Before Newton, the planets' circular motion was regarded as natural. But Newton, following Galileo, believed the only natural motion—no external force acting—was steady speed in a straight line. A change in direction meant a change in momentum, and that meant an external force. Circular motion, with a continuously changing direction, meant a continually acting force. Newton visualized this by beginning with motion around a polygon, that is, straight-line motion along a side, then an impulsive kick at each corner, provided by bouncing off a circular container.
To the top right above we show the momentum vector of the body, it's steady for motion along a side, then the impulse adds a finite component at each corner.
Circular motion is the limiting case: a polygon with an infinite number of sides. The infinite number of small impulses become a constant inward force, generating smooth circular motion. This constant force is of course gravity for a planet, or string tension for twirling a weight around on a string.
This is all discussed in more detail in my lecture.
The approach is of course reminiscent of Newton's great mathematical invention—the calculus.
Code by Atallah Hezbor