First of all, every concern you express in this last post (#115)
has been addressed already in the one immediately previous, #114.
For instance, you say here in #115:
But in the previous post, I deliberately created two new diagrams,
showing the cone and cone centerline for you (and everyone else).
You just didn't bother to look at that post, before posting your new post.
No one is hiding anything.
The reason I didn't bother with the cone centerline is that
it is not relevant to making a simple calculation as to whether
the disk pulls off-center toward the near end.
Your question is a different question,
namely does the off-center pull or line of force track the cone-centerline.
Whether it does or does not, has no bearing on the question of
whether the direction of pull veers off the Center of Mass.
The pull direction certainly DOES veer off the Center of Mass.
For the purposes of proving that, we MUST use the line that
actually passes through the Center of Mass for comparison.
You have raised a new question, one that it turns out still has
no relevance to either the question of the failure of the Center of Mass concept,
or the question of whether or not the forces 'balance'.
Why does your point have no relevance?
Here is why:
Newton proposed that the forces balanced from opposing cone areas,
in his analogy, for two reasons:
(1) The pull from each was in the exact opposite direction.
For this purpose it is not necessary that the two forces
follow the cone center-line. And in fact they don't.
But as it turns out, the two forces DON'T pull along the same line,
and since these forces don't follow the same line,
these forces DON'T follow the cone center-line either.
The centerline for the TWO cones is the same line, by choice
in the original definition and claim.
(2) The pull from each side was exactly equal.
This was a kludge by Newton caused by a misunderstanding by him
of how the process of calculus using infinitesimals worked.
This is not really surprising, since Newton was actually 'inventing'
the Calculus as he worked, and it is very common among pioneers
and inventors that they don't fully understand their own discoveries.
In fact, it took many more mathematicians centuries to sort out what
Newton and Leibnitz had begun to uncover. But it was a long time
before mathematicians had a good grasp on what the Calculus could do,
and how to apply it, and what it meant. Newton did not have the
understanding of his own rough ideas that we have now.
Our discussion revealed already that Newton's idea fails on both counts.
(1) The pulls from each side do NOT pull in exactly opposite directions.
(2) The pulls from each side are NOT exactly equal.
This failure can be shown:
First, because of the 'twist' of angle on opposing sides,
the two disks of Newton do not pull in the same direction.
Secondly, the use of circular disks by Newton is actually
a poor approximation, so that the pull is not the same strength either.
This second cause of inaccuracy comes from two problems:
(a) The concavity of the real sections, making the flatness a mirage.
(b) The impossibility of tiling the surface of a sphere with circles.
In both cases, Newton had misunderstood his own arguments regarding
the Calculus. Reducing the size of the circular disks in his argument
had no effect on the incompleteness of the tiling problem:
When we tile a plane (or even a uniformly curved surface) with packed circles,
there is always the same percentage of area unused and unaccounted for.
Shrinking the circles makes no difference, no matter how small we make
them. The areas NOT covered by the packed circles remain uncovered,
and the relative sizes of the two areas, covered/uncovered doesn't change.
Newton mistakenly thought that they vanished when the circles were made
'vanishingly small' (i.e., infinitesimals).
There are Chaotic solutions to tiling a plane with different sized circles,
but these don't apply simply to spheres, and in any case, Newton had
no knowledge of those and made no use of them:
secondly, Newton imagined also that making the circles smaller made
the curvature of the sphere negligible. This also was a case where
Newton had fooled himself, because he did not understand his own
discovery and application of the Calculus.
In fact, since the distribution of mass of the sphere does not change,
regardless of how we chop it up, or how many pieces we chop it into,
the force errors caused by the curvature do not vanish.
Newton's error in thought was that if he made the circles small enough,
the error caused by the tilt of each circle became insignificant. In fact,
Newton was right on this, but missed the big picture. Its not the
individual tilts of each circle that matter, but the total distribution of mass
for the surface, which NEVER CHANGES, and this error never decreases
nor can it vanish.
The failure of Newton to understand the effects of the Calculus on
the problem caused him to err in using his analogy as an argument
for the Hollow Sphere Theorem.
Newton was right however, in saying that if the Hollow Sphere Theorem
held, then one could move on to the Solid Sphere Theorem, by constructing
a solid sphere out of shells.
However, a much larger caveat must be stated against your objections;
Your criticisms of the other post, far too late and irrelevant,
are not worth addressing.
has been addressed already in the one immediately previous, #114.
For instance, you say here in #115:
Quote:
|
Show us where, in either one, you have shown the cone centerline. I am not asking to see that cone centerline just because it would help me to visualize things. That cone centerline points directly at the point in the ellipse that defines the division between the “near side” and the “far side” portions of the ellipse. |
But in the previous post, I deliberately created two new diagrams,
showing the cone and cone centerline for you (and everyone else).
You just didn't bother to look at that post, before posting your new post.
No one is hiding anything.
The reason I didn't bother with the cone centerline is that
it is not relevant to making a simple calculation as to whether
the disk pulls off-center toward the near end.
Your question is a different question,
namely does the off-center pull or line of force track the cone-centerline.
Whether it does or does not, has no bearing on the question of
whether the direction of pull veers off the Center of Mass.
The pull direction certainly DOES veer off the Center of Mass.
For the purposes of proving that, we MUST use the line that
actually passes through the Center of Mass for comparison.
You have raised a new question, one that it turns out still has
no relevance to either the question of the failure of the Center of Mass concept,
or the question of whether or not the forces 'balance'.
Why does your point have no relevance?
Here is why:
Newton proposed that the forces balanced from opposing cone areas,
in his analogy, for two reasons:
(1) The pull from each was in the exact opposite direction.
For this purpose it is not necessary that the two forces
follow the cone center-line. And in fact they don't.
But as it turns out, the two forces DON'T pull along the same line,
and since these forces don't follow the same line,
these forces DON'T follow the cone center-line either.
The centerline for the TWO cones is the same line, by choice
in the original definition and claim.
(2) The pull from each side was exactly equal.
This was a kludge by Newton caused by a misunderstanding by him
of how the process of calculus using infinitesimals worked.
This is not really surprising, since Newton was actually 'inventing'
the Calculus as he worked, and it is very common among pioneers
and inventors that they don't fully understand their own discoveries.
In fact, it took many more mathematicians centuries to sort out what
Newton and Leibnitz had begun to uncover. But it was a long time
before mathematicians had a good grasp on what the Calculus could do,
and how to apply it, and what it meant. Newton did not have the
understanding of his own rough ideas that we have now.
Our discussion revealed already that Newton's idea fails on both counts.
(1) The pulls from each side do NOT pull in exactly opposite directions.
(2) The pulls from each side are NOT exactly equal.
This failure can be shown:
First, because of the 'twist' of angle on opposing sides,
the two disks of Newton do not pull in the same direction.
Secondly, the use of circular disks by Newton is actually
a poor approximation, so that the pull is not the same strength either.
This second cause of inaccuracy comes from two problems:
(a) The concavity of the real sections, making the flatness a mirage.
(b) The impossibility of tiling the surface of a sphere with circles.
In both cases, Newton had misunderstood his own arguments regarding
the Calculus. Reducing the size of the circular disks in his argument
had no effect on the incompleteness of the tiling problem:
When we tile a plane (or even a uniformly curved surface) with packed circles,
there is always the same percentage of area unused and unaccounted for.
Shrinking the circles makes no difference, no matter how small we make
them. The areas NOT covered by the packed circles remain uncovered,
and the relative sizes of the two areas, covered/uncovered doesn't change.
Newton mistakenly thought that they vanished when the circles were made
'vanishingly small' (i.e., infinitesimals).
There are Chaotic solutions to tiling a plane with different sized circles,
but these don't apply simply to spheres, and in any case, Newton had
no knowledge of those and made no use of them:
secondly, Newton imagined also that making the circles smaller made
the curvature of the sphere negligible. This also was a case where
Newton had fooled himself, because he did not understand his own
discovery and application of the Calculus.
In fact, since the distribution of mass of the sphere does not change,
regardless of how we chop it up, or how many pieces we chop it into,
the force errors caused by the curvature do not vanish.
Newton's error in thought was that if he made the circles small enough,
the error caused by the tilt of each circle became insignificant. In fact,
Newton was right on this, but missed the big picture. Its not the
individual tilts of each circle that matter, but the total distribution of mass
for the surface, which NEVER CHANGES, and this error never decreases
nor can it vanish.
The failure of Newton to understand the effects of the Calculus on
the problem caused him to err in using his analogy as an argument
for the Hollow Sphere Theorem.
Newton was right however, in saying that if the Hollow Sphere Theorem
held, then one could move on to the Solid Sphere Theorem, by constructing
a solid sphere out of shells.
However, a much larger caveat must be stated against your objections;
You have continued to make a big deal about ellipses, and how I made no mention of them and did not address them in my disproof of Newton's theorem. What you have failed to note is that Newton himself insisted on tiling the surface with circles, which results in the CONES in his argument being NON-circular and irregular, i.e., not radially symmetric. The main reason I used CIRCLES in my disproof, was because NEWTON used CIRCLES in his proof. You can attempt to write your own unique 'proof' of the Sphere Theorem, using ellipses if you want to, and I will be happy to disprove and debunk that too. I'll warn you in advance, that if you don't address the clear criticisms we have made with Newton's original "proof", you are not likely to compose a successful "proof" based on ellipses rather than circles. |
Your criticisms of the other post, far too late and irrelevant,
are not worth addressing.
DavisBJ Responded:
ReplyDeleteI want to avoid us getting into a problem with leap-frogging responses, so I am going to combine my responses to your last 2 posts into this one brief response in which I am asking for clarification rather than disputing what you said.
The first of your last two posts is a substantial improvement in graphics and explanatory text. I followed your logic most of the way, but I am unclear at a couple of points. Since our exchange is necessarily fairly technical, it behooves us to choose words that are accurate. As an example, at one point you said:
Quote:
Originally Posted by Nazaroo View Post
(3) The force itself is not dependent upon the geometry.
That is, only its direction is dependent, dependent upon the angle of the Ellipse.
Force is a vector, and its direction is an essential part of it. You can’t have the vector aspect of the force be dependent on anything without the force being dependent on the same thing.
So let’s tighten up the terminology, and if the point you are making is that the magnitude of the force is not dependent on (whatever), then specify it is the magnitude that you are referring to.
Similarly, when you speak of the “geometry”, first specifically itemize exactly those geometrical aspects of the problem that you want understood the “geometry” to include. For example, when speaking of the angle of the ellipse, do you mean the angle between the ellipse and the cone axis, or the angle between the ellipse and the GC axis?
I would appreciate it if you would take a few minutes and make those types of clarifications in your last two posts (maybe combining the ideas into a new consolidated post) so we are both discussing oranges, and not oranges and apples.
I will take a technical look at your response.
I am not surprised you have chosen to remain silent regarding my inquiry asking if the open literature from NASA indicates they are aware of your contention about gravity inside a shell. No need, I literally needed less than 30 seconds (using Google) to find where NASA isn’t on your side. Years ago you were probably that young soldier marching in a parade whose mother proudly pointed out was the only soldier who was in step.