f - n - m. Here you have the portion of the hexagon and the portion of the square, and the hexagon and the square are born of circles each double the other. Now if you wish to see what proportion there is from the portion of the hexagon to the portion of the square, bring together the circles from which such portions are cut, which formerly were double, and see in what proportion they remain, and by this you will see how much they differ from the doubling; and so those portions will vary among themselves, that ...

Since these circles are double, take away the portions from each; then subtract the square from the hexagon, and by as much as the remainder of the hexagon is less than the square, by so much are the six portions joined together less than the 4 portions of the square. Half the circle of the hexagon equals the whole circle of the square.

a; b. a and b equal.

Half the circle of the hexagon equals the whole circle of the square. Remove the portions from all and subtract the remaining squares one from the other, that is, subtract the square of the half-hexagon, and the remainder of that half-hexagon will be equal to that part by which the 4 portions of the square exceed the 3 portions of the hexagon. Divide that excess by 3 and yield the difference, and you will have the lune n of the difference which lies between the single portion of the hexagon and the portion of the square. And because the triangle m equals the lune f, add the quadrature of the lune n that you made to the triangle m, and you will have the quadrature of the lune of the hexagon.

c - d e - a - b - g f. Cast off c on one side and b on the other, which is attached to the lune, and square e, and you will have the quadrature of the lune. f is greater than the lune by the whole of b.

The equalization made of the sector of the sixth of the circle with the semicircle removes the whole portion and a lune remains. Then a piece of the triangle f g is removed and f remains with a, which a is common to the lune as to f, and because a f makes a quarter of the semicircle and the lune remains without b ... a f is the fourth part of the semicircle; a b e is like a f, but because b was taken from the sector b a e, when the whole portion a b was removed from both sectors, I restore it to it in e, so that a e c equals a f, which is the quarter of the semicircle; and here is the portion a b taken from the hexagon, and here is the arc of the lune a b. a b g f is one sixth of the circle and is equal to a b e c d. I remove a b from the one and the other sector; the equal parts remain. I remove d from one and g from the other; f remains equal to e c; and f a was the fourth part of the semicircle, and e c was the lune of the hexagon, equal to the fourth part of that semicircle.

I take a doubled and b single.