Lunes, Quadruple Portions, and the Root of Ten
Sickle-shaped figures equal to the portions they embrace; geometric construction of square roots
Folio 117v gathers Leonardo's conclusions on quadruple proportionality, drawn as a large fan of nested circular sectors and a semicircle scaled by successive roots. He states that successive portions of circles are all quadruple one another to infinity, and that every sickle-shaped lune equals in area the portion it embraces. A numbered scale gives the seven square roots, and a construction shows how to build the root of ten: extend a line divided into ten by one part, make a circle on half of eleven, and raise a perpendicular to cut it.
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Quadruple portions and lunes equal to their portions
The portions a b d K, a m f d, a n h f, a o p h are all quadruple one another successively and proceed infinitely, as do their half-portions, since whole is to whole as part is to part. Leonardo concludes that all the sickle-shaped falcate (lunes) are equal in quantity to the portions they embrace.
The seven roots and circles in double or quadruple ratio
n is the root of 1, a of 2, b of 3, c of 4, d of 5, e of 6, f of 7; of these seven roots six are needed. Circles exceeding one another by the smaller circle's semidiameter are double to infinity, and those exceeding by the whole diameter of the smaller are quadruple successively to infinity.
Geometric construction of the root of ten
For the root of 10, take a line divided into 10, add one more equal part to make 11, and draw a circle whose semidiameter is half of 11. Then raise a line at right angles upon the ten, and where it cuts the circle the whole cut line is the root of 10.