The Principia

What Halley coaxed from Newton is one of the greatest masterpieces in scientific
literature. It is also one of the most inaccessible books ever written. Arguments
in the Principia are presented formally as propositions with (sometimes sketchy)
demonstrations. Some propositions are theorems and others are developed as
illustrative calculations called “problems.” The reader must meet the challenge
of each proposition in sequence to grasp the full argument.
Modern readers of the Principia are also burdened by Newton’s singular mathematical
style. Propositions are stated and demonstrated in the language of geometry,
usually with reference to a figure. (In about five hundred pages, the Principia
has 340 figures, some of them extremely complicated.) To us this seems an
anachronism. By the 1680s, when the Principia was under way, Newton had
already developed his fluxional method of calculus. Why did he not use calculus
to express his dynamics, as we do today?
Partly it was an aesthetic choice. Newton preferred the geometry of the “ancients,”
particularly Euclid and Appolonius, to the recently introduced algebra
of Descartes, which played an essential role in fluxional equations. He found the
geometrical method “much more elegant than that of Descartes . . . [who] attains
the result by means of an algebraic calculus which, if one transcribed it in words
(in accordance with the practice of the Ancients in their writings) is revealed
to be boring and complicated to the point of provoking nausea, and not be
understood.”
There was another problem. Newton could not use the fluxion language he
had invented twenty years earlier for the practical reason that he had never published
the work (and would not publish it for still another twenty years). As the
science historian Franc¸ois De Gandt explains, “[The] innovative character [of the
Principia] was sure to excite controversy. To combine with this innovative character
another novelty, this time mathematical, and to make unpublished procedures
in mathematics the foundation for astonishing physical assertions, was to
risk gaining nothing.”
So Newton wrote the Principia in the ancient geometrical style, modified when
necessary to represent continuous change. But he did not reach his audience.
Only a few of Newton’s contemporaries read the Principia with comprehension,
and following generations chose to translate it into a more transparent, if less
elegant, combination of algebra and the Newton-Leibniz calculus. The fate of the
Principia, like that of some of the other masterpieces of scientific literaturethermodynamics, and Einstein on general relativity), was
to be more admired than read.
The fearsome challenge of the Principia lies in its detailed arguments. In outline,
free of the complicated geometry and the maddening figures, the work is
much more accessible. It begins with definitions of two of the most basic concepts
of mechanics:
Definition 1: The quantity of matter is the measure of the same arising from its
density and bulk conjointly.
Definition 2: The quantity of motion is the measure of the same, arising from
the velocity and quantity of matter conjointly.
By “quantity of matter” Newton means what we call “mass,” “quantity of motion”
in our terms is “momentum,” “bulk” can be measured as a volume, and “density”
is the mass per unit volume (lead is more dense than water, and water more
dense than air). Translated into algebraic language, the two definitions read
m ρV, (12)
and
p mv, (13)
in which mass is represented by m, density by ρ, volume by V, momentum by
p, and velocity by v.
Following the definitions are Newton’s axioms, his famous three laws of motion.
The first is Galileo’s law of inertia:
Law 1: Every body continues in its state of rest, or of uniform motion in a right
[straight] line, unless it is compelled to change that state by forces impressed
upon it.
The second law of motion has more to say about the force concept:
Law 2: The change of motion is proportional to the motive force impressed; and
is made in the direction of the right line in which the force is impressed.
By “change of motion” Newton means the instantaneous rate of change in the
momentum, equivalent to the time derivative . In the modern convention, force
dp
dt
is defined as this derivative, and the equation for calculating a force f is simply
dp
f , (14)
dt
or, with the momentum p evaluated by equation (13),f . (15)
dt
The first two laws convey simple physical messages. Imagine that your car is
coasting on a flat road with the engine turned off. If the car meets no resistance
(for example, in the form of frictional effects), Newton’s first law tells us that the
car will continue coasting with its original momentum and direction forever.
With the engine turned on, and your foot on the accelerator, the car is driven by
the engine’s force, and Newton’s second law asserts that the momentum increases
at a rate ( ) equal to the force. In other words: increase the force by depressing
dp

dt
the accelerator and the car’s momentum increases.
Newton’s third law asserts a necessary constraint on forces operating mutually
between two bodies:
Law 3: To every action there is always opposed an equal reaction: or, the mutual
actions of two bodies upon each other are always equal, and directed to contrary
parts.
Newton’s homely example reminds us, “If you press on a stone with your finger,
the finger is also pressed by the stone.” If this were not the case, the stone would
be soft and not stonelike.
Building from this simple, comprehensible beginning, Newton takes us on a
grand tour of terrestrial and celestial dynamics. In book 1 he assumes an inversesquare
centripetal force and derives Kepler’s three laws. Along the way (in proposition
41), a broad concept that we now recognize as conservation of mechanical
energy emerges, although Newton does not use the term “energy,” and does not
emphasize the conservation theme.
Book 1 describes the motion of bodies (for example, planets) moving without
resistance. In book 2, Newton approaches the more complicated problem of motion
in a resisting medium. This book was something of an afterthought, originally
intended as part of book 1. It is more specialized than the other two books,
and less important in Newton’s grand scheme.
Book 3 brings the Principia to its climax. Here Newton builds his “system of
the world,” based on the three laws of motion, the mathematical methods developed
earlier, mostly in book 1, and empirical raw material available in astronomical
observations of the planets and their moons.
The first three propositions put the planets and their moons in elliptical orbits
controlled by inverse-square centripetal forces, with the planets orbiting the Sun,
and the moons their respective planets. These propositions define the centripetal
forces mathematically but have nothing to say about their physical nature.
Proposition 4 takes that crucial step. It asserts “that the Moon gravitates towards
the earth, and is always drawn from rectilinear [straight] motion, and held
back in its orbit, by the force of gravity.” By the “force of gravity” Newton means
the force that causes a rock (or apple) to fall on Earth. The proposition tells us
that the Moon is a rock and that it, too, responds to the force of gravity.
Newton’s demonstration of proposition 4 is a marvel of simplicity. First, from
the observed dimensions of the Moon’s orbit he concludes that to stay in its orbit
the Moon falls toward Earth 15.009 “Paris feet” ( 16.000 of our feet) everysecond. Then, drawing on accurate pendulum data observed by Huygens, he
calculates that the number of feet the Moon (or anything else) would fall in one
second on the surface of Earth is 15.10 Paris feet. The two results are close
enough to each other to demonstrate the proposition.
Proposition 5 simply assumes that what is true for Earth and the Moon is true
for Jupiter and Saturn and their moons, and for the Sun and its planets.
Finally, in the next two propositions Newton enunciates his universal law of
gravitation. I will omit some subtleties and details here and go straight to the
algebraic equation that is equivalent to Newton’s inverse-square calculation of
the gravitational attraction force F between two objects whose masses are m1 and
m2,
m m F G 1 2, (16) r2
where r is the distance separating the centers of the two objects, and G, called
the “gravitational constant,” is a universal constant. With a few exceptions, involving
such bizarre objects as neutron stars and black holes, this equation applies
to any two objects in the universe: planets, moons, comets, stars, and galaxies.
The gravitational constant G is always given the same value; it is the
hallmark of gravity theory. Later in our story, it will be joined by a few other
universal constants, each with its own unique place in a major theory.
In the remaining propositions of book 3, Newton turns to more-detailed problems.
He calculates the shape of Earth (the diameter at the equator is slightly
larger than that at the poles), develops a theory of the tides, and shows how to
use pendulum data to demonstrate variations in weight at different points on
Earth. He also attempts to calculate the complexities of the Moon’s orbit, but is
not completely successful because his dynamics has an inescapable limitation:
it easily treats the mutual interaction (gravitational or otherwise) of two bodies,
but offers no exact solution to the problem of three or more bodies. The Moon’s
orbit is largely, but not entirely, determined by the Earth-Moon gravitational attraction.
The full calculation is a “three-body” problem, including the slight effect
of the Sun. In book 3, Newton develops an approximate method of calculation
in which the Earth-Moon problem is first solved exactly and is then modified
by including the “perturbing” effect of the Sun. The strategy is one of successive
approximations. The calculations dictated by this “perturbation theory” are tedious,
and Newton failed to carry them far enough to obtain good accuracy. He
complained that the prospect of carrying the calculations to higher accuracy
“made his head ache.”
Publication of the Principia brought more attention to Newton than to his
book. There were only a few reviews, mostly anonymous and superficial. As De
Gandt writes, “Philosophers and humanists of this era and later generations had
the feeling that great marvels were contained in these pages; they were told that
Newton revealed truth, and they believed it. . . . But the Principia still remained
a sealed book.”