Abstract
The concept of ‘force’ is formulated in Isaac Newton’s Principia in a series of propositions that define ‘force’ in terms of Galileo Galilei’s theorem on the fall of bodies. We claim that the distinctive characteristic of the concept of force as it stands in the Principia is reminiscent of a method to treat non rectilinear orbits suggested to Newton by Robert Hooke.
According to an opinion, Newton had a method to draw physical orbits prior to his interaction with Hooke, and he used this method to draw an orbit in a letter to Hooke; according to a different opinion, Newton used Hooke’s method to draw the orbit. We draw it by three different methods of numerical computation: one corresponds to Hooke’s method, another corresponds to the above mentioned prior method, and the third is a standard higher order method of numerical computation; these solutions are then compared with the solution obtained by Newton. The conclusion is that if Newton had any method whatever to draw curves, and if he made any mistake in the application of the method, this mistake would have been in the sense of diminishing the angle of the pericenter, and not of increasing it, as it is in Newton’s drawing.
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Notes
- 1.
So that \(\vec{v}_0=v_r\;\hat{r}+v_{\theta}\;\hat{\theta}\); \(\vec{v}_0=\textrm{constant}\).
- 2.
We shall not discuss the debate between Hooke and Newton concerning the discovery of universal gravitation. The relevant point to us is that following the “legend of the apple”, those who minimize the import of Hooke in the shaping of Newton’s later thought in mechanics place Newton’s discovery in the 1660s; therefore he would have to have a correct understanding of centripetal forces by the mid 1660s.
- 3.
To explain the orbit, Borelli makes an analogy with a vertical cylinder that floats in a liquid (Koyré, 1961, p.497–498): if the cylinder is let down from a distance above the surface of the fluid, it sinks, and as it sinks the buoyant force (the vis centrifuga, in the analogy) increases till it overcomes the weight (attraction), and the cylinder starts to rise; if there is no loss of motion, the motion goes on forever. The orbit is described by the up and down oscillations of the satellite (cylinder) in a (perhaps cartesian) fluid.
- 4.
Newton uses the word ‘centripetal’ for both central and centripetal forces. Because the modern reader is used to two different words, this may be confusing sometimes; clearly, Newton is not confused, whenever he considers one or the other.
- 5.
Proof. By definition, \(\hbox{``force of motion''}\propto ab\) and \(\hbox{``pression''}\propto \frac{db}{2}\perp fg.\) From Fig. 4.7: \(\hbox{triangle}\; abd\sim \hbox{triangle}\; afb\Longrightarrow \frac{2fa}{ab}=\frac{ab}{fa}\). Or: \(\frac{\hbox{``pression'' of the sphere}} {\hbox{``force of motion''}}= \frac{\hbox{side}\; (=ab)}{\hbox{radius}\; (=\,fa)} \stackrel{\hbox{\tiny after 4 collisions}}\longrightarrow \frac{\hbox{perimeter}}{\hbox{radius}\; (r)} \stackrel{\hbox{\tiny number of sides}\rightarrow \infty}{\longrightarrow}\; \frac{\mathcal{C}}{r}\equiv\frac{2\pi r}{r}=2\pi.\)
- 6.
If τ is the period, r is the radius and v is the speed, then \(\frac{\Delta\left(mv\right)}{\tau}=\frac{2\pi \left(mv\right)}{\tau}=m\left(\frac{2\pi r}{\tau}\right)\frac{v}{r}\equiv m\times\left(\frac{v^2}{r}\right)\).
- 7.
Newton finds another consequence of the “endeavour of rescending” from (not toward) the Sun (Waste Book, in: Herivel, p.197): “Finally since in the primary planets the cubes of their distancs from the Sun are reciprocally as the squares of the numbers of revolutions in a given time: the endeavours of receding from the Sun will be reciprocally as the squares of the distances from the Sun” or, in modern language, \(g\propto\left(2r\right) \times \nu^2\propto\left(2r\right)\times \frac{1}{r^3}\propto\frac{1}{r^2}\). Some people interpret the calculation as an indication that Newton extends to the moon the gravitation fall on the surface of the Earth, and that by 1669 Newton knew the “universal gravitation”. This need not be so, as shown by Cohen (1981).
- 8.
Proof. From the geometry of the circle: \(\left(\overline{AB}\right)^2= \left(\overline{BD}\right)\times\left(\overline{BE}\right)\). For a small arc \(\widetilde {AD}\), it can be considered that \(\overline{BE}\approx \overline{DE}=2r\, \Longrightarrow\) ⟹ \(\left(\overline{AB}\right)^2\approx \left( \overline{BD}\right)\times\left(\overline{DE}\right)\). Hence: \(\frac{\left(\overline{AB}\right)^2}{{\mathcal C}^2}\approx \frac{\left(\overline{BD}\right)\times\left(\overline{DE}\right)}{{\mathcal C}^2}\equiv\frac{\left(\overline{BD}\right)}{\frac{{\mathcal C}^2}{\left(\overline{DE}\right)}}\Longrightarrow x\equiv \frac{{\mathcal C}^2}{\left(\overline{DE}\right)}\). In modern terms, \(x\equiv \frac{{\mathcal C}^2}{\left( \overline{DE}\right)}\approx\frac{\left(v\tau\right)^2}{2r}=\frac{1}{2}\left(\frac{v^2}{r}\right)\tau^2\Longrightarrow \overline{BD}\approx x\times\left(\frac{t}{\tau}\right)^2=\frac{1}{2}\left(\frac{v^2}{r}\right)t^2.\)
- 9.
\(x=\frac{{\mathcal{C}}^2}{2r}=\pi^2\left(2r\right)\), hence \(g\propto\frac{x}{\tau^2}\propto\frac{2r}{\tau^2}\propto\left(2r\right)\times \nu^2\), where ν is the frequency.
- 10.
This calculation has similarities with Huygens’s calculation of the vis centrifuga. Huygens makes an analogy between the centrifugal force and the weight, based on a physical system: he considers an observer standing on the rim, at the top of a rotating vertical wheel, holding a thread at the bottom of which hangs a small sphere; for this observer the weight of the sphere is balanced by the vis centrifuga — which is correct! Huygens uses this result to invoke the expressions for the “free fall”, \(v^2=2gh\) and \(h=\frac{1}{2}gt^2\); then he can prove a series of theorems that together mean: \(\hbox{\tiny centrifugal force}\propto\frac{v^2}{r}\).
- 11.
If r and φ denote the usual polar coordinates, the solutions in spiral are: (1) \(r\phi=\alpha\), \(\dot{\phi}\not=\hbox{\tiny constant}\); (2) \(\frac{1}{r}=\frac{1}{r_0}\cosh\left[\beta\left(\phi-\phi_0\right)\right]\), where α, β, r 0, φ 0 are constants.
- 12.
Even if we used a constant path of integration.
- 13.
It can be verified: (1) the inferior and superior radii of the inscribing circle (Earth) have different values, as do the left and right radii; (2) the vertical (AD) is not exactly perpendicular to the horizontal (BC); (3) the values of the angles of the pericenter and second apocenter can be measured with a protractor, but only approximately, because points O and H are themselves not precisely marked; \(\widehat {ACO}\) seems to be 130°, and \(\widehat {OCH}\) (corresponding to arc \(\widetilde {AFOH}\)) seems to be 237°; (4) point O seems to be at the center of the radius, if one takes CA for the true value of the radius (the radius where C lays is not exactly equal to CA). Given the precision of the drawing, the values for the angles could have been an attempt at naked eye to place C and H at 120°, and 240°, respectively.
- 14.
For instance: Let CO and AH be placed respectively at 120° and 240°, but at naked eyes (this makes the orbit symmetric inside the circle), and draw CO and CH; because a protractor has not been used, the angles become 130° and 237°. Let CO be half the radius AC (because, then, the body falls in half the time \(\frac{1}{4}\) of the distance, as in an uniformly accelerated motion), but because the circle is drawn by hand, O is not exactly at the middle of CK. Then the curve is drawn by the Hooke/SE method up to, say, a little past point F; then symmetry is used to reproduce arc AF from H in the direction of G, using the position of O as an orientation of where ends must meet; the rest of the drawing is adjusted by hand (the part of the curve between F and G has scrawls, indicating that Newton adjusted it). This drawing “looks like” the correct curve, because it starts with a correct method (Hooke’s), and a correct principle of symmetry is used (symmetry by reflection around CO); everything else is wrong.
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Dias, P.M.C., Stuchi, T.J. (2011). Isaac Newton, Robert Hooke and the Mystery of the Orbit. In: Krause, D., Videira, A. (eds) Brazilian Studies in Philosophy and History of Science. Boston Studies in the Philosophy of Science, vol 290. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9422-3_4
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