Numbers, proportions and the harmony of spheres

Acoustic experiments
Harmony of the spheres
Pythagorean women
3 pythagoras

Papyrus Berlin: "You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares?"

In modern terms we would express this as x² + y² = 100 and
x = (3/4)y. What are x and y?

A modern solution in this form might be ((3/4)y)² + y² = 100 implies
(1 + 9/16)y² = (25/16)y² = 100 implies
y² =(16/25)100 = 64 implies y=8 and x= (3/4)8 = 6.

However, most translators believe the egyptians viewed this problem the way we do the simultaneous equations
x² + y² = 100
4x - 3y = 0

What are x and y?
Here was their solution. Assume the square of the first side (y) to be 1 cubit. Then the other side (x) will be 1/2 + 1/4. Then y² = 1, and using egyptian multiplication we determine x² with
1 1/2 + 1/4
1/2* 1/4 + 1/8
1/4* 1/8 + 1/16
1/2 + 1/4 1/4 + 1/8 + 1/8 + 1/16 = 1/2 + 1/16

So x² = 1/2 + 1/16. Thus, x² + y² = 1 + 1/2 + 1/16. Now
(1 + 1/2 + 1/16)1/2 = 1 + 1/4 and (100)1/2 = 10. Divide 10 by 1 + 1/4 and you get 8. So we get y=8. The Berlin Papyrus contains damage here so we can at best assume the solution for x was to divide 8 by 1/2 + 1/4 to achieve x=6.