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NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.

illustrate the general principles of the motions of
bodies, then again in book III, in application to the
motions of planets and their satellites or of comets.
While this mode of presentation makes the Principia
more difficult for the reader, it does have the decided
advantage of separating the Newtonian principles as
they apply to the physical universe from the details
of the mathematics from which they derive.

As an example of this separation, proposition 1 of
book III states that the satellites of Jupiter are
“continually drawn off from rectilinear motions, and
are retained in their proper orbits” by forces that
“tend to Jupiter's centre” and that these forces vary
inversely as the square of their distances from that
center. The proof given in this proposition is short
and direct; the centripetal force itself follows from
“Phen. I [of book III], and Prop. II or III, Book I.”
The phenomenon cited is a statement, based upon
“astronomical observations,” that a radius drawn
from the center of Jupiter to any satellite sweeps out
areas “proportional to the times of descriptions”;
propositions 2 and 3 of book I prove by mathematics
that under these circumstances the force
about which such areas are described must be
centripetal and proportional to the times. The
inverse-square property of this force is derived
from the second part of the phenomenon, which
states that the distances from Jupiter's center are
as the 3/2th power of their periods of revolution, and
from corollary 6 to proposition 4 of book I, in which
it is proved that centripetal force in uniform circular
motion must be as the inverse square of the distance
from the center.

Newton's practice of introducing a particular
instance repeatedly, with what may seem to be only
minor variations, may render the Principia difficult
for the modern reader. But the main hurdle for any
would-be student of the treatise lies elsewhere, in
the essential mathematical difficulty of the main
subject matter, celestial mechanics, however presented.
A further obstacle is that Newton's mathematical
vocabulary became archaic soon after the Principia
was published, as dynamics in general and celestial
mechanics in particular came to be written in the
language of differentials and integrals still used today.
The reader is thus required almost to translate for
himself Newton's geometrical-limit mode of proof and
statement into the characters of the analytic algorithms
of the calculus. Even so, dynamics was taught directly
from the Principia at Cambridge until well into the
twentieth century.

In his “Mathematical Principles” Whiteside
describes the Principia as “slipshod, its level of verbal
fluency none too high, its arguments unnecessarily
diffuse and repetitive, and its content on occasion markedly
irrelevant to its professed theme: the theory of
bodies moving under impressed forces.” This view is
somewhat extreme. Nevertheless, the work might
have been easier to read today had Newton chosen
to rely to a greater extent on general algorithms.

The Principia is often described as if it were a
“synthesis,” notably of Kepler's three laws of
planetary motion and Galileo's laws of falling bodies
and projectile motion; but in fact it denies the validity
of both these sets of basic laws unless they be modified.
For instance, Newton showed for the first time the
dynamical significance of Kepler's so-called laws of
planetary motion; but in so doing he proved that
in the form originally stated by Kepler they apply
exactly only to the highly artificial condition of a point
mass moving about a mathematical center of force,
unaffected by any other stationary or moving masses.
In the real universe, these laws or planetary
“hypotheses” are true only to the limits of ordinary
observation, which may very well have been the reason
that Newton called them “Hypotheses” in the first
edition. Later, in the second and third editions, he
referred to these relations as “Phaenomena,” by
which it may be assumed that he now meant that they
were not simply true as stated (that is, not strictly
deducible from the definitions and axioms), but were
rather valid only to the limit of (or within the limits of)
observation, or were phenomenologically true. In
other words, these statements were to be regarded as
not necessarily true, but only contingently (phenomenologically)
so.

In the Principia, Newton proved that Kepler's
planetary hypotheses must be modified by at least two
factors: (1) the mutual attraction of each of any pair
of bodies, and (2) the perturbation of a moving body
by any and all neighboring bodies. He also showed
that the rate of free fall of bodies is not constant, as
Galileo had supposed, but varies with distance from
the center of the earth and with latitude along the
surface of the earth.144 In a scholium at the end of
section 2, book I, Newton further pointed out that it is
only in a limiting case, not really achieved on earth,
that projectiles (even in vacuo) move in Galilean
parabolic trajectories, as Galileo himself knew full
well. Thus, as Karl Popper has pointed out, although
“Newton's dynamics achieved a unification of
Galileo's terrestrial and Kepler's celestial physics,”
it appears that “from a logical point of view, Newton's
theory, strictly speaking, contradicts both Galileo's
and Kepler's.”145

The “Principia”: Definitions and Axioms.

The
Principia opens with two preliminary presentations:
the “Definitions” and the “Axioms, or Laws of