Probably many another man since Anaxagoras had asked the same question, but assuredly Newton was the first man to find an answer.
The thought that suggested itself to Newton's mind was this: If we make a diagram illustrating the orbital course of the moon for any given period, say one minute, we shall find that the course of the moon departs from a straight line during that period by a measurable distance--that: is to say, the moon has been virtually pulled towards the earth by an amount that is represented by the difference between its actual position at the end of the minute under observation and the position it would occupy had its course been tangential, as, according to the first law of motion, it must have been had not some force deflected it towards the earth.

Measuring the deflection in question--which is equivalent to the so-called versed sine of the arc traversed--we have a basis for determining the strength of the deflecting force.

Newton constructed such a diagram, and, measuring the amount of the moon's departure from a tangential rectilinear course in one minute, determined this to be, by his calculation, thirteen feet.
Obviously, then, the force acting upon the moon is one that would cause that body to fall towards the earth to the distance of thirteen feet in the first minute of its fall.

Would such be the force of gravitation acting at the distance of the moon if the power of gravitation varies inversely as the square of the distance?
That was the tangible form in which the problem presented itself to Newton.

The mathematical solution of the problem was simple enough.