Proportion of Circles and Squares from Their Diameters
Doubling a diameter quadruples both square and circle; lunes between inscribed circles
A geometry sheet setting out a numbered series of demonstrations (Prima to Quinta) proving that the ratio between two circles equals the ratio between the squares built on their diameters, so that doubling the diameter quadruples both the square and the circle. Further figures show two circles inscribed and touching within a larger circle, the curved-sided triangles (lunes) left between them, and the squaring of these curvilinear areas into equivalent squares. A large interlaced rosette built of circles is drawn at the lower left, with a column of small paired-circle and inscribed-square figures down the right margin.
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Circle and square grow as the square of the diameter (Prima)
Multiplying the diameter of the smaller circle a b by itself gives the smaller square a b c d; doubling that diameter to b e and squaring it gives the larger square a e f g, four times the smaller. Hence the larger circle a e is worth four times the smaller circle a b, and the greatest circle is divided into four parts equal in value but various in figure (the two curved-sided triangles n m).
Two circles touching within a third
If two circles touch one another and are touched by the circumference of a surrounding circle, the more unequal they are the more space they occupy within that circumference; conversely, the more alike they are, the less space they occupy. A pair of inscribed circles illustrates the rule.
The field between the outer circle and two inner circles
The greatest space contained between the circumference of the larger circle and the two smaller circles it encloses is worth the two smaller circles themselves. With the aid of the first demonstration the two circles a b are shown equal to the value of their field c d.
Three sorts of curvature: circle, bisangle, and lune (Quarta)
The larger circle has curvature double that of the two bisangles, which in turn is double that of the two smaller circles. Removing the smaller circles halves the great circle's value; removing the two bisangles leaves half of its greatest square, as shown in the marginal fifth figure where square e f n m is half of the enclosing square a b c d.
Doubled squares, one to the other (Quinta)
The marginal fifth figure pairs a square inscribed in a square with an analogous figure, captioned simply that these are squares double one to the other, tying the whole series to the doubling ratio of the diameters.
Interlaced rosette of circles
At the lower left a large ring of overlapping circles forms a rosette of lune-shaped petals around a hatched centre, a rendering of the same overlapping-circle geometry that generates the lunes discussed in the text.