Squares and circles that hold four equal copies of themselves
Quartering a square and a circle; a circle equal to the ring between two circles
A sequence of area propositions on halving and quartering. Leonardo notes that any square contains four equal squares and any circle four equal circles, and that halving a radius makes the inner circle enter four times into the outer. Rotated-square constructions prove that square b f d e is half of a b c d, and that a circle inscribed in the inner square (b c) equals the area standing free inside the larger circle. A closing proposition builds, on line c b, a circle whose area equals the ring between two circles and is half of the larger.
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A square holds four equal squares
The area of any square contains within itself four equal squares. A square bisected both vertically and horizontally shows the four quarters.
A circle holds four equal circles
The area of any circle contains within itself four circles equal to one another. Four smaller circles are drawn on the four perpendicular radii of the larger circle.
Half the radius, one fourth of the circle (a, b, c)
If the semidiameter a b is half of semidiameter a c, the smaller circle enters four times into the larger. Two concentric circles with the outer radius twice the inner illustrate the ratio.
Square b f d e is half of a b c d
Square b f d e is half the size of square a b c d, because line b d divides the smaller square into two parts, each matching the four equal parts into which the larger square is split by its diagonals. A fifth triangle equal to the four inside is added at vertex f.
Proof that the inscribed circles are equal
Circle b c holds as much area as stands free within the larger circle, provided square a e b f equals square s g r h and each circle touches the four sides of its square. Therefore the one circle equals the other, and the outer square is half the size of the other.
A circle equal to the ring between two circles
Drawing line c b where the larger circle meets the square's two diameters at a and b, and marking where c b crosses diameter e f at n, gives the start and finish of a circle equal in area to the ring enclosed between the two circles and half the size of the larger. The proof is shown above on the right.