Dividing the Lune and Squaring a Fraction of a Square
Quadrature of curvilinear figures and the bisection of triangles
This leaf continues the study of the lune (falcata): Leonardo divides the sickle a b c into equal parts by dividing the segment a f, treating the figure as similar to the semicircle b a. A second demonstration removes the 97th part of a given square while keeping the remainder square, nesting a square of 96 within one of 97. A closing note squares a curvilinear figure by subtracting equal circular portions and states that any triangle is halved by the line from the opposite vertex that bisects its base. Small framed diagrams of quarter-circles, squares and a semicircle accompany each step.
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Dividing the lune a b c into equal parts
The sickle a b c, similar to the semicircle b a, is divided into equal parts by dividing the segment a f into equal parts.
Removing the 97th part of a square
To subtract the 97th part of a square and keep the remainder square, a square of 96 is set inside one of 97; the removed part is then squared and the remainder stays square, or both are squared by the method g shown below.
Quadrature of a curved figure; bisecting a triangle
Because a o equals o n as portions of circles double one to another, subtracting the equal part o from each leaves equal remainders, so n squares the curvilinear a. Every triangle is halved by the line from the opposite angle that bisects the base.