How did Newton make use of Kepler’s laws to deduce the law of gravitational force? An important thing to understand first is why Newton was convinced that a mechanical explanation of planetary motion was needed. For Aristotle and even for Copernicus more than eighteen centuries later, circular motion was the “natural” motion in the heavens. It was the way stars or planets move when left to themselves (and obviously humans had no opportunity to do otherwise than leave them to themselves). It required no further explanation. Kepler and his English contemporary William Gilbert were perhaps the first to think seriously about a force to explain planetary motion. Both imagined that the force might be magnetic. But their speculations could not lead far, for neither understood the principle of inertia, that an undisturbed object continues to move uniformly in a straight line. They imagined that a force should act along the planetary orbit, to impel the planet continually along its path through space. Galileo’s attitude toward planetary motion was most interesting, for while he took giant strides forward in understanding mechanical principles and in understanding gravity, he reverted to a view of natural circular motion scarcely more sophisticated than Aristotle’s. To a full understanding of inertial motion, now expressed by Newton’s first law, Galileo came perilously close. But he did not reach it. He was the first to state that undisturbed motion (“natural” motion) is motion in a straight line with constant velocity. This indeed sounds like Newton’s first law. However, Galileo believed this law to be limited in validity to the human-sized domain. He pointed out that an object set sliding on an enormous frictionless plane in contact with the Earth at one point would not continue indefinitely at constant velocity. As it moved farther from the center of the Earth it would slow down; what started out as horizontal motion would become uphill motion. On the other hand, on an imaginary perfectly smooth sphere surrounding the Earth, an object set sliding would continue at constant speed in a great circle around the earth. Because he could not free himself mentally from the shackles of the Earth’s gravitational field, Galileo failed to extend the principle of inertia to the heavens. He succeeded only in advancing new arguments for the ancient view that natural celestial motion is circular motion. Perhaps if Galileo had accepted Kepler’s ellipses, he would have reached different views about inertial motion in the cosmos.
Newton’s imagination was able to cut loose from the bonds of earthly gravity that had held back Galileo. He unhesitatingly extended Galileo’s principle of inertial motion on to infinity and stated it as his first law of mechanics. This was a necessary first step to the discovery of the law of universal gravitation. If the “natural” motion of a planet in the absence of force is straight-line motion, the actual orbital motion requires an explanation. A force must be acting to deflect the planet from its otherwise straight course into its curved path around the Sun.
In what follows, I trace a line of reasoning from Kepler’s laws to the law of gravitational force. This should be taken as plausible reconstruction of Newton’s work, not as authoritative history. There is some evidence about the early development of Newton’s ideas in his later correspondence, but Newton himself may have been rewriting the events. Insights in science are much more likely to occur in a patchwork of intuition, deduction, guesswork, and calculation than in any logical orderly chain. Not the least important element is the preconception, the belief that the solution must be found along a certain path.
(1) Kepler’s Second Law and the Direction of the Force
Before the time of Newton, the kinematics of uniform circular motion was not understood. The essential facts are that the acceleration is directed inward toward the center of the circle and has a constant magnitude given by
where v is the speed of the object and r is the radius of the circle. These facts about uniform circular motion were first published by Christian Huygens in 1673, but they were probably known to Newton in 1666. They mean that if an object moves uniformly in a circle, it must experience a force directed toward the center of the circle. (Here I use the proportionality of force to acceleration.) This conclusion was all-important to the progress in astronomy, for it suggested that the planets move as they do because of a force pulling them toward the Sun. Newton put this reasonable guess on a firm footing by proving a theorem not just about uniform circular motion, but about any orbital motion at all that is executed in a plane. As Newton proved, an object acted upon by a central force (that is, a force directed toward a fixed center) obeys Kepler’s second law (the law of areas). In modern terminology, we can say that because it experiences no torque with respect to the center of force, it has constant angular momentum with respect to this point, and constant angular momentum in turn implies that the radial line is sweeping out area at a constant rate. To put it in planetary terms: If the Sun exerts a central force on a planet, the planet moves in such a way as to satisfy Kepler’s second law. This is the theorem that Newton proved, along with its converse: If a line drawn from the Sun to an accelerated planet sweeps out area at a constant rate, the force acting on the planet must be directed toward the Sun. The essential point is the connection between the central force and Kepler’s second law. Using only one of Kepler’s laws of planetary motion, Newton could prove the vastly important result that the planets are all acted upon by a central force directed toward the Sun.
(2) Kepler’s Third Law and the Dependence on Distance and Mass
Kepler’s second law (together with Newton’s second law) revealed the direction of the gravitational force, but no other property of the force. The next problem was: How does the strength of the sun’s gravitational pull vary as the distance from the Sun varies? There were two ways to get at this problem, both of which Newton employed. The first, my concern in this subsection, was to compare the motion of different planets at different distances from the Sun. The second, discussed in the next subsection, was to study the motion of a single planet as its distance from the sun varies during the course of its orbit. For comparing the motion of different planets, Newton had at hand Kepler’s third law, stating that the squares of the periods of the planets are proportional to the cubes of the semi-major axes of their elliptical orbits. Introducing a constant of proportionality K, we can write this law in the form
Tp2 = Kap3 .
The subscript p denotes any particular planet. The proportionality constant K carries no such subscript since it is the same for all the planets.
Now let’s make one very important approximation, which Newton undoubtedly made in his earliest calculations. We take advantage of the fact that none of the planetary ellipses differs greatly from a circle, and pretend that the planets all move precisely in circles of radius rp. Kepler’s third law for these circles may be written
Tp2 = Krp3 .
This approximation enables us to carry out a simple and significant mathematical derivation. Each planet, moving approximately at constant speed vp in its approximate circle of radius rp, has inward acceleration a (not to be confused with the distance ap in the equation above) given by
In order to take advantage of Kepler’s third law, we replace the speed vp in this expression by the orbital circumference divided by the period:
(Speed equals distance divided by time.) The formula for the planet’s acceleration then becomes
This is so far a kinematic statement about uniform circular motion in general. Kepler’s third law, on the other hand, is an observational fact about the planets. Substituting the expression for Kepler’s third law, Tp2 = Krp3, into the denominator of the equation just above gives a formula for planetary acceleration:
The combination 4π2/K is a constant; the quantities a and rp are variables. This equation states that the acceleration of a planet toward the Sun is inversely proportional to the square of its distance from the Sun. Multiplication of the acceleration of a planet by its mass gives the force acting on it:
Compare this with the law of gravitational force,
From Kepler’s third law one gets a major part of the final form of the law of force: The gravitational force experienced by a planet is proportional to its mass and inversely proportional to the square of its distance from the sun. All that is missing is the proportionality of this force to the mass of the Sun.
Two comments about the result just achieved. First, recall that the derivation is based on the approximation of circular motion. Therefore we cannot be certain that the result is exactly correct. It turns out, remarkably enough, that the result is unchanged for elliptical orbits. Newton was able to demonstrate the connection between Kepler’s third law and the inverse square radial dependence of the acceleration for the true elliptical orbits, not just for the circular approximation.
The second comment is more subtle. It appears that we have proved—at least in circular approximation—that the Sun’s force on a planet is proportional to the planet’s mass and inversely proportional to the square of its distance from the Sun. This is not quite true. Suppose, for example, that the mass of a planet were proportional to its distance from the sun. Then the combination mp/rp2 would also be proportional to 1/rp. For this hypothetical situation, it would be equally true to say that the force varies inversely with the distance and not at all with the mass. Newton probably assumed that there is no particular connection between planetary mass and distance (as indeed there is not), so that the form mp/rp2 is not masking some different radial dependence. The clinching evidence for the direct proportionality of gravitational force to mass (and hence for the correctness of the equation above) comes from the idea of the universality of the gravitational force. Near the surface of the Earth, every object experiences a downward force proportional to its own mass, F = mg. Newton assumed this dependence on mass to be a universal feature of gravitation, not limited to gravity on Earth. The equation for the force exerted by the Sun on the planets confirms and strengthens this assumption. Still another bit of evidence on the same subject appears in the next subsection.
(3) Kepler’s First Law: More Evidence on the Radial Dependence
The idea of an inverse square law of gravitational force acting on the planets was in the air around Newton’s time. His genius lay not so much in thinking of it as in demonstrating its validity mathematically and weaving it into a coherent theory of universal gravitation. The really crucial test of the inverse square law is provided by Kepler’s first law, the statement that planets move in elliptical orbits with the Sun at one focus. In any orbit that is not a circle, a planet periodically alters its distance from the Sun, sampling stronger and weaker regions of the Sun’s gravitational field as it moves around. The precise form of its orbit, therefore, depends on the law of force, on exactly how the force weakens as the distance increases. In London in 1684, Edmund Halley, Robert Hooke, and Sir Christopher Wren worked at the problem of connecting the elliptical orbits of the planets to the law of force emanating from the Sun. Although they believed in the inverse square law, they failed to connect it to Kepler’s first law of elliptical motion. Finally Halley journeyed up to Cambridge to ask Newton about the problem. Here is an account of their meeting written soon afterward by John Conduitt. “Without mentioning either his own speculations, or those of Hooke and Wren, he [Halley] at once indicated the object of the visit by asking Newton what would be the curve described by the planets on the supposition that gravity diminished as the square of the distance. Newton immediately answered, an Ellipse. Struck with joy and amazement, Halley asked him how he knew it? ‘Why,’ replied he, ‘I have calculated it.’ ” Three years later, in 1687, under the auspices of Halley, Newton’s monumental Principia was published.
1 Law 1: The planetary orbits are ellipses. Law 2: A line drawn from the Sun to a planet “sweeps out” area at a constant rate. Law 3: The squares of the periods of the planets are proportional to the cubes of their average distances from the Sun.