goldeneagle

I have seen many flat earth models / flat earthers depict the firmament as a curved half-round dome shape.

1) What empirical evidence or evidence from the senses do flat earthers have that aid in the depiction of a curved dome shaped firmament?

2) If it be that flat earthers don't have empirical evidence or conclusive evidence of the shape of the firmament, than why violate the rules of empirical evidence or evidence from the senses to depict something that might not be what it is?   

3) If there be conclusive evidence of a firmement AND a firmement that is shaped like a curved dome, than where can I find this information of conclusive evidence on this site?

4) What are Flat Earthers theories in terms of what the firmament is made of? How thick is it? Is it rigid or flexible? If rigid, than how does the firmament cope with movement of tectonic plates?

5) How is the firmament "attached" to the perimeter of the flat earth?     

Thanks.
« Last Edit: June 24, 2019, 04:23:30 AM by goldeneagle »

The empirical evidence is provided by Newton's quotes:

Newton believed that there are TWO GRAVITATIONAL FORCES AT WORK:

1. Terrestrial gravity

2. Planetary/stellar gravity

Newton still thought that the planets and Sun were kept apart by 'some secret principle of unsociableness in the ethers of their vortices,' and that gravity was due to a circulating ether.

Isaac Newton speculated that gravity was caused by a flow of ether, or space, into celestial bodies. He discussed this theory in letters to Oldenburg, Halley, and Boyle.

Circulating ether = force that keeps the planets/stars orbiting

Flow of ether = terrestrial gravity, a force of pressure

Obviously there must be a shield/barrier between the two different gravitational forces: the dome.

It consists of aether/ether, at a very high density (the Nazis tried to go beyond the dome with their mercury gyro UFOs, but they could not pass through the barrier).

More details on the dome, from the book of Enoch:

https://www.johnpratt.com/items/docs/enoch.html#Enoch_71

https://www.johnpratt.com/items/docs/enoch.html#Enoch_74

https://www.johnpratt.com/items/docs/enoch.html#Enoch_75

https://www.theflatearthsociety.org/forum/index.php?topic=80589.msg2163748#msg2163748

Offline reer

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The empirical evidence is provided by Newton's quotes:

Newton believed that there are TWO GRAVITATIONAL FORCES AT WORK:

1. Terrestrial gravity

2. Planetary/stellar gravity

Newton still thought that the planets and Sun were kept apart by 'some secret principle of unsociableness in the ethers of their vortices,' and that gravity was due to a circulating ether.

Isaac Newton speculated that gravity was caused by a flow of ether, or space, into celestial bodies. He discussed this theory in letters to Oldenburg, Halley, and Boyle.

Circulating ether = force that keeps the planets/stars orbiting

Flow of ether = terrestrial gravity, a force of pressure

Obviously there must be a shield/barrier between the two different gravitational forces: the dome.

It consists of aether/ether, at a very high density (the Nazis tried to go beyond the dome with their mercury gyro UFOs, but they could not pass through the barrier).

More details on the dome, from the book of Enoch:

https://www.johnpratt.com/items/docs/enoch.html#Enoch_71

https://www.johnpratt.com/items/docs/enoch.html#Enoch_74

https://www.johnpratt.com/items/docs/enoch.html#Enoch_75

https://www.theflatearthsociety.org/forum/index.php?topic=80589.msg2163748#msg2163748
You are saying that "quotes" (from Newton) are "empirical evidence". That sure is a new definition of empirical!

And then you throw in the clincher:  the book of Enoch. As I pointed out in another post, you do not seem to accept any evidence less than 100 years old, but this takes it to another level: conclusive evidence that's only 2200 years old!

The book of Enoch was written some 250 years ago, but let us take the official chronology of history at face value.

Then, you are going to have to explain why the ancient Egyptians were using advanced calculus (the extended arctangent infinite series) at Giza:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1834389#msg1834389 (four consecutive messages)

Offline reer

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The book of Enoch was written some 250 years ago, but let us take the official chronology of history at face value.

Then, you are going to have to explain why the ancient Egyptians were using advanced calculus (the extended arctangent infinite series) at Giza:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1834389#msg1834389 (four consecutive messages)
The Round Earth conspiracy has spread all the way to Wikipedia. How did they know to falsify the age of the Book of Enoch, just in case Sandokhan would use it to prove a flat Earth? See

https://en.wikipedia.org/wiki/Book_of_Enoch

As for your link about the pyramids, I stopped reading when I noticed you don't know tangent is a ratio, not a length:

TAN 51.8554 DEGREES = TWO SACRED CUBITS

And I also noticed the Egyptians could measure with sub-micron accuracy:

ONE SACRED CUBIT = 0.6356621 meters

Just tell me which particular atoms on that rock were use to give that distance.

Reference #1

http://davidpratt.info/pyramid.htm

For example, the angle of slope of the Pyramid’s outer casing was 51.85 degrees.


Reference #2

The Pyramid Age, E.J. Sweeney

Chapter 1, page 4

This ratio provides a slope of 51.85 degrees (calculated).


Reference #3

http://stochasticprojectmanagement.com/?p=105

ratio of height to width: 1.571 (one half of pi)  slope: 51.85 degrees


Reference #4

http://www.numberscience.me.uk/Giza.html

The slant angle of the face of the pyramid approximates to 51.85 degrees.


Then, you have a huge problem.

The angle of the slope of the Pyramid’s outer casing is 51.85 degrees.

However, in order to reach/know this value, the architects of the Gizeh Pyramid must have had at their disposal the extended arctangent series:



The VALUE of Tan 51.8554 degrees = the VALUE of two sacred cubits.

The projection on the granite leaf has some two decimal points accuracy, at best: 0.636 meters in length.

However, in 1985 the true height of the Gizeh pyramid was calculated to THE CENTIMETER:



Each one of those circles has a radius of 60 sacred cubits, to the centimeter.

One sacred cubit = 2/π or 7/22, as needed.


Here is the problem you have to solve.

TAN X = 1.27330478216 = 0.636652 x 2

In order to build the pyramid you need the slope angle to at least two decimal places accuracy.

There is no other way to calculate the inverse tangent function of a certain angle (without using a pocket calculator/computer) other than resorting to power series, that is, utilizing calculus. Moreover, one would need a clear understanding of the concept of the radian measure.

The architects of the Giza Pyramid had these choices at their disposal in order to solve the following equation:

TAN X = 1.27330478216 = 0.636652 x 2


1. Maclaurin series in conjunction with the arctan reciprocal formula







(equation #3)


51.8554° = 0.907045 radians

1/1.27330478216 = 0.78535

Substituting the value of 0.78535 in the Maclaurin arctan series and solving the reciprocal arctan equation, up to the O(x11) term we get:

0.905045

This corresponds to a 51.983° value.

Therefore, the builders of the Pyramid must have had at their disposal the notion of the derivative (either the Newton-Leibniz or the Madhava definitions) in order to obtain the arctan Maclaurin series, not to mention the reciprocal arctan equation; even in that case, they had to be able to compute powers of certain numbers, going perhaps all the way to the O(x17) term (in the Maclaurin series) or even beyond, to obtain a meaningful accuracy.


2. Extended arctangent series



This is a result from advanced calculus.


3. Gauss-Pfaff-Borchardt-Carlson iterative formula

http://www.ams.org/journals/mcom/1972-26-118/S0025-5718-1972-0307438-2/S0025-5718-1972-0307438-2.pdf

This formula necessitates the use of the concept of derivatives for its mathematical proof.

https://books.google.ro/books?id=cGnSMGSE5Y4C&pg=PR20&lpg=PR20&dq=numerical+methods+that+work+forman+acton&source=bl&ots=_TWAL76eh8&sig=UoUEc2xjUGxLP0awbJv64HXJG14&hl=ro&sa=X&ved=0ahUKEwjCsci5h4_QAhUJaRQKHcR6CkoQ6AEIXTAH#v=onepage&q=numerical%20methods%20that%20work%20forman%20acton&f=false (pg 6-9)

Other variants of this formula:

http://files.ele-math.com/articles/jmi-09-73.pdf

A more advanced look at this approach:

https://www.math.ust.hk/~machiang/education/enhancement/arithmetic_geometric.pdf


4. My formula

ARCTAN v =  2n x ((2- {2+ [2+ (2+ 2{1/(1+ v2)}1/2)1/2]...1/2}))1/2 (n+1 parentheses to be evaluated)


And there is more.

SIN X = 116.712/188.962 = 0.617648

X = 38.144 degrees, exactly the radius of the three circles used to make the initial calculations for the pyramid

Now, try and solve for X, using only basic arithmetic as a guide.

The controversy created by the measurements of the base of the Giza Pyramid made by C. Piazzi Smyth and W.F. Petrie was solved by D. Davidson who discovered the 286.1 displacement factor of the four sides of the base of the pyramid:



I = 35.8 Pyramid Inches
J = 35.8 Pyramid Inches

286.1/8 = 35.7625

http://thegreatpyramidofgiza.ca/content/#relationship-of-inner-square-circuit-and-outer-square-circuit

http://www.gizapyramid.com/beford%20article%202.htm

http://the-ultimate-frontier.org/history/Pyramid.htm


5.23 + 136.1 + 7.28 = 148.61

5.23 = masonry base = width of the queen's chamber

136.1 = 53.47 x 2.5454

7.28 = 286.1 pyramid inches

[148.612 + (233.424/2)2]1/2 = 188.962

SIN X = 116.712/188.962 = 0.617648

The architects of the Gizeh Pyramid must have had at their disposal the arcsin power series, in order to attain at least three significant digits accuracy.



Not even Bhaskara's formula (or Ptolemy's less accurate table of chords with interpolation approximations) will provide the accuracy needed for the final result (moreover, the second formula involves the radian measure and a very precise approximation of π, using at least 355/113).





Substituting the value of 0.617648 on the left side of the equations and solving for x, will lead to an inaccurate result.

In the official chronology of history, ancient Egyptians could barely work out very simple fractions; the 355/113 approximation to π, not to mention raising a number to the 17th power was way beyond their computational capabilities.


Now, let us put everything together.

51.8554/14.134725 = 11/3

1400/11 = 127.27272727

127.272727 = 63.63636363 x 2

51.8554 x 27 = 1400


51.8554 x 1.618034 = 83.904

1.618034 = PHI

83.904 x 0.6366 = 53.413

53.413 x 0.2548 = 13.61

0.02544 = one sacred inch (0.636/25)

136.1 = height of Gizeh Pyramid without the masonry base


Relationship between the two angles:

The other angle of the triangle, 38.145 degrees, is also closely related to the sacred cubit:

38.13 = 60 sacred cubits

And 51.85/38.1 = 1.361 - therefore, all these measurements/dimensions must have been known well ahead of time to the arhitects of the Gizeh Pyramid; but in order to have the actual angle values, they needed to calculate the arctangent of two sacred cubits.


Now, explain to your readers how the ancient Egyptians knew, ahead of time, that 51.85/38.1 = 1.361, where 51.85 and 38.1 are the values of the two main angles and 136.1 is the height of the pyramid without the masonry base.

How did they solve for the values of 51.85 and 38.1 degrees?
« Last Edit: June 28, 2019, 07:36:19 AM by sandokhan »

Offline reer

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I get it. You start with some fierce looking maths, and then you suddenly throw in:
  I = 35.8 Pyramid Inches
  J = 35.8 Pyramid Inches
Why are they those values? What is a "pyramid inch"? Is it measured with the same ultra-precise tool that measures the distance between some atoms on a rock to give us the "sacred cubit"?

If Egyptians could measure things that accurately, why has no evidence ever been found of their high-tech tools?  I suppose Egyptologists carefully removed all evidence of those tools. And I suppose someone screwed up, so the Antikythera mechanism became public. But, of course, that is nowhere in the same class as the tools you reckon the Egyptians must have had 2000 years earlier.

And, you still ignore the fact that tangent is a RATIO, not a LENGTH. Tan = length/length, so it is dimensionless. Why do you equate a dimensionless number with another that has the dimension of length? Your ancient Egyptians should have known better.
« Last Edit: June 28, 2019, 10:13:58 AM by reer »

The length of the projection is one sacred cubit, 0.636 m. The width of the projection is 2.54 cm. 1sc = 25si (sacred inches).

If Egyptians could measure things that accurately, why has no evidence ever been found of their high-tech tools?

You still don't get it.

You have to explain to your readers HOW the Egyptians knew the values of the angle of the slope and of the complementary angle ahead of time.

TAN X1 = 1.27330478216 = 0.636652 x 2

SIN X2 = 116.712/188.962 = 0.617648


Please explain how to find the values of X1 and X2 using only basic arithmetic.

Using a pocket calculator, an online web calculator, or the tools of advanced calculus, we obtain:

X1 = 51.8554 degrees

X2 = 38.144 degrees


How could they possibly have known that the BEST and ONLY possible choice for the slope angle is 90 - 38.144 degrees?

38.15 = 60 sacred cubits, the radius of the three circles used to construct the Gizeh pyramid.



As our drawing clearly shows, not only the pyramid's envelope but also everything inside
it was determined with the aid of three equal circles. Theodolitic equipment placed within shaft D beamed upward a key vertical line whose function we shall soon describe. But first this equipment beamed out the horizontal rock/masonry line, on which the centers of the three circles were placed. The first of these (Point 1) was at D; Points 2 and 3, where its circle intersected the line, served as centers for the other two, overlapping circles. To draw these circles the pyramid's architects, of course, had to decide on the proper radius.

Our own calculations show that the radius adopted for the three circles envisioned by us was equal to 60 such Sacred Cubits; the number 60 being, not accidentally, the base number of the Sumerian sexagesimal mathematical system. This measure of 60 Sacred Cubits is dominant in the lengths and heights of the pyramid's inner structure as well as in the dimensions of its base.



However, the builders MUST have had ALL of the other measurements of the pyramid (angles, lengths, displacement factor) at their disposal BEFORE proceeding with the drawing of the circles, which must have been a geometrical aid meant to correctly place the other features of the Gizeh Pyramid on a drawing. And this fact, of course, takes us right back to the core the problem: calculation of arctangents and arcsines, using power series approximations.

As if this wasn't enough, 51.8554/38.13 = 1.361, where the height of the pyramid without the masonry base is 136.1 meters.

Here it is not the units that matter, but the values of the various measurements of the pyramid (angles, lengths, height).

How could they have known, IN ADVANCE, that TAN 51.8554 degrees = the value of two sacred cubits (2 x 0.6366)? How could they have known, IN ADVANCE, that the sine of the complementary angle (38.144 degrees) will equal 0.617648?

Do a very simple calculation: 1/(0.617648 x 1sc) = 100si (1.5723/0.617468 = 2.545).

You see, they used the very same VALUE of 38.144 (a dimensionless number) for both the complementary angle (degrees) and the radius of the main three circles (meters).

How could the Egyptians have known that SIN 38.144 = 0.617468, where 38.15 = 60 sacred cubits, and 1/(0.617648 x 1sc) = 100si?

« Last Edit: June 28, 2019, 11:05:36 AM by sandokhan »

TAN X1 = 1.27330478216 = 0.636652 x 2
SIN X2 = 116.712/188.962 = 0.617648
Please explain how to find the values of X1 and X2 using only basic arithmetic.
Using a pocket calculator, an online web calculator, or the tools of advanced calculus, we obtain:
X1 = 51.8554 degrees
X2 = 38.144 degrees

I do Tan(x) and Sin(x) calculations in assembly language, in a 8 bits microcontroller in less than 300 bytes of code, without using any "pocket or online calculator".
Taylor, Maclaurin and my own algorithms can do it easily.
Egyptians had the same brain capacity as Taylor and myself.
Basic Arithmetic:
sin(x) = x - x^3/3! + x^5/5! - x^7/7! ... 
tan(x) = x + x^3/3 + 2x^5/15 + ...

Anyway, this is very far off from the original post, about the evidence that proves the "dome".

Offline reer

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@spherical
Sandokhan always seems to sidetrack you into weird and wonderful maths that are irrelevant to the issue. But he'll prove their relevance by pointing you at some ancient scientific papers. Or to the wiki pages, which is rather incestuous, I think.

He would have to explain how the Egyptians did their advanced maths though, as they only used integers and fractions. Real numbers were introduced at around 800 CE by the Indians. Maybe the Egyptians  used gear driven pocket calculators? And Egyptologists destroy those whenever they are found, to make sure round earth theories get validated. The point is though that, to them as to the ancient Greeks, numbers like 0.617468 did not even exist.