From his analysis of balls (or apples) that fall toward the earth Newton went on to consider gravitation in wider terms. His line of thought is demonstrated by a very interesting discussion in his Principia. Suppose, he said, we shoot a bullet horizontally from the top of a mountain so high that it rises above the atmosphere. The bullet will follow a curved trajectory and hit the surface of the earth some distance away from the base of the mountain. The greater the muzzle velocity, the farther away from the mountain the bullet will land. At a sufficiently high initial velocity the bullet will come to earth at a point directly opposite the mountain; at still higher velocity it will never hit the ground but will continue to revolve around the earth like a little moon. If, Newton argued, it is possible in this way to make an artificial satellite, why not assume that the motion of the natural moon is also a free fall? And if the moon revolves around the earth because of the earth's gravitational attraction, is it not logical to assume that the earth itself is held in orbit around the sun by the force of the sun's gravity? Then is this not also true for all the other planets and their satellites? So originated the profoundly important idea of universal gravitation, which states that all material bodies in the universe attract one another with forces determined by their masses and mutual distances.
To establish the exact relation of force to mass and distance, Newton began by assuming that, since the force between the earth and each body near its surface is proportional to the inertial mass of the body, the force should also be proportional to the inertial mass of the earth. This immediately explained why the gravitational attraction between bodies of small mass, such as two apples, had never been noticed. It was too weak. Not until half a century after Newton's death was the existence of such a force demonstrated experimentally by another British genius, Henry Cavendish.
Having postulated that the gravitational attraction between two bodies is proportional to the product of their masses, Newton then investigated the dependence on distance. He compared the force necessary to hold the moon in its orbit at the distance of 60 earth radii with the force on an apple at the distance of only one radius from the center of the earth. It is important to realize here that the great difference in mass between the two bodies does not affect the validity of the comparison. As a matter of fact, an apple placed at the m oon's distance and given its orbital velocity will move around the earth exactly as the moon does; by the same token, if one could suspend the moon from a branch, it would fall to the ground exactly as fast as apples do. Newton's mathematical analysis showed that the force of gravity decreases as the square of the distances between the attracting bodies.
He could now write the formula for gravitational force: F = G (M1 M2)/d2• G is the constant of proportionality, or the gravitational constant. It is a very small number; if the masses are measured in grams and the distance in centimeters, G is approximately .000000066. This means that a pair of one-gram weights separated by one centimeter attract each other with a force a little more than six hundred-millionths of a dyne, or about six hundred-billionths of the weight of a gram.
Combining the law of gravitation with his laws of motion, Newton was able to derive mathematically the rules governing planetary motion that had been discovered by Johannes Kepler. In the memorable era that followed, Newton and his successors explained the motions of celestial bodies down to the most minute details. But the nature of gravitational interaction, and in particular the reason for the mysterious proportionality between gravitational mass and inertial mass, remained completely hidden for more than 200 years.