Squaring lunes and curvilinear triangles
A crowded geometry sheet on transforming double-curvature figures and lunes into equal squares.
The sheet is crowded with small geometric figures, curvilinear triangles, overlapping circles, sectors and lune shapes (falcate), surrounded by columns of notes on their quadrature, the classical problem of converting a curved figure into an equal square. Leonardo argues that a curvilinear triangle of 'double curvature' can be squared by adding and subtracting known squarable triangles, citing a proposition that a known part taken from a known whole leaves a known remainder. Repeated constructions pair 'double' and 'subdouble' portions of circles to build squarable lunes. Small sketches of structures at the upper right are not part of the transcribed geometry.
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Squaring a curvilinear triangle of double curvature
The triangle b a c has sides of 'double curvature', born of double circles. Leonardo squares it through the squarable triangle b a d, then removes the triangle c a d, which is squarable because its sides are equal in curvature and length. Invoking the rule that a known part taken from a known whole leaves a known remainder, he concludes the quadrature of b a c can be found.
Contact of two equal double surfaces
If two 'double' surfaces are laid entirely one upon the other, the part that touches equals the part that does not touch. This equality underlies the sheet's repeated method of adding and subtracting equal circle-portions. Small figures of overlapping circles and sectors illustrate the claim.
Pairing double and subdouble portions into a squarable lune
Using his rule for producing circle-portions at will, Leonardo takes two equal portions and makes them the sides of a lune (falcata), which is necessarily squarable, then builds a rectilinear triangle from it. Taking a portion double the first and joining it to the subdouble portion, he squares the result and subtracts the lesser triangle from the greater. The remainder is squared and equals the lesser portion, itself a portion of a quarter circle.
Curvilinear figures equal by equal sides
Several small triangles are labelled to show that a b c d are equal when each side is equal in both curvature and length. Related figures pair a short concave line with a long convex one, and the reverse. Two sectors o n p and n m p are declared equal because their bases o n and n m are equal.