Squaring Lunes by Transposing Surfaces
Curvilinear figures reduced to equal rectilinear ones by cutting and relocating parts
A sheet of short geometrical demonstrations in which curved figures are made 'squarable' (reducible to a square) by removing shared parts and shifting them into new positions. Leonardo repeatedly sets one figure equal to another, cancels a common piece such as bc, and moves labelled fragments (a into n, d into b, h into e) until a rectilinear equivalent remains. A lune abc is likewise shown equal to efbc and thereby squarable.
On this page
Reducing a squarable figure to a rectilinear one
The squarable abc is set equal to dbchft; removing bc from each leaves a equal to dhft. Leonardo then moves d into b and h into e, since each place can receive the surface, leaving bfte equal to the squarable a, which is finally placed as a rectilinear figure at K.
A lune equal to a two-portion figure
The lune abc is stated equal to efbc; removing bc from both leaves the squarable a equal to ef, so ef too is squarable. Then e is moved into d, which can hold it, and fd remains equal to a.
Relocating part a into position n
In a smaller figure ab is stated equal to c, with the instruction to remove a and set it into n. The move preserves the total area while changing the outline, a recurring device on this leaf.