At some point Newton also realized that the orbital motion of celestial bodies was very similar to the continuous and compounded falling motion of a projectile near the surface of the Earth. The greater the lateral force applied to the projectile, the greater its inertial motion, and the longer it takes for the projectile to fall toward Earth. If the initial lateral force was great enough, if there was no resistance of the air, and if the distance of the projectile from Earth was just right, then a priori the projectile should inertially fall around the curved Earth forever. Voilà! Newton finally realized that the Moon was perpetually falling around the curved Earth at velocity v (Figure 15).
Toward the end of 1684, Newton prepared a paper known as Du Moto (on motion), and he lectured on all of the above subjects at Cambridge. In early 1685, Halley encouraged Newton to expand Du Moto and his lectures into a book. The result was Newton’s three volume Principia, written entirely in Latin, which Halley had published in 1687 at his own expense.
During the writing of the Principia, Newton extrapolated his great discoveries to the universe as a whole. In Book 3 of the Principia, entitled The System of the World, Newton described four axioms of logic that should guide a philosopher in the study of natural phenomena, such as gravity. We shall describe these axioms in detail in the next section.
Newton then used these axioms in Book 3 of the Principia to logically demonstrate that the same centripetal force which exists in all bodies and masses:
1. causes all of the planets to orbit around the Sun,
2. causes the planetary moons to orbit around their planets,
3. causes the Moon to orbit around the Earth,
4. causes the moon and the apple to fall toward Earth,
5. holds objects on the surface of the Earth,
6. holds all of the parts of the Earth together,
7. causes the tides on Earth to rise and fall twice each day, and
8. causes pendulum clocks to run slower or faster at different places on Earth.
If all of these things occur, then logically every body in the heavens must mutually attract and be attracted by every other body. From all of the above, Newton ultimately generalized these concepts into his law of Universal Gravitational Attraction. Feynman referred to Newton’s law of Universal Gravitational Attraction as “the greatest generalization achieved by the human mind.”