M

a b – c d

Make the circles in fourteenth proportion and remove at most the 6 greatest portions, and do the same to the smallest circle. Beyond this you must understand that such proportion is similar from part to part just as it is similar from whole to whole; and for this reason, the parts being similar, the portion a b is similar to the smaller portion c o. But because the proportion is given in fourteenth proportionality, it is necessary that 14 portions similar to c o be worth one greatest portion similar to a b. Therefore, the 84 smallest portions similar to c o being equal to the value of the 6 greatest portions similar to the portion a b of the largest circle, we find that to remove the 84 smallest portions inside the largest circle is as much as to remove the 6 greatest portions similar to a b, the 84 being of such value against the 6 that it comes to 14 against one, because the said proportion is given in fourteenth proportionality, the manner of making which is shown here below. And by the [...] conception the proposition is concluded, namely that the part removed from its whole does not vary in quantity by varying its position or figure.

a

Here, the sixfold proportionality being in force, the portion a is worth 12 of the smaller portions.

First – i h – d a – b – e – g – c f h m

b a – m p d c n – f e h g

a b c d is the largest circle spoken of below, which is worth 12 of the smallest circles below, e f g h. And the seventy-two smallest portions are similar in figure to the portion m of the largest circle, because each is made of the sixth part of the circumference of its circle.

72

Second – m
Such proportion is from circle to circle as it is from square to square, made by the multiplication of its diameter by itself. The proportion here from one of the smallest circles to the largest circle is twelvefold, or duodecuple; and the same holds from one of the smallest portions to the greatest, m. It follows that the 12 smallest portions of each of these 6 roses are worth one of the 6 greatest portions similar to the portion m. Therefore, if you want the squaring of this circle, you will do it as much by removing the 6 greatest portions similar to m as by removing the 72 smallest portions of the 36 double-angles; and they make the 6 roses.

From this circle one obtains a rectilinear octagon by removing the 6 greatest portions that are similar to the portion m. And if you do not wish to remove such greatest portions, remove the 72 smallest of the 6 roses, which are of equal value to the said 6 greatest portions.

b a – d c

6 times 12 makes 72. Therefore 6 of these stars are worth 72 smallest portions; it follows that the greatest portion of the largest circle is worth 12 of these smallest portions. Now, of these 72 portions I would like to make 84. Therefore I must diminish these 72 portions in such a way that they become 84. And we will do thus: we will place above here the 6 stars with 72 portions, and lay out below the 6 semicircles that denote the 6 diameters of these 6 stars. Then along the length of these same diameters below, c d, we will set the diameters of the seven stars, in which is contained the value of the said 6 stars, which contain the 84 portions. And thus we shall have reduced the 6 stars of 72 rays into 7 stars of 84 rays.

m o

6

14

84