Dividing one circle into nine equal circles
A geometric construction using the penultimate proposition of Euclid's Book I
Two mounted fragments carry a geometric problem: from a given circle to make a required number of circles equal in value to it. Leonardo circumscribes the given circle g with a square a b c d, builds an equal square c d e f beneath it, divides it into nine equal strips, and by squaring each strip with the penultimate proposition of Euclid's Book I obtains nine equal circles, of which n m is one. The lower fragment shows the accompanying figures, labelled first through fifth: a double square with an inscribed circle and rectangles, a rectangle turned into a square, and a circle inscribed in a square.
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Reducing one circle to nine equal circles by squaring
Given the circle g, to be turned into nine equal circles: circumscribe it with the square a b c d, then build an equal square c d e f beneath it and divide that square into nine equal parallel strips. Square each strip by the penultimate proposition of the first book of the Elements, and within each squaring draw one of the nine required circles, of which n m is an example. The same method, applied to various numbers of squares, yields larger or smaller squares as required.
Sequence of construction figures, first through fifth
The lower strip presents the figures keyed to the demonstration: the first, a double square with an inscribed circle g and rectangles labelled a b, c d, e f; the second, a rectangle transformed into a square; the third, a circle m n inscribed in a square; and two further figures, the fourth and fifth, without letters.