MHB Can a Philologist Crack Complex Math in Plato's Music Theory?

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A philologist with basic math knowledge seeks assistance on mathematical concepts in Plato's music theory while translating a related text. The discussion reveals an equation that needs rewriting and substitution, leading to a polynomial that can be solved using the quadratic formula. Errors in the original equations are identified, particularly in the numerator and denominator of a derived expression. Participants provide corrections and guidance on the mathematical process. The conversation emphasizes collaboration in understanding complex mathematical ideas within philosophical contexts.
Kobzar
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Hello, everybody:

I am a philologist who is fond of mathematics, but who unfortunately has just an elementary high school knowledge of them. I am translating La leçon de Platon, by Dom Néroman (La Bégude de Mazenc, Arma Artis, 2002), which deals with music theory and mathematics in the works of Plato. The problem which brings me here is not about translation, but about mathematics. Please see attached document.

Any help will be welcome. Thank you very much in advance!

Best regards.

Kobzar.
 

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Rewrite the first equation as $y=\dfrac{M-x^2}{2x}$ and then substitute into the second equation. After quite a bit of multiplying, you will get $5x^4-(2M+4N)x^2+m^2=0$. Use the quadratic formula to solve for $x^2$ and you will get the value for $x^2$ given in the statement. The value for $y^2$ is incorrect. In addition to the error in the denominator that you pointed out, there is another error in the numerator.
 
mrtwhs said:
Rewrite the first equation as $y=\dfrac{M-x^2}{2x}$ and then substitute into the second equation. After quite a bit of multiplying, you will get $5x^4-(2M+4N)x^2+m^2=0$. Use the quadratic formula to solve for $x^2$ and you will get the value for $x^2$ given in the statement. The value for $y^2$ is incorrect. In addition to the error in the denominator that you pointed out, there is another error in the numerator.
Thank you very much for your useful and quick answer!
 
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