I'll happily address the oval shape now.
(1) Assume a flat plane which is tilted.
(2) The cone is correctly noted to cut an ellipse rather than a circle, when the plane is tilted.
(3) This makes no difference to the argument that the near end of the (ellipse) has greater change in pull than the far end of the (ellipse).
(4) That is, it makes no difference whether the flat cut-out is a circle tilted or becomes an ellipse.
(5) In either case, due to the Inverse-Square law, the INCREASE in pull from the near side of the object, cannot balance the DECREASE in pull from the far side.
(6) Remember that when the (circle or ellipse) tilts on an axis through its center,
the exact same amount of mass will be on each side of the line of hinging.
(7) Since the masses ARE identical on the near and far halves of the object,
the FORCES CANNOT be identical, due to the non-linear action of the distance law.
No flaw in the argument ensues.
Yes, a circular cone cuts an ellipse on a tilted plane.
No, this cannot save Newton's argument and reasoning (which was actually only
an analogy) in support of the Hollow Sphere Theorem.
note:
I'll try and make this even easier.
We don't need to consider the complex argument of Newton at all.
The Center of Mass hypothesis naturally suggests that a disk or a ring, like a sphere,
acts the same as if all the mass were concentrated at a point at its geometrical center.
Tilting a disk, a ring, or even a flat (uniform density) ellipse on an axis through its center
changes the force it has on a nearby stationary object.
That alone is enough to destroy the Hollow Sphere Theorem Hypothesis.
It is in fact empirically known that rotating a charged or mass ring or disk changes the force on a stationary object outside its perimeter.
Newton's hypothesis (and Gauss's "proof") is literally a shell game.
Remember also that the fundamental disproof for Newton's argument is this:
This will not be affected by the fact that you wish to replace disks with ellipses, or
displace the 'effective center of mass' off-center on an ellipse.
Had the ellipses been twisting in opposite directions, this argument would be clinching,
in that any tilt of the line passing through the centers of both ellipses and through the point inside the sphere,
would be moved in opposing directions, keeping the line straight, and the pulls in direct opposition.
But since in BOTH cases (circles or ellipses) the 'line of sight' through the particle is "bent" toward
the nearest inside of the hollow sphere, and the NET pull is not opposing but toward the near side,
no amount of 'adjusting' the strengths of the pulls from each disk will save the situation.
This really is a 'qualitative' aspect of the problem of balancing forces, namely a NET Direction error,
not a 'quantitative' problem, i.e., 'balancing forces from opposing disks/ellipses/caps.
Look at the diagram again: If the Left side DOES balance, the Right side CANNOT.
(1) Assume a flat plane which is tilted.
(2) The cone is correctly noted to cut an ellipse rather than a circle, when the plane is tilted.
(3) This makes no difference to the argument that the near end of the (ellipse) has greater change in pull than the far end of the (ellipse).
(4) That is, it makes no difference whether the flat cut-out is a circle tilted or becomes an ellipse.
(5) In either case, due to the Inverse-Square law, the INCREASE in pull from the near side of the object, cannot balance the DECREASE in pull from the far side.
(6) Remember that when the (circle or ellipse) tilts on an axis through its center,
the exact same amount of mass will be on each side of the line of hinging.
(7) Since the masses ARE identical on the near and far halves of the object,
the FORCES CANNOT be identical, due to the non-linear action of the distance law.
No flaw in the argument ensues.
Yes, a circular cone cuts an ellipse on a tilted plane.
No, this cannot save Newton's argument and reasoning (which was actually only
an analogy) in support of the Hollow Sphere Theorem.
note:
I'll try and make this even easier.
We don't need to consider the complex argument of Newton at all.
The Center of Mass hypothesis naturally suggests that a disk or a ring, like a sphere,
acts the same as if all the mass were concentrated at a point at its geometrical center.
Tilting a disk, a ring, or even a flat (uniform density) ellipse on an axis through its center
changes the force it has on a nearby stationary object.
That alone is enough to destroy the Hollow Sphere Theorem Hypothesis.
It is in fact empirically known that rotating a charged or mass ring or disk changes the force on a stationary object outside its perimeter.
Newton's hypothesis (and Gauss's "proof") is literally a shell game.
Remember also that the fundamental disproof for Newton's argument is this:
This will not be affected by the fact that you wish to replace disks with ellipses, or
displace the 'effective center of mass' off-center on an ellipse.
Had the ellipses been twisting in opposite directions, this argument would be clinching,
in that any tilt of the line passing through the centers of both ellipses and through the point inside the sphere,
would be moved in opposing directions, keeping the line straight, and the pulls in direct opposition.
But since in BOTH cases (circles or ellipses) the 'line of sight' through the particle is "bent" toward
the nearest inside of the hollow sphere, and the NET pull is not opposing but toward the near side,
no amount of 'adjusting' the strengths of the pulls from each disk will save the situation.
This really is a 'qualitative' aspect of the problem of balancing forces, namely a NET Direction error,
not a 'quantitative' problem, i.e., 'balancing forces from opposing disks/ellipses/caps.
Look at the diagram again: If the Left side DOES balance, the Right side CANNOT.
-------------------------
Lets try again from another angle:
Yes, you are absolutely right that if you merely tilt the intersection of the plane
with the cone, the "far half" of the ellipse grows quite large (ignore a spherical surface here).
Now, where do you want to draw the line of force for this ellipse?
Suppose you don't move the center-line (of force) of the cone at all.
You assume that the pull from each side of the ellipse is balanced,
namely by the accelerating increase in the size of the 'far half' of
the ellipse, and the (slower accelerating area) of the near side.
Now, and this must be your argument,
the force cannot change at all.
The planes can tilt INDEPENDENTLY as long as they cut the central line of force
at the same point.
That is, although an ellipse changes in size and mass as you tilt the plane,
there is no change in the NET force (in either strength or direction!)
as you tilt your plane.
That would be wonderful, but is a far stronger assertion than the
usual Euclidean assertion that the 'force' balances between parallel planes
when double-cones intersect them as in the diagram above/below.
Your argument in essence is that the forces balance in BOTH sides of the picture below,
because the LINES OF FORCE don't change direction or strength, when you change the tilt.
Remember that for the forces to balance, they must stay on a SINGLE line pulling in opposing directions.
If that line is not the center-line of the cone, then where is it? It MUST pass through the cone point(s).
If it goes off-center of the cone-center line, on one side, it MUST go in the OPPOSITE direction in the other cone,
to stay straight.
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