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July 3rd, 2017, 09:24 PM   #1
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Square circumference.......c
Square diagonal............... d

Circle circumference .......... c
Circle diameter .................. d

Subject - Squares: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 \ c 2 = d 1 \ d 2

Subject - Circles: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 \ c 2 = d 1 \ d 2

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July 3rd, 2017, 11:24 PM   #2
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Look cool in my head maybe will put it on paper
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July 4th, 2017, 01:29 AM   #3
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So your problem statement is basically this:

If the circumferences of a square and circle are defined as $\displaystyle c_1$ and $\displaystyle c_2$, the diagonal of a square is defined as $\displaystyle d_1$ and the diameter of a circle is defined as $\displaystyle d_2$, then

$\displaystyle \frac{c_1}{c_2} = \frac{d_1}{d_2}$


This can be disproved by counter-example.

Select $\displaystyle d_1$ and $\displaystyle d_2$ to be a value $\displaystyle a$. We should therefore expect the ratio $\displaystyle \frac{c_1}{c_2}$ to be 1. However:

$\displaystyle c_1 = 4\frac{d_1}{\sqrt{2}} = 2 \sqrt{2} a$

$\displaystyle c_2 = \pi d_2 = \pi a$

Therefore the ratio

$\displaystyle \frac{c_1}{c_2} = \frac{2 \sqrt{2} a}{ \pi a} = \frac{2 \sqrt{2}}{\pi} \approx 0.900316$, which is not equal to 1.
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July 4th, 2017, 03:02 AM   #4
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These are separate proofs

This is separate proof.
Square circumference.......c
Square diagonal............... d

Subject - Squares: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 \ c 2 = d 1 \ d 2


and this is separate proof
Circle circumference .......... c
Circle diameter .................. d

Subject - Circles: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 \ c 2 = d 1 \ d 2

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July 4th, 2017, 07:13 AM   #5
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The proof of the squares

Square circumference.......c
Square diagonal............... d

Subject - Squares: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 : c 2 = d 1 : d 2

c 1 = 3 , d 1 = root of 2 *( 3 : 4 )
c 2 = 137 , d 2 = root of 2 *( 137 : 4 )

c 1 : c 2 = 3 : 137
d 1 : d 2 = root of 2 *( 3 : 4 ) : root of 2 *(137 : 4) = 3 : 137

Now remains to present the proof of circles
Circle circumference.......c
Circle diameter............... d

Subject - Circles: c 1 = 3 mm , c 2 = 137 mm
Prove that: c 1 : c 2 = d 1 : d 2
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July 4th, 2017, 07:28 AM   #6
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Euclid and Archimedes each proved both of those statements.
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July 4th, 2017, 07:40 AM   #7
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Euclid proved about a squares

Archimedes did not prove about circles.
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July 4th, 2017, 07:57 AM   #8
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I understand that he did, via similar triangles and limits. Euclid certainly proved it anyway.
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July 4th, 2017, 08:14 AM   #9
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Right, Euclid proved about squares

but Archimedes did not prove, about circles

The proof of circles c 1 : c 2 = d 1 : d 2 , simply does not exist.
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July 4th, 2017, 08:46 AM   #10
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Yes it does. I linked an example in one of your threads the other day. I'm not surprised that you don't accept it because you refuse to accept $\pi$ as a constant, but that's your problem not anyone else's.
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