Squaring a curvilinear triangle from lunes and segments
Restoring a rectilinear triangle by exchanging portions of circles double one another
A geometry page on the quadrature of curvilinear figures. Leonardo shows how to square a curvilinear triangle a by removing a circular portion n and giving back two smaller portions m and o, arguing that the circle behind portion n is double the circle behind the two lesser portions, so the triangle is restored to its original quantity. A second construction makes a rectilinear triangle a b c equal to a sickle-shaped falcata d n, and a triangle d e f equal to the lune n d m; a wedge n m o f is then divided into two equal parts born of doubled circles.
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Squaring the curvilinear triangle a by exchanging circular portions
To square a curvilinear triangle such as triangle a, portion n is removed from the similar triangle n f and the two small portions m and o are given back. Leonardo justifies the equality by noting that the circle from which portion n derives is double the circle from which the two lesser portions derive, so the triangle is restored to its first state.
Rectilinear triangle equal to a falcata and a lune
The rectilinear triangle a b c is made equal to the sickle-shaped falcata d n, and the triangle d e f is made equal to the lune n d m. The note states that d n and d e f are equal to one another, that the lune d m n behaves the same way, and that d e and d n are equal to one another.
A wedge divided by circles in double ratio
The whole wedge n m o f is divided into two equal parts, n m and o f, equal because they are born in circles double one to the other. Removing portion n above and portion o below takes twice as much space above as below, so the smaller portion must be restored to part m to equalize the losses.