newtonian mechanics - How did Newton figure out the law of gravity?

How did Newton figure out the law of gravity? I know there are two other questions answering the same question but I did not understand anything from the answers

Is there a third way of knowing how he did it without using complex formula and derivation.

I am dumb but curious :-)

Answer

To answer the question in your title, he used his newly found fluxions (calculus) to prove that Kepler's laws of planetary motion imply a radial, inverse square law.

Feynman's Lost Lecture is a mixture both of Feynman's attempts to give the simplest possible explanation of how one goes about this derivation and his insights into the history of how Newton did it grounded on Feynman's reading of Newton's Principia. Feynman's treatment is about as simple as you can get, but, as Feynman warns, it's still "God damned hard, there's no question of that." But I'd still recommend this work to you.

In summary:

1. 
   

   Kepler's equal area law implies that the force on a celestial body must always be directed towards the Sun;

   
   



2. 
   

   Feynman "reconstructs" Newton's proof that Kepler's first law - that the body's path is an ellipse with Sun at one focus - can be derived from the equal area law and the law that the period $T$ is proportional to the semimajor axis length $L$ to the power of $3/2$, whence

   
   


3. 
   

   One can derive the inverse square law from Kepler's laws as well.

   
   

In my opinion, though, at least as impressive as Newton's analysis of the problem was his unifying realization that the same force that accounted for falling apples might account for the motion of celestial bodies. Although its thorny at first, the maths is the easy part. Insights like this unifying one are in comparison dazzling. Apparently the story of the apple is not altogether apocryphal: Newton seemed to have followed a thought experiment along the lines of "what if this force reached to the top of the highest apple trees, so tall that they might reach the Moon". I explain this in more detail in my answer here. One of Newton's first biographers, William Stukeley, recorded a conversation with Newton in "Memoirs of Sir Isaac Newton's Life". Newton told the apple story to Stukeley, who relayed it as such:

"After dinner, the weather being warm, we went into the garden and drank thea, under the shade of some apple trees...he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself..."

and, in Henry Pemberton's 1728 "A View of Sir Isaac Newton's Philosophy" we find the following passage:

"....that as this power [force of gravity] is not found diminished at the remotest distance from the center of the Earth, to which we can rise, neither at the tops of the loftiest buildings nor ... the highest mountains ... it appeared to him [Newton] reasonable to conclude that this power must extend much farther than was usually thought; why not as high as the moon, said he to himself?"