As the story goes, when Newton was a young man in the mid-1660’s, he began to think about orbital motion and its forces, masses and accelerations. By 1665, Newton was theorizing about the uniform circular motion of a mass (m) moving at velocity (v) around a circle of radius (r). Based on Kepler’s laws, Newton deduced and calculated that there must be an acceleration of such orbiting mass of magnitude v2/r away from the center of such circle, with a centrifugal (outward) force of 1/r2.
Based on his analysis of Kepler’s Third Law of Harmony, Newton also deduced during this early period that the forces which kept the planets in orbit around the Sun must vary as the inverse square of their distances from the Sun (Figures 11 and 12). One dramatic illustration of this phenomenon of “weakening of force with distance,” is that the Sun at a distance of 93 million miles only exerts about one-third as much gravitational force on the Moon as does the Earth from about 240,000 miles, even though the Sun has 330,000 times the mass (gravitational force) of the Earth.
As a young man, Newton also began to think of the force that caused the apple to fall to the ground as “extending to the orb of the Moon.” In an attempt to check this hypothesis, he “compared the force required to keep the Moon in her orb with the force of gravity at the surface of the earth and found them [to agree] pretty nearly.”
By the mid-1670s, Dutch scientist Christian Huygens (1629 – 1695), English scientist Robert Hooke (1635 – 1703), and others, had also deduced from Kepler’s laws that when heavenly bodies (such as the planets) move in an elliptical orbiting motion, the orbiting body is being “attracted” towards the center of another gravitating body (i.e. the Sun) by a force. For example, Hooke wrote:
“All celestial bodies have an attraction or gravitating power towards their own centers, whereby they attract not only their own parts…[but also] all other celestial bodies that are within the sphere of their activity.”
In 1679, Hooke wrote a letter to Newton and referred to his hypothesis that the orbital motions of the planets were actually compound motions comprised “of a direct [straight] motion by the tangent and [an] attractive motion towards the central body.” In January 1680 Hooke again wrote to Newton, this time referring to his “supposition” concerning the force of attraction that keeps planets in their orbits.” Newton later acknowledged that the seminal ideas in Hooke’s letters were the key to his correct analysis of curvilinear orbital motion.
Newton had previously been thinking that the critical force in orbital motion was centrifugal (away from the central body), but after reading Hooke’s letters he realized that the critical force was actually toward the central body, which he later called “centripetal force”’ Newton also realized from Hooke’s letters that orbital motion is really a compounding of two separate motions: the straight inertial motion of the orbiting body which is tangent (transverse) to the central body, and the motion of attraction of the orbiting body towards the central body (Figure 13). As a result, Newton reversed his prior calculations: v2/r now became the magnitude of the acceleration of the orbiting body toward the central body, and 1/r2 became the magnitude of the central body’s centripetal force which pulled the orbiting body toward the center (Figure 14).
By January of 1684, English astronomer Edmund Halley (1656 – 1742), English architect Christopher Wren (1632 – 1723), and Hooke had deduced the inverse square law of gravity, but over the next six months they could not derive Kepler’s elliptical orbits from this hypothesis. So, in August of 1684, Halley traveled to Cambridge for a meeting with Newton. Newton informed Halley that, years before, he had deduced the inverse square law of gravity from Kepler’s third law.
Halley’s visit undoubtedly inspired Newton to continue and complete his work on gravity. Relying heavily on Kepler’s laws, Hooke’s suggestions, and Galileo’s experiments, Newton thereafter demonstrated mathematically that: 1) if a body exhibiting inertial motion constantly accelerates toward a center, the body must move in this orbit around equal areas in equal times; and 2) if this occurs, it must result from a centripetal force which results in an elliptical orbit.