals either into the centre of force or out to infinity, depending on the sign of \(\dot{r_0}\).
We note that circular trajectories are possible when \(2\alpha =h^2\) and \(E=0\), but they are unstable. The reader may consider the effective potential energy for inverse-cube forces, \(U_\mathrm{eff}=h^2/(2r^2)+U(r)=h^2/(2r^2)-\alpha /r^2\). When \(2\alpha =h^2\), the effective potential energy \(U_\mathrm{eff}\) is flat and any r is a possible radius for an unstable circular trajectory (\(\dot{r}=0\)) with velocity \(|v|=\sqrt{2\alpha }/r\).
The choice of the auxiliary conics is crucial. Conics with different parameters give rise to different trajectories.
The trajectory is expressed in polar coordinates. So \(\theta \) is an angle. Instead, we take VCR as the area of the circle and hyperbolic sectors, respectively.
This must have been evident for Newton from Apollonius, Conics, I.37. Apollonius of Perga (2000), 65–7.
This is an innovative element of my paper, since below I will detail the relationships between Corollary 3 and Newton’s quadrature techniques of the De methodis and De quadratura. These techniques date back to work Newton carried out in the 1660s.
This is Leibniz’s term, as Newton would use the Cartesian terminology “mechanical curves,” and this (indeed) is a terminological distinction of great significance for the historian.
Actually, Newton introduced the dot notation in the 1690s.
In modern notation, we provide the following example: \(\int \text {d}x/\sqrt{b^2-x^2}=(1/b)\int \text {d}x/\sqrt{1-(x/b)^2}= \int \text {d}\lambda /\sqrt{1-\lambda ^2}=\arcsin \lambda \), for \(\lambda =x/b\).
I will not discuss the variants between the Catalogi divided into ordines of the De methodis and the Tabulae divided into formae of De quadratura, since they are not relevant for the thesis defended in this paper. According to an expert judge such as Whiteside: “The tables of integrals which Gregory saw were in fact [...] those which Newton introduced more than twenty years earlier into his general 1671 tract on fluxions and infinite series [the De methodis], rather than their lightly refashioned equivalent which he had much more recently appended to his revised text ‘De quadratura curvarum”’ Newton (1967–1981), 7, 197.
This might be missed by a superficial reading.
In Leibnizian notation: \(\tau =\int y\text {d}z=(1/\eta )\int vdx=(1/\eta ) s\).
In the second Tabula of De quadratura, this corresponds to the first case of the fourth Forma.
Newton did not use the modern symbol for the absolute value \(|\frac{1}{2}xv - s|\) but rather one that he found in Barrow’s works. Newton wrote \(\div \) for “the Difference of two Quantities, when it is uncertain whether the latter should be subtracted from the former, or the former from the latter”.
To recapitulate. The first case of the seventh order translated into Leibnizian notation is as follows. For \(\eta =2\), Newton evaluates the integral \(\int \delta /(z\sqrt{e+fz^2})\,\text {d}z\) (\(\delta \), e, f constants). By substitution of variables \(z=x^{-1}\), he reduces it to the conic area \(s=\int vdx= \int \sqrt{f+ex^2}\,\text {d}x\). Namely,
$$\begin{aligned} \tau = \int \frac{\delta }{z\sqrt{fz^2+e}}\,\text {d}z=\frac{2\delta }{f}\left| \frac{1}{2}xv-s\right| +C=\frac{2\delta }{f}\left| \frac{1}{2}x\sqrt{f+ex^2}-\int \sqrt{f+ex^2}\,\text {d}x\right| +C. \end{aligned}$$Whiteside notes that the given constant should be \( Qc ^2/2a^4\) (Newton 1967–1981, 6, 438).
In symbolic terms: \(r=\epsilon ^2/\zeta =\epsilon ^2(\cos k\theta )^{-1}\) and \(r=\epsilon ^2/\zeta =\epsilon ^2(\mathrm {cosh}\, k\theta )^{-1}\).
The notion of a “Newtonian style” characterizing the Principia was forcefully defended by François De Gandt in his enjoyable and learned monograph (Gandt 1986).
Gregory’s Notæ are a treasure trove of information on the relationships between the method of fluxions and Newton’s Principia, something that might be missed upon superficial inspection. In many places Gregory refers to quadrature methods. Further, the Aberdeen exemplar of the Notæ includes pages on Newton’s celebrated treatment of the solid of least resistance, an extraordinary application of fluxions to a proposition of the second book of the Principia that Gregory also included in the manuscript treatise on fluxions he circulated in the mid-1690s (“Isaaci Newtoni Methodus Fluxionum; ubi Calculus Differentialis Leibnitij, et Methodus Tangentium Barrovij explicantur, et exemplis plurimis omnis generis illustrantur. Auctore Davide Gregorio M. D. Astronomiæ Professore Saviliano Oxoniæ”. A fair copy is in Christ Church (Oxford). The original is in St. Andrews University Library (MS QA 33G8/D12). Other copies are in the Cambridge University Library, Lucasian Papers [Res. 1894]: No. 13 and in the Macclesfield Collection, Add. 9597.9.3 and Add. 9597.9.4). Other relevant information on Newton’s use of fluxions in the Principia can be gathered, most notably, from Newton’s manuscripts and his correspondence with Nicolas Fatio de Duillier and Roger Cotes.
As a matter of fact, Newton contemplated ending the Principia with an appendix on quadratures in the 1690s, when revising the text, and again in the 1710s, when preparing the second edition of 1713. But in the end he resolved not to do so. It is only in the posthumous translation of 1729 due to Andrew Motte that we find two quadrature methods (for the attraction of an ellipsoid of revolution and for the solid of least resistance) printed as appendices “communicated by a friend,” who may have been David Gregory.
For example, in Thomas Simpson’s treatise on the method of fluxions, first printed in 1750, we read that “the Time wherein the Space \(\dot{x}\) would be uniformly described is known to be as \(\dot{x}/v\),” (where v is the velocity) and a few pages later this is applied to rectilinear accelerated motion in order to calculate the time T “by finding the fluent” of “\(\dot{T}=\dot{x}/v\)”. This is, of course, in Leibnzian notation equivalent to \(\text {d}t=\text {d}x/v\). See Simpson (1750), pp. 244–6.
The absence of a symbol equivalent to Leibniz’s \(\int \) is a well-known weakness of Newton’s notation. In a few instances, Newton used to draw a square \(\Box \); in most cases, he used words such as “the fluent of” or the “area of”. Some eighteenth-century British mathematicians mixed the two notations by writing, for example, \(\int y\dot{x}\) for \(\int ydx\). The use of an elongated f for “fluent of,” rather than Leibniz’s elongated s, \(\int \), for “summa” is also documented. Cajori (1929), 2, 244–6.
On the “dual” character, algebraic and geometric, of differentiation in the early Leibnizian calculus, see Fraser (2003).
We note that angular momentum is a vector quantity, so it is also the direction of the vector (or equivalently the fact that the motion is planar) that must be taken into consideration. Before the advent of vector notation at the beginning of the nineteenth century (another conceptual shift to which corresponds the introduction of yet another mathematical language), the constancy in orientation of the angular momentum for central force motion could only be expressed in terms that in some crucial way referred to the geometrical features of the orbit.
I have discussed this issue in Guicciardini (2009).
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Acknowledgments
I wish to thank Antonio Giorgilli, Massimo Galuzzi (Dipartimento di Matematica, Università di Milano), Tom Archibald (SFU, Vancouver), Craig Fraser (University of Toronto), Alain Albouy (Observatoire de Paris), and Michael Nauenberg (Santa Cruz) for their helpful comments.
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Communicated by : Jed Buchwald.
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Guicciardini, N. Lost in translation? Reading Newton on inverse-cube trajectories. Arch. Hist. Exact Sci. 70, 205–241 (2016). https://doi.org/10.1007/s00407-015-0170-z
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DOI: https://doi.org/10.1007/s00407-015-0170-z