I remember seeing Newton's force equation written in a 19th century textbook as $F=(M+m)/R^2$ (instead of $Mm$). I don't remember if they used $G$. Do you know a collection of 19th century physics textbooks available online? I'm trying to locate when physicists switched to multiplying two masses which makes no physical sense.

  • $\begingroup$ You can find many old books scanned on archive.org. But you would have to know which books to look for. $\endgroup$ Jun 20 at 14:41
  • $\begingroup$ @MichaelBächtold Thanks. I see that there is the option of searching date ranges. I tried "newton" between 1800 and 1900 but could not find any text with the force equation. $\endgroup$
    – zeynel
    Jun 20 at 18:46
  • $\begingroup$ This question is could benefit from further context. As it is, it remains unclear how one would be able to find old books with an equation that differs from the original, is false and is not popular. I can post a question to look for any (false) equation but it is too much effort of a task unless more context or proof of its existence is provided. $\endgroup$
    – Mauricio
    Jun 21 at 12:34
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Danu
    Jun 27 at 8:41

1 Answer 1


This question asks for book-references to what it calls "Newton's force equation", written in a certain way (involving the sum of two masses).

Normally, an answer could just be book-references. But here, the question also provides a context: "I'm trying to locate when physicists switched to multiplying two masses which makes no physical sense."

This context modifies the question rather heavily. As is well-known, the accepted law of gravitational attractive force between two bodies with masses $M$ and $m$ (usually formulated as $G M m / r^2$ ), involves two masses as multiplying factors. This is what the question declares to make no physical sense. So it seems clear that the questioner has some kind of problem with the law of gravitation itself. It seems unlikely that there are books that could be found that would conform to this point of view.

Further context emerges in a comment by the questioner, citing I B Cohen's 1999 translation of Newton's Book 3 Prop.7: "Gravity exists in all bodies universally and is proportional to the quantity of matter in each." The questioner goes on to say "I don't see any reference to multiplication here". But proportions are notoriously known to involve multiplication.

In principle, no accepted physical law is exempt and immune from criticism and replacement. But when an accepted law is supported by many and classic observations, then a criticism, to be credible, does have to account for those observations at least as well as the criticised theory did.

In the circumstances of the present question, it may therefore help to offer two things (and it seems difficult to go beyond),

(i) brief explanation how it comes about that results of the laws of motion and gravitation involve (without inconsistency) relations of both kinds, i.e. some involving the product $M m$, and some the sum $M+m$, of two masses; and

(ii) an indication of some of the (many) incompatibilities with observations, including the classic observations that supported the laws in the first place, that would result from the alteration that the questioner seems to want to make. (Apparently, questioner wishes to substitute the sum $M+m$ for the product $M m$ in the statement of the law of gravitation).

(i) The accepted law of the gravitational attractive force between two point-masses $M_1$ and $m_2$ at distance $d$ is as well-known, $$ F = G M_1 m_2 / d^2 .$$ Then the second law of motion, often written nowadays as $$ F = m a ,$$ describes how an acceleration $a$ and mass $m$ together relate to the force $F$ that induces the acceleration. From the second law it follows that acceleration here is force/mass , $$ a = F/m .$$ In the case of the two bodies already mentioned, the gravitational acceleration on mass $M_1$ is $$ F/M_1 = G m_2 / d^2 ,$$ and the acceleration on mass $m_2$ is $$ F/m_2 = G M_1 / d^2 .$$ But each of these two accelerations tends to bring the masses towards each other. So the total relative acceleration is the sum of the two accelerations, $$ G (M_1 + m_2) / d^2 .$$ There is no incompatibility between such an expression for the total relative acceleration involving the sum of the masses and the expression for gravitional force involving the product.

A more detailed derivation of similar kind can be seen in A E Roy 'Orbital Motion', esp. at p.62-65. (The law of gravitation appears in its usual form in section 4.3, and the derived equation showing the relative acceleration involving the sum of the masses at eqns. 4.10-4.12.)

It seems likely that the formula that the questioner remembers seeing, involving the sum of the masses, was a formula that related to the accelerations, or otherwise to the motions, and not to the forces themselves.

Nothing essential is altered if instead of acceleration one considers a finite change in the velocity or in the momentum of the bodies: the resulting relation still involves the sum of the masses, but it also derives, as shown above, from the gravitational force-relation that involves the product of the masses.

Books that reflect the point tend to submerge it under masses of detail in the equations. But Newton did deal with the point fairly straightforwardly in Book 1 of the 'Principia', in Propositions 57-59. One only needs a bit of patience and understanding for the archaic material and forms of expression, basically over 300 years old in the original. (The questioner already quoted from the (very good) Cohen modern translation from 1999, so it may be that he has easy access to this source for propositions 1.57-59 also.)

(ii) Anyone who wishes to deny the proportionality of force to two different masses in the law of gravitation is left with the problem of accounting for the equality of descending accelerations found in Galileo's classic experiments in dropping weights off a tower and many similar observations. If the earth's gravitational force on an attracted falling body had no proportionality to its mass, then bodies of different mass would have to accelerate in their fall at different rates, light faster than heavy, which is not what Galileo or anybody else found. (Complementary attractions of the falling bodies on the earth itself are negligibly tiny, as small as the relations of mass between such bodies and the whole earth itself.)

If reconciliation is sought instead by rejecting the second law of motion as well, then the lack of the second law generates other incompatibilities with the everyday realities that it reflects -- for example, that a force or shove against a very heavy object produces little effect in motion, where the same force against a lighter object produces more motion.

If it is the proportionality of gravitational force to the earth's mass that is put in question, then we can't experiementally change the earth's mass for something else, and astronomical rather than terrestrial observations have to be called in aid to settle such questions. The phenomena include Kepler's closely approximate 3/2 power relation between distances and periods of the planetary motions around the sun, and the same power relation for the distances and periods of Jupiter's major satellaites around Jupiter. But the coefficient of the power relation for the motions around the sun is different from the coefficient of the power relation for the motions around Jupiter, reflecting the different attracting masses of Jupiter and the Sun. If there were no proportionailty of gravitation to the mass of the attracting body, Jupiter's satellite system and the main solar system would have to show very different relations to each other in their relative speed/period versus distance, than what is actually observed. (The point was first handled in 'Principia' Book 3, Prop.8.)

It is highly doubtful whether any amendment of the laws of motion and gravitation could be achieved without entailing false predictions of many effects that just do not happen. Arguments about multiplying apples are beside the point because they exist in words and rhetoric only, with no connection to physical situations relevant to the problem that has been posed here. Credible answers to questions about whether proportions are physically meaningful are not couched in rhetoric, but in terms of correctly measured physical observations and correct mathematics applied to the observations.


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