b-a-cd-b-c-fea

fe g -cah-dbi
It is 5/12 of the lune b e f, as is shown below in g f i d, the quadrature of that lune, in which square is the quadrature of the smaller lune c, which is a b c d; and e f a c is as much a square as a b e d, and the remainder of the whole square, that is g e i b, is 2/5 of the quadrature e f a c. Therefore the whole is divided into 12 parts similar to g e h a, and the quadrature of c, which is e f a c (or I mean a c b d, which is the same), comes to be divided into 5, which, as I first said, are the 5/12 of the whole square g f i d, that is of the lune above, b e f.

b a -c

a c is a lune of which the quadrature is c b.

c b a

The curves a b and b c are equal to each other, and equal and similar to the opposite curve.

b a - c- d

I have given the quadrature of a surface of three curved sides, which is b c d, equal to a c, whose sides are of similar curvature but not equal.

If you know the half of c d, you have your aim exactly.
Because, the whole being known, the half remains known.

The quadrature of curvilinear surfaces is nothing other than being able to give back on one side that which is taken away from the other, without interval.

o - a c- g b d - r - e f

a b is the half of a b e f by the sixth, whence a b e is equal to e; a b c d is equal to a b e f. I will take d b from a c, and as much I will take from a b e, which is f. There will remain c a, equal to b e. Now I square the whole a c, which in itself is wholly squarable, but c alone, nor a alone, is not squarable, but the whole together is. There remains for us to square b e, which in itself is not squarable, because its sides are curved, and although they are of equal nature of curvature, they are not equal in length.
c a d b is in itself equal to a b e f by the second. I take from c a d b the part d b, and as much I take from a b e f, which is the part f. And by the eighth, taking equal parts from equal quantities, the quantities remain equal, and c a remains equal to a b e. And again I will take a from c and from b e, so that this a comes to be double and a superposition of the two surfaces; and by the seventh, which said: if two equal surfaces be superposed one upon the other etc., therefore by the said eighth, taking a doubled from c and from b e, there remains c equal to b e; but because c is unsquarable, I give back to it a d b, and it remains thus (Figure), and thus I give back to b e its parts. And what has been done up to here is only to mark such parts, and it remains thus (Figure). Having reduced the sums separately, I square the whole c a and b e, and I square e f; there remains a b known, but a is not by itself, nor b by f; but if I can divide it in half with the straight line that is born in the middle of the curve and terminates at the angle g, I have accomplished what was needed.

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