Hmm, I think we are seeing it differently :) I was just using it as an idea for how to construct different shells, building up a mass. And there I used the arrows for getting my poles to construct my shells on. You shouldn't take it as the theorem itself. Newton first decided to treat Earth as an evenly distributed mass, and that decision he made looking at the direction of falling objects. They all seemed to point to the middle of the Earth, which then gave him the direction to work from.

Then he thought of earth as one thin shell only, and decided that, if you looked at a 'moon' close to this thin Earth-shell, the Earth's gravitational attraction on the moon had to be the sum of all vectors that could bear on it. Right under the moon the Earth-moon attraction would be strongest, and the further out to the 'rim' of our shell-Earth you went, as seen from that moon (in 2-D, sort of), the weaker that attraction should be.

Thinking like that he created an equation stating that the sum of all those vectors had to be equal to if this shell-earth had all its mass placed at/in the center of itself. And if that moon had been inside the shell all those vectors would cancel themselves out leaving no measurable 'gravity'. And that's because the shells 'gravity vectors' would now work on that moon in the middle from all 'sides/points' of that Earth-shell, if we think of how he defined those first vectors.

I was wrong thinking of it as a twofold arrow. To him all mass of the Earth could be seen as laying in that center, meaning that if you dug a shaft into the earth you could ignore the mass above you as you went down. After all, according to his idea the mass would behave as if it was concentrated at that center of the earth.

Now we take my idea of all those shells again, and define each one as if all their 'mass' all was placed in the center. Phieew, I should have looked at it before telling you how I thought of it.. I actually though I knew it :) But its a long time since I looked at that one.

But you can look at it this way too, when you dig your way down the 'forces' of the gravitation coming from the sides, acting at you, take themselves out as they meet at your position, so the the force left is the one pointing down, possibly?

I'm not even sure if that's correct actually :) But it seems like it would fit this idea.

He was one very smart guy.

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But I think you can use the arrows and a twofold direction too? You just need to see where they meet the moon, there you will have the 'gravity arrows' working both directions 'attracting'. That is, if you imagine that first 'point' from where they go out to be the center of the earth. I'm not sure but I think it would be the same, as if you think of the arrows hitting the moon and then move the moon closer and closer to that point until they merge. In the merge there can't be any arrows pointing anywhere.

Does that make any sense? You would need to have a decided equilibrium between the moon and the Earth though, and then move that as a lever, reaching a maximum on the surface to then thin off, getting into a 'negative phase' as it continues past? Nah, it doesn't make sense. How would one get to that idea from my arrows?

Hah, Maybe you could, that is if you defined gravity as needing something to act on.. Then you could suspect that as the objects merged there wouldn't be anything to ... Naaah :)

But your concept of the arrows getting thicker as they reach the surface to then get thinner as gravity weaken again makes sense. The only problem being that we know it does so :)

Newton reasoned his way without knowing.

I'm getting jealous here :)