Lunes, triangles and circle-to-square proportion
Proving a lune equal to a semicircle by superposition of equal surfaces
The sheet carries a large central semicircle laid out with radiating construction lines, together with cones or pyramids, superimposed triangles and further lunes. Leonardo argues that circle-to-circle proportion equals square-to-square proportion built on the diameter, then proves that a lune equals a semicircle by superposing surfaces of equal quantity and reasoning about the parts that do and do not overlap. A left-hand group of superimposed triangles extends the same overlap argument to rectilinear figures of equal area and doubled angle.
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Circle-to-circle proportion equals square-to-square
The proportion between two circles is the same as that between two squares built on the multiplication of the diameter. Conversely, the proportion between two squares equals that between circles built on the multiplication of the side. The rule links circular and square areas through their linear dimensions.
A lune equal to a semicircle by superposition
It is proved that the lune m f n a p o q is worth the semicircle f a q, and that the portion a o q is worth the lune a n o p q. The proof superposes two surfaces of equal quantity: the touching parts are equal in quantity and figure, and the non-touching parts equal in quantity and sometimes in figure, as with circles. Since the smaller semicircle equals a quarter of the greater, portion f g a is worth the lune.
Superimposed triangles of equal area
For triangles equal in value and double in angle laid one over the other so that angle and side coincide, what is uncovered of the wider one equals, in figure and quantity, its part covered by the narrower. And the part of the narrow triangle exceeding the wide one in length equals in quantity the part of the wide one exceeding the contact of the narrower.