Squaring the circle: lunes, circles and squares
Quadrature demonstrations, with a rule derived from Euclid's Elements, Book II.
This densely drawn sheet gathers demonstrations on the quadrature of lunes and circles, laid out (as the notes say) in a right half and a left half. Figures include a hexagon enclosing a star-polygon within a circle, sets of concentric and cross-arranged circles, and tangent-circle diagrams whose crescent 'lunes' are shown equal in area to squares or to half the greatest circle. The left half develops a construction in which a length a b equals the root of 3, a rule Leonardo attributes to the fourteenth and last proposition of the second book of Euclid's Elements.
On this page
Four lunes converted into a quarter-square
Because each of the four whole lunes is worth a quarter of its whole circle, removing a single portion leaves a square; the square into which the horns a b are converted is worth a quarter of the greatest square drawn in the greatest circle.
Six lunes equal to half the greatest circle
In the figure of twelve internally tangent circles a b c d e f, the six lunes are said to equal the value of their hatched field. It follows that these lunes together equal half of the greatest circle.
The length a b as the root of 3, after Euclid II.14
On the left half, a b is stated to be the root of 3. Leonardo notes that this rule is born from the fourteenth and last proposition of the second book of Euclid's Elements, and that a second figure gave rise to the first.
Five squares in an equal-value progression
Five squares are said to exceed one another successively by the value of the smallest square a, so that a b c d e are of equal value among themselves; the same holds for the circles that touch these squares within or without, and from this the lunes of the whole circumference arise.