Euclid Book 6 Definition 3

[Speedlearner]

According to the definition, A/B = B/C and A = B + C. I don't know any set of numbers that satisfy both equations. I could really use some help.

 

[lookagain]

According to the definition, A/B = B/C and A = B + C. I don't know any set of numbers that satisfy both equations. I could really use some help.

What if I were looking for a particular case where A = B = C?

Then:


A/B =

A - (A intersect B) =

A - A =

{ }

------------------------------------

B/C =

B - (B intersect C) =

B - B =

{ }

------------------------------------

So, here, A/B = B/C = the empty set



And for A = B + C,

A = (B union C)

A = B (because C = B)

A = A (because B = A)

because all of the sets are equal to each other

 

[Speedlearner]

Euclid B6 Definition 3

What if I were looking for a particular case where A = B = C?

Then:


A/B =

A - (A intersect B) =

A - A =

{ }

------------------------------------

B/C =

B - (B intersect C) =

B - B =

{ }

------------------------------------

So, here, A/B = B/C = the empty set



And for A = B + C,

A = (B union C)

A = B (because C = B)

A = A (because B = A)

because all of the sets are equal to each other

The problem with this reasoning becomes apparent when you draw the line. A, B, and C are all lines. A is the length of B + C. So the empty set won't work because you have to draw lines A, B, and C. That's why I need numbers that can do this job. Please try again.

 

[pka]

According to the definition, A/B = B/C and A = B + C. I don't know any set of numbers that satisfy both equations. I could really use some help.

Look at this page. But be careful, they switch C & B.
Read carefully the definition statement: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

In your terms that means \(\displaystyle \dfrac{B+C}{B}=\dfrac{B}{C}\).

For example, let \(\displaystyle B=4\) then \(\displaystyle C=2\sqrt{5}-2\)

 

[lookagain]

The problem with this reasoning becomes apparent when you draw the line.


A, B, and C are all lines. \(\displaystyle \text{No, they are line segments.}\)


A is the length of B + C.

\(\displaystyle \text{You mean the length of line segment A equals the sum}\)
\(\displaystyle \text{of the lengths of line segments B and C.}\)

Note: I thought you were talking about sets, hence my wrong notation for what A/B means.


There is a common ratio for these fractions. Let's call it k and assume k > 0,
because we want positive lengths.


\(\displaystyle \text{*** Warning: There are many missing steps below.}\)


Suppose I choose A to be 100, just to anchor it.

(This is some arbitrary perfect square I chose for some apparent
relative convenience.)


Then 100k = B and Bk = C.



\(\displaystyle So \ 100k^2 = C\)


\(\displaystyle Then \ k = \dfrac{\sqrt{C}}{10}\)


\(\displaystyle And \ B \ = 10\sqrt{C}\)


For A = B + C, we have


\(\displaystyle 100 = 10\sqrt{C} + C **\)


Solve for C:

One of the values is

\(\displaystyle C \ = \150 - 50\sqrt{5}\)


\(\displaystyle And \ then \ B \ = \ 10\sqrt{150 - 50\sqrt{5}}\)


\(\displaystyle A \ = \ 100\)

\(\displaystyle B \ \approx \ 61.8034\)

\(\displaystyle C \ \approx \ 38.1966\)



Using A = 100 and the two approximations for B and C, test the following:


A = B + C ?


A/B and B/C to see how close those ratios are.





\(\displaystyle \text{** Or, I could have worked with the other C value }\)
\(\displaystyle \text{from solving this equation.}\)


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*** Edit: I typed so much work that I did not want to delete this,
despite any posts above mine.

 

[Speedlearner]

My solution

I used Euclid Book 2 Proposition 11 to solve it. The best solution I came up with is A = 2, B = 1.25 and C = .75. I know those numbers aren't exact, but they are close enough to work. I'm sure with a few adjustments, more exact numbers could be found. Thanks for the help.