On the Squaring of the Semicircle
A curvilinear triangle proved equal to a rectilinear one, giving the area of the semicircle and the circle
Titled in Latin 'On the squaring of the semicircle', this folio proves that the curvilinear triangle c e a equals the rectilinear triangle, so that the square c d a n is worth the semicircle. Leonardo grounds the argument on curved bases being equal to straight bases through motion, and on the theorem that all triangles on equal bases between parallel lines are equal among themselves. A second figure shows that a semicircle and a triangle of equal height are of equal value, and that the enclosing rectangle is therefore worth a whole circle. Diagrams of a quarter-circle in a rectangle and a triangle with a bowtie figure occupy the right margin.
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Curvilinear triangle equal to a rectilinear triangle
The curvilinear triangle c e a is worth the rectilinear triangle by the motion of its periphery a e, and the triangle a d S is worth the semicircle, so the square c d a n is worth the semicircle. The four sectors of the quarter-circle c a e equal the four triangles into which triangle a d n resolves, because side c a equals side d n as sides lying between the equidistant lines c a d n. The general conclusion is that all triangles built on equal bases between parallel lines are equal among themselves.
Rectangle equal to a whole circle
The extension of the periphery e g f forms the base of the triangle b a c, equal in height by lying between parallel lines, so the semicircle and the triangle are of equal value. Because the rectangle is worth two such triangles b a c, it is worth two semicircles; therefore the rectangle c b h l is worth a whole circle.