To Explain the World: The Discovery of Modern Science (34 page)

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
m
₁
and
m
₂
. Newton had shown that the gravitational force exerted by body 1 on body 2 is proportional to
m
₂
, and that the force that body 2 exerts on body 1 is proportional to
m
₁
. But according to the third law, these forces are equal in magnitude, and so they must each be proportional to
both m
₁
and
m
₂
.
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
m
₁
and
m
₂
, separated by a distance
r
:

F
=
G
×
m
₁
×
m
₂
/
r
²

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.

Even without knowing the value of
G
, Newton could use his theory of gravitation to calculate the
ratios
of the masses of various bodies in the solar system. (See
Technical Note 35
.) For instance, knowing the ratios of the distances of Jupiter and Saturn from their moons and from the Sun, and knowing the ratios of the orbital periods of Jupiter and Saturn and their moons, he could calculate the ratios of the centripetal accelerations of the moons of Jupiter and Saturn toward their planets and the centripetal acceleration of these planets toward the Sun, and from this he could calculate the ratios of the masses of Jupiter, Saturn, and
the Sun. Since the Earth also has a Moon, the same technique could in principle be used to calculate the ratio of the masses of the Earth and the Sun. Unfortunately, although the distance of the Moon from the Earth was well known from the Moon’s diurnal parallax, the Sun’s diurnal parallax was too small to measure, and so the ratio of the distances from the Earth to the Sun and to the Moon was not known. (As we saw in
Chapter 7
, the data used by Aristarchus and the distances he inferred from those data were hopelessly inaccurate.) Newton went ahead anyway, and calculated the ratio of masses, using a value for the distance of the Earth from the Sun that was no better than a lower limit on this distance, and actually about half the true value. Here are Newton’s results for ratios of masses, given as a corollary to Theorem VIII in Book III of the
Principia
, together with modern values:
¹⁰

As can be seen from this table, Newton’s results were pretty good for Jupiter, not bad for Saturn, but way off for the Earth, because the distance of the Earth from the Sun was not known. Newton was quite aware of the problems posed by observational uncertainties, but like most scientists until the twentieth century, he was cavalier about giving the resulting range of uncertainty in his calculated results. Also, as we have seen with Aristarchus and al-Biruni, he quoted results of calculations to a much greater precision than was warranted by the accuracy of the data on which the calculations were based.

Incidentally, the first serious estimate of the size of the solar system was carried out in 1672 by Jean Richer and Giovanni Domenico Cassini. They measured the distance to Mars by observing the difference in the direction to Mars as seen from Paris and
Cayenne; since the ratios of the distances of the planets from the Sun were already known from the Copernican theory, this also gave the distance of the Earth from the Sun. In modern units, their result for this distance was 140 million kilometers, reasonably close to the modern value of 149.5985 million kilometers for the mean distance. A more accurate measurement was made later by comparing observations at different locations on Earth of the transits of Venus across the face of the Sun in 1761 and 1769, which gave an Earth–Sun distance of 153 million kilometers.
¹¹

In 1797–1798 Henry Cavendish was at last able to measure the gravitational force between laboratory masses, from which a value of
G
could be inferred. But Cavendish did not refer to his measurement this way. Instead, using the well-known acceleration of 32 feet/second per second due to the Earth’s gravitational field at its surface, and the known volume of the Earth, Cavendish calculated that the average density of the Earth was 5.48 times the density of water.

This was in accord with a long-standing practice in physics: to report results as ratios or proportions, rather than as definite magnitudes. For instance, as we have seen, Galileo showed that the distance a body falls on the surface of the Earth is proportional to the square of the time, but he never said that the constant multiplying the square of the time that gives the distance fallen was half of 32 feet/second per second. This was due at least in part to the lack of any universally recognized unit of length. Galileo could have given the acceleration due to gravity as so many
braccia
/second per second, but what would this mean to Englishmen, or even to Italians outside Tuscany? The international standardization of units of length and mass
¹²
began in 1742, when the Royal Society sent two rulers marked with standard English inches to the French Académie des Sciences; the French marked these with their own measures of length, and sent one back to London. But it was not until the gradual international adoption of the metric system, starting in 1799, that scientists had a universally understood system of units. Today we cite a value for
G
of 66.724 trillionths of a meter/second
²
per kilogram: that is, a small body of mass 1 kilogram at a distance of 1 meter produces a gravitational acceleration of 66.724 trillionths of a meter/second per second.