Everything with mass attracts everything else. The size of that force is , where is the gravitational constant, the same value at all times and in all places.
That it is proportional to the product of the two masses is straightforward. The denominator is what needs care: the distance enters squared. Twice as far apart makes the force ; three times as far makes it . It is not inversely proportional to the distance.
The figure moves the planet nearer and farther. The arrow should look as though it grows and shrinks far more violently than the distance changes. That is what entering squared means.
The force has neither an upper nor a lower bound. However far apart, it never becomes , and wherever you are in the universe every heavenly body is pulling on you. But the pull of distant things falls off as the square, so only the massive bodies nearby have any real effect.
Notice that and enter the equation in the same way. There is no distinction between the one that pulls and the one that is pulled. The two attract each other with forces of equal size. This is the law of action and reaction itself.
The force with which the Sun pulls the Earth equals the force with which the Earth pulls the Sun. An apple pulls the Earth just as hard as the Earth pulls the apple. The two arrows in the figure stay the same length however the distance is changed.
That the Sun and the apple nevertheless move so differently is not because the forces differ. It is because the masses differ. The equation of motion gives : under the same force, the accelerations separate by exactly the inverse ratio of the masses.
As an apple falls, the Earth moves toward the apple as well, but the mass of the Earth is larger by an enormous factor, so its acceleration is far too small to measure. It is not that it does not move; it is that the motion cannot be seen.
Gravitation is not a matter of the sky alone. An object released from the hand falls under one and the same force.
Take an object of mass at the surface. With the mass of the Earth and its radius , the gravitational force on it is . That same force is also the weight . These are one thing written two ways, so they may be set equal: .
The cancels on both sides, leaving . The mass of the falling object has gone from the equation. This is why heavy and light objects fall with the same acceleration.
It also means that is a property of the Earth, not of the falling object. The more massive the body and the smaller its radius, the larger its . Things fall slowly on the Moon because the Moon is light.
So why does the Moon not fall down? It is not that it does not fall. It is falling all the time.
Newton drew a picture of it. Fire a ball horizontally from the top of a high mountain. Fire it weakly and it lands close by; the harder you fire it the farther off it lands. Here the roundness of the ground begins to tell. While the ball falls, the ground falls away too, so the landing point keeps retreating.
What happens if it is fired fast enough? The curving of the fall and the curving away of the ground come into exact balance, and the ball never reaches the ground at all. That is a circular orbit.
The ball is not travelling straight, and it has not escaped the pull. It is falling the whole time. The Moon and an artificial satellite are the same: they trace circles because they keep falling. That the fall of an apple and the motion of the Moon can be written with one and the same law is the content of the law of universal gravitation.