Squaring the sectors of quadruple circles
Quarter-circle portions related to a fourfold larger circle
The sheet works on the quadrature of curved figures using circles in fourfold ratio: since two circles stand as quadruple to one another, a quarter of the one equals a sixteenth of the other. Leonardo argues that the curvature of a quarter-circle contains four portions, each a quarter of a larger portion o, so that all four together equal o. He shows that a circular sector, once its curved portions are removed, leaves a rectilinear figure that is 'quadrabile' (squarable), and that four rectilinear cones equal a single one. Diagrams of sectors, cones and quarter-circles lettered with a, b, c, n, m, o, p accompany the argument; further untranscribed writing shows faintly on the sheet.
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Quarter of one circle equals a sixteenth of a fourfold circle
Two circles are taken as quadruple, one to the other. Under that ratio the quarter of the one is made equal to a sixteenth of the other, the arithmetical key that lets curved parts of unequal circles be equated.
Four portions of a quarter-circle equal one whole portion o
Leonardo states that a b is equal to c, and that removing a from c leaves c equal to b. The curvature of the quarter-circle contains four portions, each of which is by itself the fourth part of the portion o, so all four joined together equal the single portion o.
Squaring the difference of two circular sectors n m and o p
The sector n m is set equal to the sector o p; removing the portions n o leaves p greater than m by exactly the amount n exceeds o. Taking m from the greater p leaves a part equal to that excess, and it is squarable because it is a rectilinear surface.
Rectilinear cones equal to a single cone; the rule never fails
The portion a is the fourth part of the portion o, being born of a circle subquadruple to that of o, so the part keeps the same ratio to the like part as the whole to the like whole. Thus cone a is a quarter of cone o f, all four portions equal the single portion o, and all four rectilinear cones e n m r equal the single rectilinear cone f.