Dividing a circular segment into four equal parts
Areas transformed on the arc a d b e c using radii to the centre f
The upper diagram shows a portion of a circle bounded by the arc a d b e c above and a straight base below, with lines drawn from points of the arc down to the centre f so the figure fans into triangular sectors. The note explains how to divide such a segment, one side curved and the others straight, into four parts equal in both shape and area. Leonardo splits the arc into the four equal pieces a, d, b, e, c, trims the excess of the two larger sectors with the lines b L and b g, then cuts the leftover triangle g b L into equal strips to redistribute. It is a study in the Euclidean transformation of areas.
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Quartering the segment by radii to the centre f
The curve of the segment is divided into four equal arcs a, d, b, e, c, and the lines d f, b f and e f are drawn to the centre f of the circle, which cut the figure into four unequal parts. The excess of the two larger parts over the smaller is then trimmed away with the line b L for one and the line b g for the other, leaving four parts equal in shape and quantity.
Redistributing the leftover triangle g b L
What remains is the triangle g b L, which Leonardo divides into four equal strips along its base. One strip is added back to each of the four sectors so that all four finish as equal figures, a d h, d h i b, b i K e and e K c, all of the same area.