Squaring Lunes and Dividing Circular Portions into Sevenths
With an invention Leonardo notes he received as a gift on Christmas morning 1504
The leaf is filled with Leonardo's geometry of the lune (falcata): a doubled triangle whose areas are shown to be equal, and the squaring of a sickle-shaped figure by taking away half of a rectilinear area. Further passages set out how to divide a circular portion with certainty, building squares in 6/7 proportion so that the circles stand in that same ratio, and how to resolve a portion into seven sevenths. A framed figure at center carries Leonardo's note that this device was given to him as a gift on the morning of Christmas 1504. Small diagrams of triangles, lunes, an inscribed polygon and a fan-shaped sector accompany the demonstrations.
On this page
Doubled triangle and the equal remaining areas
Triangle a b f is twice the area of a c d; subtracting the smaller from the greater leaves a field equal to the whole smaller triangle. Since o p equals q r, the residual n equals m.
Squaring the lune (falcata)
The curved figure m n o is set equal to the rectilinear a c. Taking half of a c away from the lune leaves a remainder equal to the other half, reducing the sickle-shaped area to a straight-sided one.
Circles in 6/7 ratio built from squares
Because circles stand in the same proportion as the squares on their diameters, Leonardo builds a square that is 6/7 of another, inscribes the tangent circle, and so obtains two circles and their quarter-portions in 6/7 ratio to subtract one from the other.
Resolving a portion into seven sevenths
To extract any requested part, he takes away 1/7 of the portion by embedding one equal to 6/7, then 5/7, 4/7, 3/7, 2/7 and 1/7 in turn, until the whole is resolved into seven sevenths.
A gift on Christmas morning 1504
Beside a framed figure Leonardo records that this invention was given to him as a gift on the morning of Christmas 1504.
