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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.

Section 12 of book I contains Newton's results on
the attractions of spheres, or of spherical shells. He
dealt first with homogeneous, then nonhomogeneous
spheres, the latter being composed of uniform and
concentric spherical shells so that the density is the
same at any single given distance from the center.
In proposition 71 he proved that a “corpuscle”
situated outside such a nonhomogeneous sphere is
“attracted towards the centre of the sphere with a
force inversely proportional to the square of its
distance from the centre.” In proposition 75, he
reached the general conclusion that any two such
spheres will gravitationally attract one another as if
their masses were concentrated at their respective
centers--or, in other words, that the distance required
for the inverse-square law is measured from their
centers. A series of elegant and purely mathematical
theorems follow, including one designed to find the
force with which a corpuscle placed inside a sphere
may be “attracted toward any segment of that sphere
whatsoever.” In section 13, Newton, with a brilliant
display of mathematics (which he did not fully reveal
for the benefit of the reader) discussed the “attractive
forces” of nonspherical solids of revolution, concluding
with a solution in the form of an infinite series for
the attraction of a body “towards a given
plane.”155

Book I concludes with section 14, on the “motion of
very small bodies” acted on by “centripetal forces
tending to the several parts of any very great body.”
Here Newton used the concept of “centripetal forces”
that act under very special conditions to produce
motions of corpuscles that simulate the phenomena
of light--including reflection and refraction (according
to the laws of Snell and Descartes), the inflection of
light (as discovered by Grimaldi), and even the action
of lenses. In a scholium, Newton noted that these
“attractions bear a great resemblance to the reflections
and refractions of light,” and so

. . . because of the analogy there is between the propagation
of the rays of light and the motion of bodies,
I thought it not amiss to add the following Propositions
for optical uses; not at all considering the nature of the
rays of light, or inquiring whether they are bodies or
not; but only determining the curves of [the paths of]
bodies which are extremely like the curves of the rays.

A similar viewpoint with respect to mathematical
analyses (or models and analogies) and physical
phenomena is generally sustained throughout books I
and II of the Principia.

Newton's general plan in book I may thus be seen
as one in which he began with the simplest conditions
and added complexities step by step. In sections 2
and 3, for example, he dealt with a mass-point moving
under the action of a centripetal force directed
toward a stationary or moving point, by which the
dynamical significance of each of Kepler's three laws
of planetary motion is demonstrated. In section 6,
Newton developed methods to compute Keplerian
motion (along an ellipse, according to the law of
areas), which leads to “regular ascent and descent”
of bodies when the force is not uniform (as in Galilean
free fall) but varies, primarily as the inverse square of
the distance, as in Keplerian orbital motion. In
section 8 Newton considered the general case of
“orbits in which bodies will revolve, being acted upon
by any sort of centripetal force.” From stationary
orbits he went on, in section 9, to “movable orbits;
and the motion of the apsides” and to a mathematical
treatment of two (and then three) mutually attractive
bodies. In section 10 he dealt with motion along
surfaces of bodies acted upon by centripetal force;
in section 12, the problems of bodies that are not mere
points or point-masses and the question of the
“attractive forces of spherical bodies”; and in
section 13, “the attractive forces of bodies that are not
spherical.”

Book II of the “Principia.”

Book II, on the motion
of bodies in resisting mediums, is very different from
book I. It was an afterthought to the original treatise,
which was conceived as consisting of only two books,
of which one underwent more or less serious modifications
to become book I as it exists today, while the
other, a more popular version of the “system of the
world,” was wholly transformed so as to become
what is now book III. At first the question of motion
in resisting mediums had been relegated to some
theorems at the end of the original book I; Newton
had also dealt with this topic in a somewhat similar
manner at the end of his earlier tract De motu. The
latter parts of the published book II were added only
at the final redaction of the Principia.

Book II is perhaps of greater mathematical than
physical interest. To the extent that Newton proceeded
by setting up a sequence of mathematical conditions
and then exploring their consequences, book II
resembles book I. But there is a world of difference
between the style of the two books. In book I Newton
made it plain that the gravitational force exists in the
universe, varying inversely as the square of the
distance, and that this force accordingly merits our
particular attention. In book II, however, the reader
is never certain as to which of the many conditions of
resistance that Newton considers may actually occur
in nature.156

Book II enabled Newton to display his mathematical
ingenuity and some of his new discoveries. Occasionally,
as in the static model that he proposed to