The Lune of Hippocrates and circle-doubling
Squaring lunes and dividing semicircles into sectors
Read from right to left, this sheet centres on a large circle inscribed with a fan-like construction and is surrounded by lunes and circular segments used for squaring problems. Leonardo works the classical Lune of Hippocrates, lettered m-n-a-o-p-b, showing that the lune 'a a n m' is squarable and that portion 'a a o' can be made equal to it by removing 'n m' and returning 'o' of equal value. Other figures divide a semicircle into ten and into sixteen sectors and note doubling ratios, such as the quarter of a circle being double an eighth. The proofs rest throughout on comparing the areas of portions and lunes.
On this page
The Lune of Hippocrates
Around the lettered figure Leonardo states that 'a a n m is the lune' and that 'a n m is squarable because it is a lune'. He then squares 'a a o' by removing 'n m' and giving back 'o' of the same value, making it equal to the lune, the classical result that a lune can be squared exactly.
Quarter-circle double an eighth
A sector set over a semicircle carries the note that the quarter of a circle, set against an eighth of a circle, is double to it. It records the simple doubling ratio underlying several of the sector divisions on the sheet.
Circle on a semicircle: a b worth c
In the right-hand figure Leonardo writes that 'a b is worth c because e f is worth d and m is common'. The equality is derived by cancelling the shared part m between two overlapping areas.