d c b a - L K – m g f i – n h u t S e – o n.

b a. Here the greater circle is double the lesser, and likewise its eighth-portions are double the eighth-portions of the lesser circle. Therefore the two portions of the greater are worth 4 of the lesser circle. Therefore, removing a b c d — that is, the four half-portions — from the greater circle on the outside, and restoring to it the value from within with the portions e f g h, that circle i K L m a b c d remains at its first value, by the seventh [proposition], which says: to that which is given back what was taken from it, it returns to its first value. Therefore we have squared the part of the curvilinear parallel i a b K L c d m, because it is reduced into the rectilinear figure e i K f L g m h. Again I will go further: I will remove the rectilinear triangle and put it at n, and by such a change it will not have altered the value of its first whole. And if I lent that whole the triangle o, I would have made a square surface equal to 3/4 of the lesser semicircle e f g s t u, from which I will then remove the value of that triangle o in the angle; the remainder of such a circle will remain squared, that is, equal to the aforesaid surface i e K f L g m h — or, if you will, by removing i e and putting it at n.

b a; n - t o p s. Again a b n m is equal to S p o t. I will remove above the half-portions a b, and remove below the portions S t; there will remain p o squared, equal to the unsquared n m. But to return to the first proposition, I say that the parts i m of the figure above are parts added to accompany the portions e h, and could be greater or lesser at our pleasure; but they have taken the quantity i e m, in order to lend it another surface n that makes it of the width of the parallel K f, or if you will L g. But in truth i m must not be reckoned in this quadrature, because only e h, which goes together with f g, is what serves to restore the portions removed above, and all the addition of i or of m is superfluous. It is true that this addition of i m is common alike to the curvilinear parallel and to the rectilinear parallel.