# Newton's Law of Gravitation

• Opus_723

#### Opus_723

I've worked out the inverse square law using centripetal acceleration and Kepler's law. But I can't quite see how Newton worked out for sure that the force is directly proportional to both masses. I know it's pretty intuitive, but I thought maybe there was a mathematical way of confirming that assumption. The closest I've gotten is by accounting for the fact that two bodies would orbit about a common center of mass if the force acts on both of them. This gave me F$\propto$mM/(m+M)R^2. So that's pretty close, but now I've got extra stuff in the bottom. I know that if I use Newton's modification of Kepler's Law, the (m+M) in the denominator goes away. And I can see why this modification works, since for the sun and any planet in solar masses (m+M) is essentially 1 and doesn't change the values. But I don't see how he would know for sure what modifications Kepler's Law needed without just assuming his law of gravity and then modifying Kepler's law to fit.

Any suggestions? Did he just assume his law and go from there, or is there a way to derive it?

But I can't quite see how Newton worked out for sure that the force is directly proportional to both masses.
I am currently picking my way forward through the Principia bit by bit, so I may have already run across the answer to this. He explains his reasoning on nearly every point in detail. However, I am not completely sure what you are asking for. Are you asking how he reasoned that the force of gravity is proportional to both masses as opposed to proportional to one of the masses?

I suppose I should just read Principia, but I was trying to do as much on my own as I could. I was wondering how he knew that the force was proportional to Mm, as opposed to anything else, like (Mm)^2, or M+m, or whatever. Did he just assume that it was Mm and then confirm that, or does it fall out of the math somewhere?

It falls out of the observational data, Newton's laws of motion, and his philosophical "rules for doing science" stated at the start of Principia part 3 (which is where the book really starts - the first two parts are mostly developing the math tools needed for part 3).

The "general principles" include
1. The laws of physics are the same for everything in the universe, and
2. You should not invent hypotheses without a good reason (the quote "Hypotheses non fingo", in Latin).

From terrestrial observations (falling bodies, pendulums, etc) the gravitational acceleration acting on body is indepedent of its own mass. So from Newton's laws of motion, the gravitational force on an object must be proportional to its own mass, though from experiments on the surface of the Earth we can't tell how the force depends on the mass of the earth.

From Kepler's law relating the periods of planetary orbits to their distances, the central force must follow an inverse square law, and be proportional to the mass of the planet (unless you make an extra hypothesis that the mass of each planet must necessarily be some function of their distance from the sun, which would violate principle #2).

Newton also had observations of the Galilean satellites orbiting Jupiter, which also satisfied Kepler's laws. So he could argue

1. The force between Jupiter and the sun is proportional to the mass of Jupiter.
2. The force between Jupiter and one of its satellites is proportional to the mass of the satellite.
3. Since both sets of forces give orbits that satisfy Kepler's laws, they are the the same phenomenon.
4. Therefore, the force is proportional to both masses, and the inverse square of the distance.

..."rules for doing science" ...
Do you actually have a translation that renders "Rules of Reasoning in Philosophy" as "rules for doing science"? Hehe.

This question has set me thinking! All this while, I knew that Principia existed. Now, I have the courage to pick it up and read it.

Thanks Opus 723, zoobyshoe and AlephZero!

This question has set me thinking! All this while, I knew that Principia existed. Now, I have the courage to pick it up and read it.

Thanks Opus 723, zoobyshoe and AlephZero!

It has taken me about two months to get 20 pages digested: the language is very difficult! I'm happy if I fathom three sentences a day. On the up side, it's very rewarding to discover how he and his contemporaries reasoned things out from scratch, things we, today, take for granted.

...And I'm just in the 12th grade. Hah! Just torrented the book with high hopes and I'm wondering what just hit me! Thanks for the forewarning!