Squaring Curved Triangles by Ratios of Circles
Lunes and curved triangles equated to squares using quarter- and eighth-arcs
The page continues the squaring of curvilinear figures using the doubling relation between circles. Leonardo argues that a and b are equal because a n is a quarter of a small circle while n b is an eighth of a circle twice as large, so that removing the common part n leaves equal remainders. He then squares the curved triangle n b by a second method, restoring the part a in place of the part b, and finally squares the triangle a b c by replacing the portion c with two eighth-arcs at d e. Small labelled diagrams (a n b; a b, n; d a c e b) accompany each argument.
On this page
Equal areas from a quarter-arc and an eighth-arc
a and b are equal because a n is 1/4 of a circle and n b is 1/8 of a circle twice as large; an eighth of the doubled circle is worth as much as the quarter of the smaller. Removing the common part n from each leaves equal remainders, so a is made equal to b.
Squaring the curved triangle n b by another method
Leonardo squares the triangle n b by removing the part b and restoring the part a, which is equal to it because in every figure double another the half of the greater equals the lesser. Since the curvature of b is an eighth of a circle double that of a, a equals b, giving a different squaring than the one above.
Squaring the triangle a b c through d e
Taking half of the triangle a b c, he removes the portion c and restores two portions at d e; joined together these equal c, since c is an eighth of a circle double the eighths e d. The promised squaring then lies in the four parts a b e d joined together.