On the squaring of the oval figure
The oval proved double the circle, with proportional compasses
A geometry sheet devoted to squaring the oval. Leonardo argues that an oval figure set in a doubled 'parallel' band is exactly twice the circle inscribed in a square, imagining the circle sliced into hair-thin parallel strips that are each stretched to double length, so that squaring the oval reduces to squaring a circle of double area. He defines an oval as a figure bounded by a single curved line about a central point 'uniformly unequal' from the circumference, and notes that one and the same area can gain circumference toward infinity. A large square-and-circle diagram, marginal proofs and a pair of proportional compasses accompany the text.
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Squaring the oval via the squared circle
Once the circle is squared, the oval can be squared at once: the oval n m p S is double the circle a b c d, so a circle of double area, when squared, equals the oval. The same follows by the motion of the parts of a sphere composing an oval body.
The oval proved double the circle
By straight motion the oval is shown to be twice the circle set in the same parallel band. Imagine the circle divided into very narrow parallels, like thin hairs in continuous contact, each doubled in length; the circle in square a b c d thus becomes the oval in the band a b e f, made of two squares equal to a b c d, and the semicircle n m o equals the lune n m o p.
Definition of the oval figure
The oval is defined as a figure surrounded by a single curved line, in the middle of which is a point whose distance to the circumferential line is uniformly unequal.
Proportional compasses, double, for scaling
An oval of any proportion to a given circle can be made by this rule, but it requires proportional compasses that are double, sketched in the margin. A related note remarks that one and the same surface can gain circumference toward infinity — a circle a foot across reshaped into a band a mile long gaining a circumference of two miles.