Studies of lunes and semicircular portions
Equal remainders from equal subtractions; dividing a curved angle by a mean-proportional curve
The sheet is filled with rows of hatched lunes (crescent portions), semicircles, sectors and fan shapes, arranged like a catalogue of curvilinear areas. The accompanying text develops the proportions of these figures — labelling sectors as 'quadruple' and 'double' proportion, and arguing that if equal parts are removed from equal wholes the remainders stay equal. A left-hand column adds that a curved angle can be divided by a third, mean-proportional curve, with the first curvatures standing in quadruple ratio.
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Filling a semicircle with its greatest portions
In the last semicircle the void is filled with the two greatest dotted portions. Taking the middle circle equal to the semicircle and removing the smaller circle from each leaves equal remainders. The reasoning treats lunes and portions as areas that can be added and subtracted.
Equal remainders from equal subtractions
If from equal things equal parts are removed, the remainder stays equal. Therefore remove the smaller circle from the greater. Letting the middle circle equal the semicircle and removing the smaller circle from each yields equal remainders.
Dividing a curved angle by a mean-proportional curve
A circle removed from a circle double its size carries away half; the same holds for a semicircle, so the remainders are equal. The angle formed by two different curved sides is divided by a third curved line that is the mean proportional to each, the new curvature being as much less than the greater as it is greater than the lesser. These first curvatures are quadruple one to another.