Newton's Principia and Inverse-Square Orbits in a Resisting
Medium: A Spiral of Twisted Logic.
by Robert Weinstock, which has appeared in a recent issue of
Historia Mathematica:
Historia Mathematica, Vol. 25 (1998), No. 3, 281-289.
I presume that subscribers to and readers of Historia Mathematica
are well represented amongst the members of this list, so that it
makes sense to discuss the paper in this forum.
I should point out at the start that the following comments derive
from an analysis of Weinstock's paper, which I came across yesterday.
I have not consulted the actual text of the Principia in either
English or Latin.
-----
Weinstock claims that Newton's argument for Proposition XV/Theorem XII
in Book Two of Newton's Principia is `palpably fallacious',
though three distinguished commentators, King-Hele, Chandrasekhar,
and Erlichson manifestly failed to spot the logical fallacy.
Weinstock concludes his paper with the following:
`AN OPINION
`The Principia's status as the most widely and lavishly
praised of all books on physical science makes imperative,
in my opinion, exposure to the scholarly world of each
nontrivial fault found between its covers - especially when
there are recent publications that threat the fault as if it
were free of disabling error.'
There are two different interpretations of Newton's conclusion
presented by Weinstock: one which he himself propounds, and one
which is forcefully presented by Erlichson. One can summarize
the difference by saying Weinstock takes Newton to assert that
*every* orbit under the prescribed physical conditions is an
equilangular spiral, whereas Erlichson's interpretation merely
asserts that there exist orbits which are equiangular spirals.
I claim that, pace Weinstock, it is possible to establish
Newton's claim, as interpreted by Erlichson, using a proof
which takes as one of its hypotheses the requirement that
the projectile describe an orbit which is an equiangular
spiral. Whilst I have not consulted Newton's argument, I
shall present below my own argument along these lines.
-----
Weinstock quotes the offending proposition from Cajori's revision
of Motte's English translation of the Principia as follows:
PROPOSITION XV. THEOREM XII. If the density of a medium in
each place thereof be inversely as the distance of the places
from an imovable centre, and the centripetal force be as the
square of the density: I say, that a body may revolve in a
spiral which cuts all the radii drawn from that centre in a
given angle.
Weinstock considers the meaning of `may' and concludes that
Newton must have intended this to mean `must':
(The actual wording of 2XV in [13, 282] uses the verb ``may''
in place of ``must'' or ``will'' one normally expects in the
statement of a theorem. The indefinite ``may'' makes no
sense in modern usage; we must conclude that Newton intended
what we today would express as ``must'' or ``will''. It is
the Latin ``potest'' [10, 409] that is translated ``may'' in
[12, 223] by Andrew Motte and maintained by Florian Cajori
in [13, 282] for the respective English statements of 2XV,
the same ``potest'' [10, 411] is rendered as ``will'' be
Motte and is likewise kept be Cajori.)
Now my memories of the Latin I learnt at school are very hazy, but
I do recall that `potest' means `can' or `may'. (The Latin verb
corresponds to the French verb `pouvoir'.) Also I seem to recall
that the translation by Andrew Motte was in no sense an authorized
translation: if one wishes to determine what Newton meant, one
should surely go by what Newton in fact wrote.
I would therefore interpret Newton as claiming that a spiral orbit
is `possible'. (At least this is a *possible* translation of
Newton's Latin text. As an aside, I would note that the English
word `possible' presumably comes from the same Latin root as
`potest'.) Moreover, I note from Weinstock's article that I
would not be alone in making this interpretation: Weinstock
quotes Herman Erlichson as follows: ``Note carefully that Newton
is not claiming that the equiangular spiral is the only orbit, he
is saying that it is at least one of the possible orbits for the
given force condition.'' Weinstock later writes `Erlichson's
fallacy differs from Newton's, we observe,...', but it is by
no means clear to me that there is any distinction between the
respective `fallacies'.
It is worth noting in passing that in fact not every orbit will
be an equiangular spiral: to be convinced of this, one need only
conduct a simple thought-experiment with projectiles moving at
very high velocity through a very tenuous medium.
But is there a substantial fallacy in Newton, as interpreted
by Erlichson?
Could it be the case that Newton has merely noted that the
relevant condition is necessary, in an argument that in fact
demonstrates the condition to be both necessary and sufficient?
If so, is this a major fallacy?
Suppose that we require at the outset that the projectile is moving
in a spiral orbit under some combination of forces, and determine
the nature of the forces that can bring this about. Such an
investigation can show that this possibility, for a projectile moving
under gravity in a resisting medium, is realized if and only if
the density of the resisting medium takes the form E/r, where E is
some constant. One notes in fact that E is determined by the mass
of the the particle and the angle that the spiral makes with the
radius vector, and is obviously proportional to the mass. (Of
course the gravitational force on the projectile is proportional
to the mass of the projectile, and therefore the counterbalancing
frictional force must also be proportional to this mass.) It
follows that, for a resisting material whose density is of the form
E'/r, it is possible for a projectile of mass m' to describe a
spiral orbit, where m'/m = E'/E. We thus obtain the proposition
of Newton, as interpreted by Erlichson.
I claim that, pace Weinstock, there is no logical fallacy involved
in proving Newton's proposition in this fashion under Erlichson's
interpretation, even though such an argument `uses, as an *assumption*,
motion along an equiangular spiral', a type of argument whose
fallacious character, according to Weinstock, cannot be denied.
I append below the argument I constructed last night to obtain the
result in question. Note that I have not attacked such mechanics
problems of this sort since my undergraduate days. I hope that
this will excuse any errors in the argument. Moreover if even I
can come up with a simple solution in an hour or two, it is difficult
to conceive that it would have been beyond Newton's powers, though
Weinstock asks:
`Can it be possible that the Cambridge professor was aware
of his inability to prove 2XV and therefore presented an
intricately-infested counterfeit proof while entertaining
the hopeful expectation that no one would detect the fallacy?'
===================================================================
AN INVESTIGATION OF THE CIRCUMSTANCES UNDER WHICH A PARTICLE MOVING
IN A RESISTING MEDIUM UNDER AN INVERSE SQUARE GRAVITATIONAL FORCE
MAY TRAVERSE A SPIRAL ORBIT
We suppose that a projectile moves inwards along an equiangular
spiral under the action of a central inverse-square force combined
with a resisting force from the medium through which it moves,
and we investigate the nature of the resisting force that would be
necessary and sufficient to achieve this. We proceed by resolving
the forces into components tangential and normal to the spiral.
The tangent to the equiangular spiral makes a constant angle phi
with the radius vector.
Let the central force per unit mass be M / r^2, where M is a constant.
The resisting force from the medium will have no normal component.
Thus
Normal component of acceleration = (M/r^2) sin phi
However the normal acceleration is k v^2, where k is the curvature
of the spiral, and v is the speed of the projectile. The symmetry
properties of the spiral guarantee that the radius of curvature 1/k
at a point on the spiral must be proportional to the distance r of
that point from the centre of attraction. In fact I calculate that
k = (1/r) sin phi
Thus the speed of the projectile must satisfy
v^2 = M / r
The tangential component of acceleration is given by
dv dv dr
-- = -- --
dt dr dt
and moreover
dr
-- = - v cos phi
dt
since the tangential direction makes a constant angle phi with
the radius vector. Thus
dv v
-- = - (1/2) - . ( - v cos phi )
dt r
v^2 M
= --- cos phi = ---- cos phi.
2r 2r^2
On the other hand the gravitational force per unit mass has tangential
component (M/r^2) cos phi.
Thus if the orbit of the projectile is to be an equiangular spiral
then it is necessary and sufficient that the tangential component of
the gravitational force be balanced by a resisting force whose magnitude
is half that of the tangential component of the gravitational force.
We now take the resisting force to be of the form C rho v^2, and let
m be the mass of the projectile. For the projectile to move along
the equiangular spiral it is necessary and sufficient that
C M
- rho v^2 = ---- cos phi,
m 2r^2
i.e.,
m 1
rho = -- . -
2C r