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The famous equation $F = G\cfrac{m_1m_2}{r^2}$ is not in Mathematical Principles of Natural Philosophy, as it has been pointed out by a commentator on another question about where the equation is in the book. The closest one can get is Corollary 2 to Proposition V ("The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from that planet's centre"), according to the commentator.

If that's the case, then where does the famous version of the equation come from, when did it develop, and why is Newton credited with the equation if it's not in the book? Is the equation implied or found through examples? If it's in prose, please could someone quote the correct passage?

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Colin Pace
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    Newton states the law as Proposition VII of Book III - see archive.org copy of the principia. Prop VII starts on page 403 of 594 using the horizontal scroll bar, the inverse square law is stated at the top of page 404. Newton expressed it in prose. The formalism you seek did not exist at the time. – nwr Dec 16 '23 at 05:16
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    Thanks for the comment, nwr! I read Proposition VII of Book III. I don't see the equation expressed in prose. Please could you quote the correct section? Here's an easier version of the text to use: https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop1 – Colin Pace Dec 16 '23 at 05:41
  • The text I referred you to reads "[...]the weight of either sphere towards the other will be reciprocally as the square of the distance between their centers". Newton uses the word weight to mean mass. – nwr Dec 16 '23 at 16:45
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    Thanks for the quotation, nwr! Please excuse my ignorance, but I don't understand how a reciprocal mass as the square of the distance between centers is the same as the product of two two masses divided by the square of the distance between their centers.

    Please could you explain how one gets from the quote you gave above to the famous equation, F = G(m1m2/r^2)?

     – Colin Pace Dec 16 '23 at 17:24
  • Is the reciprocal the operation of division? And maybe Newton finds the product of the two masses in examples, so the multiplication operation is implied? Same with the gravitational constant… – Colin Pace Dec 16 '23 at 18:21
  • @nwr I thought "weight" meant force in this context, in modern terms. E.g. "the weights of bodies...are proportional to the quantities of matter" (Prop. VI, Bk. III). – Michael E2 Dec 16 '23 at 18:46
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    I think part of an answer is that at some point, math/science people stopped talking in terms of proportions and started using equations with constants instead. I'm not sure when that happened. I remember being taught proportions as if they were very important, including terms like "inversely", "reciprocally", and "duplicate"; and then never using them again in my life, except to read old math and science books. (Even in similar triangles, "proportional" means equations, and I dealt only with the equations.) – Michael E2 Dec 16 '23 at 18:52
  • Thanks for the comment! – Colin Pace Dec 16 '23 at 18:56
  • I think it needs to be understood in the context of the preceding propositions together with the fact that Newton is writing in Latin and in the style of the times, as pointed out by user @MichaelE2. A style unfamiliar to us today but more familiar to subsequent generations of physicists and mathematicians who formalized Newton's work into the equations we read today. – nwr Dec 16 '23 at 20:42
  • @MichaelE2 I'm sure that you are right, but I understood it in terms of a previous HSM question. – nwr Dec 16 '23 at 20:45
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    Thanks for the comment! I am deeply appreciative of your efforts to instruct me, but unfortunately, I still feel as if I need to understand how the famous equation is found in the text (more than an allusion to the fact that it’s in the propositions). Please, if anyone else understands this problem, let me know in the comments or answer sections. – Colin Pace Dec 16 '23 at 21:02
  • @ColinPace : you'll find the old practices about proportions versus equations somewhat explained in this answer : https://hsm.stackexchange.com/questions/7288/who-first-derived-a-v2-r/7289#7289 . The statements of proportion implicitly admit that there will be some constant of proportionality, but they would often be difficult or impossible to determine numerically by the techniques available in the 17th century. Use of proportions avoids having to handle their uncertain measures explicitly. The numerical value is the only part of the modern expression not in 'Principia'. – terry-s Dec 19 '23 at 19:35

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The language in Newton’s (translated) work is antiquated by modern standards. Saying that the force is “reciprocally as the square of the distance” means, in more modern language, that the force is inversely proportional to the square of the distance. In equation form,

$$ F = \frac{c}{r^2} $$ where $c$ is a quantity which does not depend on distance.

(Prop. VII, Book III) That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.

That all the planets mutually gravitate one towards another, we have proved before ; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by Prop. LXIX, Book I, and its Corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain. Moreover, since all the parts of any planet A gravitate towards any other planet B ; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole ; and (by Law III) to every action corresponds an equal re-action ; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A ; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D. COR, 1. Therefore the force of gravity towards any whole planet arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this ; for all at traction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet, as formed of a number of lesser planets, meeting together in one globe ; for hence it would appear that the force of the whole must arise from the forces of the component parts. If it is objected, that, ac cording to this law, all bodies with us must mutually gravitate one to wards another, whereas no such gravitation any where appears, I answer, that since the gravitation towards these bodies is to the gravitation towards the whole earth as these bodies are to the whole earth, the gravitation to wards them must be far less than to fall under the observation of our senses. COR. 2. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the parti cles ; as appears from Cor. 3, Prop. LXXIV, Book I.

This lengthy passage says that the gravitational force toward an object $O$ is proportional to the mass of $O$ (“proportional to the matter which [it] contain[s]”). Furthermore, because of the action-reaction law, the force must also be proportional to the mass of the object being attracted to $O$.

As a result, the force of attraction between objects $A$ and $B$ is proportional to $m_A$ and $m_B$ and inversely proportional to $r^2$, which can be expressed in equation form as

$$F = G \frac{m_A m_B}{r^2}$$ where $G$ is some universal constant.

The fact that Newton wrote in such a lengthy and opaque way is a product of his time. The mathematical formalism and terminology which enables us to express Newton’s law of gravitation in a few lines is an apparatus which has been developed in the intervening years.

J. Murray
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  • Thank you, J. Murray, for the clear and detailed answer for the question of where the equation comes from in the prose and how it developed since. – Colin Pace Dec 17 '23 at 20:34
  • Key sentences:
    1. that the gravity tending towards all the planets is proportional to the matter which they contain.
    2. and (by Law III) to every action corresponds an equal re-action ; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A.
    3. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the particles.

    Does Newton mention the constant G in the passage?

     – Colin Pace Dec 17 '23 at 20:37
  • @ColinPace No. Again, even writing the force law in the form we use today is a comparatively modern development. Vector notation, for example, was not fully fleshed out until the 1900’s. – J. Murray Dec 17 '23 at 21:32
  • Encyclopedia Britannica mentions that the gravitational constant was first measured by Henry Cavendish in 1797-1798, more than a century after the publication of Mathematical Principles of Natural Philosophy. It appears Newton might not have been responsible for that aspect of the equation. – Colin Pace Dec 17 '23 at 23:40
  • From the Wikipedia entry 'Newtonsches Gravitationsgesetz' https://de.wikipedia.org/wiki/Newtonsches_Gravitationsgesetz#cite_note-4: [img]https://i.imgur.com/mZ9Atqv.png[/img] – xxavier Dec 18 '23 at 11:46
  • Thanks for the comment, @xxavier! I don't read German, but I see the force proportionality equations with m1, m2, and r^-2, so that might be another restatement of where the equation comes from in the prose. Do you have more insight into the image? I don't see the gravitational constant there, so it's still a mystery where it comes from. – Colin Pace Dec 18 '23 at 12:35
  • @ColinPace My translation: In today's usual explicit form, the law of gravitation wasn't formulated by Newton himself, but only in 1873, so 200 years later, by Alfred Cornu and Jean Baptistin Balle. Before that, Newton's law was used in its original form F prop, to m1, F prop. to m2, F prop. to r^-2, and without the definition of a 'gravitational constant'. In the German Wikipedia entry, you can find the two links [3] and [4]. – xxavier Dec 18 '23 at 12:43
  • Thanks for the translation, @xxavier! That's a fascinating idea. – Colin Pace Dec 18 '23 at 13:09
  • @ColinPace : you'll find the old practices about proportions versus equations somewhat explained in this answer : https://hsm.stackexchange.com/questions/7288/who-first-derived-a-v2-r/7289#7289 . The statements of proportion implicitly admit that there will be some constant of proportionality, but they would often be difficult or impossible to determine numerically by the techniques available in the 17th century. Use of proportions avoids having to handle their uncertain measures explicitly. The numerical value is the only part of the modern expression not in 'Principia'. – terry-s Dec 19 '23 at 19:34
  • @terry-s, thanks for the comment and the link! That's interesting information about the use of proportions. Given the difficulty of numerical techniques in the 17th century, please, do you know if Newton demonstrates the proportions of gravity between planets in "Principia" or does he only mention the theory? – Colin Pace Dec 19 '23 at 22:23
  • @ColinPace : not quite sure what you ask here, but you can find the masses of jupiter, saturn and earth relative to the sun, in Principia Bk 3 Prop 8 cor.2 . The method of calculation, explained in earlier sections, depends on their having satellite(s) (so no result for Venus or Mars). The result for the earth was of poor accuracy because moon's distance was known only by its parallax and the sun's v small parallax was hard to measure. Discussed in https://astronomy.stackexchange.com/questions/26024/how-does-a-solar-transit-give-more-accurate-determination-of-earth-to-sun-distan/26031#26031 – terry-s Dec 20 '23 at 23:51
  • @terry-s, thanks again for the comment and the link! I apologize about the unclear question. This quote might help clarify what I'm asking: “Isaac Newton showed in 1687 that relationships like Kepler’s would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.” So, I’m asking about where this demonstration is and how he does it. (en.m.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion) – Colin Pace Dec 21 '23 at 03:26
  • @ColinPace : thanks for your latest comment. Your latest question about relations between Newton's laws and Kepler's relationships is quite different than your original question. It would need to be put in a separate new question here, and I suggest you raise it as a new question. Hint, wikipedia 'kepler's laws' article is not correct on this. Newton's Principia of 1687 showed (Bk1 sec.11 props 57-69, esp 66) that in a system with >2 masses, deviations from keplerian motion would occur, though they might be rather small if one of the bodies was of much greater mass than the others. – terry-s Dec 22 '23 at 02:49
  • @terry-s, thanks for the comment! Here's the new question: https://hsm.stackexchange.com/questions/16039/what-is-the-relationship-between-the-newtonian-laws-of-motion-and-gravity-and-th. – Colin Pace Dec 22 '23 at 17:24