Newton's Proposition LXXI
Newton's earliest speculations (in the 1660's) about universal
gravitation involved a rough comparison between the acceleration of a
falling apple near the Earth's surface and the acceleration required
to keep the Moon in its orbit. He found the accelerations to be
"pretty near" inversely proportional to the squares of the distances
from the Earth's center. This is an interesting observation, but it
isn't clear whether this is consistent with the idea of gravity as an
attractive force exerted by every particle of matter on every other
particle of matter. At great distances it's quite plausible that
the attraction between two spherical objects should be closely
expressible in terms of the distances between their centers, but it's
not intuitively obvious that the net force near the surface of a
massive sphere should be inversely proportional to the square of the
distance from the center of the sphere, as if all the mass of the
sphere were located at its center. In fact, Newton told Halley in
1686 (while still at work on the Principia) that he had only succeeded
the previous year in convincing himself that the "duplicate proportion"
(i.e., the inverse-square force of gravity) was actually valid at every
distance from a spherical mass, even down to the surface of the sphere.
The seriousness of his doubts about the validity of this proposition
(which some scholars have claimed was the cause of the 20-year delay
between Newton's first thoughts on universal gravitation and the
composition of Principia) can be judged from his remark to Halley
that
"There is so strong an objection against the accurateness
of this proposition that without my demonstrations... it
cannot be believed by a judicious philosopher to be any
where near accurate."
The problem can obviously be reduced to one involving just a thin
spherical shell of matter, since we can build up a solid sphere from
such shells. Interestingly it's fairly easy to see that the net
force exerted by an inverse-square attraction to the individual
particles of a shell on a point inside the shell is identically
zero (see Inverse Square Force Laws and Orthogonal Polynomials),
and this is suggestive of the unique correspondence between spherical
distributions of matter and inverse-square force laws. When treated
as a simple problem in integration the exterior points can be handled
just as easily as the interior points. The usual approach is to
consider a cross-section of a spherical shell as shown below:
By symmetry the forces perpendicular to the axis cancel out, so we
need consider only the forces in the direction of the axis. The
incremental force exerted by the ring of matter in the region dq,
when rotated about the horizontal axis, is
sin(q) [R - r cos(q)]
dF = 2pi r^2 --------------------- dq (1)
s^3
We also have
R^2 + r^2 - s^2
cos(q) = --------------- (2)
2Rr
and we can take the differentials of both sides to give
s
-sin(q) dq = - --- ds
Rr
Solving this for dq and substituting back into (1), along with the
expression for cos(q) from (2), gives
pi r / R^2 - r^2 \
dF = ---- ( 1 + --------- ) ds
R^2 \ s^2 /
Integrating this from s = R-r to R+r gives
R+r
pi r / / R^2 - r^2 \ 4 pi r^2
F = ---- | ( 1 + --------- ) ds = --------
R^2 / \ s^2 / R^2
R-r
For an interior point the same expression applies, except that the
limits of integration are r-R to r+R, in which case the integral
vanishes and we have F = 0.
Since Newton was in possession of his own version of calculus at this
time, we might expect to find something like this integration in the
Principia, but in fact Newton composed the Principia entirely without
the (exlicit) use of his method of "fluents and fluxions". This is
usually attributed to his preference for the synthetic methods of the
ancients, such as Archimedes, Euclid, and so on, but it's sometimes
suggested that although the Principia contains no explicit calculus
(setting aside the material the Newton added in later editions when
trying to solidify his priority of invention), it nevertheless is
based on the thought processes of calculus, merely disguised. It's
certainly true that Newton made use of reasoning about "ultimate
ratios", etc., but such reasoning has been employed since ancient
times, notably by Archimedes. What distinguishes the modern calculus
is precisely the formalized algorithms and systematic procedures
for dealing with continuous functions (with or without a rigorous
foundation), and these are almost entirely absent from the Principia.
To illustrate this point, let's consider how Newton actually proved
his Proposition 71 concerning the net attraction of a spherical
distribution of matter on an external particle. First we note that
in Proposition 70 he treated the case of interior points simply by
matching the opposite regions on the sphere and showing that they
cancel, but this doesn't apply directly to the case of an exterior
point. So, to finally settle his doubts about the attraction of
spherical bodies, in 1685 he developed a synthetic demonstration
by comparing the forces applied by a spherical shell on two external
points at different distances, and showed that the forces are
inversely proportional to the squares of their distances from the
center of the sphere.
Consider a cross-section of a spherical shell of matter, and two
external particles at P and p, as shown below.
Newton draws lines from P and p through the sphere such that they cut
off equal arcs, which implies that both lines are an equal distance,
denoted by m, from the center of the sphere. In terms of the distances
indicated on this drawing, we know that the forces along the horizontal
axis exerted on these two particles by the ring-shaped slices of the
spheres (on the near side) swept out by rotating about the axis are
W 2pi H B w 2pi h b
F = -------- --- f = ------- ---
D^2 D d^2 d
where W and w are the incremental widths of the rings. Hence the
ratio of these forces is
F W H/D B/D d
--- = --- --- --- -
f w h/d b/d D
Also we know the triangle with edges H,B,D is similar to the triangle
with edges m,D+c,R, and likewise the triangle with edges h,b,d is
similar to the triangle with edges m,d+c,r. Therefore B/D = (D+c)/R
and b/d = (d+c)/r. In addition, H/D = m/R and h/d = m/r. Making
these substitutions, we have
F W m/R (D+c)/R d r^2 W (D+c)d
--- = --- --- ------- - = ----- --- ------
f w m/r (d+c)/r D R^2 w (d+c)D
Now imagine the line through p rotating in the plane and sweeping
over the entire circle to the point of tangency, and likewise the
line through P can rotate and sweep over the entire circle to its
point of tangency. The key idea is that we correlate the rotation
rates of these two lines such that they are always at equal distance
(denoted by m) from the center of the sphere. Hence, a very small
rotation of the line through p results in a certain change in m, and
this implies a corresponding rotation of the line through P. Hence
these lines sweep through equal arc lengths at the radii d+c and D+c,
which implies that the arc lengths they sweep out at radii d and D
respectively are in the ratio d/(d+c) to D/(D+c). Also, since these
lines always make the same angle relative to the surface of the
circle, we know that the widths of the swept rings on the sphere
are in this same proportion, i.e.,
W (d+c)D
--- = ------
w (D+c)d
Substituting this into the preceding expression gives the ratio of
the forces
F r^2
--- = ---
f R^2
It's easy to show that essentially the same analysis applies to the
rings on the opposite side of the tangency point. Hence Newton was
finally able to prove to himself (twenty years after the question
first occurred to him) that, in fact, the force exerted by a spherical
mass on any external particle varies inversely as the square of the
distance of that particle from the center of the sphere, and this
remains exactly true all the way down to the surface of the sphere.
This created a logically coherent foundation for the theory of
universal gravitation.
It's hard not to admire the ingenuity that Newton displayed here and
throughout the Principia, producing time after time a clever synthetic
demonstration of a fact that would (today) be treated by calculus.
On the other hand, it's interesting to note that although Newton was
in possession of his fluents and fluxions since the mid 1660's, he
never took the occasion to settle his question about the force of
attraction of a sphere with an inverse-square law - at least not
until 20 years later, and then he resolved the question by means of
a synthetic demonstration, rather than by explicit use of fluents
and fluxions. It has sometimes been suggested that Newton wrote
two versions of the Principia, first using calculus, and then re-
writing it in the synthetic style. However, in the case of
Proposition 71, it seems to me that Newton's published method is not
very suggestive of the way this problem would be approached using
calculus. The whole strategy of taking the ratios of the forces on
two distinct points, and of splitting up the demonstration into two
parts for internal and external points, seems (to me) to imply that
this synthetic demonstration was the one that Newton referred to in
his comment to Halley, i.e., this is the thought process by which he
(after 20 years) arrived at this conclusion. Despite being one of
the inventors of modern analysis, the classical synthetic mode of
thought seems to have been more to his liking.
Return to MathPages Main Menu
Сайт управляется системой
uCoz