*Michael Fowler, Physics Dept., U.Va.*

In 1642, the year Galileo died, Isaac Newton was born in Woolsthorpe,

His mother’s brother, a clergyman who had been an undergraduate at
Cambridge, persuaded his mother that it would be better for Isaac to go to
university, so in 1661 he went up to Trinity College, Cambridge. Isaac paid his way through college for the
first three years by waiting tables and cleaning rooms for the fellows
(faculty) and the wealthier students. In
1664, he was elected a scholar, guaranteeing four years of financial support. Unfortunately, at that time the plague was
spreading across Europe, and reached

On returning to *De Analysi*, expounding his own wider
ranging results. His friend and mentor
Isaac Barrow communicated these discoveries to a

*public* scientific achievement was the invention, design and
construction of a reflecting telescope. He
ground the mirror, built the tube, and even made his own tools for the job. This was a real advance in telescope
technology, and ensured his election to membership in the Royal Society. The mirror gave a sharper image than was
possible with a large lens because a lens focusses different colors at slightly
different distances, an effect called *chromatic aberration*. This problem is minimized nowadays by using
compound lenses, two lenses of different kinds of glass stuck together, that
err in opposite directions, and thus tend to cancel each other’s shortcomings,
but mirrors are still used in large telescopes.

Later in the 1670’s,

In 1684, three members of the Royal Society, Sir Christopher Wren, Robert Hooke and Edmond Halley, argued as to whether the elliptical orbits of the planets could result from a gravitational force towards the sun proportional to the inverse square of the distance. Halley writes:

*Mr. Hook said he had had it, but
that he would conceal it for some time so that others, triing and failing might
know how to value it, when he should make it publick. *

Halley went up to *Principia*
in 1686. This was the book that really
did change man’s view of the universe, as we shall shortly discuss, and its
importance was fully appreciated very quickly.

An excellent, readable book is* The Life of Isaac Newton*, by Richard
Westfall,

A fascinating collection of articles, profusely illustrated, on *Let
Newton Be!*, edited by John Fauvel and
others,

Let us now turn to the central topic of the *Principia*, the
universality of the gravitational force.
The legend is that

The parabolic paths would become flatter and flatter, and, if we imagine
that the mountain is so high that air resistance can be ignored, and the gun is
sufficiently powerful, *eventually the point of landing is so far away that
we must consider the curvature of the earth in finding where it lands. *

In fact, the real situation is more dramatic—the earth’s curvature may mean
the projectile *never lands at all. *This
was envisioned by *Principia*. The following
diagram is from his later popularization, *A Treatise of the System of the
World*, written in the 1680’s:

The mountaintop at V is supposed to be above the earth’s atmosphere, and for
a suitable initial speed, the projectile orbits the earth in a circular path. In fact, the earth’s curvature is such that
the surface falls away below a truly flat horizontal line by about five meters
in 8,000 meters (five miles). Recall
that five meters is just the vertical distance an initially horizontally moving
projectile will fall in the first second of motion. But this implies that if the (horizontal)
muzzle velocity were 8,000 meters per second, the downward fall of the
cannonball would be just matched by the earth’s surface falling away, and it
would never hit the ground! This is just the motion, familiar to us now, of a
satellite in a low orbit, which travels at about 8,000 meters (five miles) a
second, or 18,000 miles per hour. (Actually,

For an animated version of

Newton realized that the moon’s circular path around the earth could be caused in this way by the same gravitational force that would hold such a cannonball in low orbit, in other words, the same force that causes bodies to fall.

To think about this idea, let us consider the moon’s motion, beginning at
some particular instant, as deviating downwards—falling—from some initial
“horizontal” line, just as for the cannonball shot horizontally from a high
mountain. The first obvious question is:
does the moon fall five meters below the horizontal line, that is, towards the
earth, in the first second? This was not difficult for *millimeter*! (Actually around 1.37 millimeters.)

It’s completely impossible to draw a diagram showing how far it falls in one
second, but the geometry is the same if we look how far it falls in one *day*, so here it is:

For one *second*, AB would be only
one kilometer, so since AC is 384,000 km., the triangle ABC is *really* thin, but we can still use
Pythagoras’ theorem!

Thus the “natural acceleration” of the moon towards the earth, measured by how far it falls below straight line motion in one second, is less than that of an apple here on earth by the ratio of five meters to 1.37 millimeters, which works out to be about 3,600.

What can be the significance of this much smaller rate of fall? *force*—a *gravitational
attraction* towards the earth, and that* the gravitational force became
weaker on going away from the earth*.

In fact, the figures we have given about the moon’s orbit enable us to compute how fast the gravitational attraction dies away with distance. The distance from the center of the earth to the earth’s surface is about 6,350 kilometers (4,000 miles), so the moon is about 60 times further from the center of the earth than we and the cannonball are.

From our discussion of how fast the moon falls below a straight line in one second in its orbit, we found that the gravitational acceleration for the moon is down by a factor of 3,600 from the cannonball’s (or the apple’s).

Putting these two facts together, and noting that 3,600 = 60 x 60, led
Newton to his famous *inverse square law*: *the force of gravitational
attraction between two bodies decreases with increasing distance between them
as the inverse of the square of that distance, so if the distance is
doubled, the force is down by a factor of four. *