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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.
Book I of the “Principia.”
far beyond that commonly associated with the
Principia. In the ancillary proposition 40, for example,
Newton (again under the most general conditions
of force) had sought the velocity at a point on an
orbit, finding a result that is the equivalent of an
integral, which (in E. J. Aiton's words) in “modern
terms ... expresses the invariance of the sum of the
kinetic and gravitational potential energies in an
orbit.”153
In section 11, Newton reached a level of mathematical
analysis of celestial motions that fully
distinguishes the Principia from any of its predecessors.
Until this point, he there explained, he had
been “treating of the attractions of bodies towards an
immovable centre; though very probably there is no
such thing existent in nature.” He then outlined a plan
to deal with nature herself, although in a “purely
mathematical” way, “laying aside all physical
considerations”—such as the nature of the gravitating
force. “Attractions” are to be treated here as
originating in bodies and acting toward other bodies;
in a two-body system, therefore, “neither the attracted
nor the attracting body is truly at rest, but both ...
being as it were mutually attracted, revolve about a
common centre of gravity.” In general, for any system
of bodies that mutually attract one another, “their
common centre of gravity will either be at rest,
or move uniformly” in a straight line. Under these conditions,
both members of a pair of mutually attractive
bodies will describe “similar figures about their
common centre of gravity, and about each other
mutually” (proposition 57).
By studying such systems, rather than a single body
attracted toward a point-center of force, Newton
proved that Kepler's laws (or “planetary hypotheses”)
cannot be true within this context, and hence need
modification when applied to the real system of the
world. Thus, in proposition 59, Newton stated that
Kepler's third law should not be written
T12:T22 =
a13:a23, as
Kepler, Hooke, and everybody else had
supposed, but must be modified.
A corollary that may be drawn from the proposition
is that the law might be written as (M +
m1)T12:(M +
m2)T22:a13:a23, where m1,
m2 are any two
planetary masses and M is the mass of the sun.
(Newton's expression of this new relation may be
reduced at once to the more familiar form in which
we use this law today.) Clearly, it follows from
Newton's analysis and formulation that Kepler's
own third law may safely be used as an approximation
in most astronomical calculations only because m1,
m2 are very small in relation to M. Newton's
modification
of Kepler's third law fails to take account of
any possible interplanetary perturbations. The chief
function of proposition 59 thus appears to be not to
reach the utmost generalization of that law, but
rather to reach a result that will be useful in the
problems that follow, most notably proposition 60
(on the orbits described when each of two bodies
attracts the other with a force proportional to the
square of the distance, each body “revolving about the
common centre of gravity”).
From proposition 59 onward, Newton almost at
once advanced to various motions of mutually
attractive bodies “let fall from given places” (in
proposition 62), “going off from given places in given
directions with given velocities” (proposition 63), or
even when the attractive forces “increase in a simple
ratio of their [that is, the bodies'] distances from the
centres” (proposition 64). This led him to examine
Kepler's first two laws for real “bodies,” those
“whose forces decrease as the square of their distances
from their centres.” Newton demonstrated in proposition
65 that in general it is not “possible that bodies
attracting each other according to the law supposed
in this proposition should move exactly in ellipses,”
because of interplanetary perturbations, and discussed
cases in astronomy in which “the orbits will
not much differ from ellipses.” He added that the
areas described will be only “very nearly proportional
to the times.”
Proposition 66 presents the restricted three-body
problem, developed in a series of twenty-two corollaries.
Here Newton attempted to apply the law of
mutual gravitational attraction to a body like the sun
to determine how it might perturb the motion of a
moonlike body around an earthlike body. Newton
examined the motion in longitude and in latitude, the
annual equation, the evection, the change of the
inclination of the orbit of the body resembling the
moon, and the motion on the line of apsides. He
considered the tides and explained, in corollary 22,
that the internal “constitution of the globe” (of the
earth) can be known “from the motion of the nodes.”
He further demonstrated that the shape of the globe
can be derived from the precession constant
(precession being caused, in the case of the earth,
by the pull of the moon on the equatorial bulge of the
spinning earth). He thus established, for the first time,
a physical theory, elaborated in mathematical
expression, from which some of the “inequalities” of
the motion of the moon could be deduced; and he
added some hitherto unknown “inequalities” that
he had found. Previous to Newton's work, the study
of the irregularities in the motion of the moon had
been posited on the elaboration of geometric models,
in an attempt to make predicted positions agree with
actual observations.154