Squaring the circle: circles, squares and octagons
Geometric quadrature through annular rings, with square-root number series
The sheet is covered with geometrical figures — double and triple circles, stars, squares, octagons and figure-of-eight forms — exploring the quadrature of the circle. Leonardo compares annular rings (circular 'parallels') with inscribed squares and octagons, arguing that pairs of squared portions together equal three-quarters of the greatest square of the greatest circle. Two circular arrangements of squares and triangles carry number series (9-13 and 1-10) which the Ambrosiana's cataloguing links to finding the square roots of the first integers by geometric means. Numerous faint annotations accompany the diagrams.
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Annular ring equated with the smallest circle
The smallest circle equals the surrounding circular ring (annulus). So the inner circle of the first figure is taken, stretched out and its own figure girded on the outside. The study redistributes an annulus into an equivalent squared area in the attempt to square the circle.
Greatest square and greatest octagon of the greatest circle
One figure is labelled the greatest square of the greatest circle, another the greatest octagon of the greatest circle. These inscribed polygons serve to compare areas in the effort to square the circle.
Two squares equal three-quarters of the greatest square
With the labels b a — c, the middle circle equals a quarter of the greatest circle and the greatest ring equals half of it. Removing a b leaves the ring squared and removing c leaves the middle circle squared, so the two squares together equal 3/4 of the greatest square drawn from the greatest circle.
Number series in the circular figures
Two circular structures of squares and triangles carry the number series 9 10 11 12 13 and 1 2 3 4 5 6 7 8 9 10. Per the Ambrosiana cataloguing these are tied to obtaining the square roots of the first integers by geometric means.