To Explain the World: The Discovery of Modern Science (33 page)

Next, by “Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs” Newton meant what we now call Kepler’s third law: that the square of the periods of the planets in their orbits is proportional to the cubes of the mean radii of their orbits, or in other words, the periods are proportional to the
³
/
₂
power (the “sesquialterate proportion”) of the mean radii.
*
The period of a body moving with speed
v
around a circle of radius
r
is the circumference 2π
r
divided by the speed
v
, so for circular orbits Kepler’s third law tells us that
r
²
/
v
²
is proportional to
r
³
, and therefore their inverses are proportional:
v
²
/
r
²
is proportional to 1/
r
³
. It follows that the force keeping the planets in orbit, which is proportional to
v
²
/
r
, must be proportional to 1/
r
²
. This is the inverse square law of gravity.

This in itself might be regarded as just a way of restating Kepler’s third law. Nothing in Newton’s consideration of the planets makes any connection between the force holding the planets in their orbits and the commonly experienced phenomena associated with gravity on the Earth’s surface. This connection was provided by Newton’s consideration of the Moon. Newton’s statement that he “compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly” indicates that he had calculated the centripetal acceleration of the Moon, and found that it was less than the acceleration of falling bodies on the surface of the Earth by just the ratio one would expect if these accelerations were inversely proportional to the square of the distance from the center of the Earth.

To be specific, Newton took the radius of the Moon’s orbit (well known from observations of the Moon’s diurnal parallax) to be 60 Earth radii; it is actually about 60.2 Earth radii. He used a crude estimate of the Earth’s radius,
*
which gave a crude value for the radius of the Moon’s orbit, and knowing that the sidereal period of the Moon’s revolution around the Earth is 27.3 days, he could estimate the Moon’s velocity and from that its centripetal acceleration. This acceleration turned out to be less than the acceleration of falling bodies on the surface of the Earth by a factor roughly (very roughly) equal to 1/(60)
²
, as expected if the force holding the Moon in its orbit is the same that attracts bodies on the Earth’s surface, but reduced in accordance with the inverse square law. (See
Technical Note 33
.) This is what Newton meant when he said that he found that the forces “answer pretty nearly.”

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 resting 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.