To Explain the World: The Discovery of Modern Science

by Steven Weinberg

  This was the climactic step in the unification of the celestial and terrestrial in science. Copernicus had placed the Earth among the planets, Tycho had shown that there is change in the heavens, and Galileo had seen that the Moon’s surface is rough, like the Earth’s, but none of this related the motion of planets to forces that could be observed on Earth. Descartes had tried to understand the motions of the solar system as the result of vortices in the ether, not unlike vortices in a pool of water on Earth, but his theory had no success. Now Newton had shown that the force that keeps the Moon in its orbit around the Earth and the planets in their orbits around the Sun is the same as the force of gravity that causes an apple to fall to the ground in Lincolnshire, all governed by the same quantitative laws. After this the distinction between the celestial and terrestrial, which had constrained physical speculation from Aristotle on, had to be forever abandoned. But this was still far short of a principle of universal gravitation, which would assert that every body in the universe, not just the Earth and Sun, attracts every other body with a force that decreases as the inverse square of the distance between them.

  There were still four large holes in Newton’s arguments:

  1. In comparing the centripetal acceleration of the Moon with the acceleration of falling bodies on the surface of the Earth, Newton had assumed that the force producing these accelerations decreases with the inverse square of the distance, but the distance from what? This makes little difference for the motion of the Moon, which is so far from the Earth that the Earth can be taken as almost a point particle as far as the Moon’s motion is concerned. But for an apple falling to the ground in Lincolnshire, the Earth extends from the bottom of the tree, a few feet away, to a point at the antipodes, 8,000 miles away. Newton had assumed that the distance relevant to the fall of any object near the Earth’s surface is its distance to the center of the Earth, but this was not obvious.

  2. Newton’s explanation of Kepler’s third law ignored the obvious differences between the planets. Somehow it does not matter that Jupiter is much bigger than Mercury; the difference in their centripetal accelerations is just a matter of their distances from the Sun. Even more dramatically, Newton’s comparison of the centripetal acceleration of the Moon and the acceleration of falling bodies on the surface of the Earth ignored the conspicuous difference between the Moon and a falling body like an apple. Why do these differences not matter?

  3. In the work he dated to 1665–1666, Newton interpreted Kepler’s third law as the statement that the products of the centripetal accelerations of the various planets with the squares of their distances from the Sun are the same for all planets. But the common value of this product is not at all equal to the product of the centripetal acceleration of the Moon with the square of its distance from the Earth; it is much greater. What accounts for this difference?

  4. Finally, in this work Newton had taken the orbits of the planets around the Sun and of the Moon around the Earth to be circular at constant speed, even though as Kepler had shown they are not precisely circular but instead elliptical, the Sun and Earth are not at the centers of the ellipses, and the Moon’s and planets’ speeds are only approximately constant.

  Newton struggled with these problems in the years following 1666. Meanwhile, others were coming to the same conclusions that Newton had already reached. In 1679 Newton’s old adversary Hooke published his Cutlerian lectures, which contained some suggestive though nonmathematical ideas about motion and gravitation:

  First, that all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers, whereby they attract not only their own parts, and keep them from flying from them, as we may observe the Earth to do, but that they do also attract all the other Coelestial Bodies that are within the sphere of their activity—The second supposition is this, That all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent into a Motion, describing a Circle, Ellipsis, or some other more compounded Curve Line. The third supposition is, That these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers.6

  Hooke wrote to Newton about his speculations, including the inverse square law. Newton brushed him off, replying that he had not heard of Hooke’s work, and that the “method of indivisibles”7 (that is, calculus) was needed to understand planetary motions.

  Then in August 1684 Newton received a fateful visit in Cambridge from the astronomer Edmund Halley. Like Newton and Hooke and also Wren, Halley had seen the connection between the inverse square law of gravitation and Kepler’s third law for circular orbits. Halley asked Newton what would be the actual shape of the orbit of a body moving under the influence of a force that decreases with the inverse square of the distance. Newton answered that the orbit would be an ellipse, and promised to send a proof. Later that year Newton submitted a 10-page document, On the Motion of Bodies in Orbit, which showed how to treat the general motion of bodies under the influence of a force directed toward a central body.

  Three years later the Royal Society published Newton’s Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), doubtless the greatest book in the history of physical science.

  A modern physicist leafing through the Principia may be surprised to see how little it resembles any of today’s treatises on physics. There are many geometrical diagrams, but few equations. It seems almost as if Newton had forgotten his own development of calculus. But not quite. In many of his diagrams one sees features that are supposed to become infinitesimal or infinitely numerous. For instance, in showing that Kepler’s equal-area rule follows for any force directed toward a fixed center, Newton imagines that the planet receives infinitely many impulses toward the center, each separated from the next by an infinitesimal interval of time. This is just the sort of calculation that is made not only respectable but quick and easy by the general formulas of calculus, but nowhere in the Principia do these general formulas make their appearance. Newton’s mathematics in the Principia is not very different from what Archimedes had used in calculating the areas of circles, or what Kepler had used in calculating the volumes of wine casks.

  The style of the Principia reminds the reader of Euclid’s Elements. It begins with definitions:8

  Definition I

  Quantity of matter is a measure of matter that arises from its density and volume jointly.

  What appears in English translation as “quantity of matter” was called massa in Newton’s Latin, and is today called “mass.” Newton here defines it as the product of density and volume. Even though Newton does not define density, his definition of mass is still useful because his readers could take it for granted that bodies made of the same substances, such as iron at a given temperature, will have the same density. As Archimedes had shown, measurements of specific gravity give values for density relative to that of water. Newton notes that we measure the mass of a body from its weight, but does not confuse mass and weight.

  Definition II

  Quantity of motion is a measure of motion that arises from the velocity and the quantity of matter jointly.

  What Newton calls “quantity of motion” is today termed “momentum.” It is defined here by Newton as the product of the velocity and the mass.

  Definition III

  Inherent force of matter [vis insita] is the power of resisting by which every body, so far as [it] is able, perseveres in its state either of resting or of moving uniformly straight forward.

  Newton goes on to explain that this force arises from the body’s mass, and that it “does not differ in any way from the inertia of the mass.” We sometimes now distinguish mass, in its role as the quantity that resists changes in motion, as “inertial mass.”

  Definition IV

  Impressed force is the action exerted on a body to change its state either of res
ting or of uniformly moving straight forward.

  This defines the general concept of force, but does not yet give meaning to any numerical value we might assign to a given force. Definitions V through VIII go on to define centripetal acceleration and its properties.

  After the definitions comes a scholium, or annotation, in which Newton declines to define space and time, but does offer a description:

  I. Absolute, true, and mathematical time, in and of itself, and of its own nature, without relation to anything external, flows uniformly. . . .

  II. Absolute space, of its own nature without relation to anything external, always remains homogeneous and immovable.

  Both Leibniz and Bishop George Berkeley criticized this view of time and space, arguing that only relative positions in space and time have any meaning. Newton had recognized in this scholium that we normally deal only with relative positions and velocities, but now he had a new handle on absolute space: in Newton’s mechanics, acceleration (unlike position or velocity) has an absolute significance. How could it be otherwise? It is a matter of common experience that acceleration has effects; there is no need to ask, “Acceleration relative to what?” From the forces pressing us back in our seats, we know that we are being accelerated when we are in a car that suddenly speeds up, whether or not we look out the car’s window. As we will see, in the twentieth century the views of space and time of Leibniz and Newton were reconciled in the general theory of relativity.

  Then at last come Newton’s famous three laws of motion:

  Law I

  Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

  This was already known to Gassendi and Huygens. It is not clear why Newton bothered to include it as a separate law, since the first law is a trivial (though important) consequence of the second.

  Law II

  A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

  By “change of motion” here Newton means the change in the momentum, which he called the “quantity of motion” in Definition II. It is actually the rate of change of momentum that is proportional to the force. We conventionally define the units in which force is measured so that the rate of change of momentum is actually equal to the force. Since momentum is mass times velocity, its rate of change is mass times acceleration. Newton’s second law is thus the statement that mass times acceleration equals the force producing the acceleration. But the famous equation F = ma does not appear in the Principia; the second law was reexpressed in this way by Continental mathematicians in the eighteenth century.

  Law III

  To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal, and always opposite in direction.

  In true geometric style, Newton then goes on to present a series of corollaries deduced from these laws. Notable among them was Corollary III, which gives the law of the conservation of momentum. (See Technical Note 34.)

  After completing his definitions, laws, and corollaries, Newton begins in Book I to deduce their consequences. He proves that central forces (forces directed toward a single central point) and only central forces give a body a motion that sweeps out equal areas in equal times; that central forces proportional to the inverse square of the distance and only such central forces produce motion on a conic section, that is, a circle, an ellipse, a parabola, or a hyperbola; and that for motion on an ellipse such a force gives periods proportional to the 3/2 power of the major axis of the ellipse (which, as mentioned in Chapter 11, is the distance of the planet from the Sun averaged over the length of its path). So a central force that goes as the inverse square of the distance can account for all of Kepler’s laws. Newton also fills in the gap in his comparison of lunar centripetal acceleration and the acceleration of falling bodies, proving in Section XII of Book I that a spherical body, composed of particles that each produce a force that goes as the inverse square of the distance to that particle, produces a total force that goes as the inverse square of the distance to the center of the sphere.

  There is a remarkable scholium at the end of Section I of Book I, in which Newton remarks that he is no longer relying on the notion of infinitesimals. He explains that “fluxions” such as velocities are not the ratios of infinitesimals, as he had earlier described them; instead, “Those ultimate ratios with which quantities vanish are not actually ratios of ultimate quantities, but limits which the ratios of quantities decreasing without limit are continually approaching, and which they can approach so closely that their difference is less than any given quantity.” This is essentially the modern idea of a limit, on which calculus is now based. What is not modern about the Principia is Newton’s idea that limits have to be studied using the methods of geometry.

  Book II presents a long treatment of the motion of bodies through fluids; the primary goal of this discussion was to derive the laws governing the forces of resistance on such bodies.9 In this book he demolishes Descartes’ theory of vortices. He then goes on to calculate the speed of sound waves. His result in Proposition 49 (that the speed is the square root of the ratio of the pressure and the density) is correct only in order of magnitude, because no one then knew how to take account of the changes in temperature during expansion and compression. But (together with his calculation of the speed of ocean waves) this was an amazing achievement: the first time that anyone had used the principles of physics to give a more or less realistic calculation of the speed of any sort of wave.

  At last Newton comes to the evidence from astronomy in Book III, The System of the World. At the time of the first edition of the Principia, there was general agreement with what is now called Kepler’s first law, that the planets move on elliptical orbits; but there was still considerable doubt about the second and third laws: that the line from the Sun to each planet sweeps out equal areas in equal times, and that the squares of the periods of the various planetary motions go as the cubes of the major axes of these orbits. Newton seems to have fastened on Kepler’s laws not because they were well established, but because they fitted so well with his theory. In Book III he notes that Jupiter’s and Saturn’s moons obey Kepler’s second and third laws, that the observed phases of the five planets other than Earth show that they revolve around the Sun, that all six planets obey Kepler’s laws, and that the Moon satisfies Kepler’s second law.* His own careful observations of the comet of 1680 showed that it too moved on a conic section: an ellipse or hyperbola, in either case very close to a parabola. From all this (and his earlier comparison of the centripetal acceleration of the Moon and the acceleration of falling bodies on the Earth’s surface), he comes to the conclusion that it is a central force obeying an inverse square law by which the moons of Jupiter and Saturn and the Earth are attracted to their planets, and all the planets and comets are attracted to the Sun. From the fact that the accelerations produced by gravitation are independent of the nature of the body being accelerated, whether it is a planet or a moon or an apple, depending only on the nature of the body producing the force and the distance between them, together with the fact that the acceleration produced by any force is inversely proportional to the mass of the body on which it acts, he concludes that the force of gravity on any body must be proportional to the mass of that body, so that all dependence on the body’s mass cancels when we calculate the acceleration. This makes a clear distinction between gravitation and magnetism, which acts very differently on bodies of different composition, even if they have the same mass.

  Newton then, in Proposition 7, uses his third law of motion to find out how the force of gravity depends on the nature of the body producing the force. Consider two bodies, 1 and 2, with masses m1 and m2. Newton had shown that the gravitational force exerted by body 1 on body 2 is proportional to m2, and that the force that body 2 exerts on bo
dy 1 is proportional to m1. But according to the third law, these forces are equal in magnitude, and so they must each be proportional to both m1 and m2. Newton was able to confirm the third law in collisions but not in gravitational interactions. As George Smith has emphasized, it was many years before it became possible to confirm the proportionality of gravitational force to the inertial mass of the attracting as well as the attracted body. Nevertheless, Newton concluded, “Gravity exists in all bodies universally and is proportional to the quantity of matter in each.” This is why the products of the centripetal accelerations of the various planets with the squares of their distances from the Sun are much greater than the product of the centripetal acceleration of the Moon with the square of its distances from the Earth: it is just that the Sun, which produces the gravitational force on the planets, is much more massive than the Earth.

  These results of Newton are commonly summarized in a formula for the gravitational force F between two bodies, of masses m1 and m2, separated by a distance r:

  F = G × m1 × m2 / r2

  where G is a universal constant, known today as Newton’s constant. Neither this formula nor the constant G appears in the Principia, and even if Newton had introduced this constant he could not have found a value for it, because he did not know the mass of the Sun or the Earth. In calculating the motion of the Moon or the planets, G appears only as a factor multiplying the mass of the Earth or the Sun, respectively.