Author’s note: The following is adapted from a chapter of my book in progress, “The Inductive Method in Physics.”
In tribute to his predecessors, Isaac Newton (1643–1727) once wrote: “If I have seen further it is by standing on the shoulders of giants.”1In the first half of the 17th century, two giants stood out above the rest: Galileo Galilei (1564–1642) and Johannes Kepler (1571–1630).
Galileo pioneered the use of experimental method to discover mathematical laws governing the motion of terrestrial bodies. Abstracting from the effects of friction, he proved that free bodies move horizontally with constant speed and fall vertically with constant acceleration. He also demonstrated that these two motions combine independently to produce parabolic trajectories. Both the laws themselves and the method by which they were discovered were revolutionary achievements that opened the door to modern physics. At the same time, Kepler achieved a similar revolution in astronomy by discovering causal laws governing the motion of the planets around the sun. His book, New Astronomy Based on Causes (1609), completed the overthrow of the acausal, descriptive approach to astronomy that had impeded progress for so long.
Furthermore, some preliminary steps had been taken toward integrating the sciences of physics and astronomy. Earth was identified as one of the planets, and the telescope revealed that some celestial bodies have Earth-like characteristics: Our moon has mountains and valleys, Jupiter has moons, and the sun rotates. Nevertheless, at this early stage the connection between the terrestrial and celestial realms was tenuous; Galileo’s laws of horizontal and vertical motion and Kepler’s laws of planetary motion stood apart without any known relationship.
How does one identify fundamental connections between phenomena that seem so radically different—for instance, between an apple falling or a pendulum bob swinging and a planet orbiting in an ellipse? The key was to discover a mathematical theory relating motions to the forces that cause them. This task was extraordinarily ambitious; in addition to the need for crucial new experiments and more accurate astronomical data, it required the development of new concepts and new mathematical methods. When it was finally completed, the modern science of physics had been created—and celestial bodies took their place among its subjects, ruled by its laws.
The Development of Dynamics
Isaac Newton began with a problem that was simple enough to solve, yet complex enough to yield crucial new insights. He began by analyzing the form of motion that the Greeks had regarded as perfect: uniform circular motion. In one sense, it was perfect—it was perfectly suited to expose the errors of Newton’s predecessors and illuminate the principles of a new dynamics.
Galileo had never grasped that bodies move with constant speed in a straight line in the absence of all external forces. Lacking the concept “gravity,” he suggested that horizontal motion at constant speed ultimately meant motion in a circle around Earth, which he thought could occur in the absence of an external force. Kepler, on the other hand, had never grasped that any motion could occur in the absence of a force; he assumed that every motion is the result of an external push in the direction of the motion. In his analysis of circular motion, Newton identified and rejected both of these errors.
Prior to Newton, the case of the moon circling Earth was regarded as entirely different from the case of a hawk circling its prey. Newton, however, ascended to a level of abstraction that treated these two phenomena as the same; his goal was to analyze circular motion as such, and apply what he found to any and all instances of it. His policy here is expressed in the dictum he would later identify as a “rule of reasoning”: “[T]o the same natural effects we must, as far as possible, assign the same causes.”2
A major part of Newton’s motivation for studying circular motion was the planetary orbits, which are nearly circular. But he did not begin his analysis by considering the planets; he began with cases in which the cause of the motion is much easier to identify. He considered a weight attached to the end of a rope and swung around in a circle, and a ball rolling around in a circle inside a bowl. In these cases, what is the cause of the circular motion? For the weight, it is the tension in the rope; the man holding the rope must pull inward. If he lets go, the weight will no longer move in a circle, but will fly off horizontally in a straight line (until the force of gravity pulls it to Earth). For the ball in the bowl, the circular motion is caused by the inward push exerted by the surface of the bowl. If the ball escapes the bowl, then it too will initially fly off in a straight line. In both cases, the uniform circular motion of the body is sustained by a constant force directed toward the center of the circle.
In a notebook, Newton wrote an early version of what later became his first law of motion: “A quantity will always move on in the same straight line (not changing the determination or celerity of its motion) unless some external cause divert it.”3 The external cause is a force, some push or pull.
Newton recognized that it was crucial to distinguish between the type of motion that results from a force and the type that can occur in the absence of force. The concepts of motion used by Galileo were inadequate for this purpose. Galileo’s definition of “constant acceleration” applied only to the case of motion in a constant direction; in other words, acceleration was a scalar quantity that referred only to change of speed. In the case of uniform circular motion, the speed of the body is constant and therefore its “Galilean acceleration” is zero. However, something is essentially the same about the cases of acceleration studied by Galileo and the case of uniform circular motion: In both, a change in the motion results from an applied force on the body. An expanded concept of “acceleration” was needed to integrate these instances.
In order to study and understand the effects of forces, motion had to be characterized in terms of both its magnitude and direction. Thus the concept “velocity” was formed, and “acceleration” was then defined as the rate of change of velocity. Both velocity and acceleration are vector quantities—integrations of magnitude and direction. The formation of these concepts was a revolutionary step that made possible the science of dynamics.
Armed with these concepts, Newton could ask: What is the acceleration of a body that moves with constant speed in a circle? From symmetry, he knew that the acceleration is constant and always directed toward the center of the circle. But what is its magnitude? Newton considered a short time interval in which the body moves through a small arc on the circle. During this time, the body has deviated from a straight path by a small distance. For cases of constant acceleration, Galileo had given the mathematical law relating this distance to the acceleration and the time interval. Using Galileo’s law and classical geometry, Newton was able to derive an equation that expressed the acceleration as a function of the “arc chord” (the line segment connecting the endpoints of the arc), the time interval, and the radius of the circle.
In his next step, Newton made use of a new concept—“limit”—that lies at the foundation of calculus, the branch of mathematics he had discovered. As the above time interval is made progressively shorter, the chord of the arc becomes ever more nearly equal to the arc itself. In the “limit,” as the time interval approaches zero, the ratio of the chord length to the arc length approaches one. Therefore, in this limit, the chord can be replaced with the arc. Newton made this substitution and arrived at his law of uniform circular motion: The magnitude of the acceleration at any point on the circle is equal to the speed of the body squared divided by the radius of the circle.
Newton assumed nothing about the specific nature of the force causing this acceleration. His analysis relied only upon the fact that a force causes a body to deviate from motion in a straight line at constant speed, and hence for the purpose of studying forces we must define acceleration as indicated earlier. Therefore, his law does not specify the physical causes operating in any particular case; it is applicable to any body moving uniformly in a circle.
It was at this stage that Newton turned his attention to the planets. If the orbits are approximated as circular and if we express the speed as a function of radius and period, then Newton’s law implies that a planet’s acceleration is proportional to its orbital radius divided by its period squared. He then recalled that, according to Kepler’s third law, the period squared is proportional to the radius cubed. By combining these two relationships, he derived an extraordinary result: The sun exerts an attractive force on each of the planets, causing accelerations that are inversely proportional to the square of the planet’s distance from the sun.
Next he considered the moon and its approximately circular orbit around Earth. Such motion, he knew, implies that Earth exerts an attractive force on the moon. Since he was always seeking to connect disparate but related facts, Newton thought to ask: Is Earth’s attractive force of the same nature as the solar force; does it cause accelerations that also vary as the inverse square of the distance? If Earth had multiple moons at different distances, then the question could be answered by comparing the different accelerations. But we have only the one moon—so how could Newton determine the variation of acceleration with distance?
The answer lies in the concept of acceleration itself. The concept identifies an essential similarity between uniform circular motion and free fall: A body in circular motion is continuously falling away from a straight path and accelerating toward the center of the circle. Thus the moon accelerates toward Earth at a constant rate, as does a body dropped near the surface of Earth. Galileo had studied terrestrial free fall, and it was this acceleration that Newton could compare to that of the moon. This legendary comparison between the moon and the falling apple was demanded by the (inductively reached) vector concept of acceleration.
The quantities needed to make the comparison were known. The distance of the apple from the center of Earth is one Earth radius and the distance to the moon is sixty Earth radii. If the acceleration varies as the inverse square of the distance, then the apple’s acceleration will be greater than the moon’s acceleration by the factor (60) 2. Using rough data about free fall and the size of Earth, Newton calculated the ratio of accelerations and found approximate agreement with the inverse square law. Thus terrestrial gravity seemed to be the same force that holds the moon in its orbit and that the sun exerts on the planets. Kepler’s dream of one integrated science encompassing physics and astronomy was no longer merely a dream; with this calculation, it became a real possibility.
This was the birth of the idea of universal gravitation, but it was far from being the proof of it. At this early stage, Newton had many more questions than answers. For example, what about the fact that the actual orbits are ellipses, not circles? And what is the justification for using one Earth radius as the distance between the apple and Earth? Much of Earth is closer to the apple, and much is farther away; why would Earth attract from its center? Furthermore, if gravity is truly universal and each bit of matter attracts all other matter, the implications and complexities are daunting. For example, what is the effect of the moon’s attraction of Earth, or of the sun’s attraction of the moon, or of a planet’s attraction of other planets? What about strange bodies like comets, which move so differently?
The main difficulty that Newton confronted was not that such questions were as yet unanswered. The difficulty was that they were not yet answerable—not without a much deeper understanding of the relation between force and motion. It is one thing to say that a push or pull is necessary to change a body’s velocity; it is quite another feat to identify the exact mathematical law relating the external force to the body’s acceleration, and it is still another feat to identify a law that tells us what happens to the body exerting the force. Newton was just beginning to develop the cognitive tools he would need to prove universal gravitation.
We have seen how Newton grasped that a body’s velocity remains constant in the absence of an external force, which is his first law of motion. Now let us follow the main steps of reasoning that led to his second and third laws of motion.
The concept of “force” originates from sensations of pressure that we experience directly when we hold a weight or when we push or pull a body. Force has magnitude and direction, and men learned to measure the magnitude using balances, steelyards, and spring scales. The concept of “acceleration,” on the other hand, is a more advanced development. It was Galileo who first explained how linear acceleration could be calculated from measured times and distances, and we have now seen the concept expanded from a scalar to a vector quantity. At this stage, when Newton inquires into the mathematical relation of force and acceleration, both quantities are clearly defined and independently measurable.
Furthermore, a key fact had already been discovered. Force is directly proportional to acceleration, which had been proven by experiments in which the force was varied in a known way and the resulting acceleration was measured. Galileo’s investigations of a ball rolling down an inclined plane provided the first such experiments.
Galileo described a procedure for directly measuring the force on the ball.4 First, he said, attach the ball to a known weight by means of a string and attach a pulley to the top of the inclined plane. Place the ball on the inclined plane with the string over the pulley and the weight hanging vertically over the back of the plane. Then adjust the weight until it exactly balances the ball; this weight is the force on the ball in the direction of its constrained motion down the plane. The result of this measurement is what one might expect: The force on the ball is simply the component of its weight in the direction of the incline; in other words, it is the weight of the ball multiplied by the height to length ratio of the plane.
Therefore we can quadruple the force on the ball simply by quadrupling the height of the plane (while keeping the length the same). If we do so, we find that the time of descent is half what it was before, which implies that the acceleration has quadrupled—that it has increased by the same factor as the force. Alternatively, we can demonstrate by experiment that the initial height is proportional to the square of the final speed. With a little algebra, it can be shown that this relationship also implies that force is directly proportional to acceleration.
The pendulum provides another experiment that leads to the same conclusion. The period of a pendulum swinging along the arc of a cycloid (a curve traced by a point on the rim of a rolling wheel) is independent of amplitude, and it can be demonstrated mathematically that this fact also implies a direct proportionality between force and acceleration. Because the inclined plane and pendulum experiments were so well known, Newton took this proportionality for granted and never bothered to present its inductive proof in any detail.
Of course, he did not yet have a law of motion in the form of an equation. A concept was still missing, and one can sense Newton’s frustration in some of his early notes. At one point, he wrote: “As the body A is to the body B so must the power or efficacy, vigor, strength, or virtue of the cause which begets the same quantity of velocity. . . . ”5 As he was writing, Newton must have been asking himself: As precisely what about the body A is to precisely what about the body B? Nobody had yet formed a clear concept of “mass.”
The Greeks had proposed that all matter is endowed with either “heaviness” or “lightness.” The elements earth and water were claimed to be intrinsically heavy, whereas air and fire are intrinsically light. These properties were regarded as the cause of natural, vertical motion. The invalid Greek concept of “lightness” was an obstacle that prevented anyone from discovering that all matter has the property “mass.” In 1643, Evangelista Torricelli performed a crucial experiment that removed this obstacle from the path of modern physics.
Torricelli sought to explain a fact that was well known to mining engineers: A pump cannot lift water more than thirty-four feet above its natural level. The first question Torricelli asked himself was: Why does a pump work at all? In other words, when one end of a tube is inserted into water and the air is pumped out of the tube, why does the water rise into the tube? The commonly accepted answer was that “nature abhors a vacuum,” but this answer implies that the absence of matter in the tube is the cause of the water’s movement, that “nothingness” is literally pulling the water up the tube. It was obvious to Torricelli that those who tried to explain the effect by reference to nothingness had in fact explained nothing.
Instead, Torricelli identified something that did explain the effect: the weight of the air pressing down on the water surface. When air is removed from the tube, the atmosphere outside pressing on the water surface pushes water up the tube. It is similar to the action of a lever; the weight of the air (per surface area) will raise that same weight of water. Hence the weight of the entire atmosphere above a particular surface must be equal to the weight of thirty-four feet of water over that surface.
Torricelli’s idea implied that air pressure would lift the same weight of any fluid. For example, 2.5 feet of mercury weighs the same as thirty-four feet of water; therefore, when an evacuated tube is placed in a pool of mercury, the mercury should rise 2.5 feet up the tube. Torricelli did the experiment and observed precisely this result. Note that he used the method of agreement here: The same cause (i.e., the same weight of air) leads to the same effect (i.e., raises the same weight of fluid). Later experiments by Blaise Pascal and Robert Boyle demonstrated this relationship by showing that a change in the cause leads to a change in the effect (the method of difference). These experiments showed that decreasing the amount of air above the fluid surface results in less fluid rising in the tube; in other words, as we remove the cause the effect disappears.6
Thus it was proven that even air is heavy. Contrary to the Greeks, there is no such property as absolute “lightness.” When something rises in air, it does so because it is less heavy than the air it displaces. In other words, such “natural” rising is explained by Archimedes’ principle of buoyancy, a principle that applies to air as well as to water. After the work of Torricelli, scientists accepted the fact that all matter is heavy.
The next step was to clarify the meaning of “heaviness.” The Greeks had regarded heaviness as an intrinsic property of a body. However, to weigh a body is to measure the magnitude of its “downward push,” and this depends on something other than the body itself. As we have seen, Newton realized that heaviness is a measure of Earth’s gravitational attraction, and that this force varies with the position of the body relative to Earth. Additional evidence for this conclusion was discovered in the 1670s. Two astronomers, Edmund Halley and Jean Richer, independently discovered that pendulum clocks swing more slowly near the equator than at higher latitudes, and they correctly inferred that pendulum bobs weigh less near the equator. Therefore, “heaviness” arises from three factors: the nature of the body, the nature of Earth, and the spatial relationship between the body and Earth.
But what is the property of the body that contributes to its heaviness? Newton identified it as the body’s “quantity of matter,” or “mass.” His reasoning made use of both the method of difference and the method of agreement. First, he considered two solid bodies of the same material, weighed at the same location. Their weights are found to be precisely proportional to their volumes, and the constant of proportionality is an invariant characteristic of each pure, incompressible material. Therefore the weight of a body is proportional to its “quantity of matter”; by doubling the volume we have doubled the amount of matter, and the weight has doubled (method of difference). Second, Newton considered a compressible material such as snow. We can weigh a sample of snow, then compress it to a smaller volume, and then weigh it again. The quantity of matter has remained the same, and we find that the weight is the same (method of agreement).
Newton then asked how a body’s mass affects its motion when a force is applied. It is obvious that the mass does affect the motion; in order to cause a particular acceleration, a greater force is required for a greater quantity of matter (e.g., pushing a car requires more effort than pushing a bicycle). But what is the exact relationship? In order to answer the question, he needed an experiment in which the acceleration is held constant while the mass of the body and the applied force are varied. Newton did not have to look far to find such experiments; Galileo had done them when he investigated free fall.
From the top of a tower, Galileo had dropped two lead balls that differed greatly in size and weight. Let us assume that the larger ball had a volume ten times that of the smaller ball; therefore, its quantity of matter, or mass, was ten times greater. The force on each ball is simply its weight; by using a balance or a steelyard, we can determine that the weight of the larger ball is ten times the weight of the smaller ball. So, considering the larger ball relative to the smaller ball, we have increased both the force and the mass by a factor of ten. Yet Galileo demonstrated that the acceleration of free fall remains the same. We know that acceleration is exactly proportional to force, so it must be exactly inversely proportional to mass (so that the factors of ten cancel). This result accords with our common experience; it implies that for a body of greater mass a proportionally greater force is required to achieve a particular acceleration. Newton thus arrived at his second law of motion: The applied force is equal to the product of the body’s mass and its acceleration, or F = mA.
The scope of this generalization is breathtaking. It may seem astonishing that Newton could arrive at such an all-encompassing, fundamental law from the observations and experiments that have been described. But once one has the idea of grouping together all pushes and pulls under the concept “force,” and of grouping together all changes of velocity under the concept “acceleration,” and of ascribing to all bodies a property called “mass,” and of searching for a mathematical relationship among these measured quantities—then a few well-designed experiments can give rise to a law. At this stage, however, the validation of this universal law is not yet complete. It depends not only on the foregoing, but on all the evidence presented in this section and the next; the law is part of a theory that must be evaluated as a whole.
We have seen how this law rests on Galileo’s principle that all bodies fall with equal acceleration. Because this principle was so crucial to his theory of motion, Newton demanded that it be established by experiments more accurate than those of Galileo. He wished to prove beyond any doubt that a body’s inertial mass—the property by which it resists acceleration—is exactly proportional to its weight.
Newton realized that the pendulum provided the means for such an experimental proof. He deduced from F = mA that the inertial mass of a pendulum bob is proportional to its weight multiplied by the period squared (assuming the length of the pendulum is held constant). Thus if the period is always the same for any and all pendulum bobs, then inertial mass must be exactly proportional to weight. By using a small container as a pendulum bob, Newton varied both the mass and material of the bobs; he filled the container with gold, silver, glass, sand, salt, wood, water, and even wheat. All the bobs swung back and forth with the same period, and he performed the experiment with such care that he could easily have detected a difference of one part in a thousand. (The creator of modern physics had a passion for accurate measurement.)
So far, Newton had focused on the movement of one body subject to an applied force. At this stage, he turned his attention to the force itself and its origin: It is exerted by another body. What happens to this other body?
In order to answer the question, Newton needed to study the interaction of two bodies under conditions where the forces are known and the subsequent motion of both can be accurately measured. He devised the perfect experiment using a double pendulum with colliding bobs. He used pendulums with a length of ten feet, and he carefully measured and compensated for the small effects of air resistance. He varied the mass of the bobs and their initial amplitudes, then measured their final amplitudes after the collision.
Galileo had proven that a bob’s speed at the bottom of the swing is proportional to the chord of the arc through which it has swung. At the moment of collision, therefore, Newton knows the relative speed of both bobs. Furthermore, from his measurements of the final amplitudes, he could compute the relative speed of both bobs immediately after the collision. The results of the experiment showed that the mass of the first bob multiplied by the change of its speed is equal to the mass of the second bob multiplied by the change of its speed. Since the force exerted on each bob is equal to the product of its mass and its change of speed, Newton had proven that the bobs exert forces on each other that are equal in magnitude and oppositely directed.
Newton performed this experiment with pendulum bobs made of steel, glass, cork, and even tightly wound wool. In his choice of materials, he deliberately varied the hardness of the bobs and thereby proved that his law applied to both elastic and inelastic collisions. Since all collisions fall into one of these two categories, his generalization followed: Whenever two bodies exert forces on each other by means of direct contact, the forces are equal in magnitude and oppositely directed.
Newton then investigated the case of non-contact forces—forces that act over distances by imperceptible means. He attached a magnet and some iron to a piece of wood and floated the wood in calm water. The magnet and the iron were separated by a short distance, and each exerted a strong attractive force on the other. Yet the vessel did not move—implying that the two forces were equal in magnitude and oppositely directed, thus giving rise to zero net force.
Does the law also apply to bodies that attract each other gravitationally? Newton answered that it does and gave a convincing argument. Since Earth attracts all materials on its surface, it is reasonable to suppose (and it was later proven) that every part of Earth attracts all other parts. So consider the mutual attraction, say, of Asia and South America. If these two forces were not equal and opposite, there would be a net force on Earth as a whole—and hence Earth would cause itself to accelerate. This self-acceleration would continue indefinitely and lead to disturbances in Earth’s orbit. But no such disturbances are observed; on the contrary, Earth’s acceleration is determined by its position relative to the sun. Therefore the mutual attractive forces exerted by any two parts of Earth must be equal and opposite. (Newton also could have pointed out that unbalanced forces would lead to other effects that are not observed, e.g., asymmetries in Earth’s shape and in ocean tides.)
At this point, Newton had shown that his law applies to gravitational forces, magnetic forces, elastic collisions, and inelastic collisions—he gathered evidence over the range of known forces and found no exceptions. He had thus arrived at his third law of motion: All forces are two-body interactions, and the bodies always exert forces on each other that are equal in magnitude and oppositely directed.
When considering only one body, the concept “velocity” identified that which remained the same in the absence of an external force (this is the first law). In the case of two interacting bodies, Newton now identified a total “quantity of motion” that remains the same before and after the interaction. This quantity, which we now call “momentum,” is the product of a body’s mass and its velocity. Newton’s third law implies that the total momentum of two interacting bodies always remains the same, provided there is no external force. Furthermore, this “conservation of momentum” principle applies even to a complex system of many interacting bodies; since it is true for each individual interaction, it is also true of the sum.
After forming the concept of “momentum,” Newton could give a more general formulation of his second law. In its final form, which is applicable to a body or system of bodies, the law states that the net external force is equal to the rate of change of the total momentum. This form of the law can be applied in a straightforward way to more complex cases (e.g., imagine two bodies that collide and explode into many bodies).
Newton recognized that his three laws of motion are intimately related. We have seen that the third law prohibits the self-acceleration of Earth—but notice that such a phenomenon is also prohibited by the first and second laws, which identify the cause of acceleration as an external force. Given the fact that forces are two-body interactions, consistency with the second law demands that these interactions conform to the third law. The laws name related aspects of one integrated theory of motion; indeed, when the second law is given its general formulation, both the first and third laws can be viewed as its corollaries. Hence the laws mutually reinforce one another: The experimental evidence for the third law also counts as evidence for the second law.
I have outlined the main steps by which Newton induced his laws of motion. In their final statement, the laws appear deceptively simple. But we can now appreciate that they are very far from self-evident. In order to reach them, Newton needed complex, high-level concepts that did not exist prior to the 17th century, concepts such as “acceleration,” “limit,” “gravity,” “mass,” and “momentum.” He needed a variety of experiments that studied free fall, inclined plane motion, pendulums, projectiles, air pressure, double pendulums, and floating magnets. He relied upon the observations that had led to the heliocentric theory of the solar system, upon the experience of pulling inward in order to swing a body in a circle, upon the observations that determined the distance to the moon, upon the instruments invented for measuring force, and even upon chemical knowledge of how to purify materials (as this played a role in forming the concept “mass”). His laws apply to everything that we observe in motion, and he induced them from knowledge ranging across that enormous database.
For the past century, however, many philosophers, physicists, and historians of science have claimed that the laws of motion are not really laws at all; rather, they are merely definitions accepted by convention. This view derives from empiricist philosophy and was famously advocated by Ernst Mach.7 The empiricists regard the second law as a convenient definition of the concept “force,” which allegedly has no meaning except as a name for the product of mass and acceleration; similarly, they argue that the third law amounts to a convenient definition of “mass.” Those advocating such views have left themselves the inconvenient task of answering some obvious questions. Why is a particular definition “convenient,” whereas any alternative definition would be cognitively disastrous? What about static forces that exist and can be measured in the absence of acceleration? How is it possible that the concept “force” was formed millennia before the concepts “mass” and “acceleration”? No answers have been forthcoming from Mach’s disciples.
Newton did not anticipate the skepticism that became rampant in the post-Kantian era. He regarded as obvious the fact that the laws of motion are general truths reached by induction, and therefore he did not go out of his way to emphasize the point. Indeed, he regarded the laws of motion as uncontroversial, which is why his discussion of them in the Principia is so concise.
He viewed these laws as the means to his end, not the end itself. The laws enable us to reason from observed motions to the forces that cause them, and then reason from these known forces to all their diverse effects. As Newton put it: “[T]he whole burden of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.”8 He made his meaning clear by providing a grand-scale example of this program.
The Discovery of Universal Gravitation
The Principia presents a long and complex argument for the law of universal gravitation. Today, the law itself is familiar to any educated person. Newton’s argument, however, is less familiar, and its epistemological implications are largely ignored or unknown. I will outline the steps of his reasoning in this section, and discuss some of the implications in the next.
Newton began by inferring the nature of the solar force from Kepler’s laws of planetary motion. Of course, this had been Kepler’s goal—but without an understanding of dynamics and without the methods of differential calculus, it had been unattainable. Three quarters of a century later, Newton had the right tools for the job.
His first step was to prove a result that is initially somewhat surprising: Kepler’s area law (that a line from the sun to a planet sweeps out equal areas in equal times) is true even in the absence of a force. Using trigonometry, Newton showed that a line from any fixed point to a body moving with constant velocity will sweep out equal areas in equal times. So he immediately established a connection between his dynamics and Kepler’s planetary theory: Inertial motion conforms to the area law. From this result alone, it was clear that this law has broad application beyond planetary motion.
In his next step, Newton assumed the body is subject to a series of impact forces that are always directed toward a fixed point. He showed that the area law is true for this case as well. He then let the time interval between these impacts approach zero, thereby proving that Kepler’s law holds for any continuous force that is always directed along the line connecting the body to some fixed point (such forces are called “central forces”). It makes no difference how the force varies with distance, or whether it is attractive or repulsive. So long as the force has no tangential (or sideways) component, the area law is valid.
So the area law tells us the direction of the solar force, but it contains no information about the magnitude. It was Kepler’s law of elliptical orbits that enabled Newton to prove that the magnitude of the solar force varies as the inverse square of the distance. Those who study the details of this proof will be impressed by Newton’s mathematical genius. For our purposes, however, we can pass over these details and merely identify the essential elements of the proof.
First, the solar force is related to the planet’s acceleration by Newton’s second law of motion. Second, for any short time interval during which the acceleration may be regarded as constant, Galileo gave the law relating the acceleration to the time interval and to the distance that the body falls (in this case, the planet’s “fall” is its movement away from a straight path and toward the sun). Third, Kepler’s area law enabled Newton to replace time intervals with areas, thereby transforming a problem of dynamics into a problem of geometry. Finally, certain theorems about ellipses (discovered in antiquity by Apollonius) enabled Newton to relate the small distance the planet “falls” during the interval to other distances defining its location on the ellipse. Therefore he had all the pieces he needed: He knew the relation between force and acceleration, and he could express the acceleration in terms of the geometric properties of the ellipse. In the end, this mathematical complexity led to a simple result: The sun exerts an attractive force on the planets that varies as the inverse square of the distance.
Just as with the area law, Newton recognized that the law of elliptical orbits is a special case of a more general truth. The geometric properties that Newton had used are not unique to ellipses; they are general properties of conic sections that also apply to parabolas and hyperbolas. Therefore, the solar force will not necessarily cause a body to move in an elliptical orbit; the path may be a parabola or a hyperbola instead. In general, an inverse square attractive force causes a body to move in a conic section; the particular conic section is determined by the initial position and velocity of the body. If the initial conditions are such that the body is captured by the sun’s gravitational field, then the orbit will be an ellipse (or a circle). However, if the body’s velocity is too great, then it will pass through our solar system in a parabolic or hyperbolic path. Newton presented the details, showing how to calculate the path of a body from any set of initial conditions.
Finally, Newton considered Kepler’s third law. For an elliptical orbit, he showed that this relationship between the orbital period and the major radius follows from the nature of the solar force. In the proof, he used all the facts that had entered into his proof of the inverse square law, plus he used the inverse square law itself and the well-known expression for the area of an ellipse. Here we see yet another example of an astounding connection established by means of mathematics. There is no way to guess that the orbital period is proportional to the three-halves power of the major radius and yet that it is entirely independent of the minor radius. This fact is implicit in the premises of Newton’s argument, but advanced mathematics is required to make the deduction.
This mathematical analysis had another implication: It showed that Kepler’s third law is not exact. In his proof, Newton assumed that the sun is not accelerating. However, his third law of motion implies that the planet exerts an equal and opposite force on the sun, causing it to move in a very small orbit of its own around the center of mass of the two bodies. Newton proved that this effect leads to a slight modification of Kepler’s third law; the correction, he showed, depends on the ratio of the planet’s mass to the sun’s mass. In the case of Jupiter, the most massive planet, the magnitude of this correction is about one part in a thousand.
Newton induced the nature of the solar force from Kepler’s laws, and, in the process, he gained a much deeper understanding of those laws. He proved that the area law applies to any two bodies that attract or repel each other, that the law of elliptical orbits can be expanded to a law of conic sections describing the movements of any two bodies attracting by an inverse square law, and that Kepler’s third law is very nearly true because the mass of the sun is so much greater than the mass of the planets.
In Newton’s analysis, we can see three interrelated aspects of the power of mathematics. First, mathematics enables us to discover new facts by deriving further implications of that which is already known. Second, it connects known facts that would otherwise stand apart with no relation. Third, it gives crucial insight into the domain over which a generalization is valid—by making clear what the generalization depends upon and what it does not depend upon.
After inferring the sun’s inverse square attraction from the observed planetary motions, Newton investigated force laws that can be inferred from other types of motion. For example, he considered the eccentric circles that astronomers had traditionally used to model planetary orbits. He proved that the force required to produce such motion is physically absurd; the attractive force exerted by the sun on a planet would have to depend not only on where the planet is, but also on where it will be at a later time. So, even in cases where such a model is consistent with available data, it is ruled out because it violates the law of causality.
Interestingly, Newton proved that an attractive force proportional to distance would cause an elliptical orbit. In this case, however, the sun must be at the center rather than at a focus of the ellipse. Furthermore, all the planets would revolve around the sun with the same period, in marked contrast to the observations. Newton then considered the case of an attractive inverse cube solar force and showed that the resulting orbit would be spiral with a constant angle between the radius and the velocity vector.
The most important of these “counterfactual cases” that Newton analyzed was that of an inverse square force with a small inverse cube term added. Here he showed that the resulting orbits could obey Kepler’s laws to a very close approximation. However, Newton was still able to identify a difference between these orbits and those actually observed. When a small inverse cube term is added, the major axis of the ellipse does not remain fixed in space; instead, it slowly rotates at a rate that depends on the magnitude of the inverse cube term.
Newton was not merely flexing his mathematical muscles in this calculation. He realized that if the idea of universal gravitation is correct, then the planetary orbits are not exactly elliptical; a planet’s motion will be slightly disturbed by bodies other than the sun. Furthermore, he was keenly aware of the fact that we are always reasoning from data of limited accuracy. Although he had already shown that an exactly elliptical orbit implies a solar force that is exactly inverse square, it does not necessarily follow that an approximately elliptical orbit implies such a solar force. Newton was cautious about making such inferences, so he decided to investigate the effects of a small deviation from the inverse square law. By proving that even a small inverse cube term would change the planetary orbits in a way that contradicts the observations, he removed any lingering doubts about the nature of the solar force.
At this stage, Newton turned his attention to bodies other than the sun. Galileo had discovered four moons orbiting Jupiter, and later Christian Huygens and Gian Cassini had discovered five moons orbiting Saturn. Because astronomers had made remarkable improvements in the design of telescopes, Newton had accurate data about these lunar orbits. He found that Kepler’s laws applied to these moons as well as to the planets. Most impressively, he showed that for both sets of moons the orbital period squared was precisely proportional to the orbital radius cubed. It follows that Jupiter and Saturn attract their moons with the same type of inverse square force that the sun exerts on the planets.
Newton also cited evidence that the planets attract each other. Astronomers had noticed disturbances in the orbit of Saturn when it is in conjunction with Jupiter (i.e., when Jupiter is between the sun and Saturn). Of course, the idea of universal gravitation explains such disturbances. Jupiter is the most massive of the planets, and at its point of closest approach it exerts a significant pull on Saturn. Newton identified a simple way to improve the model of Saturn’s orbit: The focus of the ellipse should be placed at the sun-Jupiter center of mass, rather than at the sun itself. In this way, Newton reduced the maximum errors in Saturn’s angular position to only two minutes arc (one thirtieth of a degree).
Finally, Newton’s focus returned to Earth—and to the origin of his great idea. He once again calculated the relative accelerations of the moon and the apple. In the rough calculation he had performed many years earlier, there had been about a 10 percent discrepancy with the inverse square law. He was now prepared to eliminate the assumptions, the approximations, the inaccuracies in the data—and to see whether Earth’s attraction really does vary exactly as the inverse square of the distance.
In his original calculation, Newton had used one Earth radius as the distance between the apple and Earth. This is equivalent to assuming that the spherical Earth attracts as if all of its mass is at the center. But why would it attract in this way? Universal gravitation implies that every bit of matter is independently pulling the apple toward it, with a force that is inversely proportional to the square of its distance from the apple. It seems almost miraculous that all of these independent pulls from every part of Earth are exactly the same as the entire mass of Earth pulling from the center. Without a mathematical proof, Newton was not inclined to believe it.
Today, with the full power of integral calculus available, this proof can be performed by any competent student of physics. It was more challenging for Newton, but he succeeded by constructing a very clever geometrical proof that took full advantage of the symmetry of the sphere. The Earth attracts from its center, he showed, provided that two conditions are satisfied: First, the attractive force must vary as the inverse square of the distance; and second, the mass density of Earth must depend only on the distance from the center.
Earlier, his use of one Earth radius for the distance to the apple had been a dubious assumption; now, with his mathematical proof, it was demanded by the idea of universal gravitation. The radius of Earth had been measured accurately, and the value Newton used was very close to the one accepted today. For the distance to the moon, Newton carefully reviewed the independent measurements of several researchers and adopted sixty Earth radii as the best available value. So, if Earth’s attraction varies in the same way as the attraction of Jupiter and Saturn and the sun, then the moon’s acceleration multiplied by (60) 2 should equal the apple’s acceleration.
The period of the moon’s orbit was known very precisely. Since the orbit is nearly circular, Newton could use his law of circular acceleration (as he had years before). However, this time he added a small correction for the attraction of the sun, which slightly decreased (by one part in 179) his estimate of the acceleration caused by Earth. He also estimated the minor effect of the moon’s reciprocal pull on the Earth. When he at last arrived at his final answer and multiplied by (60) 2, his predicted value for the gravitational acceleration on Earth’s surface was 32.2 ft/sec 2. Years earlier, Huygens had used pendulums to measure this acceleration very accurately. The measured value matched Newton’s calculated value: It was 32.2 ft/sec 2. Newton had proven that terrestrial “heaviness”—the ubiquitous phenomenon known to every toddler—is the same force that moves planets and moons.
Note that it would be invalid if Newton had merely said: “The moon accelerates toward Earth, the apple accelerates toward Earth, therefore the cause is the same in both cases.” Different causes can sometimes lead to qualitatively similar effects (e.g., a magnet with electric charge on its surface will attract both straw and iron filings, but for different reasons). However, when Newton proved that the moon and the apple fall at rates that are precisely in accordance with a force that varies as the inverse square of the distance from Earth’s center—then there can be no doubt that the same cause is at work.
Once Newton proved that the attraction between celestial bodies is the familiar force of terrestrial gravity, then everything known about gravity on Earth was applicable to the celestial force. Therefore, the experimental proof that terrestrial gravity is proportional to mass also serves as a proof that the attractive force of any celestial body is proportional to its mass. Furthermore, by Newton’s third law, the gravitational interaction must depend on the mass of both attracting bodies in the same way. Thus Newton arrived at the complete law of gravitation: The force varies directly as the product of the masses and inversely as the square of the distance between them.
For the purpose of comparing the acceleration of the moon to that of the apple, Newton could approximate the moon’s orbit as circular without introducing any significant error. The orbit can be modeled more accurately, of course, as an ellipse. However, the observational data of astronomers proved that the orbit of the moon is quite complex; it does not precisely obey Kepler’s laws. The reason for the anomalies in the orbit is the gravitational pull of the sun; the moon-sun distance differs slightly from the Earth-sun distance, which causes a small relative acceleration between the moon and Earth.
Newton made an enormous effort to explain the lunar anomalies. Starting from the fact that the sun’s gravitational attraction varies as the inverse square of the distance, he showed that the perturbing accelerations caused by the sun vary as the inverse cube. We have already seen that such a force causes the major axis of the orbit to rotate; from the magnitude of the term, Newton was able to explain the three-degree annual rotation in the moon’s orbit that had been observed by astronomers. He also explained the variations in the eccentricity of the orbit, the movement of the points at which the moon crosses the ecliptic (the plane of Earth’s orbit around the sun), and the annual variations in these anomalies. His analysis provided very impressive evidence for the law of universal gravitation; in addition to explaining Kepler’s laws, he could explain the observed deviations from Kepler’s laws.
Newton continued to exploit the moon when he turned his attention to the ocean tides. The correlation of the tides with the position of the moon had been noticed by the first Greek explorers who ventured out into the Atlantic Ocean. Prior to the formation of the concept “gravity,” however, the idea that the moon could influence our seas was often dismissed as equivalent to a belief in magic. Newton’s discoveries brought about a radical change by identifying the physical cause and thereby making it clear that such an influence is a necessary consequence of natural laws.
Ocean tides are caused by the fact that the moon does not attract all parts of Earth equally. The side of Earth nearest the moon is attracted slightly more than Earth’s center (pulling it toward the moon), whereas the opposite side of Earth is attracted less than the center (leaving it farther from the moon). This causes the oceans to bulge on both sides, giving rise to high tides. The bulges are fixed with respect to the moon, but the daily rotation of Earth causes the tides to rise and fall at any particular location. If the moon were stationary, the time between high tides would be 12 hours; the moon’s movement increases this time interval to 12.5 hours. Newton pointed out that the sun also causes ocean tides, but he showed that the sun’s effect is less than one third that of the moon.
He was able to explain all the main features of the tides. The tides are greatest when the lunar and solar tides coincide, which happens when the moon is in opposition or conjunction with the sun (i.e., when the moon is full or new). During half moons, the tides are least because the sun partially cancels the effect of the moon. He explained the observed variations in the tides that are caused by variations in the distance to the sun, in the distance to the moon, and in the inclination of the moon with respect to the equator. Furthermore, he analyzed the reverse tidal effect on the moon caused by the attraction of Earth. This attraction causes a bulge of almost a hundred feet on the side of the moon facing Earth. Newton pointed out that Earth’s pull on this bulge explains why the same side of the moon always faces Earth.
The tides affect the shape of Earth by raising our oceans by a mere ten feet (at most). Newton realized that his dynamics implied another effect on the shape of Earth that is much greater in magnitude. If Earth’s matter is (or once was) sufficiently mobile, then the daily rotation of Earth will cause it to bulge at the equator and flatten at the poles. This must be the case, because otherwise matter that is mobile would move to the equator and the Sahara desert would be at the bottom of a very deep ocean. Given the rate at which Earth spins and some assumption about the distribution of mass within Earth, Newton could use his laws of motion and gravitation to calculate the size of the equatorial bulge.
Of course, no data was available regarding the variation of mass density within Earth. Newton decided to perform the calculation using a constant density, while explicitly noting that ignorance of this factor caused some uncertainty in the result. He estimated that the equatorial radius exceeded the polar radius by seventeen miles, which is reasonably close to the actual difference of thirteen miles. Thus Earth is an oblate spheroid rather than a sphere, and the size of the effect is such that there should be observable consequences.
In fact, astronomers had already observed some of the consequences. Recall that it had been discovered in the 1670s that pendulum clocks swing more slowly near the equator than in Paris or London. Newton explained that the pendulum bobs move slower at the equator for two reasons: Gravity is weaker at the equator because it is farther from Earth’s center and because of the centrifugal effect of Earth’s rotation. He proved that the mathematical expression for the weight of a terrestrial body contains a small variable term that is proportional to the square of the sine of the latitude, and his analysis accounted for the observed changes in the clock rates.
Further evidence for Newton’s theory came from observations of Jupiter. Astronomers had discovered that Jupiter’s equatorial radius is greater than its polar radius by about one part in thirteen. By observing the spots on Jupiter, they knew the rate at which the large planet rotated. Newton calculated its equatorial bulge, and his result was close to the measured value. He was demonstrating the explanatory power of his dynamics on an ever-increasing scale.
Newton’s calculation of Earth’s shape enabled him to clear up another mystery, which had perplexed astronomers for eighteen hundred years. In the second century BC, Hipparchus discovered that the stars move in a peculiar way. In addition to the apparent daily rotation, the center of this celestial rotation (i.e., the location of the “north star”) also moves slowly around in a small circle. Given the heliocentric theory, this implies a precession of Earth’s spin axis; in other words, the axis sweeps out a cone just as we can observe in the case of a spinning top. Newton’s laws of motion and gravitation explained this effect. The moon and sun attract the mass of Earth’s equatorial bulge, causing a torque that moves Earth’s spin axis in a cone with an angular radius of 23 degrees (equal to the angle between the plane of the equator and that of the ecliptic). The torque is small, and therefore the precession is very slow. Newton carefully estimated the gravitational pull on the equatorial bulge and calculated the precession rate. He arrived at a value very close to the one measured by astronomers, who had determined that Earth’s axis completes one revolution in about twenty-six thousand years. With nearly every turn of a page in the Principia, another phenomenon was explained.
The Principia was a tour de force demonstration of the intelligibility of the universe. The grand finale of this demonstration was Newton’s analysis of comets, the mysterious and previously unpredictable objects that were widely regarded as signs of God’s anger. Newton dispelled such fears by proving that comets were ruled by the force of gravitation, not by a moody God.
Astronomers had collected accurate data on the movements of a comet that had appeared in 1680. Newton analyzed these data with great care and concluded that the comet moved in an extremely elongated ellipse. Its speed was observed to change rapidly, but always in perfect conformity to Kepler’s area law. The orbit is inclined at an angle of 61 degrees with the plane of Earth’s orbit. The comet approaches the sun very closely every 575 years, and its maximum distance from the sun is 138 times greater than the mean Earth-sun distance. One can hardly imagine an orbit that differs more dramatically from the planetary orbits, and yet Newton proved that the comet is moving in accordance with the same laws. He drew the only possible conclusion: “The theory which justly corresponds with a motion so unequable, and through so great a part of the heavens, which observes the same laws with the theory of the planets, and which accurately agrees with accurate astronomical observations, cannot be otherwise than true.”9
Another comet was observed in late 1682. Edmund Halley, with the aid of Newton’s theory, processed the data and calculated the orbit. He showed that the comet approaches inside the orbit of Venus every seventy-five years, its maximum distance from the sun is about thirty-five times greater than the Earth-sun distance, and the orbit is inclined by 18 degrees with respect to the plane of Earth’s orbit. Halley predicted that the comet would appear again in 1758—and it did return almost exactly on schedule, delayed only slightly by the influence of Jupiter.
The law of universal gravitation integrated and explained diverse observations on an unprecedented scale. Even so, there were scientists who found the Principia unsatisfying. They raised the same criticism that they had directed earlier at Newton’s theory of colors.10 Again, they complained, Newton had failed to identify the first cause. At bottom, they charged, his explanations were empty because he had not explained the physical means by which bodies attract one another.11
This criticism derives from the idea that we must deduce knowledge from “first causes” rather than induce it from experience. Newton’s opponents could not grasp that knowledge is gained by starting with observations and proceeding step-by-step to the discovery of causes, eventually to the discovery of fundamental causes. Instead, they wished to start with imagined first causes and deduce the entire science of physics from them (this method is referred to as “rationalism”). Newton knew that such an approach leads only to the indulgence of fantasy, not to scientific knowledge. In responding to his critics, he repeated the point he had made years earlier:
I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not [inferred] from the phenomena is to be called a hypothesis, and hypotheses . . . have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena and afterward rendered general by induction. Thus it was that . . . the laws of motion and gravitation were discovered. And to us it is enough that gravity really does exist and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies and of our sea.12
Scientists who follow the rationalist method attempt to bypass the process of discovery. Using nothing more than imagination and deduction, they fabricate whole sciences—discovering no knowledge, while leaving no questions unanswered. Newton’s inductive method leads to the opposite result: an enormously expanded context of knowledge, with each discovery giving rise to further questions.
Today, Newton’s refusal to speculate about an underlying mechanism of gravitational attraction is often misinterpreted in a way that would have been inconceivable to him. The modern empiricists, in an effort to claim Newton as one of their own, argue that he advocated a non-causal, descriptive approach to physics.13 But the attempt to portray Newton as a post-Kantian empiricist is laughable. In conjunction with the laws of motion, the law of gravitation is the very archetype of a causal law: It states a necessary relationship between a property of an entity (mass) and its action. Throughout the Principia, Newton was focused on identifying causal relationships.
The final book of the Principia is appropriately titled “The System of the World.” Newton had inherited a wealth of knowledge from his predecessors, but it was knowledge that consisted of separate laws belonging to separate sciences. Newton induced the fundamental causal relationships that connected this knowledge into a systematic whole. With this achievement, the science of physics reached maturity.
Here are the nature and result of the inductive method in their full glory.
Discovery Is Proof
The “problem of induction” is usually posed in a way that seems to preclude a solution. It is described as the problem of justifying an inductive “leap” from a relatively few observations to a universal truth. It is then asked: How can we be certain of a conclusion that transcends the evidence in this way?
This perspective is strangely detached from the actual discovery process that culminated in Newton’s laws of motion and gravitation. We have followed the reasoning that led to these laws and yet we have not encountered any steps that could reasonably be described as “leaps” beyond the evidence. On the contrary, every new step followed from the evidence, given the prior context of knowledge. So now the question is: How did Newton manage to arrive at universal laws without making any illogical leaps?
A large part of the answer lies in the objectivity of the concepts themselves. For example, at the turn of the 17th century, there was nothing arbitrary about the expansion of the concept “force” to include pushes and pulls exerted across a distance by imperceptible means; this was necessitated by observations of electric and magnetic phenomena. Similarly, there was nothing arbitrary about the expansion of the concept “acceleration” to include changes in a body’s direction as well as its speed; this was necessary in order to distinguish motion caused by a force from motion that can occur in the absence of force. The concepts of “force” and “acceleration” then made it possible to identify that both the sun and Earth exerted an attractive force of the same nature, denoted by the concept “gravitation.” This concept, in turn, made it possible to identify weight as a measure of gravitational force, and it became necessary to isolate the property of bodies that causes this force; experiments then determined that a body’s weight and inertia are proportional to its “quantity of matter,” or “mass.”
Likewise, there was nothing arbitrary in the reasoning that identified the causal connections made accessible by these concepts. The variables were systematically isolated and measured in a series of experiments involving free fall, inclined planes, pendulums, and double pendulums. By the time Newton announced his mathematical laws, he had studied mechanical, gravitational, and even magnetic forces; he had studied masses that ranged in magnitude from that of a pebble to that of the sun and included a wide variety of different materials; he had studied motions that ranged in speed from a bob swinging slowly at the end of a long pendulum to a comet streaking across the night sky, and ranged in shape from linear to circular to parabolic to elliptical. Thus the laws were truly integrations of data, not leaps of faith.
A rigorous process of inductive logic enabled Newton to climb from narrower generalizations to his fundamental laws. For example, he did not leap to the law of universal gravitation and then search for confirming instances. Rather, as we saw, he began by identifying the nature of the solar force on the planets. In the Principia, he then showed that a similar force is exerted by Jupiter and Saturn on their respective moons—and he therefore had a law pertaining to both planets and moons. He next showed that a similar force is exerted by Earth on both terrestrial bodies and our moon—and he therefore had a law that applied to all bodies on Earth’s surface as well as planets and moons. He then showed that the attractive force is not merely exerted by Earth as a whole, but it is exerted independently by every bit of matter making up Earth (his analysis of Earth’s shape and precession, and the ocean tides, provided important evidence for this conclusion). Finally, he showed that the law applied even to comets, the celestial bodies that were legendary for their mysterious behavior and appearance. This was the genesis of Newton’s discovery that all bodies have the property “mass” and thus attract in accordance with his law of gravitation.
If, at the end, Newton had been asked, “Now that you have this theory, how are you going to prove it?” he could answer simply by pointing to the discovery process itself. The step-by-step logical sequence by which he arrived at his theory is the proof. Each step was the grasp of a causal connection by the mathematical processing of observational data. Since there were no arbitrary leaps, there is no problem of justifying them.
To state this point negatively: In order to ask the above question, one has to drop the relevant context. The question does not arise if one keeps clearly in mind the whole sequence that led from observations to the fundamental laws. If, however, one assumes that the theory was created from the resources of Newton’s imagination, then the issue of proof becomes an insolvable problem. The mere process of deducing consequences of a theory that are confirmed by observations never does or can lead to a proof. Such a process is insufficient even when the predictions range over a wide variety of different phenomena. The inevitable counterargument, offered by all those who take concepts and generalizations as given, without inquiring into their source, is: Perhaps someone else, possessing an equally rich imagination, can dream up an entirely different theory that accounts for the same facts. Without grasping the way in which Newton’s conceptual framework emerged from and is necessitated by the observations, there is no answer to this objection. Given the inductive proof, however, one can and must answer simply by dismissing this suggestion as an arbitrary fantasy.
Today, it is almost universally held that the process of theory creation is non-objective. According to the most common view, which is institutionalized in the so-called “hypothetico-deductive method,” it is only the testing of theories (i.e., comparing predictions to observations) that gives science any claim to objectivity. Unfortunately, say the advocates of this method, such testing cannot result in proof—and it cannot result even in disproof, since any theory can be saved from an inconvenient observation merely by adding more arbitrary hypotheses. So the hypothetico-deductive method leads inevitably to skepticism.
Despite its implicit denial of scientific knowledge, this view of method strikes many scientists as plausible. One reason can be found in the way that science is taught; fundamental truths about nature are handed out like Halloween candy to young students, who are given only random snippets of the evidence from which the theories were induced. The education of a scientist today is focused on developing his proficiency in deducing consequences of the theories. Thus the scientist emerges from his training with memorized floating abstractions and a great deal of expertise in applying them. When he hears a description of the hypothetico-deductive method, he then recognizes it as an accurate description of his own state of mind. In this way, an embarrassing failure of education becomes a standard theory of scientific method.
The difference between a scientist who induces a theory and one who “freely creates” a theory is the difference between a man standing on solid ground and a cartoon character hovering in midair over an abyss. It is little wonder that those who believe theories are “free creations” sense the impending disaster: They typically believe that all theories are doomed to fall and be replaced by other imaginative constructs. In contrast, the inductive method leads to the opposite conclusion: A theory reached and validated by this method is never overthrown. Thus, for example, Newton’s laws have not been contradicted by any discoveries made since the publication of the Principia. Rather, all subsequent discoveries in physics have presupposed his theory and built on it. His laws have been the rock-solid foundation for the work of every physicist of the past three centuries, and they continue to be applied today in countless ways.
The widespread confusion regarding this point is caused by treating scientific laws as out-of-context dogmas rather than as integrations of concretes. Newton himself, however, never said: “My laws apply without modification not only to all that is currently known in physics and astronomy, but also to every phenomenon that will ever be studied, no matter how far removed it is from any phenomenon studied to date. I give these laws as commandments, to be applied independent of cognitive context and without thought.” He made no such statement because he knew that the process of inductive reasoning that led to his laws established the context within which they are proven. Further evidence is required if the laws are to be extended into previously unstudied realms.
The cases in which Newton’s laws are said to fail are all the same: They are cases where his laws have been torn from the context in which they were discovered and applied to a realm far removed from anything he ever considered. The cases pertain to bodies moving at near light-speed, which is about ten thousand times the speed of Earth in its orbit around the sun; or they pertain to subtle effects of very strong gravitational fields, none of which could be measured until more than a century after Newton; or they pertain to the behavior of subatomic particles, a realm that physicists did not begin to study until about two centuries after Newton.
In order to clarify the relation between early theories and the later advancements that they make possible, let us examine one particular piece of evidence that is often said to refute Newton’s gravitational theory. The major axis of Mercury’s orbit is observed to rotate very slowly. As seen from Earth, the total rotation appears to be about 1.56 degrees per century. Calculations show that almost 90 percent of this apparent rotation is caused by the precession of Earth’s spin axis, which is entirely explained by Newton’s theory. Of the remaining effect, more than 90 percent is caused by the gravitational pull of other planets, which is also explained by Newton’s theory. That leaves less than 1 percent of the total observed effect, which amounts to 43 arc seconds per century, which is unexplained by Newton’s theory. This residual effect is explained by Einstein’s theory, the predictions of which differ slightly from Newton’s in the strong gravitational field near the sun.
Einstein did not refute the laws of Newton, just as Newton did not refute the laws of Kepler. In both cases, the truth of the earlier theory was presupposed and then a more general theory was developed that applied within an expanded context of knowledge. And, in both cases, the expanded context of knowledge included small discrepancies between new data and the old theory, which were then explained by the new theory. This is how science progresses.
Only one aspect of Newton’s theory was rejected rather than absorbed into Einstein’s theory (and, in this case, one can only wish that Einstein had been consistent in his rejection). Newton treated the concepts “space” and “time” as existents independent of bodies, rather than as relationships among bodies. Thus he viewed space as an infinite cosmic backdrop that exists independent of the bodies placed in it, and he claimed that this backdrop has real physical effects on the bodies that accelerate with respect to it.
Newton offered scientific arguments to support his view of space and time, but these arguments are non sequiturs.14 “Absolute” space and time played no role in the reasoning that proved his theory (thus I had no need to mention these ideas while presenting his discovery process). In fact, “absolute” space and time are intimately connected to Newton’s religious views, and are therefore arbitrary elements in his theory. He occasionally made concessions to religion and thereby departed from his explicitly stated scientific method. This is the most egregious example of such a departure.
In Newton’s theory, the frame of absolute space is identified with a coordinate system in which the fixed stars do not rotate. This frame is defined objectively, on the basis of observation. Therefore Newton could have replaced his discussion of absolute space (and time) with the following statement: “The laws of motion and gravitation presented herein are valid in the frame of the fixed stars, or in any frame that can be approximated as unaccelerated with respect to the fixed stars. Whether it is possible to develop a theory that is free from this restriction, I leave to the consideration of the reader.” Such a statement would have made clear the objective status of his theory, and it would have eliminated the impossible task of trying to establish the existence of space as a supernatural pseudo-entity.
No later discoveries in physics were required in order to identify and reject Newton’s error. Several of Newton’s contemporaries pointed out that there was no justification for reifying space and time.15 The correct relational view dates back to Aristotle, who treated space as a sum of places and explained that the concept “place” refers to a relationship among bodies. Thus the ideas of absolute space and time were identified as arbitrary two thousand years earlier; the discoveries of Einstein are irrelevant to this issue.
Later discoveries add to the cognitive whole, but they never refute it. Indeed, there is a symbiotic relationship here; the earlier knowledge makes it possible to discover the later knowledge, and the later knowledge often makes it possible for us to see profound new implications in the earlier knowledge.
As an example of seeing new implications in old knowledge, consider the relationship between Newton’s dynamics and Galileo’s kinematics. It has always perplexed historians of science that Newton credited Galileo with the second law of motion (F = mA). Galileo did not know this law, so why did Newton say that he learned it from him? The answer provides insight into the way Newton took full advantage of his predecessor’s achievements. This law was out of Galileo’s reach because he did not have the prerequisite concepts. In Newton’s context, which included the vector concept “acceleration” and the concepts “gravity” and “mass,” Galileo’s experiments do imply that F = mA. In effect, Newton could read his second law of motion between the lines of Galileo writings, even though this message was invisible to the author himself.
We have encountered other similar examples. Torricelli’s discovery that air has weight led scientists to a more general formulation of Archimedes’ principle of buoyancy. In light of Newton’s dynamics, Kepler’s area law of planetary motion was generalized to the conservation of angular momentum principle, which applies to all bodies. The acquisition of knowledge is not merely a step-by-step climb up the hierarchy, with one’s eyes always forward on the next step. Such a metaphor misses the fact that a thinker’s focus must regularly return to earlier knowledge in order to integrate it with new discoveries. A crucial aspect of cognitive integration is the task of revisiting old knowledge and extracting from it the new implications that can be seen only in the light of more recent advances.
The scientific revolution of the 17th century achieved the ambitious goal that was first pursued in ancient Greece. The Greeks attempted to identify basic principles that could integrate their knowledge of the universe into one intelligible whole. However, they lacked the necessary experimental and mathematical methods. In their impatience, they bypassed the slow, painstaking process of discovery; instead, they attempted a giant leap from observations to the fundamental principles—and they fell short. Eventually, a loss of confidence led to the pragmatic acceptance of Ptolemy’s geocentric astronomy, which abandoned the goal of understanding causes and settled for describing “appearances.”
At the outset of the scientific revolution, Copernicus commented on the lack of integration in astronomy. Regarding his predecessors, he wrote: “[T]hey are in exactly the same fix as someone taking from different places hands, feet, head, and other limbs—shaped very beautifully but not with reference to one body and without correspondence to one another—so that such parts make up a monster rather than a man.”16 Copernicus took the first steps toward transforming this monster into a man. The task was completed by Newton, who made physics and astronomy into one body of knowledge with all parts fitted together in a perfect whole.
Under Newton’s powerful influence, the inductive method rose to prominence and its arch-nemesis—the arbitrary—fell into disrepute. The method that scientists learned from the Principia, as well as his later work Optics, led to a new era in which many long-held secrets of nature were finally illuminated and new sciences were born (e.g., electricity, chemistry, and geology). Because such illumination was so characteristic of the century that followed Newton, historians have given this era an appropriate name: the Enlightenment.