This is a good example of how physics is the art of casuistry. When you want it you read M and m as “matter” to say that force acts between two pieces of matter. When you need to eliminate m because you must according to the rules of algebra then you read m as “inertial mass” and M as “gravitiation mass”. And you conveniently forget that you first read them as matter.

Your second paragraph is not about orbits, it is another topic.

But you can easily come up with a very simple system that none of the masses cancel out. For example a mass M1 which is free to move up and down attached to another mass M2 with a string (+a pulley) which is free to move in the horizontal direction. Assume there is no friction, then the acceleration (due to gravity) of this system is equal to a = M1*g/(M1+M2). This can be easily measured and the effect of each mass can be identified.

not only the assumption of an instantaneous spread of some effect, but also, more generally, any assumption of the spreading of an effect with a velocity greater than the velocity of light is incompatible with the theory of relativity.

Is this acceptable?

The way I understand it Einstein rejected Newton’s definition of force because he believed that instantaneous action at a distance did not exist in nature and that nothing could move faster than light. Instantaneous action at a distance is occult and Einstein wanted to eliminate this occult quality from physics.

can you please provide a source for this statement? “Einstein correctly thought that Newton’s force was occult and wanted to remove force from physics”

reading with interest,

Netizen12

Obfuscated???? It is the first chapters! Blame of you if you jump over first chapters of every book.

Questioning the first chapters of Euclid’s Elements led to non-Euclidean geometries. I don’t see your point

My oppinion is that Newton hides it because he is not sure that he can assume that the variation of angular momentum in the proofs can be as small as one wants, and he needs such smallness to have a sensible infinitesimal calculus.

Newton hides what? I am saying that Newton hides the fact that Kepler’s Rule is the definition of density. Where do you see any mention of infinitesimal calculus in Newton’s definition 1?

To use infinitesimal calculus to compute orbits is premature optimization; the same computations can be done by using average quantities because we are interested in showing that Newton’s occult force does not enter orbit calculations; we are not interested in computing precise orbits.

Newton does not use the word “momentum”

What “proofs” are you talking about?

Of course, without this assumtion, he had need a more powerful mathematical aparatus, beyond his reach: the Path Integral. Note how tha graphs in section II are amazingly similar to the ones in Feynmannish textbooks.

Why do you need “path integral” to work with the simple proportionality RRR::TT?

There are no graphs in definition 1 and Principia’s section II is a filler. In the Principia Newton makes orbital computation only in propositions III.4, III.8, and I.57-60. We are not interesting in anything else Newton dumpted in the Principia.

Still, density transmits the wrong idea.

Why?

Consider angular momentum, and remember that h is the smallest quantity of angular momentum, but then it is also the smallest are you can sweep in a dt, because such area is one half of x cross p times dt.

Working with calculus notation to describe orbits adds nothing to our understanding of orbits; average values are good enough in this case.

If you want understand Kepler, you need to read about his Wine Barrels.

Kepler discovered Kepler’s Rule; in orbits we are dealing with Kepler’s Rule and I understand Kepler’s Rule. If you want to understand orbits, understand Kepler’s Rule

My oppinion is that Newton hides it because he is not sure that he can assume that the variation of angular momentum in the proofs can be as small as one wants, and he needs such smallness to have a sensible infinitesimal calculus. Of course, without this assumtion, he had need a more powerful mathematical aparatus, beyond his reach: the Path Integral. Note how tha graphs in section II are amazingly similar to the ones in Feynmannish textbooks.

Still, density transmits the wrong idea. Consider angular momentum, and remember that h is the smallest quantity of angular momentum, but then it is also the smallest are you can sweep in a dt, because such area is one half of x cross p times dt.

If you want understand Kepler, you need to read about his Wine Barrels.