* Let there be given any known part that may be asked for in a semicircle, or in any other portion of a circle.

c - d - e - b a

* a b is a whole portion, a c is the half, a d is the quarter, a e is 1/8, and so it continues to infinity.

n - m

* The whole squarable part – that is, reduced within straight lines – which lies beneath the part of the portion, can be reduced to a square, as is shown in portion n, which has beneath it the square m.

Portion and half-portion joined, with: a – c S – t – b

* I take away a and set its value in b, and I have squared a surface of two curved and equal sides. Therefore a c is equal to c b.

* The a c S t is a part of the parallel [ring] that is equal to the circle it surrounds; and if it is an aliquot part of that ring, I will put the ring together with its like parts; and if it is not an aliquot part, I will make it by cutting away its sides S t with rectilinear parts.

o – r – S o c - t r

r o is equal to c S.

o – p n – n S - q; r t; h
I will take n o from the curvilinear parallel and put it together into the figure r t above, and there will remain the rectilinear triangle p, which I will subtract from triangle q, and there will remain of q a part equal to the squarable r t.

But if I take q from q S, there will remain the half-portion S; and if from the parallel I take a surface equal to q, which will be p, there will remain n o equal to S; and because n o is placed in r t, to which is given a triangle equal to it, namely h, it follows that h equals the half-portion S.

b a – c

If a b equals the half-portion b c, then, taking b from a b and b from b c, there remains the rectilinear triangle c equal to triangle a, which has one convex side.

Half-portion superimposed on a portion, with: f - n - m - o d – c – p e r b - a

* The portion a b c d e equals the half-portion r f p, and the two portions that project above the half-portion r f p equal the four portions of the portion a c e. Therefore the half-portion f n equals the whole portion d e; whence I will take the half-portion f n from its place and render below the same value in d e, and I shall have made the surface f p d e, which will have straightened the curve d e without diminishing its former value; which I shall divide into 4 equal parts, distributing within them the straightness of the portion that I have rendered below.

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